7.3: Development of Quantum Theory
- Extend the concept of wave–particle duality that was observed in electromagnetic radiation to matter as well
- Understand the general idea of the quantum mechanical description of electrons in an atom, and that it uses the notion of three-dimensional wave functions, or orbitals, that define the distribution of probability to find an electron in a particular part of space
- List and describe traits of the four quantum numbers that form the basis for completely specifying the state of an electron in an atom
Bohr’s model explained the experimental data for the hydrogen atom and was widely accepted, but it also raised many questions. Why did electrons orbit at only fixed distances defined by a single quantum number n = 1, 2, 3, and so on, but never in between? Why did the model work so well describing hydrogen and one-electron ions, but could not correctly predict the emission spectrum for helium or any larger atoms? To answer these questions, scientists needed to completely revise the way they thought about matter.
Behavior in the Microscopic World
We know how matter behaves in the macroscopic world—objects that are large enough to be seen by the naked eye follow the rules of classical physics. A billiard ball moving on a table will behave like a particle: It will continue in a straight line unless it collides with another ball or the table cushion, or is acted on by some other force (such as friction). The ball has a well-defined position and velocity (or a well-defined momentum, \(p = mv\), defined by mass \(m\) and velocity \(v\)) at any given moment. In other words, the ball is moving in a classical trajectory. This is the typical behavior of a classical object.
When waves interact with each other, they show interference patterns that are not displayed by macroscopic particles such as the billiard ball. For example, interacting waves on the surface of water can produce interference patters similar to those shown on Figure \(\PageIndex{1}\). This is a case of wave behavior on the macroscopic scale, and it is clear that particles and waves are very different phenomena in the macroscopic realm.
As technological improvements allowed scientists to probe the microscopic world in greater detail, it became increasingly clear by the 1920s that very small pieces of matter follow a different set of rules from those we observe for large objects. The unquestionable separation of waves and particles was no longer the case for the microscopic world.
One of the first people to pay attention to the special behavior of the microscopic world was Louis de Broglie . He asked the question: If electromagnetic radiation can have particle-like character, can electrons and other submicroscopic particles exhibit wavelike character? In his 1925 doctoral dissertation, de Broglie extended the wave–particle duality of light that Einstein used to resolve the photoelectric-effect paradox to material particles. He predicted that a particle with mass m and velocity v (that is, with linear momentum p ) should also exhibit the behavior of a wave with a wavelength value λ , given by this expression in which h is the familiar Planck’s constant
\[\lambda=\dfrac{h}{mv}=\dfrac{h}{p} \label{6.4.1} \]
This is called the de Broglie wavelength . Unlike the other values of λ discussed in this chapter, the de Broglie wavelength is a characteristic of particles and other bodies, not electromagnetic radiation (note that this equation involves velocity [ v , m/s], not the greek symbol for frequency [ \(\nu\) , Hz]. Although these two symbols are identical, they mean very different things). Where Bohr had postulated the electron as being a particle orbiting the nucleus in quantized orbits, de Broglie argued that Bohr’s assumption of quantization can be explained if the electron is considered not as a particle, but rather as a circular standing wave such that only an integer number of wavelengths could fit exactly within the orbit (Figure \(\PageIndex{2}\)).
For a circular orbit of radius r, the circumference is 2πr, and so de Broglie’s condition is:
\[2πr=nλ \]
with \(n=1,2,3,...\)
Since the de Broglie expression relates the wavelength to the momentum and, hence, velocity this expression can be rearranged to give Bohr’s formula for the quantization of the angular momentum:
\[L=\dfrac{nh}{2π}=n \hbar \]
Classical angular momentum L for a circular motion is equal to the product of the radius of the circle and the momentum of the moving particle p.
\[L=rp=rmv \;\;\; \text{(for a circular motion)} \]
Shortly after de Broglie proposed the wave nature of matter, two scientists at Bell Laboratories, C. J. Davisson and L. H. Germer , demonstrated experimentally that electrons can exhibit wavelike behavior by showing an interference pattern for electrons traveling through a regular atomic pattern in a crystal. The regularly spaced atomic layers served as slits, as used in other interference experiments. Since the spacing between the layers serving as slits needs to be similar in size to the wavelength of the tested wave for an interference pattern to form, Davisson and Germer used a crystalline nickel target for their “slits,” since the spacing of the atoms within the lattice was approximately the same as the de Broglie wavelengths of the electrons that they used. Figure \(\PageIndex{3}\) shows an interference pattern.
The wave–particle duality of matter can be seen by observing what happens if electron collisions are recorded over a long period of time. Initially, when only a few electrons have been recorded, they show clear particle-like behavior, having arrived in small localized packets that appear to be random. As more and more electrons arrived and were recorded, a clear interference pattern that is the hallmark of wavelike behavior emerged. Thus, it appears that while electrons are small localized particles, their motion does not follow the equations of motion implied by classical mechanics, but instead it is governed by some type of a wave equation that governs a probability distribution even for a single electron’s motion. Thus the wave–particle duality first observed with photons is actually a fundamental behavior intrinsic to all quantum particles.
If an electron travels at a velocity of 1.000 × 10 7 m s –1 and has a mass of 9.109 × 10 –31 kg, what is its wavelength?
Solution
We can use de Broglie’s equation to solve this problem, since 1 J = 1 kg m 2 /s 2 .
\[\begin{align*} λ&=\dfrac{h}{mv} \\[4pt] &=\mathrm{\dfrac{6.626×10^{−34} J s}{(9.109×10^{−31}\:kg)(1.000×10^7\:m/s)}}\\[4pt] &= \mathrm{7.274×10^{−11}\:m} \end{align*} \nonumber \]
This is a small value, but it is significantly larger than the size of an electron in the classical (particle) view. This size is the same order of magnitude as the size of an atom. This means that electron wavelike behavior is going to be noticeable in an atom.
Calculate the wavelength of a softball with a mass of 0.100 kg traveling at a velocity of 35 m s –1 , assuming that it can be modeled as a single particle.
- Answer
-
1.9 × 10 –34 m.
We never think of a thrown softball having a wavelength, since this wavelength is so small it is impossible for our senses or any known instrument to detect (strictly speaking, the wavelength of a real baseball would correspond to the wavelengths of its constituent atoms and molecules, which, while much larger than this value, would still be microscopically tiny). The de Broglie wavelength is only appreciable for matter that has a very small mass and/or a very high velocity.
Werner Heisenberg considered the limits of how accurately we can measure properties of an electron or other microscopic particles. He determined that there is a fundamental limit to how accurately one can measure both a particle’s position and its momentum simultaneously. The more accurately we measure the momentum of a particle, the less accurately we can determine its position at that time, and vice versa. This is summed up in what we now call the Heisenberg uncertainty principle : It is fundamentally impossible to determine simultaneously and exactly both the momentum and the position of a particle . This allows us to calculate the limit to how precisely we can know both the simultaneous position of an object and its momentum.
It should be noted that Heisenberg’s uncertainty principle is not just limited to uncertainties in position and momentum, but it also links other dynamical variables. For example, when an atom absorbs a photon and makes a transition from one energy state to another, the uncertainty in the energy and the uncertainty in the time required for the transition are similarly related, as Δ E Δ t ≥ \(\dfrac{ℏ}{2}\).
Heisenberg’s principle imposes ultimate limits on what is knowable in science. The uncertainty principle can be shown to be a consequence of wave–particle duality, which lies at the heart of what distinguishes modern quantum theory from classical mechanics. Recall that the equations of motion obtained from classical mechanics are trajectories where, at any given instant in time, both the position and the momentum of a particle can be determined exactly. Heisenberg’s uncertainty principle implies that such a view is untenable in the microscopic domain and that there are fundamental limitations governing the motion of quantum particles. This does not mean that microscopic particles do not move in trajectories, it is just that measurements of trajectories are limited in their precision. In the realm of quantum mechanics, measurements introduce changes into the system that is being observed.
The Quantum–Mechanical Model of an Atom
Shortly after de Broglie published his ideas that the electron in a hydrogen atom could be better thought of as being a circular standing wave instead of a particle moving in quantized circular orbits, as Bohr had suggested, Erwin Schrödinger extended de Broglie’s work by incorporating the de Broglie relation into a wave equation, deriving what is today known as the Schrödinger equation. When Schrödinger applied his equation to hydrogen-like atoms, he was able to reproduce Bohr’s expression for the energy and, thus, the Rydberg formula governing hydrogen spectra, and he did so without having to invoke Bohr’s assumptions of stationary states and quantized orbits, angular momenta, and energies; quantization in Schrödinger’s theory was a natural consequence of the underlying mathematics of the wave equation. Like de Broglie, Schrödinger initially viewed the electron in hydrogen as being a physical wave instead of a particle, but where de Broglie thought of the electron in terms of circular stationary waves, Schrödinger properly thought in terms of three-dimensional stationary waves, or wavefunctions , represented by the Greek letter psi, ψ . A few years later, Max Born proposed an interpretation of the wavefunction ψ that is still accepted today: Electrons are still particles, and so the waves represented by ψ are not physical waves but, instead, are complex probability amplitudes. The square of the magnitude of a wavefunction \(∣ψ∣^2\) describes the probability of the quantum particle being present near a certain location in space. This means that wavefunctions can be used to determine the distribution of the electron’s density with respect to the nucleus in an atom.
Schrödinger’s work, as well as that of Heisenberg and many other scientists following in their footsteps, is generally referred to as
quantum mechanics
. Video \(\PageIndex{2}\) describes an interesting paradox that Schrödinger used to explain how observation or measurement changes a system at the quatum mechanical level.
Understanding Quantum Theory of Electrons in Atoms
The goal of this section is to understand the electron orbitals (location of electrons in atoms), their different energies, and other properties. The use of quantum theory provides the best understanding to these topics. This knowledge is critical to understanding chemical bonding.
Electrons in atoms can exist only on discrete energy levels. The energy of an electron in an atom is quantized, it can only have certain specific values. It can change between these values but is never between these values.
The energy levels, or orbitals, are labeled with quantum numbers. The principal quantum number is n , where n = 1, 2, 3, …. Generally speaking, the energy of an electron in an atom is greater for greater values of n . The principal quantum number defines the location of the energy level. It is essentially the same concept as the n in the Bohr atom description. Another name for the principal quantum number is the shell number. The shells of an atom can be thought of concentric circles radiating out from the nucleus. The electrons that belong to a specific shell are most likely to be found within the corresponding circular area. The further we proceed from the nucleus, the higher the shell number, and so the higher the energy level (Figure \(\PageIndex{4}\)). The positively charged protons in the nucleus stabilize the electronic orbitals by electrostatic attraction between the positive charges of the protons and the negative charges of the electrons. So the further away the electron is from the nucleus, the greater the energy it has.
The next quantum number is l , the angular momentum quantum number . It is an integer that defines the shape of the orbital, and takes on the values, l = 0, 1, 2, …, n – 1. An orbital with n = 1 can have only one value of l , l = 0, but when n = 2 this permits l = 0 and l = 1, and so on. The principal quantum number defines the general size and energy of the orbital. The l value specifies the shape of the orbital. Orbitals with l = 0 are called s orbitals (or the s subshells). The value l = 1 corresponds to the p orbitals. For a given n , p orbitals constitute a p subshell (e.g., 3 p if n = 3). The orbitals with l = 2 are called the d orbitals , followed by the f orbitals for l = 3.
Together the n and l quantum numbers describe the energy and shape of an orbital. A close look at the shape of the orbitals shows there are certain distances from the nucleus at which the probability density of finding an electron located at a particular orbital is zero. This is a node like the ones described for standing waves in section 7.1. The value of the wavefunction ψ is zero at this distance for this orbital. Such a value of radius r is called a radial node. The number of radial nodes in an orbital is n – l – 1. Figure \(\PageIndex{5}\) shows cross sections of the 1s, 2s, and 3s orbitals ( n = 1, 2, and 3; while l = 0) and the probability graph showing the probability as a function of radial distance from the nucleus of the atom. These graphs show there are 1 – 0 – 1 = 0 places where the density is zero (nodes) for 1 s ( n = 1), 2 – 0 – 1 = 1 node for 2 s , and 3 – 0 – 1 = 2 nodes for the 3 s orbitals.
Figure \(\PageIndex{6}\) shows the shapes of all the possible orbitals for l = 0, 1, 2, and 3. The s subshell electron density distribution is spherical and the p subshell has a dumbbell shape. The d and f orbitals are more complex. These shapes represent the three-dimensional regions within which the electron is likely to be found. Notice that for the s orbital when l = 0 there is only one possible orientation, a sphere does not have a top or bottom. However when l = 1, 2, or 3 there are different possible orientations. These orientations are represented by the third quantum number m l , the magnetic quantum number. For an s orbital, l = 0, and the only value of m l is zero. For p orbitals, l = 1, and m l can be equal to –1, 0, or +1. Generally speaking, m l can be equal to – l , –( l – 1), …, –1, 0, +1, …, ( l – 1), l . The total number of possible orbitals with the same value of l (a subshell) is 2 l + 1. Thus, there is one s -orbital for m l = 0 , there are three p -orbitals for m l = 1 , five d -orbitals for m l = 2 , seven f -orbitals for m l = 3 , and so forth.
The principal quantum number defines the general value of the electronic energy. The angular momentum quantum number determines the shape of the orbital. And the magnetic quantum number specifies orientation of the orbital in space. This is sumarized in the Energy level diagram shown in Figure \(\PageIndex{7}\). The number before the orbital name (such as 2 s , 3 p , and so forth) stands for the principal quantum number, n . The letter in the orbital name defines the subshell with a specific angular momentum quantum number l = 0 for s orbitals, 1 for p orbitals, 2 for d orbitals. Finally, there are more than one possible orbitals for l ≥ 1, each corresponding to a specific value of m l . In the case of a hydrogen atom or a one-electron ion (such as He + , Li 2 + , and so on), energies of all the orbitals with the same n are the same. This is called a degeneracy, and the energy levels for the same principal quantum number, n , are called degenerate energy levels. However, in atoms with more than one electron, this degeneracy is eliminated by the electron–electron interactions, and orbitals that belong to different subshells have different energies. Orbitals within the same subshell (for example ns, np, nd, nf , such as 2 p , 3 s ) are still degenerate and have the same energy.
These three quantum numbers, n , l , and m l , are properties of specific atomic orbitals that also define in what part of the space an electron is most likely to be located. Orbitals are a result of solving the Schrödinger equation for electrons in atoms. The electron spin is a different kind of property. It is a completely quantum phenomenon with no analogues in the classical realm. In addition, it cannot be derived from solving the Schrödinger equation and is not related to the normal spatial coordinates (such as the Cartesian x , y , and z ). Electron spin describes an intrinsic electron “rotation” or “spinning.” Each electron acts as a tiny magnet or a tiny rotating object with an angular momentum, even though this rotation cannot be observed in terms of the spatial coordinates.
The Pauli Exclusion Principle
While the three quantum numbers discussed in the previous section work well for describing electron orbitals, some experiments showed that they were not sufficient to explain all observed results. It was demonstrated in the 1920s that when hydrogen-line spectra are examined at extremely high resolution, some lines are actually not single peaks but, rather, pairs of closely spaced lines. This is the so-called fine structure of the spectrum, and it implies that there are additional small differences in energies of electrons even when they are located in the same orbital. These observations led Samuel Goudsmit and George Uhlenbeck to propose that electrons have a fourth quantum number. They called this the spin quantum number , or m s . Any electron, regardless of the atomic orbital it is located in, can only have one of two values for the spin quantum number, \(m_s=−\dfrac{1}{2}\) or \(m_s=\dfrac{1}{2}\) as shown in Figure \(\PageIndex{8}\).
An electron in an atom is completely described by four quantum numbers: n , l , m l , and m s . The first three quantum numbers define the orbital and the fourth quantum number describes the intrinsic electron property called spin. An Austrian physicist Wolfgang Pauli formulated a general principle that gives the last piece of information that we need to understand the general behavior of electrons in atoms. The Pauli exclusion principle can be formulated as follows: No two electrons in the same atom can have exactly the same set of all the four quantum numbers. What this means is that electrons can share the same orbital (the same set of the quantum numbers n , l , and m l ), but only if their spin quantum numbers m s have different values. Since the spin quantum number can only have two values \(\left(±\dfrac{1}{2}\right)\), no more than two electrons can occupy the same orbital (and if two electrons are located in the same orbital, they must have opposite spins). Therefore, any atomic orbital can be populated by only zero, one, or two electrons. The properties and meaning of the quantum numbers of electrons in atoms are briefly summarized in Table \(\PageIndex{1}\).
| Name | Symbol | Allowed values | Physical meaning |
|---|---|---|---|
| principal quantum number | n | 1, 2, 3, 4, …. | shell, the general region for the value of energy for an electron on the orbital |
| angular momentum or azimuthal quantum number | l | 0 ≤ l ≤ n – 1 | subshell, the shape of the orbital |
| magnetic quantum number | m l | – l ≤ m l ≤ l | orientation of the orbital |
| spin quantum number | m s | \(+\dfrac{1}{2},\:−\dfrac{1}{2}\) | direction of the intrinsic quantum “spinning” of the electron |
Indicate the number of subshells, the number of orbitals in each subshell, and the values of l and m l for the orbitals in the n = 4 shell of an atom.
Solution
For n = 4, l can have values of 0, 1, 2, and 3. Thus, s , p , d , and f subshells are found in the n = 4 shell of an atom. For l = 0 (the s subshell), m l can only be 0. Thus, there is only one 4 s orbital. For l = 1 ( p -type orbitals), m can have values of –1, 0, +1, so we find three 4 p orbitals. For l = 2 ( d -type orbitals), m l can have values of –2, –1, 0, +1, +2, so we have five 4 d orbitals. When l = 3 ( f -type orbitals), m l can have values of –3, –2, –1, 0, +1, +2, +3, and we can have seven 4 f orbitals. Thus, we find a total of 16 orbitals in the n = 4 shell of an atom.
Identify the subshell in which electrons with the following quantum numbers are found:
- n = 3, l = 1;
- n = 5, l = 3;
- n = 2, l = 0.
- Answer a
-
3 p
- Answer b
-
5 f
- Answer c
-
2s
Calculate the maximum number of electrons that can occupy a shell with (a) n = 2 and (b) n = 5. Note you are only looking at the orbitals with the specified n value, not those at lower energies.
Solution
(a) When n = 2, there are four orbitals (a single 2 s orbital, and three orbitals labeled 2 p ). These four orbitals can contain eight electrons.
(b) When n = 5, there are five subshells of orbitals that we need to sum:
&\phantom{+}\textrm{1 orbital labeled }5s\\
&\phantom{+}\textrm{3 orbitals labeled }5p\\
&\phantom{+}\textrm{5 orbitals labeled }5d\\
&\phantom{+}\textrm{7 orbitals labeled }5f\\
&\underline{+\textrm{9 orbitals labeled }5g}\\
&\,\textrm{25 orbitals total}
\end{align*}\)
If a shell contains a maximum of 32 electrons, what is the principal quantum number, n ?
- Answer
-
n = 4
Complete the following table for atomic orbitals:
| Orbital | n | l | m l degeneracy | Radial nodes (no.) |
|---|---|---|---|---|
| 4 f | ||||
| 4 | 1 | |||
| 7 | 7 | 3 | ||
| 5 d |
Solution
The table can be completed using the following rules:
- The orbital designation is nl , where l = 0, 1, 2, 3, 4, 5, … is mapped to the letter sequence s, p , d , f , g , h , …,
- The m l degeneracy is the number of orbitals within an l subshell, and so is 2 l + 1 (there is one s orbital, three p orbitals, five d orbitals, seven f orbitals, and so forth).
- The number of radial nodes is equal to n – l – 1.
| Orbital | n | l | m l degeneracy | Radial nodes (no.) |
|---|---|---|---|---|
| 4 f | 4 | 3 | 7 | 0 |
| 4 p | 4 | 1 | 3 | 2 |
| 7 f | 7 | 3 | 7 | 3 |
| 5 d | 5 | 2 | 5 | 2 |
How many orbitals have l = 2 and n = 3?
- Answer
-
The five degenerate 3 d orbitals
Summary
Macroscopic objects act as particles. Microscopic objects (such as electrons) have properties of both a particle and a wave. Their exact trajectories cannot be determined. The quantum mechanical model of atoms describes the three-dimensional position of the electron in a probabilistic manner according to a mathematical function called a wavefunction, often denoted as ψ . Atomic wavefunctions are also called orbitals. The squared magnitude of the wavefunction describes the distribution of the probability of finding the electron in a particular region in space. Therefore, atomic orbitals describe the areas in an atom where electrons are most likely to be found.
An atomic orbital is characterized by three quantum numbers. The principal quantum number, n , can be any positive integer. The general region for value of energy of the orbital and the average distance of an electron from the nucleus are related to n . Orbitals having the same value of n are said to be in the same shell. The angular momentum quantum number, l , can have any integer value from 0 to n – 1. This quantum number describes the shape or type of the orbital. Orbitals with the same principal quantum number and the same l value belong to the same subshell. The magnetic quantum number, m l , with 2 l + 1 values ranging from – l to + l , describes the orientation of the orbital in space. In addition, each electron has a spin quantum number, m s , that can be equal to \(±\dfrac{1}{2}\). No two electrons in the same atom can have the same set of values for all the four quantum numbers.
Glossary
- angular momentum quantum number ( l )
- quantum number distinguishing the different shapes of orbitals; it is also a measure of the orbital angular momentum
- atomic orbital
- mathematical function that describes the behavior of an electron in an atom (also called the wavefunction), it can be used to find the probability of locating an electron in a specific region around the nucleus, as well as other dynamical variables
- d orbital
- region of space with high electron density that is either four lobed or contains a dumbbell and torus shape; describes orbitals with l = 2. An electron in this orbital is called a d electron
- electron density
- a measure of the probability of locating an electron in a particular region of space, it is equal to the squared absolute value of the wave function ψ
- f orbital
- multilobed region of space with high electron density, describes orbitals with l = 3. An electron in this orbital is called an f electron
- Heisenberg uncertainty principle
- rule stating that it is impossible to exactly determine both certain conjugate dynamical properties such as the momentum and the position of a particle at the same time. The uncertainty principle is a consequence of quantum particles exhibiting wave–particle duality
- magnetic quantum number ( m l )
- quantum number signifying the orientation of an atomic orbital around the nucleus; orbitals having different values of m l but the same subshell value of l have the same energy (are degenerate), but this degeneracy can be removed by application of an external magnetic field
- p orbital
- dumbbell-shaped region of space with high electron density, describes orbitals with l = 1. An electron in this orbital is called a p electron
- Pauli exclusion principle
- specifies that no two electrons in an atom can have the same value for all four quantum numbers
- principal quantum number ( n )
- quantum number specifying the shell an electron occupies in an atom
- quantum mechanics
- field of study that includes quantization of energy, wave-particle duality, and the Heisenberg uncertainty principle to describe matter
- s orbital
- spherical region of space with high electron density, describes orbitals with l = 0. An electron in this orbital is called an s electron
- shell
- set of orbitals with the same principal quantum number, n
- spin quantum number ( m s )
- number specifying the electron spin direction, either \(+\dfrac{1}{2}\) or \(−\dfrac{1}{2}\)
- subshell
- set of orbitals in an atom with the same values of n and l
- wavefunction ( ψ )
- mathematical description of an atomic orbital that describes the shape of the orbital; it can be used to calculate the probability of finding the electron at any given location in the orbital, as well as dynamical variables such as the energy and the angular momentum


5.6: Reading- Web Browser
Introduction
A web browser (commonly referred to as a browser ) is a software application for retrieving, presenting and traversing information resources on the World Wide Web. An information resource is identified by a Uniform Resource Identifier (URI/URL) and may be a web page, image, video or other piece of content. Hyperlinks present in resources enable users easily to navigate their browsers to related resources.
Although browsers are primarily intended to use the World Wide Web, they can also be used to access information provided by web servers in private networks or files in file systems.
The major web browsers are Firefox, Internet Explorer, Google Chrome, Opera, and Safari.
History
The first web browser was invented in 1990 by Sir Tim Berners-Lee. Berners-Lee is the director of the World Wide Web Consortium (W3C), which oversees the Web’s continued development, and is also the founder of the World Wide Web Foundation. His browser was called WorldWideWeb and later renamed Nexus.
The first commonly available web browser with a graphical user interface was Erwise. The development of Erwise was initiated by Robert Cailliau.
In 1993, browser software was further innovated by Marc Andreessen with the release of Mosaic, “the world’s first popular browser”, which made the World Wide Web system easy to use and more accessible to the average person. Andreesen’s browser sparked the internet boom of the 1990s. The introduction of Mosaic in 1993 – one of the first graphical web browsers – led to an explosion in web use. Andreessen, the leader of the Mosaic team at National Center for Supercomputing Applications (NCSA), soon started his own company, named Netscape, and released the Mosaic-influenced Netscape Navigator in 1994, which quickly became the world’s most popular browser, accounting for 90% of all web use at its peak (see usage share of web browsers).
Microsoft responded with its Internet Explorer in 1995, also heavily influenced by Mosaic, initiating the industry’s first browser war. Bundled with Windows, Internet Explorer gained dominance in the web browser market; Internet Explorer usage share peaked at over 95% by 2002.
In 1998, Netscape launched what was to become the Mozilla Foundation in an attempt to produce a competitive browser using the open source software model. That browser would eventually evolve into Firefox, which developed a respectable following while still in the beta stage of development; shortly after the release of Firefox 1.0 in late 2004, Firefox (all versions) accounted for 7% of browser use. As of August 2011, Firefox has a 28% usage share.
Apple’s Safari had its first beta release in January 2003; as of April 2011, it had a dominant share of Apple-based web browsing, accounting for just over 7% of the entire browser market.
The most recent major entrant to the browser market is Chrome, first released in September 2008. Chrome’s take-up has increased significantly year by year, by doubling its usage share from 8% to 16% by August 2011. This increase seems largely to be at the expense of Internet Explorer, whose share has tended to decrease from month to month. In December 2011, Chrome overtook Internet Explorer 8 as the most widely used web browser but still had lower usage than all versions of Internet Explorer combined. Chrome’s user-base continued to grow and in May 2012, Chrome’s usage passed the usage of all versions of Internet Explorer combined. By April 2014, Chrome’s usage had hit 45%.
Internet Explorer will be deprecated in Windows 10, with Microsoft Edge replacing it as the default web browser.
Business models
The ways that web browser makers fund their development costs has changed over time. The first web browser, WorldWideWeb, was a research project.
Netscape Navigator was sold commercially, as was Opera.
Internet Explorer, on the other hand, was bundled free with the Windows operating system (and was also downloadable free), and therefore it was funded partly by the sales of Windows to computer manufacturers and direct to users. Internet Explorer also used to be available for the Mac. It is likely that releasing IE for the Mac was part of Microsoft’s overall strategy to fight threats to its quasi-monopoly platform dominance – threats such as web standards and Java – by making some web developers, or at least their managers, assume that there was “no need” to develop for anything other than Internet Explorer. In this respect, IE may have contributed to Windows and Microsoft applications sales in another way, through “lock-in” to Microsoft’s browser.
In January 2009, the European Commission announced it would investigate the bundling of Internet Explorer with Windows operating systems from Microsoft, saying “Microsoft’s tying of Internet Explorer to the Windows operating system harms competition between web browsers, undermines product innovation and ultimately reduces consumer choice.”
Safari and Mobile Safari were likewise always included with OS X and iOS respectively, so, similarly, they were originally funded by sales of Apple computers and mobile devices, and formed part of the overall Apple experience to customers.
Today, most commercial web browsers are paid by search engine companies to make their engine default, or to include them as another option. For example, Google pays Mozilla, the maker of Firefox, to make Google Search the default search engine in Firefox. Mozilla makes enough money from this deal that it does not need to charge users for Firefox. In addition, Google Search is also (as one would expect) the default search engine in Google Chrome. Users searching for websites or items on the Internet would be led to Google’s search results page, increasing ad revenue and which funds development at Google and of Google Chrome.
Many less-well-known free software browsers, such as Konqueror, were hardly funded at all and were developed mostly by volunteers free of charge.
Function
The primary purpose of a web browser is to bring information resources to the user (“retrieval” or “fetching”), allowing them to view the information (“display”, “rendering”), and then access other information (“navigation”, “following links”).
This process begins when the user inputs a Uniform Resource Locator (URL), for example http://en.wikipedia.org/ , into the browser. The prefix of the URL, the Uniform Resource Identifier or URI, determines how the URL will be interpreted. The most commonly used kind of URI starts with http: and identifies a resource to be retrieved over the Hypertext Transfer Protocol(HTTP). Many browsers also support a variety of other prefixes, such as https: for HTTPS, ftp: for the File Transfer Protocol, and file: for local files. Prefixes that the web browser cannot directly handle are often handed off to another application entirely. For example, mailto: URIs are usually passed to the user’s default e-mail application, and news: URIs are passed to the user’s default newsgroup reader.
In the case of http , https , file , and others, once the resource has been retrieved the web browser will display it. HTML and associated content (image files, formatting information such as CSS, etc.) is passed to the browser’s layout engine to be transformed from markup to an interactive document, a process known as “rendering”. Aside from HTML, web browsers can generally display any kind of content that can be part of a web page. Most browsers can display images, audio, video, and XML files, and often have plug-ins to support Flash applications and Java applets. Upon encountering a file of an unsupported type or a file that is set up to be downloaded rather than displayed, the browser prompts the user to save the file to disk.
Information resources may contain hyperlinks to other information resources. Each link contains the URI of a resource to go to. When a link is clicked, the browser navigates to the resource indicated by the link’s target URI, and the process of bringing content to the user begins again.
Features
Available web browsers range in features from minimal, text-based user interfaces with bare-bones support for HTML to rich user interfaces supporting a wide variety of file formats and protocols. Browsers which include additional components to support e-mail, Usenet news, and Internet Relay Chat (IRC), are sometimes referred to as “Internet suites” rather than merely “web browsers”.
All major web browsers allow the user to open multiple information resources at the same time, either in different browser windows or in different tabs of the same window. Major browsers also include pop-up blockers to prevent unwanted windows from “popping up” without the user’s consent.
Most web browsers can display a list of web pages that the user has bookmarked so that the user can quickly return to them. Bookmarks are also called “Favorites” in Internet Explorer. In addition, all major web browsers have some form of built-in web feed aggregator. In Firefox, web feeds are formatted as “live bookmarks” and behave like a folder of bookmarks corresponding to recent entries in the feed. In Opera, a more traditional feed reader is included which stores and displays the contents of the feed.
Furthermore, most browsers can be extended via plug-ins, downloadable components that provide additional features.
User interface
Most major web browsers have these user interface elements in common:
- Back and forward buttons to go back to the previous resource and forward respectively.
- A refresh or reload button to reload the current resource.
- A stop button to cancel loading the resource. In some browsers, the stop button is merged with the reload button.
- A home button to return to the user’s home page.
- An address bar to input the Uniform Resource Identifier (URI) of the desired resource and display it.
- A search bar to input terms into a search engine. In some browsers, the search bar is merged with the address bar.
- A status bar to display progress in loading the resource and also the URI of links when the cursor hovers over them, and page zooming capability.
- The viewport , the visible area of the webpage within the browser window.
- The ability to view the HTML source for a page.
Major browsers also possess incremental find features to search within a web page.
Privacy and security
Most browsers support HTTP Secure and offer quick and easy ways to delete the web cache, cookies, and browsing history. For a comparison of the current security vulnerabilities of browsers, see comparison of web browsers.
Standards support
Early web browsers supported only a very simple version of HTML. The rapid development of proprietary web browsers led to the development of non-standard dialects of HTML, leading to problems with interoperability. Modern web browsers support a combination of standards-based and de facto HTML and XHTML, which should be rendered in the same way by all browsers.
Extensibility
A browser extension is a computer program that extends the functionality of a web browser. Every major web browser supports the development of browser extensions.
Components
Web browsers consist of a user interface, layout engine, rendering engine, JavaScript interpreter, UI backend, networking component and data persistence component. These components achieve different functionalities of a web browser and together provide all capabilities of a web browser.
- Web Browser. Provided by : Wikipedia. Located at : https://en.wikipedia.org/wiki/Web_browser . License : CC BY-SA: Attribution-ShareAlike
- Web browswers. Authored by : Geralt. Located at : https://pixabay.com/en/browser-web-www-computer-773215/ . License : CC0: No Rights Reserved
- Clockwork Oyyla. Authored by : Anna Jumped. Located at : https://www.flickr.com/photos/17665603@N02/4023601737/ . License : CC BY-SA: Attribution-ShareAlike


6.2: The Trigonometric Series
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Periodic Functions and Motivation
As motivation for studying Fourier series, suppose we have the problem
\[\label{eq:1} x'' + \omega^2_0 x=f(t), \]
for some periodic function \( f(t)\). We have already solved
\[\label{eq:2} x''+ \omega^2_0 x=F_0 \cos( \omega t). \]
One way to solve \(\eqref{eq:1}\) is to decompose \(f(t)\) as a sum of cosines (and sines) and then solve many problems of the form \(\eqref{eq:2}\). We then use the principle of superposition, to sum up all the solutions we got to get a solution to \(\eqref{eq:1}\).
Before we proceed, let us talk a little bit more in detail about periodic functions. A function is said to be periodic with period \(P\) if \( f(t)\) for all \(t\). For brevity we will say \( f(t)\) is \(P-\)periodic. Note that a \(P-\)periodic function is also \(2P-\)periodic, \(3P-\)periodic and so on. For example, \( \cos(t)\) and \( \sin(t)\) are \( 2 \pi -\)periodic. So are \( \cos(kt)\) and \( \sin(kt)\) for all integers \( k\). The constant functions are an extreme example. They are periodic for any period (exercise).
Normally we will start with a function \(f(t)\) defined on some interval \( [-L, L]\) and we will want to extend \(f(t)\) periodically to make it a \( 2L-\)periodic function. We do this extension by defining a new function \(F(t)\) such that for \(t\) in\( [-L, L]\), \( F(t)=f(t)\). For \(t\) in \( [L, 3L]\), we define \( F(t)=f(t-2L)\), for \(t\) in \( [-3L, -L]\), \( F(t)=f(t+2L)\), and so on. We assumed that \( f(-L)=f(L)\). We could have also started with \(f\) defined only on the half-open interval \( (-L, L]\) and then define \( f(-L)=f(L)\).
Example \(\PageIndex{1}\)
Define \( f(t)=1-t^2\) on \([-1, 1]\). Now extend \(f(t)\) periodically to a \(2\)-periodic function. See Figure \(\PageIndex{1}\) on the facing page.
You should be careful to distinguish between \( f(t)\) and its extension. A common mistake is to assume that a formula for \( f(t)\) holds for its extension. It can be confusing when the formula for \( f(t)\) is periodic, but with perhaps a different period.
Exercise \(\PageIndex{1}\)
Define \( f(t)= \cos t\) on \([\dfrac{ - \pi}{2}, \dfrac{ \pi}{2} ]\) . Take the \(\pi -\) periodic extension and sketch its graph. How does it compare to the graph of \( \cos t\) ?
Inner Product and Eigenvector Decomposition
Suppose we have a symmetric matrix , that is \( A^T=A\). We have said before that the eigenvectors of \( A\) are then orthogonal. Here the word orthogonal means that if \( \vec{v}\) and \( \vec{w}\) are two distinct (and not multiples of each other) eigenvectors of \( A\), then \( \left \langle \vec{v}, \vec{w} \right \rangle=0\). In this case the inner product \(\left \langle \vec{v}, \vec{w} \right \rangle\) is the dot product , which can be computed as \( \vec{v}^T \vec{w}\).
To decompose a vector \( \vec{v}\) in terms of mutually orthogonal vectors \( \vec{w}_1\) and \( \vec{w}_2\) we write
\[ \vec{v} = a_1 \vec{w}_1+a_2 \vec{w}_2. \nonumber \]
Let us find the formula for \(a_1\) and \(a_2\). First let us compute
\[\langle \vec{v} , \vec{w_1} \rangle = \langle a_1 \vec{w}_1 + a_2 \vec{w}_2 , \vec{w_1} \rangle = a_1 \langle \vec{w}_1 , \vec{w_1} \rangle + a_2 \underbrace{\langle \vec{w}_2 , \vec{w_1} \rangle}_{=0} = a_1 \langle \vec{w}_1 , \vec{w_1} \rangle . \nonumber \]
Therefore,
\[ a_1= \dfrac{\left \langle \vec{v}, \vec{w}_1 \right \rangle}{\left \langle \vec{w}_1, \vec{w}_1 \right \rangle}. \nonumber \]
Similarly
\[ a_2= \dfrac{\left \langle \vec{v}, \vec{w}_2 \right \rangle}{\left \langle \vec{w}_2, \vec{w}_2 \right \rangle}. \nonumber \]
You probably remember this formula from vector calculus.
Example \(\PageIndex{2}\)
Write \( \vec{v}=\left[ \begin{array}{c} 2 \\ 3 \end{array} \right] \) as a linear combination of \( \vec{w}_1=\left[ \begin{array}{c} 1 \\ -1 \end{array} \right] \) and \( \vec{w}_2=\left[ \begin{array}{c} 1 \\ 1 \end{array} \right] \).
First note that \( \vec{w}_1\) and \( \vec{w}_2\) are orthogonal as \( \left \langle \vec{w}_1, \vec{w}_2 \right \rangle = 1(1)+(-1)1=0\). Then
\[\begin{align}\begin{aligned} a_1&= \dfrac{\left \langle \vec{v}, \vec{w}_1 \right \rangle}{\left \langle \vec{w}_1, \vec{w}_1 \right \rangle}= \dfrac{2(1)+3(-1)}{1(1)+(-1)(-1)}=\dfrac{-1}{2}, \\ a_2&= \dfrac{\left \langle \vec{v}, \vec{w}_2 \right \rangle}{\left \langle \vec{w}_2, \vec{w}_2 \right \rangle}=\dfrac{2+3}{1+1}= \dfrac{5}{2}.\end{aligned}\end{align} \nonumber \]
Hence
\[ \left[ \begin{array}{c} 2 \\ 3 \end{array} \right]=\dfrac{-1}{2} \left[ \begin{array}{c} 1 \\ -1 \end{array} \right]+ \dfrac{5}{2} \left[ \begin{array}{c} 1 \\ 1 \end{array} \right]. \nonumber \]
Trigonometric Series
Instead of decomposing a vector in terms of eigenvectors of a matrix, we will decompose a function in terms of eigenfunctions of a certain eigenvalue problem. The eigenvalue problem we will use for the Fourier series is
\[ x'' + \lambda x=0,~~~~ x(- \pi)=x(\pi)~~~~x'(- \pi)=x'( \pi). \nonumber \]
We have previously computed that the eigenfunctions are \(1, \cos(kt), \sin(kt)\). That is, we will want to find a representation of a \( 2 \pi -\)periodic function \( f(t)\) as
\[ f(t)= \dfrac{a_0}{2}+ \sum^{\infty}_{n=1}a_n \cos(nt)+b_n \sin(nt). \nonumber \]
This series is called the Fourier series \(^{1}\) or the trigonometric series for \(f(t)\). We write the coefficient of the eigenfunction \(1\) as \( \dfrac{a_0}{2}\) for convenience. We could also think of \( 1= \cos(0t)\), so that we only need to look at \( \cos(kt)\) and \( \sin(kt)\).
As for matrices we want to find a projection of \(f(t)\) onto the subspace generated by the eigenfunctions. So we will want to define an inner product of functions . For example, to find \( a_n\) we want to compute \( \left \langle f(t), \cos(nt) \right \rangle \). We define the inner product as
\[ \left \langle f(t), g(t) \right \rangle = \int^{\pi}_{-\pi} f(t)g(t)dt. \nonumber \]
With this definition of the inner product, we have seen in the previous section that the eigenfunctions \( \cos(kt)\)(including the constant eigenfunction), and \( \sin(kt)\) are orthogonal in the sense that
\[\begin{align}\begin{aligned} \langle \, \cos (mt)\, , \, \cos (nt) \, \rangle = 0 & \qquad \text{for } m \not= n , \\ \langle \, \sin (mt)\, , \, \sin (nt) \, \rangle = 0 & \qquad \text{for } m \not= n , \\ \langle \, \sin (mt)\, , \, \cos (nt) \, \rangle = 0 & \qquad \text{for all } m \text{ and } n .\end{aligned}\end{align} \nonumber \]
By elementary calculus for \( n=1,2,3, \ldots .\) we have \( \left \langle \cos(nt), \cos(nt) \right \rangle = \pi\) and \( \left \langle \sin(nt), \sin(nt) \right \rangle = \pi\) . For the constant we get
\[ \left \langle 1, 1 \right \rangle = \int_{\pi}^{\pi}1\cdot 1\: dt= 2 \pi. \nonumber \]
The coefficients are given by
\[\begin{align}\begin{aligned} a_n &= \dfrac{ \left \langle f(t), \cos(nt) \right \rangle}{\left \langle \cos(nt), \cos(nt) \right \rangle}= \dfrac{1}{ \pi} \int^{\pi}_{-\pi} f(t) \cos(nt)dt, \\ b_n &= \dfrac{ \left \langle f(t), \sin(nt) \right \rangle}{\left \langle \sin(nt), \sin(nt) \right \rangle}= \dfrac{1}{ \pi} \int^{\pi}_{-\pi} f(t) \sin(nt)dt.\end{aligned}\end{align} \nonumber \]
Compare these expressions with the finite-dimensional example. For \( a_0\) we get a similar formula
\[ a_0 = 2 \dfrac{ \left \langle f(t), 1 \right \rangle}{\left \langle 1, 1 \right \rangle} \dfrac{1}{ \pi} \int^{\pi}_{-\pi} f(t)dt. \nonumber \]
Let us check the formulas using the orthogonality properties. Suppose for a moment that
\[ f(t)= \frac{a_0}{2}+ \sum^\infty_{n=1}a_n \cos(nt)+b_n \sin(nt). \nonumber \]
Then for \( m \geq 1\) we have
\[\begin{align}\begin{aligned} \left \langle f(t), \cos(mt) \right \rangle &= \left \langle \frac{a_0}{2}+ \sum^\infty_{n=1}a_n \cos(nt)+b_n \sin(nt), \cos(mt) \right \rangle \\ &=\frac{a_0}{2}\left \langle 1, \cos(mt) \right \rangle +\sum^\infty_{n=1}a_n \left \langle \cos(nt), \cos(mt) \right \rangle + b_n \left \langle \sin(nt), \sin(mt) \right \rangle \\ &= a_m \left \langle \cos(mt), \cos(mt) \right \rangle . \end{aligned}\end{align} \nonumber \]
And hence \( a_m=\frac{\left \langle f(t), \cos(mt) \right \rangle}{\left \langle \cos(mt), \cos(mt) \right \rangle}.\)
Exercise \(\PageIndex{2}\)
Carry out the calculation for \(a_0\) and \(b_m\) .
Example \(\PageIndex{3}\)
Take the function
\[ f(t)=t \nonumber \]
for \(t\) in \((- \pi, \pi]\). Extend \( f(t)\) periodically and write it as a Fourier series. This function is called the sawtooth .
The plot of the extended periodic function is given in Figure \(\PageIndex{2}\). Let us compute the coefficients.
Solution
We start with \(a_0\),
\[ a_0 = \frac{1}{\pi} \int^\pi_{-\pi} tdt=0. \nonumber \]
We will often use the result from calculus that says that the integral of an odd function over a symmetric interval is zero. Recall that an odd function is a function \( \varphi(t)\) such that \( \varphi(-t) = - \varphi(t)\). For example the functions \( t, \sin t\), or (importantly for us) \( t \cos(nt)\) are all odd functions. Thus
\[ a_n=\frac{1}{\pi} \int^\pi_{-\pi} t \cos(nt)dt=0. \nonumber \]
Let us move to \( b_n\). Another useful fact from calculus is that the integral of an even function over a symmetric interval is twice the integral of the same function over half the interval. Recall an even function is a function \(\varphi(t)\) such that \( \varphi(-t) = \varphi(t)\). For example \( t \sin(nt)\) is even.
\[\begin{align}\begin{aligned} b_n &= \frac{1}{\pi} \int^\pi_{-\pi} t \sin(nt)dt \\ &= \frac{2}{\pi} \int^\pi_{0} t \sin(nt)dt \\ &= \frac{2}{\pi} \left( \left[ \frac{-t \cos(nt)}{n}\right] ^\pi_{t=0}+\frac{1}{n} \int^\pi_{0} \cos(nt)dt\right) \\ &= \frac{2}{\pi} \left( \frac{- \pi \cos(n \pi)}{n}+0\right) \\ &= \frac{-2 \cos(n \pi)}{n}=\frac{2(-1)^{n+1}}{n}.\end{aligned}\end{align} \nonumber \]
We have used the fact that
\[ \cos(n \pi)=(-1)^n= \left\{ \begin{array}{c} 1~~~~ {\rm{~if~}} n {\rm{~even,~}}\\ -1~~~~ {\rm{~if~}} n {\rm{~odd.~}} \end{array} \right. \nonumber \]
The series, therefore, is
\[ \sum^\infty_{n=1} \frac{2(-1)^{n+1}}{n} \sin(nt). \nonumber \]
Let us write out the first \(3\) harmonics of the series for \(f(t)\).
\[ 2 \sin(t)- \sin(2t)+ \frac{2}{3} \sin(3t)+ \cdots \nonumber \]
The plot of these first three terms of the series, along with a plot of the first \(20\) terms is given in Figure \(\PageIndex{3}\).
Example \(\PageIndex{4}\)
Take the function
\[ f(t)= \left\{ \begin{array}{cc} 0&~~~~ {\rm{~if~}} - \pi < t \leq 0, \\ \pi& {\rm{~if~}} 0 < t \leq \pi. \end{array} \right. \nonumber \]
Extend \(f(t)\) periodically and write it as a Fourier series. This function or its variants appear often in applications and the function is called the square wave.
T he plot of the extended periodic function is given in Figure \(\PageIndex{4}\). Now we compute the coefficients. Let us start with \(a_0\)
\[ a_0= \frac{1}{\pi} \int^\pi_{-\pi} f(t)dt= \frac{1}{\pi} \int^\pi_{0} \pi dt= \pi . \nonumber \]
Next,
\[ a_n = \frac{1}{\pi} \int^\pi_{-\pi} f(t) \cos(nt)dt= \frac{1}{\pi} \int^\pi_{0} \pi \cos(nt)dt= 0 . \nonumber \]
And finally
\[\begin{align}\begin{aligned} b_n &= \frac{1}{\pi} \int^\pi_{-\pi} f(t) \sin(nt)dt \\ &= \frac{1}{\pi} \int^\pi_{0} \pi \sin(nt)dt \\ &= \left[ \frac{- \cos(nt)}{n}\right]_{t=0}^{\pi} \\ &=\frac{1- \cos( \pi n)}{n}=\frac{1-(-1)^n}{n}= \left\{ \begin{array}{c} \frac{2}{n}~~~~ {\rm{~if~}} n {\rm{~is~odd,~}}\\ 0~~~~ {\rm{~if~}} n {\rm{~is~even.~}} \end{array} \right. \end{aligned}\end{align} \nonumber \]
The Fourier series is
\[ \frac{\pi}{2} + \sum^{\infty}_{n=1~ n~ \rm{odd}} \frac{2}{n} \sin(nt)+\sum^{\infty}_{k=1} \frac{2}{2k-1} \sin((2k-1)t). \nonumber \]
Let us write out the first 3 harmonics of the series for \(f(t)\).
\[ \frac{\pi}{2}+2 \sin(t)+ \frac{2}{3} \sin(3t)+ \cdots \nonumber \]
The plot of these first three and also of the first 20 terms of the series is given in Figure \(\PageIndex{5}\).
We have so far skirted the issue of convergence. For example, if \(f(t)\) is the square wave function, the equation
\[ f(t)= \frac{\pi}{2} + \sum_{k=1}^{\infty}\frac{2}{2k-1} \sin((2k-1)t). \nonumber \]
is only an equality for such \(t\) where \(f(t)\) is continuous. That is, we do not get an equality for \(t= - \pi, 0, \pi\) and all the other discontinuities of \(f(t)\). It is not hard to see that when \(t\) is an integer multiple of \(\pi\) (which includes all the discontinuities), then
\[ \frac{\pi}{2} + \sum_{k=1}^{\infty}\frac{2}{2k-1} \sin((2k-1)t)=\frac{\pi}{2}. \nonumber \]
We redefine \(f(t)\) on \([- \pi, \pi]\) as
\[ f(t)=\left\{ \begin{array}{cc} 0 &~~~~ {\rm{~if~}} - \pi<t<0, \\ \pi & {\rm{~if~}} 0<t<\pi, \\ \pi/2& ~~~~~~~~~~~~~~~~~~ {\rm{~if~}} t= - \pi, t=0, {\rm{~or~}} t=\pi, \end{array} \right. \nonumber \]
and extend periodically. The series equals this extended \(f(t)\) everywhere, including the discontinuities. We will generally not worry about changing the function values at several (finitely many) points.
We will say more about convergence in the next section. Let us however mention briefly an effect of the discontinuity. Let us zoom in near the discontinuity in the square wave. Further, let us plot the first 100 harmonics, see Figure \(\PageIndex{6}\). You will notice that while the series is a very good approximation away from the discontinuities, the error (the overshoot) near the discontinuity at \( t= \pi\) does not seem to be getting any smaller. This behavior is known as the Gibbs phenomenon . The region where the error is large does get smaller, however, the more terms in the series we take.
We can think of a periodic function as a “signal” being a superposition of many signals of pure frequency. For example, we could think of the square wave as a tone of certain base frequency. This base frequency is called the fundamental frequency . The square wave will be a superposition of many different pure tones of frequencies that are multiples of the fundamental frequency. In music, the higher frequencies are called the overtones . All the frequencies that appear are called the spectrum of the signal. On the other hand a simple sine wave is only the pure tone (no overtones). The simplest way to make sound using a computer is the square wave, and the sound is very different from a pure tone. If you ever played video games from the 1980s or so, then you heard what square waves sound like.


6.7: PayPal Case Study- Ensuring Your Workers Are Not Just Surviving
Accommodation and Enhancement in Action
Since 2015, I’ve been President and CEO at PayPal, a company that focuses on the digital transfer of money and e-commerce. Our business model is driven by our mission: We want everyone — individuals, families, small businesses, retailers of all sizes — to have access to affordable, convenient and secure financial services. As a company, our employees and I have worked hard to deliver on this goal for customers and communities around the world. And we’ve felt even greater urgency now during the COVID-19 pandemic.
But while we were focused on fulfilling our mission in the larger world, we missed something that was going on inside our company. At PayPal, we make it a point to pay above-market salaries and offer generous benefits, and we assumed this was sufficient. But when we looked more closely, we saw that in some places and for some people, it just wasn’t enough.
For example, 65 percent of our customer-service and entry-level employees reported they were frequently running out of money between paydays. As a result, they had to make difficult trade-offs in educating their kids, paying for healthcare, and planning for the long term.
At times, they faced more heartbreaking situations. In 2017, we created an Employee Relief Fund to help people in case of a significant or unforeseen financial shock. One sign of trouble came from a trend that we saw in applications to this fund: We found urgent requests for help were increasingly the result of everyday events, like an unexpectedly steep medical bill, a student-loan payment or a car breaking down.
Knowing that our employees were struggling in this way was simply unacceptable. So, we decided to take some significant steps to shore up their financial security and health. There were four key elements to the Employee Financial Wellness Initiative that we announced at the end of 2019: 1) We reduced the cost of healthcare; 2) we reviewed and raised wages where appropriate; 3) we made everyone a shareholder and long-term beneficiary of our collective success as a company; and 4) we wrapped everything with a long-term financial education and planning program.
At the core of this program was a calculation called Net Disposable Income, or NDI . NDI is the amount of money that a person has left over after paying taxes and necessary living expenses, such as taxes, food, housing and transportation.
We began by reviewing current starting pay for our more than 23,000 global employees by location, taking into account their estimated cost of living and taxes. But we also wanted to go further and understand the impact of other key factors that were within our control as a company — factors like health insurance, long-term savings opportunities, even timing of when people were paid.
We looked at NDI in each of the different geographic locations in which we have employees, and we found that many of our entry-level and hourly employees had an NDI of 4-6 percent. In other words, at the end of every month, they had only this small percentage of their pay that wasn’t automatically going toward their essentials. If we want employees to fulfill our company’s mission and innovate and effectively deliver services, we simply can’t have them worrying how they’ll make ends meet every month or how they’ll cover the long-term needs of their families.
Ultimately, we decided to target an NDI of at least 20 percent. To drive an increase this significant, we needed to think more holistically and look at both their income and their costs. By far, the biggest cost driver for this portion of our workforce was healthcare. While we offered a generous health plan, it was generally the same price for everyone at the company. But this meant the lower a person’s income, the more expensive it was for them — and a higher percentage of their income. So, we lowered the cost of US healthcare benefits for hourly wage earners by an average of nearly 60 percent for employees for whom the cost of healthcare was too high.
After tackling wages and healthcare costs , a third step was to help employees save more and better prepare for the future. We chose to make everyone a shareholder and owner of PayPal, irrespective of level or position. We made one-time PayPal stock grants to anyone who hadn’t already received one, and we made everyone eligible to earn PayPal stock awards annually. I fundamentally believe that employee ownership of our company through stock is foundational to our success, because it gives every employee the opportunity to benefit from our performance and puts them all on the path toward building long-term savings.
Finally, we’ve launched a full suite of financial learning programs and counseling. As we all know, building and maintaining fiscal health requires continual learning, and it’s not a linear journey. Financial matters are complex, and we all need support along the way. We now offer coaching services, personalized consultations and other resources for all of our employees, and we’re continually exploring, evaluating and introducing new tools. Last but not least, we continue to maintain the Employee Relief Fund for urgent situations.
Of course, it will be a while before we learn the real impact of these changes and whether they’re sufficient in meeting people’s needs. We announced this program at the end of 2019, and we’re monitoring its progress.
But we’ve already started to see promising results, such as:
- A decrease in the number of employees who report they are running out of funds between pay periods
- An increase in healthcare benefit enrollment and employees upgrading their plans
- A higher rate of 401(k) and employee stock purchase plan enrollment
This has been a major undertaking, but it’s something we’d absolutely do again and encourage other companies to do as well. When we began this effort, I certainly didn’t expect to discover such a large gap between what we were providing for our employees and what they needed. Closing this gap has required a significant financial investment, but I believe it’s worth every dollar and every resource and that it’s been essential to making our workforce more resilient in the face of COVID-19.
Capitalism needs an upgrade — while I still think it’s the best system to stimulate innovation and commerce, it’s not working for a lot of people. The next generation of business leaders has an obligation to take action and do more, and prioritizing and investing in our employees is tangible and doable. If enough leaders and companies can make this difference for their own people, we can start to fix what has been broken.
About the author
Dan Schulman is the CEO of PayPal. He is focused on democratizing and transforming financial services and e-commerce to improve the financial health of billions of people, families and businesses around the world. With extensive experience in payments and mobile technology, Schulman is leading PayPal to reimagine how people move and manage money, and how merchants and consumers interact and transact.
Source: "How can you ensure your workers are not just surviving — but thriving? A CEO shares a new approach", Ideas.ted.com. May 29, 2020.


3: Congruences
A congruence is nothing more than a statement about divisibility. The theory of congruences was introduced by Carl Friedreich Gauss. Gauss contributed to the basic ideas of congruences and proved several theorems related to this theory. We start by introducing congruences and their properties. We proceed to prove theorems about the residue system in connection with the Euler \(\phi\)-function. We then present solutions to linear congruences which will serve as an introduction to the Chinese remainder theorem. We present finally important congruence theorems derived by Wilson, Fermat and Euler.
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- 3.4: The Chinese Remainder Theorem
- In this section, we discuss the solution of a system of congruences having different moduli. An example of this kind of systems is the following; find a number that leaves a remainder of 1 when divided by 2, a remainder of 2 when divided by three and a remainder of 3 when divided by 5. This kind of question can be translated into the language of congruences. As a result, in this chapter, we present a systematic way of solving this system of congruences.
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- 3.5: Theorems of Fermat, Euler, and Wilson
- In this section we present three applications of congruences. The first theorem is Wilson’s theorem which states that (p−1)!+1 is divisible by p , for p prime. Next, we present Fermat’s theorem, also known as Fermat’s little theorem which states that ap and a have the same remainders when divided by p where p∤a . Finally we present Euler’s theorem which is a generalization of Fermat’s theorem and it states that for any positive integer m that is relatively prime to an integer a.
Contributors and Attributions
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Dr. Wissam Raji, Ph.D., of the American University in Beirut. His work was selected by the Saylor Foundation’s Open Textbook Challenge for public release under a Creative Commons Attribution ( CC BY ) license.


12.4: Kentucky v. King
PETITIONER RESPONDENT
Kentucky Hollis Deshaun King
LOCATION
DOCKET NO. DECIDED BY
09-1272 Roberts Court
LOWER COURT
Kentucky Supreme Court
CITATION
GRANTED
Sep 28, 2010
ARGUED
Jan 12, 2011
DECIDED
May 16, 2011
ADVOCATES
Joshua D. Farley for the petitioner
Ann O'Connell Assistant to the Solicitor General, Department of Justice, as amicus curiae, supporting the petitioner
Jamesa J. Drake for the respondent
Facts of the case
Police officers in Lexington, Ky., entered an apartment building in pursuit of a suspect who sold crack cocaine to an undercover informant. The officers lost sight of the suspect and mistakenly assumed he entered an apartment from which they could detect the odor of marijuana. After police knocked on the door and identified themselves, they heard movements, which they believed indicated evidence was about to be destroyed. Police forcibly entered the apartment and found Hollis King and others smoking marijuana. They also found cash, drugs and paraphernalia. King entered a conditional guilty plea; reserving his right to appeal denial of his motion to suppress evidence obtained from what he argued was an illegal search.
The Kentucky Court of Appeals affirmed the conviction, holding that exigent circumstances supporting the warrantless search were not of the police's making and that police did not engage in deliberate and intentional conduct to evade the warrant requirement. In January 2010, the Kentucky Supreme Court reversed the lower court order, finding that the entry was improper. The court held that the police were not in pursuit of a fleeing suspect when they entered the apartment, since there was no evidence that the original suspect even knew he was being followed by police.
Question
Does the exclusionary rule, which forbids the use of illegally seized evidence except in emergency situations, apply when the emergency is created by lawful police actions?
Co n c l u s i on
8–1 Decision for Kentucky Majority Opinion by Samuel A. Alito,
| FOR | AGAINST |
|---|---|
|
Sotomayor Kagan Breyer Kennedy Roberts Alito Scalia Thomas |
Ginsburg |
The Supreme Court reversed and remanded the lower court order in a decision by Justice Samuel Alito. "The exigent circumstances rule applies when the police do not create the exigency by engaging or threatening to engage in conduct that violates the Fourth Amendment," Alito wrote for the majority. Justice Ruth Bader Ginsburg dissented, contending that "the Court today arms the police with a way routinely to dishonor the Fourth Amendment's warrant requirement in drug cases."


8.2: Introduction to Estimating a Population Proportion
What you’ll learn to do: Construct a confidence interval to estimate a population proportion.
In this section, we will continue studying the estimation of a proportion with a confidence interval. We will recognize situations that call for testing a claim about a population or estimating a population proportion. We will learn to construct a confidence interval to estimate a population proportion when certain conditions are met and interpret this interval in context. We also will learn how to interpret the meaning of a confidence level and relate it to the margin of error. These tools can be used in many applications from sports attendances to online medical tools.
Contributors and Attributions
CC licensed content, Shared previously
- Concepts in Statistics. Provided by : Open Learning Initiative. Located at : http://oli.cmu.edu . License : CC BY: Attribution


25.1D: Water Balance Disorders
Dehydration is the excessive loss of body fluid.
- Describe disorders due to water balance problems
Key Points
- There are three types of dehydration: hypotonic or hyponatremic, hypertonic or hypernatremic, and isotonic or isonatremic.
- Hypotonic dehydration is primarily a loss of electrolytes, sodium in particular.
- Hypertonic dehydration is primarily a loss of water.
- Isotonic dehydration is an equal loss of water and electrolytes.
- Hypvolemia is a loss of blood volume, and may cause hypovolemic shock. In humans, the most common type of dehydration by far is isotonic (isonatraemic) dehydration.
- Water balance disorders are generally treated by increasing water intake and reducing or stopping fluid loss.
Key Terms
- isotonic : When comparing solutions, an isotonic solution has the same osmolarity (ion concentration) as the solution it is being compared to.
- plasma : The straw-colored/pale-yellow, liquid component of blood that normally holds the blood cells of whole blood in suspension.
- electrolyte : Any of the various ions (such as sodium or chloride) that regulate the electric charge on cells and the flow of water across their membranes.
Water Balance Disorders
In physiology and medicine, dehydration (hypohydration) is defined as the excessive loss of body fluid. It is literally the removal of water from an object. However, in physiological terms, it entails a deficiency of fluid within an organism.
Much of the physiological effects of dehydration is due to the changes in ion concentration that may occur as a result of the dehydration. Alternatively, hypovolemia may occur due to loss of blood volume itself.
Dehydration
There are three types of dehydration that differ based on the type of change in ion concentrations:
- Hypotonic—primarily a loss of electrolytes, sodium in particular. Hypotonic dehydration causes decreased plasma osmolarity.
- Hypertonic—primarily a loss of water. Hypertonic dehydration causes increased plasma osmolarity.
- Isotonic—an equal loss of water and electrolytes. Isotonic dehydration will not change plasma osmolarity, but it will reduce overall plasma volume. Isotonic dehydration is the most common type of dehydration.
Further complications may also occur. In hypotonic dehydration, intravascular water shifts to the extravascular space and exaggerates intravascular volume depletion for a given amount of total body water loss.
Neurological complications can occur in hypotonic and hypertonic states. The former can lead to seizures, while the latter can lead to osmotic cerebral edema upon rapid rehydration.
Hypovolemia
Hypovolemia is specifically a decrease in the volume of blood plasma. Furthermore, hypovolemia defines water deficiency in terms of blood volume rather than the overall water content of the body.
IV fluid and electrolyte administration : Intravenous administration of fluid is one effective treatment of dehydration in humans.
Hypovolemia is a cause of hypovolemic shock. Shock is any condition in which the body’s fluids are unable to properly circulate and oxygenate the major organs of the human body; this causes compensatory mechanisms to activate that cause further bodily harm as the body’s metabolism is maintained for a while longer.
In the case of hypovolemic shock, the tissue metabolism is impaired due to a lack of blood volume and makes it difficult for red blood cells to reach all of the tissues of the body. It is most often caused by severe vomiting, diarrhea, blood loss, or hemorrhage. Other forms of shock with similar symptoms may be due to problems in the heart (cardiogenic) or bacterial infection (septic).
Treatment Options
To treat minor dehydration water intake must be increased, while the source of fluid loss must be reduced or stopped altogether. Plain water restores only the volume of the blood plasma and inhibits the thirst mechanism before solute levels can be replenished.
Solid foods can contribute to fluid loss from vomiting and diarrhea. In more severe cases, correction of a dehydrated state is accomplished by the replenishment of necessary water and electrolytes through oral rehydration therapy or fluid replacement by intravenous therapy (an IV drip).
As oral rehydration is easier to provide, it is the treatment of choice for mild dehydration. Solutions used for intravenous rehydration must be isotonic or hypotonic. Pure water injected into the veins will cause the breakdown (lysis) of red blood cells that could cause other problems.
LICENSES AND ATTRIBUTIONS
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- Free water clearance. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Free_water_clearance . License : CC BY-SA: Attribution-ShareAlike
- Osmoreceptors. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Osmoreceptors . License : CC BY-SA: Attribution-ShareAlike
- Fluid balance. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Fluid_balance . License : CC BY-SA: Attribution-ShareAlike
- dehydration. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/dehydration . License : CC BY-SA: Attribution-ShareAlike
- thirst. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/thirst . License : CC BY-SA: Attribution-ShareAlike
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- Fluid balance. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Fluid_balance . License : CC BY-SA: Attribution-ShareAlike
- aldosterone. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/aldosterone . License : CC BY-SA: Attribution-ShareAlike
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- OpenStax College, Biology. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44811/latest...ol11448/latest . License : CC BY: Attribution
- Urea. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Urea . License : CC BY-SA: Attribution-ShareAlike
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2.5.8: Derivability and Consistency
We will now establish a number of properties of the derivability relation. They are independently interesting, but each will play a role in the proof of the completeness theorem.
If \(\Gamma \Proves A\) and \(\Gamma \cup \{A\}\) is inconsistent, then \(\Gamma\) is inconsistent.
Proof. Let the derivation of \(A\) from \(\Gamma\) be \(\delta_1\) and the derivation of \(\lfalse\) from \(\Gamma \cup \{A\}\) be \(\delta_2\) . We can then derive:
In the new derivation, the assumption \(A\) is discharged, so it is a derivation from \(\Gamma\) . ◻
\(\Gamma \Proves A\) iff \(\Gamma \cup \{\lnot A\}\) is inconsistent.
Proof. First suppose \(\Gamma \Proves A\) , i.e., there is a derivation \(\delta_0\) of \(A\) from undischarged assumptions \(\Gamma\) . We obtain a derivation of \(\lfalse\) from \(\Gamma \cup \{\lnot A\}\) as follows:
Now assume \(\Gamma \cup \{\lnot A\}\) is inconsistent, and let \(\delta_1\) be the corresponding derivation of \(\lfalse\) from undischarged assumptions in \(\Gamma \cup \{\lnot A\}\) . We obtain a derivation of \(A\) from \(\Gamma\) alone by using \(\FalseCl\) :
◻
Prove that \(\Gamma \Proves \lnot A\) iff \(\Gamma \cup \{A\}\) is inconsistent.
If \(\Gamma \Proves A\) and \(\lnot A \in \Gamma\) , then \(\Gamma\) is inconsistent.
Proof. Suppose \(\Gamma \Proves A\) and \(\lnot A \in \Gamma\) . Then there is a derivation \(\delta\) of \(A\) from \(\Gamma\) . Consider this simple application of the \(\Elim{\lnot}\) rule:
Since \(\lnot A \in \Gamma\) , all undischarged assumptions are in \(\Gamma\) , this shows that \(\Gamma \Proves \lfalse\) . ◻
If \(\Gamma \cup \{A\}\) and \(\Gamma \cup \{\lnot A\}\) are both inconsistent, then \(\Gamma\) is inconsistent.
Proof. There are derivations \(\delta_1\) and \(\delta_2\) of \(\lfalse\) from \(\Gamma \cup \{ A \}\) and \(\lfalse\) from \(\Gamma \cup \{ \lnot A \}\) , respectively. We can then derive
Since the assumptions \(A\) and \(\lnot A\) are discharged, this is a derivation of \(\lfalse\) from \(\Gamma\) alone. Hence \(\Gamma\) is inconsistent. ◻


|
\(\ce{Ag+ + e- ⟶Ag}\)
|
+0.7996
|
|
\(\ce{AgCl + e- ⟶Ag + Cl-}\)
|
+0.22233
|
|
\(\ce{[Ag(CN)2]- + e- ⟶Ag + 2CN-}\)
|
−0.31
|
|
\(\ce{Ag2CrO4 + 2e- ⟶2Ag + CrO4^2-}\)
|
+0.45
|
|
\(\ce{[Ag(NH3)2]+ + e- ⟶Ag + 2NH3}\)
|
+0.373
|
|
\(\ce{[Ag(S2O3)2]^3+ + e- ⟶Ag + 2S2O3^2-}\)
|
+0.017
|
|
\(\ce{[AlF6]^3- + 3e- ⟶Al + 6F-}\)
|
−2.07
|
|
\(\ce{Al^3+ + 3e- ⟶Al}\)
|
−1.662
|
|
\(\ce{Am^3+ + 3e- ⟶Am}\)
|
−2.048
|
|
\(\ce{Au^3+ + 3e- ⟶Au}\)
|
+1.498
|
|
\(\ce{Au+ + e- ⟶Au}\)
|
+1.692
|
|
\(\ce{Ba^2+ + 2e- ⟶Ba}\)
|
−2.912
|
|
\(\ce{Be^2+ + 2e- ⟶Be}\)
|
−1.847
|
|
\(\ce{Br2}(aq) + \ce{2e-} ⟶\ce{2Br-}\)
|
+1.0873
|
|
\(\ce{Ca^2+ + 2e- ⟶Ca}\)
|
−2.868
|
|
\(\ce{Ce^3 + 3e- ⟶Ce}\)
|
−2.483
|
|
\(\ce{Ce^4+ + e- ⟶Ce^3+}\)
|
+1.61
|
|
\(\ce{Cd^2+ + 2e- ⟶Cd}\)
|
−0.4030
|
|
\(\ce{[Cd(CN)4]^2- + 2e- ⟶Cd + 4CN-}\)
|
−1.09
|
|
\(\ce{[Cd(NH3)4]^2+ + 2e- ⟶Cd + 4NH3}\)
|
−0.61
|
|
\(\ce{CdS + 2e- ⟶Cd + S^2-}\)
|
−1.17
|
|
\(\ce{Cl2 + 2e- ⟶2Cl-}\)
|
+1.35827
|
|
\(\ce{ClO4- + H2O + 2e- ⟶ClO3- + 2OH-}\)
|
+0.36
|
|
\(\ce{ClO3- + H2O + 2e- ⟶ClO2- + 2OH-}\)
|
+0.33
|
|
\(\ce{ClO2- + H2O + 2e- ⟶ClO- + 2OH-}\)
|
+0.66
|
|
\(\ce{ClO- + H2O + 2e- ⟶Cl- + 2OH-}\)
|
+0.89
|
|
\(\ce{ClO4- + 2H3O+ + 2e- ⟶ClO3- + 3H2O}\)
|
+1.189
|
|
\(\ce{ClO3- + 3H3O+ + 2e- ⟶HClO2 + 4H2O}\)
|
+1.21
|
|
\(\ce{HClO + H3O+ + 2e- ⟶Cl- + 2H2O}\)
|
+1.482
|
|
\(\ce{HClO + H3O+ + e- ⟶\dfrac{1}{2}Cl2 + 2H2O}\)
|
+1.611
|
|
\(\ce{HClO2 + 2H3O+ + 2e- ⟶HClO + 3H2O}\)
|
+1.628
|
|
\(\ce{Co^3+ + e- ⟶Co^2+ \:(2\:mol\,//\,H2SO4)}\)
|
+1.83
|
|
\(\ce{Co^2+ + 2e- ⟶Co}\)
|
−0.28
|
|
\(\ce{[Co(NH3)6]^3+ + e- ⟶[Co(NH3)6]^2+}\)
|
+0.1
|
|
\(\ce{Co(OH)3 + e- ⟶Co(OH)2 + OH-}\)
|
+0.17
|
|
\(\ce{Cr^3 + 3e- ⟶Cr}\)
|
−0.744
|
|
\(\ce{Cr^3+ + e- ⟶Cr^2+}\)
|
−0.407
|
|
\(\ce{Cr^2+ + 2e- ⟶Cr}\)
|
−0.913
|
|
\(\ce{[Cu(CN)2]- + e- ⟶Cu + 2CN-}\)
|
−0.43
|
|
\(\ce{CrO4^2- + 4H2O + 3e- ⟶ Cr(OH)3 + 5OH-}\)
|
−0.13
|
|
\(\ce{Cr2O7^2- + 14H3O+ + 6e- ⟶2Cr^3+ + 21H2O}\)
|
+1.232
|
|
\(\ce{[Cr(OH)4]- + 3e- ⟶Cr + 4OH-}\)
|
−1.2
|
|
\(\ce{Cr(OH)3 + 3e- ⟶Cr + 3OH-}\)
|
−1.48
|
|
\(\ce{Cu^2+ + e- ⟶Cu+}\)
|
+0.153
|
|
\(\ce{Cu^2+ + 2e- ⟶Cu}\)
|
+0.34
|
|
\(\ce{Cu+ + e- ⟶Cu}\)
|
+0.521
|
|
\(\ce{F2 + 2e- ⟶2F-}\)
|
+2.866
|
|
\(\ce{Fe^2+ + 2e- ⟶Fe}\)
|
−0.447
|
|
\(\ce{Fe^3+ + e- ⟶Fe^2+}\)
|
+0.771
|
|
\(\ce{[Fe(CN)6]^3- + e- ⟶[Fe(CN)6]^4-}\)
|
+0.36
|
|
\(\ce{Fe(OH)2 + 2e- ⟶Fe + 2OH-}\)
|
−0.88
|
|
\(\ce{FeS + 2e- ⟶Fe + S^2-}\)
|
−1.01
|
|
\(\ce{Ga^3+ + 3e- ⟶Ga}\)
|
−0.549
|
|
\(\ce{Gd^3+ + 3e- ⟶Gd}\)
|
−2.279
|
|
\(\ce{\dfrac{1}{2}H2 + e- ⟶H- }\)
|
−2.23
|
|
\(\ce{2H2O + 2e- ⟶H2 + 2OH-}\)
|
−0.8277
|
|
\(\ce{H2O2 + 2H3O+ + 2e- ⟶4H2O}\)
|
+1.776
|
|
\(\ce{2H3O+ + 2e- ⟶H2 + 2H2O}\)
|
0.00
|
|
\(\ce{HO2- + H2O + 2e- ⟶3OH-}\)
|
+0.878
|
|
\(\ce{Hf^4+ + 4e- ⟶Hf}\)
|
−1.55
|
|
\(\ce{Hg^2+ + 2e- ⟶Hg}\)
|
+0.851
|
|
\(\ce{2Hg^2+ + 2e- ⟶Hg2^2+}\)
|
+0.92
|
|
\(\ce{Hg2^2+ + 2e- ⟶2Hg}\)
|
+0.7973
|
|
\(\ce{[HgBr4]^2- + 2e- ⟶Hg + 4Br-}\)
|
+0.21
|
|
\(\ce{Hg2Cl2 + 2e- ⟶2Hg + 2Cl-}\)
|
+0.26808
|
|
\(\ce{[Hg(CN)4]^2- + 2e- ⟶Hg + 4CN-}\)
|
−0.37
|
|
\(\ce{[HgI4]^2- + 2e- ⟶Hg + 4I-}\)
|
−0.04
|
|
\(\ce{HgS + 2e- ⟶Hg + S^2-}\)
|
−0.70
|
|
\(\ce{I2 + 2e- ⟶2I-}\)
|
+0.5355
|
|
\(\ce{In^3+ + 3e- ⟶In}\)
|
−0.3382
|
|
\(\ce{K+ + e- ⟶K}\)
|
−2.931
|
|
\(\ce{La^3+ + 3e- ⟶La}\)
|
−2.52
|
|
\(\ce{Li+ + e- ⟶Li}\)
|
−3.04
|
|
\(\ce{Lu^3+ + 3e- ⟶Lu}\)
|
−2.28
|
|
\(\ce{Mg^2+ + 2e- ⟶Mg}\)
|
−2.372
|
|
\(\ce{Mn^2+ + 2e- ⟶Mn}\)
|
−1.185
|
|
\(\ce{MnO2 + 2H2O + 2e- ⟶Mn(OH)2 + 2OH-}\)
|
−0.05
|
|
\(\ce{MnO4- + 2H2O + 3e- ⟶MnO2 + 4OH-}\)
|
+0.558
|
|
\(\ce{MnO2 + 4H+ + 2e- ⟶Mn^2+ + 2H2O}\)
|
+1.23
|
|
\(\ce{MnO4- + 8H+ + 5e- ⟶Mn^2+ + 4H2O}\)
|
+1.507
|
|
\(\ce{Na+ + e- ⟶Na}\)
|
−2.71
|
|
\(\ce{Nd^3+ + 3e- ⟶Nd}\)
|
−2.323
|
|
\(\ce{Ni^2+ + 2e- ⟶Ni}\)
|
−0.257
|
|
\(\ce{[Ni(NH3)6]^2+ + 2e- ⟶Ni + 6NH3}\)
|
−0.49
|
|
\(\ce{NiO2 + 4H+ + 2e- ⟶Ni^2+ + 2H2O}\)
|
+1.593
|
|
\(\ce{NiO2 + 2H2O + 2e- ⟶Ni(OH)2 + 2OH-}\)
|
+0.49
|
|
\(\ce{NiS + 2e- ⟶Ni + S^2-}\)
|
+0.76
|
|
\(\ce{NO3- + 4H+ + 3e- ⟶NO + 2H2O}\)
|
+0.957
|
|
\(\ce{NO3- + 3H+ + 2e- ⟶HNO2 + H2O}\)
|
+0.92
|
|
\(\ce{NO3- + H2O + 2e- ⟶NO2- + 2OH-}\)
|
+0.10
|
|
\(\ce{Np^3+ + 3e- ⟶Np}\)
|
−1.856
|
|
\(\ce{O2 + 2H2O + 4e- ⟶4OH-}\)
|
+0.401
|
|
\(\ce{O2 + 2H+ + 2e- ⟶H2O2}\)
|
+0.695
|
|
\(\ce{O2 + 4H+ + 4e- ⟶2H2O}\)
|
+1.229
|
|
\(\ce{Pb^2+ + 2e- ⟶Pb}\)
|
−0.1262
|
|
\(\ce{PbO2 + SO4^2- + 4H+ + 2e- ⟶PbSO4 + 2H2O}\)
|
+1.69
|
|
\(\ce{PbS + 2e- ⟶Pb + S^2-}\)
|
−0.95
|
|
\(\ce{PbSO4 + 2e- ⟶Pb + SO4^2-}\)
|
−0.3505
|
|
\(\ce{Pd^2+ + 2e- ⟶Pd}\)
|
+0.987
|
|
\(\ce{[PdCl4]^2- + 2e- ⟶Pd + 4Cl-}\)
|
+0.591
|
|
\(\ce{Pt^2+ + 2e- ⟶Pt}\)
|
+1.20
|
|
\(\ce{[PtBr4]^2- + 2e- ⟶Pt + 4Br-}\)
|
+0.58
|
|
\(\ce{[PtCl4]^2- + 2e- ⟶Pt + 4Cl-}\)
|
+0.755
|
|
\(\ce{[PtCl6]^2- + 2e- ⟶[PtCl4]^2- + 2Cl-}\)
|
+0.68
|
|
\(\ce{Pu^3 + 3e- ⟶Pu}\)
|
−2.03
|
|
\(\ce{Ra^2+ + 2e- ⟶Ra}\)
|
−2.92
|
|
\(\ce{Rb+ + e- ⟶Rb}\)
|
−2.98
|
|
\(\ce{[RhCl6]^3- + 3e- ⟶Rh + 6Cl-}\)
|
+0.44
|
|
\(\ce{S + 2e- ⟶S^2-}\)
|
−0.47627
|
|
\(\ce{S + 2H+ + 2e- ⟶H2S}\)
|
+0.142
|
|
\(\ce{Sc^3+ + 3e- ⟶Sc}\)
|
−2.09
|
|
\(\ce{Se + 2H+ + 2e- ⟶H2Se}\)
|
−0.399
|
|
\(\ce{[SiF6]^2- + 4e- ⟶Si + 6F-}\)
|
−1.2
|
|
\(\ce{SiO3^2- + 3H2O + 4e- ⟶Si + 6OH-}\)
|
−1.697
|
|
\(\ce{SiO2 + 4H+ + 4e- ⟶Si + 2H2O}\)
|
−0.86
|
|
\(\ce{Sm^3+ + 3e- ⟶Sm}\)
|
−2.304
|
|
\(\ce{Sn^4+ + 2e- ⟶Sn^2+}\)
|
+0.151
|
|
\(\ce{Sn^2+ + 2e- ⟶Sn}\)
|
−0.1375
|
|
\(\ce{[SnF6]^2- + 4e- ⟶Sn + 6F-}\)
|
−0.25
|
|
\(\ce{SnS + 2e- ⟶Sn + S^2-}\)
|
−0.94
|
|
\(\ce{Sr^2+ + 2e- ⟶Sr}\)
|
−2.89
|
|
\(\ce{TeO2 + 4H+ + 4e- ⟶Te + 2H2O}\)
|
+0.593
|
|
\(\ce{Th^4+ + 4e- ⟶Th}\)
|
−1.90
|
|
\(\ce{Ti^2+ + 2e- ⟶Ti}\)
|
−1.630
|
|
\(\ce{U^3+ + 3e- ⟶U}\)
|
−1.79
|
|
\(\ce{V^2+ + 2e- ⟶V}\)
|
−1.19
|
|
\(\ce{Y^3+ + 3e- ⟶Y}\)
|
−2.37
|
|
\(\ce{Zn^2+ + 2e- ⟶Zn}\)
|
−0.7618
|
|
\(\ce{[Zn(CN)4]^2- + 2e- ⟶Zn + 4CN-}\)
|
−1.26
|
|
\(\ce{[Zn(NH3)4]^2+ + 2e- ⟶Zn + 4NH3}\)
|
−1.04
|
|
\(\ce{Zn(OH)2 + 2e- ⟶Zn + 2OH-}\)
|
−1.245
|
|
\(\ce{[Zn(OH)4]^2 + 2e- ⟶Zn + 4OH-}\)
|
−1.199
|
|
\(\ce{ZnS + 2e- ⟶Zn + S^2-}\)
|
−1.40
|
|
\(\ce{Zr^4 + 4e- ⟶Zr}\)
|
−1.539
|


12.1: The Spinal Cord
The description of the CNS is concentrated on the structures of the brain, but the spinal cord is another major organ of the system. Whereas the brain develops out of expansions of the neural tube into primary and then secondary vesicles, the spinal cord maintains the tube structure and is only specialized into certain regions. As the spinal cord continues to develop in the newborn, anatomical features mark its surface. The anterior midline is marked by the anterior median fissure , and the posterior midline is marked by the posterior median sulcus . Axons enter the posterior side through the dorsal (posterior) nerve root , which marks the posterolateral sulcus on either side. The axons emerging from the anterior side do so through the ventral (anterior) nerve root . Note that it is common to see the terms dorsal (dorsal = “back”) and ventral (ventral = “belly”) used interchangeably with posterior and anterior, particularly in reference to nerves and the structures of the spinal cord. You should learn to be comfortable with both.
Gray Horns
In cross-section, the gray matter of the spinal cord has the appearance of an ink-blot test, with the spread of the gray matter on one side replicated on the other—a shape reminiscent of a bulbous capital “H.” As shown in Figure \(\PageIndex{1}\) , the gray matter is subdivided into regions that are referred to as horns.
The posterior horn is responsible for sensory processing. The anterior horn sends out motor signals to the skeletal muscles. The lateral horn , which is only found in the thoracic, upper lumbar, and sacral regions, is the central component of the sympathetic division of the autonomic nervous system.
Cranial Nerves
The nerves attached to the brain are the cranial nerves, which are primarily responsible for the sensory and motor functions of the head and neck (one of these nerves targets organs in the thoracic and abdominal cavities as part of the parasympathetic nervous system). There are twelve cranial nerves, which are designated CNI through CNXII for “Cranial Nerve,” using Roman numerals for 1 through 12. They can be classified as sensory nerves, motor nerves, or a combination of both, meaning that the axons in these nerves originate out of sensory ganglia external to the cranium or motor nuclei within the brain stem. Sensory axons enter the brain to synapse in a nucleus. Motor axons connect to skeletal muscles of the head or neck. Three of the nerves are solely composed of sensory fibers; five are strictly motor; and the remaining four are mixed nerves.


5.5: Homogeneous Coordinates
In the previous section we were able to combine rotation and scaling into a single composite operation by matrix multiplication. Unfortunately, translation cannot yet be included in the composite operator since we do it by addition rather than by multiplication.
Suppose we wish to rotate the image \(G\) by \(\frac{\pi}{3}\) about the point (−10,10). Our rotation matrix R(\(\theta\)) always rotates about the origin, so we must combine three transformations to accomplish this:
- translate the point (−10,10) to the origin;
- rotate \(\frac{\pi}{3}\) radians about the origin; and
- translate the origin back to (−10,10).
For step (i), we have \(b_{0}=\left[\begin{array}{l}
10 \\
-10
\end{array}\right]\) and
\[\mathrm{G}_{1}=\mathrm{G}+\mathrm{b}_{0} 1^{T} \nonumber \]
For step (ii),
\[\begin{align}
\mathrm{G}_{2} &=\mathrm{R}\left(\frac{\pi}{3}\right) \mathrm{G}_{1} \nonumber \\
&=\mathrm{R}\left(\frac{\pi}{3}\right)\left[\mathrm{G}+\mathrm{b}_{0} 1^{T}\right] \nonumber \\
&=\mathrm{R}\left(\frac{\pi}{3}\right) \mathrm{G}+\mathrm{R}\left(\frac{\pi}{3}\right) \mathrm{b}_{0} 1^{T} .
\end{align} \nonumber \]
For step (iii), we can use −\(b_0\) from step (i):
\[\begin{align}
\mathrm{G}_{\text {new }} &=\mathrm{G}_{2}-\mathrm{b}_{0} 1^{T} \nonumber \\
&=\mathrm{R}\left(\frac{\pi}{3}\right) \mathrm{G}+\mathrm{R}\left(\frac{\pi}{3}\right) \mathrm{b}_{0} 1^{T}-\mathrm{b}_{0} 1^{T} \nonumber \\
&=\mathrm{R}\left(\frac{\pi}{3}\right) \mathrm{G}+\left[\left(\mathrm{R}\left(\frac{\pi}{3}\right)-\mathrm{I}\right) \mathrm{b}_{0}\right] 1^{T} .
\end{align} \nonumber \]
In this example we were unable to find a single matrix operator \(A\) to do the entire job. The total transformation took the form
\[\mathrm{G}_{\text {new }}=\mathrm{AG}+\mathrm{b} 1^{T} \nonumber \]
This is called an affine transformation because it involves both multiplication by \(A\) and addition of a constant matrix. This is in contrast to the more desirable linear transformation , which involves only multiplication by \(A\).
We will now move toward a modified representation of the image and the operators by rewriting the last equation as
\[\mathrm{G}_{\text {new }}=[\mathrm{Ab}]\left[\begin{array}{l}
\mathrm{G} \\
1^{T}
\end{array}\right] \nonumber \]
where in the example we had \(A = \mathrm{R}\left(\frac{\pi}{3}\right)\) and \(b=\left(\mathrm{R}\left(\frac{\pi}{3}\right)-\mathrm{I}\right) b_{0}\).
Show that, for any matrices A,B,C,D of compatible sizes,
\[\mathrm{AB}+\mathrm{CD}=\left[\begin{array}{ll}
A & \mathrm{C}
\end{array}\right]\left[\begin{array}{l}
\mathrm{B} \\
\mathrm{D}
\end{array}\right] \nonumber \]
The matrix \(\left[\begin{array}{l}
\mathrm{G} \\
1^{T}
\end{array}\right]\) looks like
\[\left[\begin{array}{lll}
x_{1} & x_{2} & x_{n} \\
y_{1} & y_{2} & y_{n} \\
1 & 1 & 1
\end{array}\right] \nonumber \]
and the points \((x_i,y_i,1)\) are called homogeneous coordinates . We can modify Equation 5 so that the new point matrix is also in homogeneous coordinates:
\[\left[\begin{array}{l}
\mathrm{G}_{\text {new }} \\
1^{T}
\end{array}\right]=\left[\begin{array}{ll}
\mathrm{A} & \mathrm{b} \\
\mathrm{O}^{T} & 1
\end{array}\right]\left[\begin{array}{l}
\mathrm{G} \\
1^{T}
\end{array}\right] \nonumber \]
In the new representation, each point in the image has a third coordinate, which is always a 1. The homogeneous transformation is a 3×3 matrix,
\[\mathrm{A}_{h}=\left[\begin{array}{ll}
\mathrm{A} & \mathrm{b} \\
0^{T} & 1
\end{array}\right] \text { , } \nonumber \]
which is capable of translation, rotation, and scaling all by matrix multiplication. Thus, using homogeneous coordinates, we can build composite transformations that include translation.
In homogeneous coordinates, we have
\[\mathrm{R}(\theta)=\left[\begin{array}{lll}
\cos \theta & -\sin \theta & 0 \\
\sin \theta & \cos \theta & 0 \\
0 & 0 & 1
\end{array}\right] \nonumber \]
\[\mathrm{S}\left(s_{x}, s_{y}\right)=\left[\begin{array}{lll}
s_{x} & 0 & 0 \\
0 & s_{y} & 0 \\
0 & 0 & 1
\end{array}\right] \nonumber \]
\[\mathrm{T}\left(t_{x}, t_{y}\right)=\left[\begin{array}{lll}
1 & 0 & t_{x} \\
0 & 1 & t_{y} \\
0 & 0 & 1
\end{array}\right] \nonumber \]
The composite transformation to triple the size of an image and then move it 2 units to the left is
\[\mathrm{A}=\left[\begin{array}{lll}
1 & 0 & -2 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]\left[\begin{array}{lll}
3 & 0 & 0 \\
0 & 3 & 0 \\
0 & 0 & 1
\end{array}\right]=\left[\begin{array}{lll}
3 & 0 & -2 \\
0 & 3 & 0 \\
0 & 0 & 1
\end{array}\right] \nonumber \]
On the other hand, the composite transformation to move an image 2 units to the left and then triple its size is
\[\mathrm{B}=\left[\begin{array}{lll}
3 & 0 & 0 \\
0 & 3 & 0 \\
0 & 0 & 1
\end{array}\right]\left[\begin{array}{lll}
1 & 0 & -2 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]=\left[\begin{array}{lll}
3 & 0 & -6 \\
0 & 3 & 0 \\
0 & 0 & 1
\end{array}\right] \nonumber \]
In the latter case, the distance of the translation is also tripled.
Find a single composite transformation in homogeneous coordinates that rotates an image by angle \(\theta\) about point \((x_i,y_i)\) as in Example 1 .


2.8: Discussion
Over coffee, Xing said that he had been experimenting with the SageMath software discussed in Chapter 1. He understood that SageMath was treating a big integer as a string. Xing enthusiastically reported that he had asked SageMath to find the sum \(a+b\) of two large integers \(a\) and \(b\), each having more than 800 digits. The software found the answer about as fast as he could hit the enter key on his netbook. “That's not so impressive,” Alice interjected. “A human, even Bob, could do this in a couple of minutes using pencil and paper.”
“Thanks for your kind remarks,” replied Bob, with the rest of the group noting that that Alice was being pretty harsh on Bob and not for any good reason.
Dave took up Bob's case by remarking, “Very few humans, not even you Alice, would want to tackle finding the product of \(a\) and \(b\) by hand.” Xing jumped back in with, “That's the point. Even a tiny netbook can find the product very, very quickly. In fact, I tried it out with two integers, each having more than one thousand digits. It found the product in about one second.” Ever the skeptic, Zori said, “You mean you carefully typed in two integers of that size?” Xing quickly replied “Of course not. I just copied and pasted the data from one source to another.” Yolanda said, “What a neat trick that is. Really cuts down the chance of an error.”
Dave said “What about factoring? Can your netbook with its fancy software for strings factor big integers?” Xing said that he would try some sample problems and report back. Carlos said “Factoring an integer with several hundred digits is likely to be very challenging, not only for a netbook, but also for a super computer. For example, suppose the given integer was either a prime or the product of two large primes. Detecting which of these two statements holds could be very difficult.”
Undeterred, Dave continued, “What about exponentiation? Can your software calculate \(a^b\) when \(a\) and \(b\) are large integers?” Xing said “That shouldn't be a problem. After all, \(a^b\) is just multiplying \(a\) times itself a total of \(b\) times, and if you can do multiplication quickly, that's just a loop.” Yolanda said that the way Xing was describing things, he was actually talking about a program with nested loops so it might take a long time for such a program to halt. Carlos was quiet but he thought there might be ways to speed up such computations.
By this time, Alice reinserted herself into the conversation: “Hey guys. While you were talking, I was looking for big integer topics on the web and found this problem. ‘Is \(838200020310007224300\) a Catalan number?’ How would you answer this? Do you have to use special software?”
Zori was not happy. She gloomily envisioned a future job hunt in which she was compelled to use big integer arithmetic as a job skill. Arrgghh.