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2.1: Introduction to Functions

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• Page ID 19685

• David Arnold
• College of the Redwoods

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Our development of the function concept is a modern one, but quite quick, particularly in light of the fact that today’s definition took over 300 years to reach its present state. We begin with the definition of a relation.

We use the notation (2, 4) to denote what is called an ordered pair. If you think of the positions taken by the ordered pairs (4, 2) and (2, 4) in the coordinate plane (see Figure \(\PageIndex{1}\)), then it is immediately apparent why order is important. The ordered pair (4, 2) is simply not the same as the ordered pair (2, 4).

Figure \(\PageIndex{1}\)

The first element of an ordered pair is called its abscissa. The second element of an ordered pair is called its ordinate. Thus, for example, the abscissa of (4, 2) is 4, while the ordinate of (4, 2) is 2.

A collection of ordered pairs is called a relation . (2)

For example, the collection of ordered pairs \[R=\{(0,1),(0,2),(3,4)\} \nonumber \] is a relation.

The domain of a relation is the collection of all abscissas of each ordered pair.

Thus, the domain of the relation R in (2) is \[\text { Domain }=\{0,3\} \nonumber \]

Note that we list each abscissa only once.

The range of a relation is the collection of all ordinates of each ordered pair.

Thus, the range of the relation R in (2) is \[\text { Range }=\{1,2,4\} \nonumber \]

Let’s look at an example.

Example \(\PageIndex{1}\)

Consider the relation T defined by \[T=\{(1,2),(3,2),(4,5)\} \nonumber \]

What are the domain and range of this relation?

The domain is the collection of abscissas of each ordered pair. Hence, the domain of T is \[\text { Domain }=\{1,3,4\} \nonumber \]

The range is the collection of ordinates of each ordered pair. Hence, the range of T is \[\text { Range }=\{2,5\} \nonumber \]

Note that we list each ordinate in the range only once.

In Example \(\PageIndex{1}\), the relation is described by listing the ordered pairs. This is not the only way that one can describe a relation. For example, a graph certainly represents a collection of ordered pairs.

Example \(\PageIndex{2}\)

Consider the graph of the relation S shown in Figure \(\PageIndex{2}\).

Figure \(\PageIndex{2}\) The graph of a relation.

What are the domain and range of the relation S?

There are five ordered pairs (points) plotted in Figure \(\PageIndex{2}\). They are \[S=\{(1,2),(2,1),(2,4),(3,3),(4,4)\} \nonumber \]

Therefore, the relation S has Domain = {1, 2, 3, 4} and Range = {1, 2, 3, 4}. In the case of the range, note how we’ve sorted the ordinates of each ordered pair in ascending order, taking care not to list any ordinate more than once.

A function is a very special type of relation. We begin with a formal definition.

A relation is a function if and only if each object in its domain is paired with one and only one object in its range.

This is not an easy definition, so let’s take our time and consider a few examples. Consider, if you will, the relation R in (2), repeated here again for convenience.

\[R=\{(0,1),(0,2),(3,4)\} \nonumber \]

The domain is {0, 3} and the range is {1, 2, 4}. Note that the number 0 in the domain of R is paired with two numbers from the range, namely, 1 and 2. Therefore, R is not a function.

There is a construct, called a mapping diagram, which can be helpful in determining whether a relation is a function. To craft a mapping diagram, first list the domain on the left, then the range on the right, then use arrows to indicate the ordered pairs in your relation, as shown in Figure \(\PageIndex{3}\).

Figure \(\PageIndex{3}\) A mapping diagram for R.

It’s clear from the mapping diagram in Figure \(\PageIndex{3}\) that the number 0 in the domain is being paired (mapped) with two different range objects, namely, 1 and 2. Thus, R is not a function.

Let’s look at another example.

Example \(\PageIndex{3}\)

Is the relation described in Example \(\PageIndex{1}\) a function?

First, let’s repeat the listing of the relation T here for convenience.

\[T=\{(1,2),(3,2),(4,5)\} \nonumber \]

Next, construct a mapping diagram for the relation T. List the domain on the left, the range on the right, then use arrows to indicate the pairings, as shown in Figure \(\PageIndex{4}\).

From the mapping diagram in Figure \(\PageIndex{4}\), we can see that each domain object on the left is paired (mapped) with exactly one range object on the right. Hence, the relation T is a function.

Figure \(\PageIndex{4}\). A mapping diagram for T.

Example \(\PageIndex{4}\)

Is the relation of Example \(\PageIndex{2}\), pictured in Figure \(\PageIndex{2}\), a function?

First, we repeat the graph of the relation from Example \(\PageIndex{2}\) here for convenience. This is shown in Figure \(\PageIndex{5}\)(a). Next, we list the ordered pairs of the relation S.

\[S=\{(1,2),(2,1),(2,4),(3,3),(4,4)\} \nonumber \]

Then we create a mapping diagram by first listing the domain on the left, the range on the right, then using arrows to indicate the pairings, as shown in Figure \(\PageIndex{5}\)(b).

Figure \(\PageIndex{5}\) A graph of the relation S and its corresponding mapping diagram

Each object in the domain of S gets mapped to exactly one range object with one exception. The domain object 2 is paired with two range objects, namely, 1 and 4. Consequently, S is not a function.

This is a good point to summarize what we’ve learned about functions thus far.

A function consists of three parts:

• a set of objects which mathematicians call the domain ,
• a second set of objects which mathematicians call the range ,
• and a rule that describes how to assign a unique range object to each object in the domain.

The rule can take many forms. For example, we can use sets of ordered pairs, graphs, and mapping diagrams to describe the function. In the sections that follow, we will explore other ways of describing a function, for example, through the use of equations and simple word descriptions.

Function Notation

We’ve used the word “mapping” several times in the previous examples. This is not a word to be taken lightly; it is an important concept. In the case of the mapping diagram in Figure \(\PageIndex{5}\)(b), we would say that the number 1 in the domain of S is “mapped” (or “sent”) to the number 2 in the range of S.

There are a number of different notations we could use to indicate that the number 1 in the domain is “mapped” or “sent” to the number 2 in the range. One possible notation is

\[S : 1 \longrightarrow 2 \nonumber \]

which we would read as follows: “The relation S maps (sends) 1 to 2.” In a similar vein, we see in Figure \(\PageIndex{5}\)(b) that the domain objects 3 and 4 are mapped (sent) to the range objects 3 and 4, respectively. In symbols, we would write

\[\begin{array}{l}{S : 3 \longrightarrow 3, \text { and }} \\ {S : 4 \longrightarrow 4}\end{array} \nonumber \]

A difficulty arises when we examine what happens to the domain object 2. There are two possibilities, either

\[S : 2 \longrightarrow 1 \nonumber \] or \[S : 2 \longrightarrow 4 \nonumber \]

Which should we choose? The 1? Or the 4? Thus, S is not well-defined and is not a function, since we don’t know which range object to pair with the domain object 1.

The idea of mapping gives us an alternative way to describe a function. We could say that a function is a rule that assigns a unique object in its range to each object in its domain. Take for example, the function that maps each real number to its square. If we name the function f, then f maps 5 to 25, 6 to 36, −7 to 49, and so on. In symbols, we would write

\[f : 5 \longrightarrow 25, \quad f : 6 \longrightarrow 36, \quad \text { and } \quad f :-7 \longrightarrow 49 \nonumber \]

In general, we could write

\[f : x \longrightarrow x^{2} \nonumber \]

Note that each real number x gets mapped to a unique number in the range of f, namely, \(x^{2}\). Consequently, the function f is well defined. We’ve succeeded in writing a rule that completely defines the function f.

As another example, let’s define a function that takes a real number, doubles it, then adds 3. If we name the function g, then g would take the number 7, double it, then add 3. That is,

\[g : 7 \longrightarrow 2(7)+3 \nonumber \]

Simplifying, \(g : 7 \longrightarrow 17\). Similarly, g would take the number 9, double it, then add 3. That is,

\[g : 9 \longrightarrow 2(9)+3 \nonumber \]

Simplifying, \(g : 9 \longrightarrow 21\). In general, g takes a real number x, doubles it, then adds three. In symbols, we would write

\[g : x \longrightarrow 2 x+3 \nonumber \]

Notice that each real number x is mapped by g to a unique number in its range. Therefore, we’ve again defined a rule that completely defines the function g.

It is helpful to think of a function as a machine. The machine receives input, processes it according to some rule, then outputs a result. Something goes in (input), then something comes out (output). In the case of the function described by the rule \(f : x \longrightarrow x^{2}\), the “f-machine” receives input x, then applies its “square rule” to the input and outputs \(x^{2}\), as shown in Figure \(\PageIndex{6}\)(a). In the case of the function described by the rule \(g : x \longrightarrow 2 x+3\), the “g-machine” receives input x, then applies the rules “double,” then “add 3,” in that order, then outputs \(2x + 3\), as shown in Figure \(\PageIndex{6}\)(b).

Figure \(\PageIndex{6}\) Function machines.

Example \(\PageIndex{5}\)

Suppose that f is defined by the following rule. For each real number x,

\[f : x \longrightarrow x^{2}-2 x-3 \nonumber \]

Where does f map the number −3? Is f a function?

We substitute −3 for x in the rule for f and obtain

\[f :-3 \longrightarrow(-3)^{2}-2(-3)-3 \nonumber \]

Simplifying,

\[f :-3 \longrightarrow 9+6-3 \nonumber \]

\[f :-3 \longrightarrow 12 \nonumber \]

Thus, f maps (sends) the number −3 to the number 12. It should be clear that each real number x will be mapped (sent) to a unique real number, as defined by the rule \(f : x \longrightarrow x^{2}-2 x-3\). Therefore, f is a function.

Example \(\PageIndex{6}\)

Suppose that g is defined by the following rule. For each real number x that is greater than or equal to zero,

\[g : x \longrightarrow \pm \sqrt{x} \nonumber \]

Where does g map the number 4? Is g a function?

Again, we substitute 4 for x in the rule for g and obtain

\[g : 4 \longrightarrow \pm \sqrt{4} \nonumber \]

\[g : 4 \longrightarrow \pm 2 \nonumber \]

Thus, g maps (sends) the number 4 to two different objects in its range, namely, 2 and −2. Consequently, g is not well-defined and is not a function.

Let’s look at another example

Example \(\PageIndex{7}\)

Suppose that we have functions f and g, defined by

\[f : x \longrightarrow x^{4}+11 \quad \text { and } \quad g : x \longrightarrow(x+2)^{2} \nonumber \]

Where does g send 5?

In this example, we see a clear advantage of function notation. Because our functions have distinct names, we can simply reference the name of the function we want our readers to use. In this case, we are asked where the function g sends the number 5, so we substitute 5 for x in

\[g : x \longrightarrow(x+2)^{2} \nonumber \]

\[g : 5 \longrightarrow(5+2)^{2} \nonumber \]

Simplifying, \(g : 5 \longrightarrow 49\).

Modern Notation

Function notation is relatively new, with some of the earliest symbolism first occurring in the 17th century. In a letter to Leibniz (1698), Johann Bernoulli wrote “For denoting any function of a variable quantity x, I rather prefer to use the capital letter having the same name X or the Greek \(\xi\), for it appears at once of what variable it is a function; this relieves the memory.”

Mathematicians are fond of the notation \[f : x \longrightarrow x^{2}-2 x \nonumber \]

because it conveys a sense of what a function does; namely, it “maps” or “sends” the number x to the number \(x^{2}-2 x\). This is what functions do, they pair each object in their domain with a unique object in their range. Equivalently, functions “send” each object in their domain to a unique object in their range.

However, in common computational situations, the “arrow” notation can be a bit clumsy, so mathematicians tend to favor a slightly different notation. Instead of writing

\[f : x \longrightarrow x^{2}-2 x \nonumber \]

mathematicians tend to favor the notation

\[f(x)=x^{2}-2 x \nonumber \]

It is important to understand from the outset that these two different notations are equivalent; they represent the same function f, one that pairs each real number x in its domain with the real number \(x^{2}-2 x\) in its range.

The first notation, \(f : x \longrightarrow x^{2}-2 x\), conveys the sense that the function f is a mapping. If we read this notation aloud, we should pronounce it as “f sends (or maps) x to \(x^{2}-2 x\).” The second notation, \(f(x) = x^{2}-2 x\), is pronounced “f of x equals \(x^{2}-2 x\).”

The phrase “f of x” is unfortunate, as our readers might recall being trained from a very early age to pair the word “of” with the operation of multiplication. For example, 1/2 of 12 is 6, as in \(1 / 2 \times 12=6\). However, in the context of function notation, even though f(x) is read aloud as “f of x,” it does not mean “f times x.” Indeed, if we remind ourselves that the notation \(f(x)=x^{2}-2 x\) is equivalent to the notation \(f : x \longrightarrow x^{2}-2 x\), then even though we might say “f of x,” we should be thinking “f sends x” or “f maps x.” We should not be thinking “f times x.”

Now, let’s see how each of these notations operates on the number 5. In the first case, using the “arrow” notation,

To find where f sends 5, we substitute 5 for x as follows.

\[f : 5 \longrightarrow(5)^{2}-2(5) \nonumber \]

Simplifying,\(f : 5 \longrightarrow 15\). Now, because both notations are equivalent, to compute f(5), we again substitute 5 for x in

\[f(5)=(5)^{2}-2(5) \nonumber \]

Simplifying, \(f(5)=15\). This result is read aloud as “f of 5 equals 15,” but we want to be thinking “f sends 5 to 15.”

Let’s look at examples that use this modern notation.

Example \(\PageIndex{8}\)

Given \(f(x)=x^{3}+3 x^{2}-5,\) determine \(f(-2)\)

Simply substitute −2 for x. That is,

\[\begin{aligned} f(-2) &=(-2)^{3}+3(-2)^{2}-5 \\ &=-8+3(4)-5 \\ &=-8+12-5 \\ &=-1 \end{aligned} \nonumber \]

Thus, \(f(−2) = −1\). Again, even though this is pronounced “f of −2 equals −1,” we still should be thinking “f sends −2 to −1.”

Example \(\PageIndex{9}\)

Given \[f(x)=\frac{x+3}{2 x-5} \nonumber \] determine f(6).

Simply substitute 6 for x. That is, \[\begin{aligned} f(6) &=\frac{6+3}{2(6)-5} \\ &=\frac{9}{12-5} \\ &=\frac{9}{7} \end{aligned} \nonumber \]

Thus, \(f(6) = 9/7\). Again, even though this is pronounced “f of 6 equals 9/7,” we should still be thinking “f sends 6 to 9/7.”

Example \(\PageIndex{10}\)

Given \(f(x)=5 x-3,\) determine \(f(a+2)\).

If we’re thinking in terms of mapping notation, then \[f : x \longrightarrow 5 x-3 \nonumber \]

Think of this mapping as a “machine.” Whatever we put into the machine, it first is multiplied by 5, then 3 is subtracted from the result, as shown in Figure \(\PageIndex{7}\). For example, if we put a 4 into the machine, then the function rule requires that we multiply input 4 by 5, then subtract 3 from the result. That is,

\[f : 4 \longrightarrow 5(4)-3 \nonumber \]

Simplifying, \(f : 4 \longrightarrow 17\)

Figure \(\PageIndex{7}\). The multiply by 5 then subtract 3 machine.

Similarly, if we put an a + 2 into the machine, then the function rule requires that we multiply the input a + 2 by 5, then subtract 3 from the result. That is,

\[f : a+2 \longrightarrow 5(a+2)-3 \nonumber \]

Using modern function notation, we would write

\[f(a+2)=5(a+2)-3 \nonumber \]

Note that this is again a simple substitution, where we replace each occurrence of x in the formula \(f(x) = 5x − 3\) with the expression a + 2. Finally, use the distributive property to first multiply by 5, then subtract 3.

\[\begin{aligned} f(a+2) &=5 a+10-3 \\ &=5 a+7 \end{aligned} \nonumber \]

We will often need to substitute the result of one function evaluation into a second function for evaluation. Let’s look at an example.

Example \(\PageIndex{11}\)

Given two functions defined by \(f(x) = 3x + 2\) and \(g(x) = 5 − 4x\), find f(g(2)).

The nested parentheses in the expression f(g(2)) work in the same manner that they do with nested expressions. The rule is to work the innermost grouping symbols first, proceeding outward as you work. We’ll first evaluate g(2), then evaluate f at the result.

We begin. First, evaluate g(2) by substituting 2 for x in the defining equation \(g(x) = 5 − 4x\). Note that \(g(2) = 5 − 4(2)\), then simplify.

\[f(g(2))=f(5-4(2))=f(5-8)=f(-3) \nonumber \]

To complete the evaluation, we substitute −3 for x in the defining equation \(f(x) = 3x + 2\), then simplify.

\[f(-3)=3(-3)+2=-9+2=-7 \nonumber \]

Hence, \(f(g(2))=-7\).

It is conventional to arrange the work in one contiguous block, as follows.

\[\begin{aligned} f(g(2)) &=f(5-4(2)) \\ &=f(-3) \\ &=3(-3)+2 \\ &=-7 \end{aligned} \nonumber \]

You can shorten the task even further if you are willing to do the function substitution and simplification in your head. First, evaluate g at 2, then f at the result.

\[f(g(2))=f(-3)=-7 \nonumber \]

Let’s look at another example of this unique way of combining functions.

Example \(\PageIndex{12}\)

Given \(f(x) = 5x + 2\) and \(g(x) = 3 − 2x\), evaluate \(g(f(a))\) and simplify the result.

We work the inner function evaluation in the expression \(g(f(a))\) first. Thus, to evaluate f(a), we substitute a for x in the definition \(f(x) = 5x + 2\) to get

\[g(f(a))=g(5 a+2) \nonumber \]

Now we need to evaluate \(g(5a + 2)\). To do this, we substitute \(5a + 2\) for x in the definition \(g(x) = 3 − 2x\) to get

\[g(5 a+2)=3-2(5 a+2) \nonumber \]

We can expand this last result and simplify. Thus,

\[g(f(a))=3-10 a-4=-10 a-1 \nonumber \]

Again, it is conventional to arrange the work in one continuous block, as follows.

\[\begin{aligned} g(f(a)) &=g(5 a+2) \\ &=3-2(5 a+2) \\ &=3-10 a-4 \\ &=-10 a-1 \end{aligned} \nonumber \]

Hence, \(g(f(a))=-10 a-1\).

Extracting the Domain of a Function

We’ve seen that the domain of a relation or function is the set of all the first coordinates of its ordered pairs. However, if a functional relationship is defined by an equation such as \(f(x) = 3x − 4\), then it is not practical to list all ordered pairs defined by this relationship. For any real x-value, you get an ordered pair. For example, if x = 5, then \(f(5) = 3(5) − 4 = 11\), leading to the ordered pair (5, f(5)) or (5, 11). As you can see, the number of such ordered pairs is infinite. For each new x-value, we get another function value and another ordered pair.

Therefore, it is easier to turn our attention to the values of x that yield real number responses in the equation \(f(x) = 3x − 4\). This leads to the following key idea.

If a function is defined by an equation, then the domain of the function is the set of “permissible x-values,” the values that produce a real number response defined by the equation.

We sometimes like to say that the domain of a function is the set of “OK x-values to use in the equation.” For example, if we define a function with the rule \(f(x) = 3x − 4\), it is immediately apparent that we can use any value we want for x in the rule \(f(x) = 3x − 4\). Thus, the domain of f is all real numbers. We can write that the domain \(D=\mathbb{R}\), or we can use interval notation and write that the domain \(D=(-\infty, \infty)\).

It is not the case that x can be any real number in the function defined by the rule \(f(x)=\sqrt{x}\). It is not possible to take the square root of a negative number.2 Therefore, x must either be zero or a positive real number. In set-builder notation, we can describe the domain with \(D=\{x : x \geq 0\}\). In interval notation, we write \(D=[0, \infty)\).

We must also be aware of the fact that we cannot divide by zero. If we define a function with the rule \(f(x)=x /(x-3)\), we immediately see that x = 3 will put a zero in the denominator. Division by zero is not defined. Therefore, 3 is not in the domain of f. No other x-value will cause a problem. The domain of f is best described with set-builder notation as \(D=\{x : x \neq 3\}\).

Functions Without Formulae

In the previous section, we defined functions by means of a formula, for example, as in

\[f(x)=\frac{x+3}{2-3 x} \nonumber \]

Euler would be pleased with this definition, for as we have said previously, Euler thought of functions as analytic expressions.

However, it really isn’t necessary to provide an expression or formula to define a function. There are other forms we can use to express a functional relationship: a graph, a table, or even a narrative description. The only thing that is really important is the requirement that the function be well-defined, and by “well-defined,” we mean that each object in the function’s domain is paired with one and only one object in its range.

As an example, let’s look at a special function \(\pi\) on the natural numbers,3 which returns the number of primes less than or equal to a given natural number. For example, the primes less than or equal to the number 23 are 2, 3, 5, 7, 11, 13, 17, 19, and 23, nine numbers in all. Therefore, the number of primes less than or equal to 23 is nine. In symbols, we would write

\[\pi(23)=9 \nonumber \]

Note the absence of a formula in the definition of this function. Indeed, the definition is descriptive in nature, so we might write

\[\pi(n)=\text { number of primes less than or equal to } n \nonumber \]

The important thing is not how we define this special function \(π\), but the fact that it is well-defined; that is, for each natural number \(n\), there are a fixed number of primes less than or equal to \(n\). Thus, each natural number in the domain of \(π\) is paired with one and only one number in its range.

Now, just because our function doesn’t provide an expression for calculating the number of primes less than or equal to a given natural number n, it doesn’t stop mathematicians from seeking such a formula. Euclid of Alexandria (325-265 BC), a Greek mathematician, proved that the number of primes is infinite, but it was the German mathematician and scientist, Johann Carl Friedrich Gauss (1777-1855), who first proposed that the number of primes less than or equal to n can be approximated by the formula

\[\pi(n) \approx \frac{n}{\ln n} \nonumber \]

where ln n is the “natural logarithm” of n (to be explained in Chapter 9). This approximation gets better and better with larger and larger values of n. The formula was refined by Gauss, who did not provide a proof, and the problem became known as the Prime Number Theorem. It was not until 1896 that Jacques Salomon Hadamard (1865-1963) and Charles Jean Gustave Nicolas Baron de la Vallee Poussin (1866-1962), working independently, provided a proof of the Prime Number Theorem.

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Functions: New Essays in the Philosophy of Psychology and Biology

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In explaining aspects of the natural world, including the aspects of mind, scientists have frequently used the concept of function. But what are functions? Here, 15 leading scholars of philosophy of psychology and philosophy of biology present new essays on functions.

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Functions Assignment

Question 1. A function assigns a unique output value to every input value. This definition implies that a vertical line cannot intersect a function in more than one place (Stewart). The set of input values is known as the domain of a function. The set of output values is known as the range of a function. An example of a linear function is, f left (x right ) = 4x+3 where 4 and 3 are the slope and vertical intercept of the straight line f(x). The domain of this linear function is all values x can take on, namely, all real numbers. The range is also all real numbers. A parabola is a good example of a quadratic function, f left (x right ) = {x} ^ {2} The vertex of the parabola coincides with the origin of a Cartesian coordinate system. The domain of this function is all real numbers and its range is all positive real numbers, f(x)≥0. An example of a third degree polynomial is, f left (x right ) = {x} ^ {3} +3 {x} ^ {2} +2x+1 The domain and range of this polynomial are all positive real numbers. The exponential function is defined as, f left (x right ) = {e} ^ {x} The domain of the exponential function is all real numbers and its range corresponds to all positive real numbers. The natural logarithm is a good example of a logarithmic function and is defined as, f left (x right ) = ln {x} The domain of the natural logarithm is all positive real numbers, x>0. The range is all real numbers. The trigonometric functions are good examples of periodic functions. For instance, the sine function, f left (x right ) = sin {x} It repeats every 2π radians. The domain of the sine function is all real numbers. The range is -1 ≤ f(x) ≤ 1. Question 2. The electrical activity of the human heart can be monitored by means of an Electrocardiogram (ECG). An example of ECG signal is shown in Figure 1. The signal was retrieved from National Instruments website (“National Instruments: Test, Measurement, and Embedded Systems”). The electrical heartbeats are represented by the voltage amplitude (dependent variable) over time (independent variable). The figure shows that heartbeats exhibit nearly periodic behavior. Strong beats repeat approximately every 1.6 seconds.

Question 3. The displacement of a piston in an internal combustion engine is represented by a periodic function, f left (t right ) = A sin {ωt} where ω is referred to as the angular frequency and has units of radians per second (rad/s). The independent variable is time t (s). A stands for the amplitude and describes the magnitude of the maximum displacement of the piston. The amplitude is a constant value. One cycle takes place when the crack (a linkage connecting the engine shaft and piston) turns through 2π radians. In other words, the sine function repeats itself when ωT=2π, where T is the period, so this time is defined as, T= {2π} over {ω} The frequency is the inverse of the period and it is measured in s-1 or hertz (hz) f= {1} over {T} For example, a period of 10s means a frequency of 1 every 10 seconds. The average value of function f(x) over an interval [a,b,] is defined as, F = {1} over {b-a} int from {a} to {b} {f(x)dx} The above expression can be used to calculate the average displacement over one cycle of the piston [0,T], F = {1} over {T} int from {0} to {T} {A sin {{2πt} over {T}} dt=} {1} over {T} {left [- {T} over {2π} cos {{2πt} over {T}} right ]} rsub {0} rsup {T} =- {1} over {2π} left [cos {2π} – cos {0} right ] =0 Question 4. The inverse of a function ( f -1) takes on a value from the range of f and assigns a single value from its domain. The inverse of a function exist only if a horizontal line does not intersect the function f in more than one place. This is not satisfied for the case of periodic functions like the sinusoidal displacement of a piston, y=f left (t right ) = A sin {ωt} The domain of sin ωt has to be restricted in order to define the inverse function. It can be worked out as follows, {sin} ^ {-1} {left ({y} over {A} right ) =ωt} {{f} ^ {-1} (y) = t = {1} over {ω} sin} ^ {-1} {left ({y} over {A} right )} The inverse function holds as long as, – {π} over {2} ≤ {y} over {A} ≤ {π} over {2} In a given cycle of the piston, the inverse function provides the time for which the piston is displaced y length units. Question 5. Part 1. The Richter scale is a scientific tool to measure the intensity of an earthquake (“Khan Academy | Khan Academy”). The following facts summarize the main features of the Richter scale: The Richter scale measures the earthquake magnitude. The intensity or amplitude of the ground motion is collected at the measuring stations. The maximum ground motion detected at a measuring station can be plotted against the distance to the epicenter. The Moment Magnitude Scale (MMS) was calibrated to the Richter scale. MMS is related to the energy released by earthquakes. Unfortunately, strong and weak earthquakes do not fit on the same plot if a linear scale is used. The Richter scale employs a logarithmic scale in order to present the data on the same chart. The Richter scale can be used to compare the magnitude of two earthquakes. For instance, the 5.8 earthquake that took place on the east coast of US in 2011and the 9.0 earthquake off the coast of Japan in the same year. Both magnitudes are reported on a logarithmic scale so the difference between these two earthquakes is not 3.2 but 103.2 (3.2 powers of 10). It means the earthquake in Japan was about 1600 times stronger than the east coast earthquake. Part 2. Table 1 summarizes the facts about the earthquakes chosen to draw a comparison.

Table 1. Earthquake information.

Part 3. Quantitative comparison of the two earthquakes. The equation for Richter magnitude can be written as, M = {log} rsub {10} {A} where A is the amplitude in millimeters directly measured from a seismograph. The equation may also include a distance correction factor originally used by Richter. It is omitted for the sake of clarity. The Richter magnitude of Pakistan’s 2013 earthquake can be related to its amplitude by, 7.7 = {log} rsub {10} {{A} rsub {Pakistan}} The equation above can be written as an exponential equation, that is, the amplitude read in the seismograph, {A} rsub {Pakista n} = {10} ^ {7.7} Similarly, the magnitude of Chile’s 2010 earthquake can be mathematically written as, 8.8 = {log} rsub {10} {{A} rsub {Chile}} It can also be recast as an exponential equation, {A} rsub {Chile} = {10} ^ {8.8} The ratio of exponential expressions can be used to compare the intensities of earthquakes, {{A} rsub {Chile}} over {{A} rsub {Pakistan}} = {{10} ^ {8.8}} over {{10} ^ {7.7}} =12.6 It can be concluded that Chile’s earthquake was 12.6 times as intense as the recently ground motion in Pakistan. Task 1. The exponential function ex can be written as an infinite Taylor series about x=0. The Taylor series is, {e} ^ {x} =1+x+ {{x} ^ {2}} over {2!} + {{x} ^ {3}} over {3!} + {{x} ^ {4}} over {4!} +… {{x} ^ {n}} over {n!} +… It can be used to approximate the value of e1 up to n= 10. The results are shown in Table 2. The approximation value for e1 up to 10 decimal places is obtained when n = 15, e = 2.7182818285

Table 2. Approximation value for e.

Works Cited

“Khan Academy | Khan Academy.” N. p., n.d. Web. 4 Dec. 2013. “National Instruments: Test, Measurement, and Embedded Systems.” N. p., n.d. Web. 3 Dec. 2013. Stewart, J. Calculus. Brooks/Cole, 2009.

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What is an Essay?

10 May, 2020

11 minutes read

Author:  Tomas White

Well, beyond a jumble of words usually around 2,000 words or so - what is an essay, exactly? Whether you’re taking English, sociology, history, biology, art, or a speech class, it’s likely you’ll have to write an essay or two. So how is an essay different than a research paper or a review? Let’s find out!

Defining the Term – What is an Essay?

The essay is a written piece that is designed to present an idea, propose an argument, express the emotion or initiate debate. It is a tool that is used to present writer’s ideas in a non-fictional way. Multiple applications of this type of writing go way beyond, providing political manifestos and art criticism as well as personal observations and reflections of the author.

An essay can be as short as 500 words, it can also be 5000 words or more.  However, most essays fall somewhere around 1000 to 3000 words ; this word range provides the writer enough space to thoroughly develop an argument and work to convince the reader of the author’s perspective regarding a particular issue.  The topics of essays are boundless: they can range from the best form of government to the benefits of eating peppermint leaves daily. As a professional provider of custom writing, our service has helped thousands of customers to turn in essays in various forms and disciplines.

Origins of the Essay

Over the course of more than six centuries essays were used to question assumptions, argue trivial opinions and to initiate global discussions. Let’s have a closer look into historical progress and various applications of this literary phenomenon to find out exactly what it is.

Today’s modern word “essay” can trace its roots back to the French “essayer” which translates closely to mean “to attempt” .  This is an apt name for this writing form because the essay’s ultimate purpose is to attempt to convince the audience of something.  An essay’s topic can range broadly and include everything from the best of Shakespeare’s plays to the joys of April.

The essay comes in many shapes and sizes; it can focus on a personal experience or a purely academic exploration of a topic.  Essays are classified as a subjective writing form because while they include expository elements, they can rely on personal narratives to support the writer’s viewpoint.  The essay genre includes a diverse array of academic writings ranging from literary criticism to meditations on the natural world.  Most typically, the essay exists as a shorter writing form; essays are rarely the length of a novel.  However, several historic examples, such as John Locke’s seminal work “An Essay Concerning Human Understanding” just shows that a well-organized essay can be as long as a novel.

The Essay in Literature

The essay enjoys a long and renowned history in literature.  They first began gaining in popularity in the early 16 th century, and their popularity has continued today both with original writers and ghost writers.  Many readers prefer this short form in which the writer seems to speak directly to the reader, presenting a particular claim and working to defend it through a variety of means.  Not sure if you’ve ever read a great essay? You wouldn’t believe how many pieces of literature are actually nothing less than essays, or evolved into more complex structures from the essay. Check out this list of literary favorites:

• The Book of My Lives by Aleksandar Hemon
• Notes of a Native Son by James Baldwin
• Against Interpretation by Susan Sontag
• High-Tide in Tucson: Essays from Now and Never by Barbara Kingsolver
• Slouching Toward Bethlehem by Joan Didion
• Naked by David Sedaris
• Walden; or, Life in the Woods by Henry David Thoreau

Pretty much as long as writers have had something to say, they’ve created essays to communicate their viewpoint on pretty much any topic you can think of!

The Essay in Academics

Not only are students required to read a variety of essays during their academic education, but they will likely be required to write several different kinds of essays throughout their scholastic career.  Don’t love to write?  Then consider working with a ghost essay writer !  While all essays require an introduction, body paragraphs in support of the argumentative thesis statement, and a conclusion, academic essays can take several different formats in the way they approach a topic.  Common essays required in high school, college, and post-graduate classes include:

Five paragraph essay

This is the most common type of a formal essay. The type of paper that students are usually exposed to when they first hear about the concept of the essay itself. It follows easy outline structure – an opening introduction paragraph; three body paragraphs to expand the thesis; and conclusion to sum it up.

Argumentative essay

These essays are commonly assigned to explore a controversial issue.  The goal is to identify the major positions on either side and work to support the side the writer agrees with while refuting the opposing side’s potential arguments.

Compare and Contrast essay

This essay compares two items, such as two poems, and works to identify similarities and differences, discussing the strength and weaknesses of each.  This essay can focus on more than just two items, however.  The point of this essay is to reveal new connections the reader may not have considered previously.

Definition essay

This essay has a sole purpose – defining a term or a concept in as much detail as possible. Sounds pretty simple, right? Well, not quite. The most important part of the process is picking up the word. Before zooming it up under the microscope, make sure to choose something roomy so you can define it under multiple angles. The definition essay outline will reflect those angles and scopes.

Descriptive essay

Perhaps the most fun to write, this essay focuses on describing its subject using all five of the senses.  The writer aims to fully describe the topic; for example, a descriptive essay could aim to describe the ocean to someone who’s never seen it or the job of a teacher.  Descriptive essays rely heavily on detail and the paragraphs can be organized by sense.

Illustration essay

The purpose of this essay is to describe an idea, occasion or a concept with the help of clear and vocal examples. “Illustration” itself is handled in the body paragraphs section. Each of the statements, presented in the essay needs to be supported with several examples. Illustration essay helps the author to connect with his audience by breaking the barriers with real-life examples – clear and indisputable.

Informative Essay

Being one the basic essay types, the informative essay is as easy as it sounds from a technical standpoint. High school is where students usually encounter with informative essay first time. The purpose of this paper is to describe an idea, concept or any other abstract subject with the help of proper research and a generous amount of storytelling.

Narrative essay

This type of essay focuses on describing a certain event or experience, most often chronologically.  It could be a historic event or an ordinary day or month in a regular person’s life. Narrative essay proclaims a free approach to writing it, therefore it does not always require conventional attributes, like the outline. The narrative itself typically unfolds through a personal lens, and is thus considered to be a subjective form of writing.

Persuasive essay

The purpose of the persuasive essay is to provide the audience with a 360-view on the concept idea or certain topic – to persuade the reader to adopt a certain viewpoint. The viewpoints can range widely from why visiting the dentist is important to why dogs make the best pets to why blue is the best color.  Strong, persuasive language is a defining characteristic of this essay type.

The Essay in Art

Several other artistic mediums have adopted the essay as a means of communicating with their audience.  In the visual arts, such as painting or sculpting, the rough sketches of the final product are sometimes deemed essays.  Likewise, directors may opt to create a film essay which is similar to a documentary in that it offers a personal reflection on a relevant issue.  Finally, photographers often create photographic essays in which they use a series of photographs to tell a story, similar to a narrative or a descriptive essay.

Drawing the line – question answered

“What is an Essay?” is quite a polarizing question. On one hand, it can easily be answered in a couple of words. On the other, it is surely the most profound and self-established type of content there ever was. Going back through the history of the last five-six centuries helps us understand where did it come from and how it is being applied ever since.

If you must write an essay, follow these five important steps to works towards earning the “A” you want:

• Understand and review the kind of essay you must write
• Brainstorm your argument
• Find research from reliable sources to support your perspective
• Cite all sources parenthetically within the paper and on the Works Cited page
• Follow all grammatical rules

Generally speaking, when you must write any type of essay, start sooner rather than later!  Don’t procrastinate – give yourself time to develop your perspective and work on crafting a unique and original approach to the topic.  Remember: it’s always a good idea to have another set of eyes (or three) look over your essay before handing in the final draft to your teacher or professor.  Don’t trust your fellow classmates?  Consider hiring an editor or a ghostwriter to help out!

If you are still unsure on whether you can cope with your task – you are in the right place to get help. HandMadeWriting is the perfect answer to the question “Who can write my essay?”

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Writing a research paper on ethics is not an easy task, especially if you do not possess excellent writing skills and do not like to contemplate controversial questions. But an ethics course is obligatory in all higher education institutions, and students have to look for a way out and be creative. When you find an […]

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Home — Essay Samples — Science — Linguistics — Four Functions Of Language

Four Functions of Language

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Published: Mar 13, 2024

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Table of contents

Expressive function, informative function, directive function, aesthetic function.

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Argumentative Essay

Definition of argumentative essay.

An argumentative essay is a type of essay that presents arguments about both sides of an issue. It could be that both sides are presented equally balanced, or it could be that one side is presented more forcefully than the other. It all depends on the writer, and what side he supports the most. The general structure of an argumentative essay follows this format:

• Introduction : Attention Grabber/ hook , Background Information , Thesis Statement
• Body : Three body paragraphs (three major arguments)
• Counterargument : An argument to refute earlier arguments and give weight to the actual position
• Conclusion : Rephrasing the thesis statement , major points, call to attention, or concluding remarks .

Models for Argumentative Essays

There are two major models besides this structure given above, which is called a classical model. Two other models are the Toulmin and Rogerian models.

Toulmin model is comprised of an introduction with a claim or thesis, followed by the presentation of data to support the claim. Warrants are then listed for the reasons to support the claim with backing and rebuttals. However, the Rogerian model asks to weigh two options, lists the strengths and weaknesses of both options, and gives a recommendation after an analysis.

Five Types of Argument Claims in Essay Writing  

There are five major types of argument claims as given below.

• A claim of definition
• A claim about values
• A claim about the reason
• A claim about comparison
• A claim about policy or position

A writer makes a claim about these issues and answers the relevant questions about it with relevant data and evidence to support the claim.

Three Major Types of Argument and How to Apply Them

Classical argument.

This model of applying argument is also called the Aristotelian model developed by Aristotle. This type of essay introduces the claim, with the opinion of the writer about the claim, its both perspectives, supported by evidence, and provides a conclusion about the better perspective . This essay includes an introduction, a body having the argument and support, a counter-argument with support, and a conclusion.

Toulmin Argument

This model developed by Stephen Toulmin is based on the claim followed by grounds, warrant, backing, qualifier, and rebuttal . Its structure comprises, an introduction having the main claim, a body with facts and evidence, while its rebuttal comprises counter-arguments and a conclusion.

Rogerian Argument

The third model by Carl Rogers has different perspectives having proof to support and a conclusion based on all the available perspectives. Its structure comprises an introduction with a thesis, the opposite point of view and claim, a middle-ground for both or more perspectives, and a conclusion.

Four Steps to Outline and Argumentative Essay

There are four major steps to outlining an argumentative essay.

• Introduction with background, claim, and thesis.
• Body with facts, definition, claim, cause and effect, or policy.
• The opposing point of view with pieces of evidence.

Examples of Argumentative Essay in Literature

Example #1: put a little science in your life by brian greene.

“When we consider the ubiquity of cellphones, iPods, personal computers and the Internet, it’s easy to see how science (and the technology to which it leads) is woven into the fabric of our day-to-day activities . When we benefit from CT scanners, M.R.I. devices, pacemakers and arterial stents, we can immediately appreciate how science affects the quality of our lives. When we assess the state of the world, and identify looming challenges like climate change, global pandemics, security threats and diminishing resources, we don’t hesitate in turning to science to gauge the problems and find solutions. And when we look at the wealth of opportunities hovering on the horizon—stem cells, genomic sequencing, personalized medicine, longevity research, nanoscience, brain-machine interface, quantum computers, space technology—we realize how crucial it is to cultivate a general public that can engage with scientific issues; there’s simply no other way that as a society we will be prepared to make informed decisions on a range of issues that will shape the future.”

These two paragraphs present an argument about two scientific fields — digital products and biotechnology. It has also given full supporting details with names.

Example #2: Boys Here, Girls There: Sure, If Equality’s the Goal by Karen Stabiner

“The first objections last week came from the National Organization for Women and the New York Civil Liberties Union, both of which opposed the opening of TYWLS in the fall of 1996. The two groups continue to insist—as though it were 1896 and they were arguing Plessy v. Ferguson—that separate can never be equal. I appreciate NOW ’s wariness of the Bush administration’s endorsement of single-sex public schools, since I am of the generation that still considers the label “feminist” to be a compliment—and many feminists still fear that any public acknowledgment of differences between the sexes will hinder their fight for equality .”

This paragraph by Karen Stabiner presents an objection to the argument of separation between public schools. It has been fully supported with evidence of the court case.

Example #3: The Flight from Conversation by Sherry Turkle

“We’ve become accustomed to a new way of being “ alone together.” Technology-enabled, we are able to be with one another, and also elsewhere, connected to wherever we want to be. We want to customize our lives. We want to move in and out of where we are because the thing we value most is control over where we focus our attention. We have gotten used to the idea of being in a tribe of one, loyal to our own party.”

This is an argument by Sherry Turkle, who beautifully presented it in the first person plural dialogues . However, it is clear that this is part of a greater argument instead of the essay.

Function of Argumentative Essay

An argumentative essay presents both sides of an issue. However, it presents one side more positively or meticulously than the other one, so that readers could be swayed to the one the author intends. The major function of this type of essay is to present a case before the readers in a convincing manner, showing them the complete picture.

Synonyms of Argumentative Essay

Argumentative Essay synonyms are as follows: persuasive essays, research essays, analytical essays, or even some personal essays.

Related posts:

• Elements of an Essay
• Narrative Essay
• Definition Essay
• Descriptive Essay
• Types of Essay
• Analytical Essay
• Cause and Effect Essay
• Critical Essay
• Expository Essay
• Persuasive Essay
• Process Essay
• Explicatory Essay
• An Essay on Man: Epistle I
• Comparison and Contrast Essay

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Descriptive Essay: Function, Purpose and Process

Table of Contents

Descriptive essay writing tells a story through adjectives and details. It is more than analytical writing because it gives room for vivid scenes and strong expressions to make a point.

A descriptive essay describes anything from an object to a person. It’s a form of writing that uses sensory information to enable readers to use their five senses to understand the topic.

The objective of a descriptive essay is to present enough information for the reader to visualize the topic.

This article examines the function, purpose, and steps of crafting a stellar descriptive essay. Let’s dive in!

Why are Students Assigned Descriptive Essays?

A descriptive essay provides a full explanation of the subject and creates a mental image for the reader. It could be about any object, location, person, feeling, or circumstance. It appears similar to writing a narrative essay but is distinct and gives opposing viewpoints.

Descriptive essays are frequently assigned to high school and college students. This is because students’ writing skills are honed through descriptive essays, which in turn benefits their future academic and professional endeavors.

A descriptive essay promotes student’s ability to evoke a mental image by appealing to the reader’s senses.

Purpose of a Descriptive Essay

The objective of a descriptive essay is to describe a person, item, location, or scenario . A descriptive speech or essay paints a vivid image of the subject for the reader. The author employs vivid language to help the reader comprehend the essay’s subject.

Descriptive essays do not require convincing readers or providing evidence to substantiate a point as an argumentative essay requires. Instead, it strives to provide the reader with all the pertinent information regarding the chosen topic to facilitate comprehension.

Elements of a Descriptive Essay

There are elements to descriptive essay writing. These elements include:

1. Sensory Details

It entails evoking the readers’ emotions and establishing a connection with them. They engage the reader’s senses, such as sight, touch, smell, and taste, to construct a vivid image of the topic.

2. Figurative Language

Figurative language is a crucial component of a descriptive essay. The subject’s character profile is created using metaphors, similes, adjectives, adverbs, etc.

This sketch lets the reader experience what the author felt about the topic and envision the topic.

3. Central Theme

A descriptive essay has a central theme. The principal subject develops and leads the essay’s content and facilitates the organization of its specifics. It should be well-defined and singularly focused.

4. Precise Language

The impression of your essay is dependent on the language you have employed. The wording should underline the essay’s fundamental idea and purpose. Avoid using imprecise and unclear language.

5. Organized Structure

A well-organized structure is crucial to this essay. Also essential are the chronology, physical position, and order.

How to Write a Descriptive Essay?

Writing a successful descriptive essay requires selecting a topic, constructing an outline, organizing ideas, and including pertinent details.

Let’s view the process of descriptive writing in details:

1. Select a Topic

Selecting a strong topic for your essay is essential. The topic of the essay should be captivating so that the reader is compelled to stay with you throughout the essay.

2. Construct an Outline

Create an outline for your descriptive essay to organize your facts in a logical order. It will assist you in managing your essay and serve as a reminder to include all sensory information.

3. Create an Introduction

The essay’s first section is the introduction. It presents the main issue and contains a strong opinion that establishes the essay’s first impression. The introduction provides a concise outline of the essay’s content.

4. Create an enlightening thesis statement

A descriptive essay’s thesis statement specifies the essay’s scope and aim. It is a small topic line that must be concise and exact. Creatively write the statement and employ descriptive language.

Creating intrigue in the thesis statement draws the reader into the essay’s body.

5. Develop Body Paragraphs

The body paragraphs reinforce the introduction and purpose and adhere to the direction outlined in the thesis. Use subject statements as the opening sentence of each paragraph and provide evidence to support them.

Next, connect your paragraphs with transitional phrases. The majority of transitional words in this article are conjunctions. Use conjunctions that are suitable for your writing.

In the case of experience and memories, arrange your paragraphs in an orderly fashion.

6. Give an Effective Conclusion

As the conclusion is the final section of an essay, this is your final opportunity to wow the reader. It summarizes the essay and suggests a route ahead.

You may use phrases such as “to conclude,” “in conclusion,” and “finally” to indicate the conclusion of the essay. Your conclusion will restate the thesis statement and describe the findings if it is a research paper.

The closing sentence of an essay should summarize the purpose of the essay and leave a lasting impression.

Students are given the option of using the descriptive essay as a writing assignment to improve the quality of their writing. The objective of a descriptive essay is to present a known concept or issue from a new angle , revealing what lies under the surface.

Abir Ghenaiet

Abir is a data analyst and researcher. Among her interests are artificial intelligence, machine learning, and natural language processing. As a humanitarian and educator, she actively supports women in tech and promotes diversity.

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The Function and Structures of the Human Heart Essay

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• As a source of information (ensure proper referencing)
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Introduction

The function of the heart, types of blood circulation, the external structure of the heart, the internal structure of the heart.

The human heart is a critical organ in the nervous system, and it is responsible for sustaining life throughout its lifespan. The muscular organ is characterized by four major functionally distinct chambers and several valves responsible for regulating the normal flow of blood within the body (Litviňuková et al., 2020). It is found at the center of the chest between the lungs, underneath the thoracic cavity.

The primary function of the human heart is to ensure that blood constantly flows by pumping it throughout the body (Willie et al., 2015). It is also responsible for delivering and replenishing oxygen, hormones, glucose, nutrients, and other components to the cells and tissues in various parts of the body (Miranda, 2019; Willie et al., 2015). Maintaining a constant blood flow also helps in ensuring adequate blood pressure in the body. It also removes carbon dioxide and waste from the body system.

There are three major types of blood circulation. Pulmonary circulation involves the transfer of deoxygenated blood from the heart to the lungs and then the movement of oxygenated blood from the lungs back to the heart (Visible Body, 2020).

Systematic circulation is distinct from pulmonary circulation in the sense that oxygenated blood is carried from the heart to all other organs and tissues in the body, and deoxygenated blood is transferred back to the heart (Visible Body, 2020).

Coronary circulation is responsible for replenishing the heart with oxygen and involves the supply of oxygenated blood to the heart. Other organs, such as the brain, depend hugely on the steady supply of fresh, oxygenated blood to survive (Miranda, 2019).

The pericardium and walls of the heart make up the external structure. The pericardium is the membrane covering the heart, which comprises two layers: serous and fibrous (Saikat, 2016). It has three distinct layers (myocardium, endocardium, and epicardium) and a cavity, as illustrated above. It is responsible for preventing excessive stretching, insufflation, or inflation of the heart and protecting it by secreting a lubricant that reduces the risk of friction with adjacent organs (Saikat, 2016; Miranda, 2019). Moreover, the pericardium is responsible for holding this organ in its position (Saikat, 2016).

The internal structure of the heart is more complex and consists of four chambers (which differ in terms of their morphology and function) and several blood vessels and valves which are responsible for controlling the flow of blood in the body (Litviňuková et al., 2020).

The four chambers (the left atrium, right atrium, left ventricle, and right ventricle) receive blood and pump it away from the heart.

The heart has three major blood vessels (veins, capillaries, and arteries) which distribute blood throughout the body.

Finally, the heart has two types of valves (atrioventricular valves and semilunar valves) which ensure that the blood flows in a single direction (Miranda, 2019).

Litviňuková, Talavera-López, C., Maatz, H., Reichart, D., Worth, C., Lindberg, E., Kanda, M., Polanski,K., Heinig, M., Lee, M., Nadelmann, E., Roberts, K., Tuck, L., Fasouli, E., DeLaughter, D., McDonough, B., Wakimoto, H., Gorham, J., Samari, S., … Teichmann, S. A. (2020). Cells of the adult human heart. Nature . Web.

Miranda, M., G. (2019 ). Structure and function of the heart. New Medical. Web.

Saikat, R. (2016). What is the structure of the pericardium? What is its function ? Socratic. Web.

Visible Body (2020). Pulmonary Circulation and systemic circulation: the routes and function of blood flow . Web.

Willie, C. K., Tzeng, Y. C., Fisher, J. A., & Ainslie, P. N. (2014). Integrative regulation of human brain blood flow. The Journal of Physiology , 592 (5), 841–859.

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IvyPanda. (2023, January 23). The Function and Structures of the Human Heart. https://ivypanda.com/essays/the-function-and-structures-of-the-human-heart/

"The Function and Structures of the Human Heart." IvyPanda , 23 Jan. 2023, ivypanda.com/essays/the-function-and-structures-of-the-human-heart/.

IvyPanda . (2023) 'The Function and Structures of the Human Heart'. 23 January.

IvyPanda . 2023. "The Function and Structures of the Human Heart." January 23, 2023. https://ivypanda.com/essays/the-function-and-structures-of-the-human-heart/.

1. IvyPanda . "The Function and Structures of the Human Heart." January 23, 2023. https://ivypanda.com/essays/the-function-and-structures-of-the-human-heart/.

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Understanding the Role and Function of Somatic Motor Neurons

This essay is about the essential role of somatic motor neurons in enabling voluntary movement by transmitting signals from the central nervous system to skeletal muscles. It explains the process of signal transmission, where motor neurons convert electrical signals from the brain into chemical signals that trigger muscle contractions. The essay highlights the unique structure of these neurons, their involvement in reflex actions, and their role in motor learning and memory. It also discusses the impact of motor neuron dysfunction in diseases like ALS and SMA, and explores advances in neurorehabilitation and prosthetics. The essay underscores the significance of somatic motor neurons in human physiology and medical research.

How it works

The somatic motor neuron emerges as a captivating and indispensable constituent of the human nervous system, playing a pivotal role in our locomotion and interaction with the surroundings. These specialized nerve cells undertake the transmission of signals from the central nervous system (CNS) to skeletal muscles, thereby facilitating voluntary movement. To truly grasp the significance of somatic motor neurons, it’s imperative to delve into their function, structure, and the ramifications of their malfunction on human health.

Somatic motor neurons serve as linchpins in the process of signal transmission.

Originating in the spinal cord and brainstem, these neurons extend their axons to skeletal muscles throughout the body. When the brain initiates a movement, an electrical signal emanates from the motor cortex. This signal traverses down the spinal cord to the pertinent motor neuron. Subsequently, the motor neuron transmutes the electrical signal into a chemical one by discharging neurotransmitters at the neuromuscular junction, a specialized synapse linking the motor neuron and the muscle fiber. This chemical signal, predominantly acetylcholine, adheres to receptors on the muscle fiber’s surface, instigating an action potential that induces muscle contraction. This intricate process enables us to execute a broad spectrum of voluntary movements, ranging from the simple gesture of raising a finger to the intricate coordination requisite for playing a musical instrument.

The configuration of somatic motor neurons is intricately tailored to their function. These neurons boast elongated axons capable of spanning significant distances to reach their target muscles. Situated within the spinal cord or brainstem, the cell body shelters the nucleus and serves as the hub for signal integration and processing. The dendrites, emanating from the cell body, receive input from other neurons, including interneurons and sensory neurons. This architecture ensures the swift and efficient transmission of signals from the CNS to the muscles, facilitating precise and punctual movements.

In addition to their involvement in voluntary movement, somatic motor neurons partake in reflex actions. Reflexes, innate and involuntary responses to specific stimuli, such as the knee-jerk reaction elicited by tapping the patellar tendon, are orchestrated by these neurons. When a reflex is triggered, sensory neurons perceive the stimulus and convey the information to the spinal cord. Here, the sensory neurons directly interact with motor neurons, circumventing the brain to expedite a response. This mechanism facilitates swift, protective reactions to potentially hazardous stimuli, such as retracting one’s hand from a scorching surface, sans the delay of conscious cognition.

Impairment or malfunction of somatic motor neurons can precipitate severe motor deficits. Conditions like amyotrophic lateral sclerosis (ALS) and spinal muscular atrophy (SMA) specifically target these neurons, resulting in progressive muscle weakness and dwindling voluntary movement. ALS, colloquially known as Lou Gehrig’s disease, manifests as a neurodegenerative ailment wherein motor neurons progressively degenerate and perish. As these neurons degrade, the brain forfeits its capacity to instigate and regulate muscle movements, culminating in the gradual paralysis of voluntary muscles. In contrast, SMA, a genetic malady, entails the loss of motor neurons in the spinal cord and brainstem, leading to muscular atrophy and enfeeblement, notably in muscles proximal to the body’s core. Comprehending the root causes of these conditions is imperative for devising efficacious treatments and therapies.

Exploration into somatic motor neurons bears extensive implications, particularly in the realms of neurorehabilitation and prosthetics. Strides in neuroscience and technology are paving the way for the development of brain-computer interfaces (BCIs) capable of circumventing impaired motor pathways. BCIs decipher neural signals from the brain and translate them into directives for prosthetic limbs or other assistive devices. This innovation holds immense potential for reinstating a degree of autonomy to individuals grappling with motor neuron maladies or spinal cord injuries. By amalgamating our comprehension of motor neuron function with cutting-edge technology, we can enhance the quality of life for those beset by motor impairments.

Furthermore, the scrutiny of somatic motor neurons proffers valuable insights into fundamental human physiology and the intricate interplay between the nervous and muscular systems. These neurons constitute the cornerstone of our capacity to engage with our surroundings, execute routine tasks, and undertake complex endeavors. The intricate interplay between electrical and chemical signals within somatic motor neurons underscores the marvel of human movement and the prospect for scientific progression to fortify human health and aptitude.

Somatic motor neurons also contribute to the genesis and execution of motor learning and memory. Motor learning denotes the process by which we assimilate and refine new motor skills through practice and repetition. This process engenders alterations in the potency and efficiency of synaptic connections between neurons, a phenomenon termed synaptic plasticity. When acquiring a novel skill, such as playing the piano or cycling, somatic motor neurons collaborate with other neural circuits to encode and preserve the motor patterns associated with that skill. Iterative practice fortifies these connections, rendering movements more precise and automatic over time. This capacity to acquire and hone new motor skills attests to the remarkable adaptability of the human nervous system.

Another captivating facet of somatic motor neurons is their involvement in motor unit recruitment and regulation. A motor unit encompasses a solitary motor neuron and all the muscle fibers it innervates. The magnitude of a motor unit varies, with diminutive motor units found in muscles responsible for nuanced, precise movements, such as those in the digits and ocular muscles, and substantial motor units found in muscles responsible for robust, forceful movements, like those in the lower extremities and dorsal musculature. During movement execution, the CNS enlists motor units in a sequential manner, known as the size principle. Smaller motor units, furnishing lesser force yet superior precision, are enlisted foremost. As the exigency for force amplifies, larger motor units are gradually recruited. This methodical enlistment facilitates fluid, regulated movements and the efficient utilization of muscle fibers.

Besides their involvement in voluntary movement and reflexes, somatic motor neurons contribute to the maintenance of muscle tone. Muscle tone denotes the perpetual, passive partial contraction of muscles, which aids in upholding posture and readiness for action. This state of partial contraction is governed by the tonic activity of motor neurons, which incessantly dispatch low-level signals to the muscles. Muscle tone is imperative for stabilizing joints, preserving equilibrium, and facilitating prompt responses to abrupt changes in position or load.

Various factors, including genetics, environmental exposures, and lifestyle choices, can influence the health and functionality of somatic motor neurons. For instance, regular physical activity and exercise have been evinced to bolster motor neuron health and bolster neuromuscular function. Exercise elicits the release of neurotrophic factors, proteins that bolster the growth, viability, and differentiation of neurons. These factors aid in preserving the integrity of motor neurons and their connections with muscle fibers. Conversely, a sedentary lifestyle and inadequate nutrition can compromise motor neuron health, precipitating muscle weakness and heightening susceptibility to neurodegenerative maladies.

The arena of somatic motor neuron research is perpetually evolving, with novel revelations shedding light on the intricacies of motor control and neuromuscular function. Progressions in imaging modalities, such as functional magnetic resonance imaging (fMRI) and positron emission tomography (PET), afford scientists the ability to visualize and scrutinize the activity of motor neurons in real-time. These technologies furnish invaluable insights into the orchestration of movement by the brain and spinal cord, as well as how motor neurons adapt to diverse tasks and challenges.

Furthermore, nascent therapies and interventions hold promise for ameliorating motor neuron maladies and injuries. Gene therapy, for instance, is being explored as a prospective treatment for SMA, with the objective of delivering functional copies of the defective gene to afflicted motor neurons. Similarly, stem cell therapy is under scrutiny as a means of replenishing damaged motor neurons and fostering neural regeneration. These avant-garde approaches epitomize the vanguard of medical research and offer optimism for individuals grappling with debilitating neuromuscular conditions.

In summation, somatic motor neurons are indispensable to our capacity to move, learn, and interact with our milieu. Their specialized configuration and function enable the precise transmission of signals from the CNS to skeletal muscles, thereby facilitating voluntary movement, reflexes, and muscle tone. The scrutiny of these neurons furnishes invaluable insights into human physiology, the mechanisms of motor learning and memory, and the potential for innovative treatments and technologies to address motor impairments. As research in this sphere progresses, we can anticipate novel discoveries and interventions that will augment our comprehension of motor control and enhance the lives of those afflicted by neuromuscular conditions. The remarkable adaptability and resilience of somatic motor neurons underscore the incredible complexity and capability of the human nervous system, engendering continual exploration and innovation in the quest to unlock its full potential.

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Mathematics > Number Theory

Title: report on some papers related to the function $\mathop{\mathcal r }(s)$ found by siegel in riemann's posthumous papers.

Abstract: In a letter to Weierstrass Riemann asserted that the number $N_0(T)$ of zeros of $\zeta(s)$ on the critical line to height $T$ is approximately equal to the total number of zeros to this height $N(T)$. Siegel studied some posthumous papers of Riemann trying to find a proof of this. He found a function $\mathop{\mathcal R }(s)$ whose zeros are related to the zeros of the function $\zeta(s)$. Siegel concluded that Riemann's papers contained no ideas for a proof of his assertion, connected the position of the zeros of $\mathop{\mathcal R }(s)$ with the position of the zeros of $\zeta(s)$ and asked about the position of the zeros of $\mathop{\mathcal R }(s)$. This paper is a summary of several papers that we will soon upload to arXiv, in which we try to answer Siegel's question about the position of the zeros of $\mathop{\mathcal R }(s)$. The articles contain also improvements on Siegel's results and also other possible ways to prove Riemann's assertion, but without achieving this goal.

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Guest Essay

The Long-Overlooked Molecule That Will Define a Generation of Science

By Thomas Cech

Dr. Cech is a biochemist and the author of the forthcoming book “The Catalyst: RNA and the Quest to Unlock Life’s Deepest Secrets,” from which this essay is adapted.

From E=mc² to splitting the atom to the invention of the transistor, the first half of the 20th century was dominated by breakthroughs in physics.

Then, in the early 1950s, biology began to nudge physics out of the scientific spotlight — and when I say “biology,” what I really mean is DNA. The momentous discovery of the DNA double helix in 1953 more or less ushered in a new era in science that culminated in the Human Genome Project, completed in 2003, which decoded all of our DNA into a biological blueprint of humankind.

DNA has received an immense amount of attention. And while the double helix was certainly groundbreaking in its time, the current generation of scientific history will be defined by a different (and, until recently, lesser-known) molecule — one that I believe will play an even bigger role in furthering our understanding of human life: RNA.

You may remember learning about RNA (ribonucleic acid) back in your high school biology class as the messenger that carries information stored in DNA to instruct the formation of proteins. Such messenger RNA, mRNA for short, recently entered the mainstream conversation thanks to the role they played in the Covid-19 vaccines. But RNA is much more than a messenger, as critical as that function may be.

Other types of RNA, called “noncoding” RNAs, are a tiny biological powerhouse that can help to treat and cure deadly diseases, unlock the potential of the human genome and solve one of the most enduring mysteries of science: explaining the origins of all life on our planet.

Though it is a linchpin of every living thing on Earth, RNA was misunderstood and underappreciated for decades — often dismissed as nothing more than a biochemical backup singer, slaving away in obscurity in the shadows of the diva, DNA. I know that firsthand: I was slaving away in obscurity on its behalf.

In the early 1980s, when I was much younger and most of the promise of RNA was still unimagined, I set up my lab at the University of Colorado, Boulder. After two years of false leads and frustration, my research group discovered that the RNA we’d been studying had catalytic power. This means that the RNA could cut and join biochemical bonds all by itself — the sort of activity that had been thought to be the sole purview of protein enzymes. This gave us a tantalizing glimpse at our deepest origins: If RNA could both hold information and orchestrate the assembly of molecules, it was very likely that the first living things to spring out of the primordial ooze were RNA-based organisms.

That breakthrough at my lab — along with independent observations of RNA catalysis by Sidney Altman at Yale — was recognized with a Nobel Prize in 1989. The attention generated by the prize helped lead to an efflorescence of research that continued to expand our idea of what RNA could do.

In recent years, our understanding of RNA has begun to advance even more rapidly. Since 2000, RNA-related breakthroughs have led to 11 Nobel Prizes. In the same period, the number of scientific journal articles and patents generated annually by RNA research has quadrupled. There are more than 400 RNA-based drugs in development, beyond the ones that are already in use. And in 2022 alone, more than $1 billion in private equity funds was invested in biotechnology start-ups to explore frontiers in RNA research.

What’s driving the RNA age is this molecule’s dazzling versatility. Yes, RNA can store genetic information, just like DNA. As a case in point, many of the viruses (from influenza to Ebola to SARS-CoV-2) that plague us don’t bother with DNA at all; their genes are made of RNA, which suits them perfectly well. But storing information is only the first chapter in RNA’s playbook.

Unlike DNA, RNA plays numerous active roles in living cells. It acts as an enzyme, splicing and dicing other RNA molecules or assembling proteins — the stuff of which all life is built — from amino acid building blocks. It keeps stem cells active and forestalls aging by building out the DNA at the ends of our chromosomes.

RNA discoveries have led to new therapies, such as the use of antisense RNA to help treat children afflicted with the devastating disease spinal muscular atrophy. The mRNA vaccines, which saved millions of lives during the Covid pandemic, are being reformulated to attack other diseases, including some cancers . RNA research may also be helping us rewrite the future; the genetic scissors that give CRISPR its breathtaking power to edit genes are guided to their sites of action by RNAs.

Although most scientists now agree on RNA's bright promise, we are still only beginning to unlock its potential. Consider, for instance, that some 75 percent of the human genome consists of dark matter that is copied into RNAs of unknown function. While some researchers have dismissed this dark matter as junk or noise, I expect it will be the source of even more exciting breakthroughs.

We don’t know yet how many of these possibilities will prove true. But if the past 40 years of research have taught me anything, it is never to underestimate this little molecule. The age of RNA is just getting started.

Thomas Cech is a biochemist at the University of Colorado, Boulder; a recipient of the Nobel Prize in Chemistry in 1989 for his work with RNA; and the author of “The Catalyst: RNA and the Quest to Unlock Life’s Deepest Secrets,” from which this essay is adapted.

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Novel engineering of single-metals (TM: Cr, Mo, W) chemical tailoring of Pt-encapsulated fullerenes (Pt@C59TM) as dual sensors for H2CO and H2S gases: A theoretical study

• Original Paper
• Published: 03 June 2024

Cite this article

• Daniel Etiese 1 ,
• Ismail O. Amodu 2 ,
• Henry O. Edet   ORCID: orcid.org/0000-0002-3642-7298 3 ,
• Adedapo S. Adeyinka 4 &
• Hitler Louis 1 , 5  

This theoretical study explores the novel engineering of group 6 transition metals (Cr, Mo, W) functionalized Pt-encapsulated fullerenes (Pt@C 59 TM) as dual sensors for H 2 CO and H 2 S gases. Using appropriate density functional theory (DFT) calculations at the PBE0/GenECP/Def2svp/LanL2DZ method, the research investigates the adsorption characteristics and sensing capabilities of Pt@C 59 TM complexes. Various analyses, including adsorption energy, molecular dynamics, electronic structure, and thermodynamics among others, are employed to assess the interactions between the sensor surfaces and target gases. The study reveals unique reactivity patterns, with Pt@C 59 W identified as the most responsive (reactive) surface with an N max value of 7.056 eV. Additionally, insights into the stabilization mechanisms revealed that stabilization further increased after the adsorption of the gases. Due to the nucleophilic nature of the gases, the Pt@C 59 W surface showcased the highest adsorption energy for both gases (-590.849 kcal/mol for H 2 S and -525.629 kcal/mol for H 2 CO). The negative values of the entropy change (ΔS) for all the systems further ascertain the values of the adsorption energy, indicating that the interaction between the adsorbent and the adsorbate is chemisorption. The sensing dynamics of the systems denoted a work function value increasing for all complexes with a moderate recovery time, necessitating an excellent H 2 S and H 2 CO gas sensor materials. Two materials, H 2 S-Pt@C 59 W and H 2 CO-Pt@C 59 W, showed significantly higher levels of FET (Fraction Electron Transfer) compared to others. These values, 0.000377 and 0.003101, respectively, suggest that these materials strongly bind and are very stable during the adsorption process. The non-covalent interactions contribute to a comprehensive understanding of the engineered sensor materials. This theoretical exploration provides a foundation for the design and development of highly efficient dual sensors for H 2 CO and H 2 S gases, with potential applications in environmental monitoring and industrial safety.

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Agwamba EC, Ukono AD, Louis H, Mathias GE, Benjamin I, Ikenyirimba OJ, Etiese D, Ahauekwe EF, (2023) Adsorption mechanism of AsH 3 pollutant on metal-functionalized coronene C 24 H 12 -X (X = Mg, Al, K) quantum dots. Chem Phys Impact 6:100224

Article   Google Scholar  

Ahmed SA, Rahman AA, Elsayed KN, Abd El-Mageed HR, Mohamed HS, Ahmed SA (2021) Cytotoxic activity, molecular docking, pharmacokinetic properties and quantum mechanics calculations of the brown macroalga Cystoseira trinodis compounds. J Biomol Struct Dyn 39(11):3855–3873

Article   CAS   PubMed   Google Scholar  

Al-Zaqri N, Pooventhiran T, Rao DJ, Alsalme A, Warad I, Thomas R (2021) Structure, conformational dynamics, quantum mechanical studies and potential biological activity analysis of multiple sclerosis medicine ozanimod. J Mol Struct 1227:129685

Article   CAS   Google Scholar  

Awang Z (2014) Gas Sensors: A Review Sens Transducers 168(4):61–75

Google Scholar  

Baskar AV, Benzigar MR, Talapaneni SN, Singh G, Karakoti AS, Yi J, Al-Muhtaseb AA, Ariga K, Ajayan PM, Vinu A (2022) Self-assembled fullerene nanostructures: synthesis and applications. Adv Funct Mater 32(6):2106924

Bastuck M (2019) Improving the performance of gas sensor systems with advanced data evaluation, operation, and calibration methods. Saarlandische Universitats-Undlandesbibliothek

Batterman S, Grant-Alfieri A, Seo SH (2023) Low level exposure to hydrogen sulfide: A review of emissions, community exposure, health effects, and exposure guidelines. Crit Rev Toxicol 53(4):244–295

Bayoumi AM, Elhaes HGAA, Osman O, Hussein T, Ibrahim MA (2020) Mapping molecular electrostatic potential for heme interacting with nano metal oxides. Biointerface Res Appl Chem 1:2020

Bayoumy AM, Ibrahim M, Omar A (2020) Mapping molecular electrostatic potential (MESP) for fulleropyrrolidine and its derivatives. Opt Quant Electron 52:1–13

Boyen HG, Ziemann P, Wiedwald U, Ivanova V, Kolb DM, Sakong S, Gross A, Romanyuk A, Büttner M, Oelhafen P (2006) Local density of states effects at the metal-molecule interfaces in a molecular device. Nat Mater 5(5):394–399

Chen X, Leishman M, Bagnall D, Nasiri N (2021) Nanostructured gas sensors: From air quality and environmental monitoring to healthcare and medical applications. Nanomaterials 11(8):1927

Article   CAS   PubMed   PubMed Central   Google Scholar  

Chen Y, Yao L, Wang F (2023) Hydrogen-bonding interactions involving the Imidazol-2-ylidene and its Heavy-atom analogues. Comput Theor Chem 1220:114020

Concas F, Mineraud J, Lagerspetz E, Varjonen S, Liu X, Puolamäki K, and Tarkoma S (2021) Low-cost outdoor air quality monitoring and sensor calibration A survey and critical analysis. ACM Transactions on Sensor Networks (TOSN) 17(2):1–44

Cortés-Guzmán F, Rodríguez JI, Anderson JS (2023) Introduction to QTAIM and beyond. In: Advances in Quantum Chemical Topology Beyond QTAIM. Elsevier, pp 1–19

Daghar C, Issaoui N, Roisnel T, Dorcet V, Marouani H (2021) Empirical and computational studies on newly synthesis cyclohexylammonium perchlorate. J Mol Struct 1230:129820

Dai J, Ogbeide O, Macadam N, Sun Q, Yu W, Li Y, Huang W (2020) Printed gas sensors. Chem Soc Rev 49(6):1756–1789

Dai ZN, Sheng W, Zhou XY, Zhan J, Xu Y (2023) A novel broken-gap chemical-bonded SiC/Ti 2 CO 2 heterojunction with band to band tunneling: first-principles investigation. Phys Chem Chem Phys 25(35):23954–23962

Dennington R, Keith TA, Millam JM (2016) GaussView 6.0. 16. Semichem Inc.: Shawnee Mission, KS, USA. HyperChem, T. (2001). HyperChem 8.07, HyperChem Professional Program. Gainesville, Hypercube

Dey A (2018) Semiconductor metal oxide gas sensors: A review. Mater Sci Eng, B 229:206–217

Ding W, Liu D, Yao Y, Wang Y (2024) Electronic and optical properties of Cr/Cl co-doped in single-layer MoS2: A first principles study. Solid State Sci 148:107415

Doust Mohammadi M, Abdullah HY (2021) Adsorption of 1-chloro-1, 2, 2, 2-tetrafluoroethane on pristine, Al, Ga-doped boron nitride nanotubes: a study involving PBC-DFT, NBO analysis, and QTAIM. Can J Chem 99(1):51–62

Dugbartey GJ (2023) Hydrogen Sulfide and the Renal System. Hydrogen Sulfide in Kidney Diseases: A Novel Pharmacotherapy. Cham, Springer Nature Switzerland, pp 1–21

Chapter   Google Scholar  

Feng S, Farha F, Li Q, Wan Y, Xu Y, Zhang T, Ning H (2019) Review on smart gas sensing technology. Sensors 19(17):3760

Filippatos PP, Kelaidis N, Vasilopoulou M, Davazoglou D, Chroneos A (2021) Structural, electronic, and optical properties of group 6 doped anatase TiO2: a theoretical approach. Appl Sci 11(4):1657

Frisch ME, Trucks GW, Schlegel HB, Scuseria GE, Robb M, Cheeseman JR, Fox DJ (2016) Gaussian 16

Fu H, Xu Z (2024) Adsorption mechanism of formaldehyde, ammonia and sulfur dioxide gases on inexpensive metal-doped ZnO surface: A DFT study. Surf Sci 741:122423

Gaffney JS, Marley NA (2003) Atmospheric chemistry and air pollution. Sci World J 3:199–234

Gao D, Guo J, He H, Xiao P, Zhang Y (2022) Geometric and electronic modulation of fcc NiCo alloy by Group-VI B metal doping to accelerate hydrogen evolution reaction in acidic and alkaline media. Chem Eng J 430:133110

Gatfaoui S, Issaoui N, Roisnel T, Marouani H (2021) Synthesis, experimental and computational study of a non-centrosymmetric material 3-methylbenzylammonium trioxonitrate. J Mol Struct 1225:129132

Gauglitz G (2005) Direct optical sensors: principles and selected applications. Anal Bioanal Chem 381:141–155

Golden R (2011) Identifying an indoor air exposure limit for formaldehyde considering both irritation and cancer hazards. Crit Rev Toxicol 41(8):672–721

Gorini F, Bustaffa E, Chatzianagnostou K, Bianchi F, Vassalle C (2020) Hydrogen sulfide and cardiovascular disease: Doubts, clues, and interpretation difficulties from studies in geothermal areas. Sci Total Environ 743:140818

Grieshaber D, MacKenzie R, Vörös J, Reimhult E (2008) Electrochemical biosensors-sensor principles and architectures. Sensors 8(3):1400–1458

Guidotti TL (2015) Hydrogen sulfide intoxication. In: Handbook of clinical neurology, vol 131. Elsevier, pp 111–133

Harbison SC, Bourgeois JR (2015) Hydrogen sulfide. Hamilton & Hardy’s Industrial Toxicology 325–330

Humphrey W, Dalke A, Schulten K (1996) VMD: visual molecular dynamics. J Mol Graph 14(1):33–38

Janeoo S, Saroa A, Kumar R, Kaur H (2022) Computational investigation of bioactive 2, 3-diaryl quinolines using DFT method: FT-IR, NMR spectra, NBO, NLO, HOMO-LUMO transitions, and quantum-chemical properties. J Mol Struct 1253:132285

Jha RK (2021) Non-dispersive infrared gas sensing technology: A review. IEEE Sens J 22(1):6–15

John AM, Jose J, Thomas R, Thomas KJ, Balakrishnan SP (2020) Spectroscopic and TDDFT investigation of highly selective fluoride sensors by substituted acyl hydrazones. Spectrochim Acta Part A Mol Biomol Spectrosc 236:118329

Jumabaev A, Holikulov U, Hushvaktov H, Issaoui N, Absanov A (2023) Intermolecular interactions in ethanol solution of OABA: Raman, FTIR, DFT, M062X, MEP, NBO, FMO, AIM, NCI, RDG analysis. J Mol Liq 377:121552

Kazachenko AS, Medimagh M, Issaoui N, Al-Dossary O, Wojcik MJ, Kazachenko AS, Miroshnokova AV, Malyar YN (2022) Sulfamic acid/water complexes [SAA-H2O (1–8)] intermolecular hydrogen bond interactions: FTIR, X-ray, DFT and AIM analysis. J Mol Struct 1265:133394

Kern DJ (2021) Atmospheric chemistry: an overview-ozone, acid rain, and greenhouse gases. Building STEM Skills Through Environmental Education 172–218

Kim KH, Jahan SA, Lee JT (2011) Exposure to formaldehyde and its potential human health hazards. J Environ Sci Health C 29(4):277–299

Korotcenkov G (2013) Handbook of gas sensor materials. Conventional approaches 1

Kumar AR, Ilavarasan L, Mol GPS, Selvaraj S, Azam M, Jayaprakash P, Kesavan M, Alam M, Dhanalakshmi J, Al-Resayes SI, Ravi A (2024) Spectroscopic (FT-IR, FT-Raman, UV–Vis and NMR) and computational (DFT, MESP, NBO, NCI, LOL, ELF, RDG and QTAIM) profiling of 5-chloro-2-hydroxy-3-methoxybenzaldehyde: A promising antitumor agent. J Mol Struct 1298:136974

Lambert TW, Goodwin VM, Stefani D, Strosher L (2006) Hydrogen sulfide H2S and sour gas effects on the eye A historical perspective. Sci Total Environ 367(1):1–22

Leikauf GD (2020) Formaldehyde and other saturated aldehydes. Environmental Toxicants: Human Exposures and Their Health Effects, pp 555–626

Lewis RJ, Copley GB (2015) Chronic low-level hydrogen sulfide exposure and potential effects on human health: a review of the epidemiological evidence. Crit Rev Toxicol 45(2):93–123

Li Y, Xu Y, Li X (2022) The sensing mechanism of HCHO gas sensor based on transition metal doped graphene: Insights from DFT study. Sens Actuators, A 338:113460

Liu X, Cheng S, Liu H, Hu S, Zhang D, Ning H (2012) A survey on gas sensing technology. Sensors 12(7):9635–9665

Article   PubMed   PubMed Central   Google Scholar  

Louis H, Etiese D, Unimuke TO, Owen AE, Rajee AO, Gber TE, Nfor EN (2022) Computational design and molecular modeling of the interaction of nicotinic acid hydrazide nickel based complexes with H2S gas. RSC Adv 12(47):30365–30380

Lu T, Chen F (2012) Multiwfn: a multifunctional wavefunction analyzer. J Comput Chem 33(5):580–592

Article   PubMed   Google Scholar  

Mary SJ, Siddique MUM, Pradhan S, Jayaprakash V, James C (2021) Quantum chemical insight into molecular structure, NBO analysis of the hydrogen-bonded interactions, spectroscopic (FT–IR, FT–Raman), drug likeness and molecular docking of the novel anti COVID-19 molecule 2-[(4, 6-diaminopyrimidin-2-yl) sulfanyl]-N-(4-fluorophenyl) acetamide-dimer. Spectrochim Acta Part A Mol Biomol Spectrosc 244:118825

Mohammadi MD, Abdullah HY (2021) The adsorption of bromochlorodifluoromethane on pristine, Al, Ga, P, and As-doped boron nitride nanotubes: a study involving PBC-DFT, NBO analysis, and QTAIM. Comput Theor Chem 1193:113047

Nielsen GD, Larsen ST, Wolkoff P (2013) Recent trend in risk assessment of formaldehyde exposures from indoor air. Arch Toxicol 87:73–98

Nikolic MV, Milovanovic V, Vasiljevic ZZ, Stamenkovic Z (2020) Semiconductor gas sensors: Materials, technology, design, and application. Sensors 20(22):6694

Odey MO, Okon GA, Al-Sehemi AG, Adindu EA, Kavil YN, Agwamba EC, Bakheet AM, Pembere AM, Louis H (2023) Single-atoms (B, P, S, Si) doping of Pt-coordinated graphitic carbon nitride (Pt@ g-C3N4) nanostructured as sensors for Buprofezin (BPF) insecticide: Outlook from computational study. J Organomet Chem 1002:122923

Paul A, Thomas R (2022) Evidences for sulfur centered hydrogen bond with sulfur atoms as a donor in aromatic thiols and aliphatic thiols in aqueous solution. J Mol Liq 348:118078

Preston W, Kelly C (eds) (2016) Respiratory nursing at a glance. John Wiley & Sons

Rajimon KJ, Elangovan N, Khairbek AA, Thomas R (2023) Schiff bases from chlorine substituted anilines and salicylaldehyde: Synthesis, characterization, fluorescence, thermal features, biological studies and electronic structure investigations. J Mol Liq 370:121055

Ranjan A, Zhao Y, Sahu HB, Misra P (2019) Opportunities and challenges in health sensing for extreme industrial environment: perspectives from underground mines. IEEE Access 7:139181–139195

Rubright SLM, Pearce LL, Peterson J (2017) Environmental toxicology of hydrogen sulfide. Nitric Oxide Biol Chem 71:1

Saini J, Dutta M, Marques G (2021) Sensors for indoor air quality monitoring and assessment through Internet of Things: a systematic review. Environ Monit Assess 193(2):66

Salthammer T, Mentese S, Marutzky R (2010) Formaldehyde in the indoor environment. Chem Rev 110(4):2536–2572

Samudro H, Samudro G, Mangkoedihardjo S (2022) Prevention of indoor air pollution through design and construction certification: A review of the sick building syndrome conditions. Journal of Air Pollution and Health 7(1):81–94

Sharma S, Madou M (2012) A new approach to gas sensing with nanotechnology. Philosophical Transactions of the Royal Society a: Mathematical, Physical and Engineering Sciences 370(1967):2448–2473

Shetti NP, Mishra A, Basu S, Aminabhavi TM (2021) Versatile fullerenes as sensor materials. Materials Today Chemistry 20:100454

Stuyver T, Danovich D, Joy J, Shaik S (2020) External electric field effects on chemical structure and reactivity. Wiley Interdisciplinary Reviews: Computational Molecular Science 10(2):e1438

CAS   Google Scholar  

Suresh CH, Remya GS, Anjalikrishna PK (2022) Molecular electrostatic potential analysis: A powerful tool to interpret and predict chemical reactivity. Wiley Interdisciplinary Reviews: Computational Molecular Science 12(5):e1601

Wehrhold M (2022) Investigations of electron transfer at graphene and graphene sandwiches. Humboldt-Universitat Zuberlin

Yi X, Chen W, Xiao Y, Liu F, Yu X, Zheng A (2023) Spectroscopically visualizing the evolution of hydrogen-bonding interactions. J Am Chem Soc

Zhu X, Wang Z, Zhong K, Li Q, Ding P, Feng Z, Yang J, Du Y, Song Y, Hua Y, Yuan J, She Y, Li H, Xu H (2022) Mo-O-Bi bonds as interfacial electron transport bridges to fuel CO2 photoreduction via in-situ reconstruction of black Bi2MoO6/BiO2-x heterojunction. Chem Eng J 429:132204

Zhurko G, Zhurko D (2012) Chemcraft program revision 1.6. Zhurko, GA, Ed

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Etiese, D., Amodu, I.O., Edet, H.O. et al. Novel engineering of single-metals (TM: Cr, Mo, W) chemical tailoring of Pt-encapsulated fullerenes (Pt@C59TM) as dual sensors for H2CO and H2S gases: A theoretical study. Chem. Pap. (2024). https://doi.org/10.1007/s11696-024-03525-z

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The state of AI in early 2024: Gen AI adoption spikes and starts to generate value

If 2023 was the year the world discovered generative AI (gen AI) , 2024 is the year organizations truly began using—and deriving business value from—this new technology. In the latest McKinsey Global Survey  on AI, 65 percent of respondents report that their organizations are regularly using gen AI, nearly double the percentage from our previous survey just ten months ago. Respondents’ expectations for gen AI’s impact remain as high as they were last year , with three-quarters predicting that gen AI will lead to significant or disruptive change in their industries in the years ahead.

About the authors

This article is a collaborative effort by Alex Singla , Alexander Sukharevsky , Lareina Yee , and Michael Chui , with Bryce Hall , representing views from QuantumBlack, AI by McKinsey, and McKinsey Digital.

Organizations are already seeing material benefits from gen AI use, reporting both cost decreases and revenue jumps in the business units deploying the technology. The survey also provides insights into the kinds of risks presented by gen AI—most notably, inaccuracy—as well as the emerging practices of top performers to mitigate those challenges and capture value.

AI adoption surges

Interest in generative AI has also brightened the spotlight on a broader set of AI capabilities. For the past six years, AI adoption by respondents’ organizations has hovered at about 50 percent. This year, the survey finds that adoption has jumped to 72 percent (Exhibit 1). And the interest is truly global in scope. Our 2023 survey found that AI adoption did not reach 66 percent in any region; however, this year more than two-thirds of respondents in nearly every region say their organizations are using AI. 1 Organizations based in Central and South America are the exception, with 58 percent of respondents working for organizations based in Central and South America reporting AI adoption. Looking by industry, the biggest increase in adoption can be found in professional services. 2 Includes respondents working for organizations focused on human resources, legal services, management consulting, market research, R&D, tax preparation, and training.

Also, responses suggest that companies are now using AI in more parts of the business. Half of respondents say their organizations have adopted AI in two or more business functions, up from less than a third of respondents in 2023 (Exhibit 2).

Gen AI adoption is most common in the functions where it can create the most value

Most respondents now report that their organizations—and they as individuals—are using gen AI. Sixty-five percent of respondents say their organizations are regularly using gen AI in at least one business function, up from one-third last year. The average organization using gen AI is doing so in two functions, most often in marketing and sales and in product and service development—two functions in which previous research  determined that gen AI adoption could generate the most value 3 “ The economic potential of generative AI: The next productivity frontier ,” McKinsey, June 14, 2023. —as well as in IT (Exhibit 3). The biggest increase from 2023 is found in marketing and sales, where reported adoption has more than doubled. Yet across functions, only two use cases, both within marketing and sales, are reported by 15 percent or more of respondents.

Gen AI also is weaving its way into respondents’ personal lives. Compared with 2023, respondents are much more likely to be using gen AI at work and even more likely to be using gen AI both at work and in their personal lives (Exhibit 4). The survey finds upticks in gen AI use across all regions, with the largest increases in Asia–Pacific and Greater China. Respondents at the highest seniority levels, meanwhile, show larger jumps in the use of gen Al tools for work and outside of work compared with their midlevel-management peers. Looking at specific industries, respondents working in energy and materials and in professional services report the largest increase in gen AI use.

Investments in gen AI and analytical AI are beginning to create value

The latest survey also shows how different industries are budgeting for gen AI. Responses suggest that, in many industries, organizations are about equally as likely to be investing more than 5 percent of their digital budgets in gen AI as they are in nongenerative, analytical-AI solutions (Exhibit 5). Yet in most industries, larger shares of respondents report that their organizations spend more than 20 percent on analytical AI than on gen AI. Looking ahead, most respondents—67 percent—expect their organizations to invest more in AI over the next three years.

Where are those investments paying off? For the first time, our latest survey explored the value created by gen AI use by business function. The function in which the largest share of respondents report seeing cost decreases is human resources. Respondents most commonly report meaningful revenue increases (of more than 5 percent) in supply chain and inventory management (Exhibit 6). For analytical AI, respondents most often report seeing cost benefits in service operations—in line with what we found last year —as well as meaningful revenue increases from AI use in marketing and sales.

Inaccuracy: The most recognized and experienced risk of gen AI use

As businesses begin to see the benefits of gen AI, they’re also recognizing the diverse risks associated with the technology. These can range from data management risks such as data privacy, bias, or intellectual property (IP) infringement to model management risks, which tend to focus on inaccurate output or lack of explainability. A third big risk category is security and incorrect use.

Respondents to the latest survey are more likely than they were last year to say their organizations consider inaccuracy and IP infringement to be relevant to their use of gen AI, and about half continue to view cybersecurity as a risk (Exhibit 7).

Conversely, respondents are less likely than they were last year to say their organizations consider workforce and labor displacement to be relevant risks and are not increasing efforts to mitigate them.

In fact, inaccuracy— which can affect use cases across the gen AI value chain , ranging from customer journeys and summarization to coding and creative content—is the only risk that respondents are significantly more likely than last year to say their organizations are actively working to mitigate.

Some organizations have already experienced negative consequences from the use of gen AI, with 44 percent of respondents saying their organizations have experienced at least one consequence (Exhibit 8). Respondents most often report inaccuracy as a risk that has affected their organizations, followed by cybersecurity and explainability.

Our previous research has found that there are several elements of governance that can help in scaling gen AI use responsibly, yet few respondents report having these risk-related practices in place. 4 “ Implementing generative AI with speed and safety ,” McKinsey Quarterly , March 13, 2024. For example, just 18 percent say their organizations have an enterprise-wide council or board with the authority to make decisions involving responsible AI governance, and only one-third say gen AI risk awareness and risk mitigation controls are required skill sets for technical talent.

Bringing gen AI capabilities to bear

The latest survey also sought to understand how, and how quickly, organizations are deploying these new gen AI tools. We have found three archetypes for implementing gen AI solutions : takers use off-the-shelf, publicly available solutions; shapers customize those tools with proprietary data and systems; and makers develop their own foundation models from scratch. 5 “ Technology’s generational moment with generative AI: A CIO and CTO guide ,” McKinsey, July 11, 2023. Across most industries, the survey results suggest that organizations are finding off-the-shelf offerings applicable to their business needs—though many are pursuing opportunities to customize models or even develop their own (Exhibit 9). About half of reported gen AI uses within respondents’ business functions are utilizing off-the-shelf, publicly available models or tools, with little or no customization. Respondents in energy and materials, technology, and media and telecommunications are more likely to report significant customization or tuning of publicly available models or developing their own proprietary models to address specific business needs.

Respondents most often report that their organizations required one to four months from the start of a project to put gen AI into production, though the time it takes varies by business function (Exhibit 10). It also depends upon the approach for acquiring those capabilities. Not surprisingly, reported uses of highly customized or proprietary models are 1.5 times more likely than off-the-shelf, publicly available models to take five months or more to implement.

Gen AI high performers are excelling despite facing challenges

Gen AI is a new technology, and organizations are still early in the journey of pursuing its opportunities and scaling it across functions. So it’s little surprise that only a small subset of respondents (46 out of 876) report that a meaningful share of their organizations’ EBIT can be attributed to their deployment of gen AI. Still, these gen AI leaders are worth examining closely. These, after all, are the early movers, who already attribute more than 10 percent of their organizations’ EBIT to their use of gen AI. Forty-two percent of these high performers say more than 20 percent of their EBIT is attributable to their use of nongenerative, analytical AI, and they span industries and regions—though most are at organizations with less than $1 billion in annual revenue. The AI-related practices at these organizations can offer guidance to those looking to create value from gen AI adoption at their own organizations.

To start, gen AI high performers are using gen AI in more business functions—an average of three functions, while others average two. They, like other organizations, are most likely to use gen AI in marketing and sales and product or service development, but they’re much more likely than others to use gen AI solutions in risk, legal, and compliance; in strategy and corporate finance; and in supply chain and inventory management. They’re more than three times as likely as others to be using gen AI in activities ranging from processing of accounting documents and risk assessment to R&D testing and pricing and promotions. While, overall, about half of reported gen AI applications within business functions are utilizing publicly available models or tools, gen AI high performers are less likely to use those off-the-shelf options than to either implement significantly customized versions of those tools or to develop their own proprietary foundation models.

What else are these high performers doing differently? For one thing, they are paying more attention to gen-AI-related risks. Perhaps because they are further along on their journeys, they are more likely than others to say their organizations have experienced every negative consequence from gen AI we asked about, from cybersecurity and personal privacy to explainability and IP infringement. Given that, they are more likely than others to report that their organizations consider those risks, as well as regulatory compliance, environmental impacts, and political stability, to be relevant to their gen AI use, and they say they take steps to mitigate more risks than others do.

Gen AI high performers are also much more likely to say their organizations follow a set of risk-related best practices (Exhibit 11). For example, they are nearly twice as likely as others to involve the legal function and embed risk reviews early on in the development of gen AI solutions—that is, to “ shift left .” They’re also much more likely than others to employ a wide range of other best practices, from strategy-related practices to those related to scaling.

In addition to experiencing the risks of gen AI adoption, high performers have encountered other challenges that can serve as warnings to others (Exhibit 12). Seventy percent say they have experienced difficulties with data, including defining processes for data governance, developing the ability to quickly integrate data into AI models, and an insufficient amount of training data, highlighting the essential role that data play in capturing value. High performers are also more likely than others to report experiencing challenges with their operating models, such as implementing agile ways of working and effective sprint performance management.

About the research

The online survey was in the field from February 22 to March 5, 2024, and garnered responses from 1,363 participants representing the full range of regions, industries, company sizes, functional specialties, and tenures. Of those respondents, 981 said their organizations had adopted AI in at least one business function, and 878 said their organizations were regularly using gen AI in at least one function. To adjust for differences in response rates, the data are weighted by the contribution of each respondent’s nation to global GDP.

Alex Singla and Alexander Sukharevsky  are global coleaders of QuantumBlack, AI by McKinsey, and senior partners in McKinsey’s Chicago and London offices, respectively; Lareina Yee  is a senior partner in the Bay Area office, where Michael Chui , a McKinsey Global Institute partner, is a partner; and Bryce Hall  is an associate partner in the Washington, DC, office.

They wish to thank Kaitlin Noe, Larry Kanter, Mallika Jhamb, and Shinjini Srivastava for their contributions to this work.

This article was edited by Heather Hanselman, a senior editor in McKinsey’s Atlanta office.

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