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The **Cole–Cole equation** is a relaxation model that is often used to describe dielectric relaxation in polymers.

In the physical sciences, **relaxation** usually means the return of a perturbed system into equilibrium. Each relaxation process can be categorized by a **relaxation time** τ. The simplest theoretical description of relaxation as function of time *t* is an exponential law exp(-*t*/τ).

A **polymer** is a large molecule, or macromolecule, composed of many repeated subunits. Due to their broad range of properties, both synthetic and natural polymers play essential and ubiquitous roles in everyday life. Polymers range from familiar synthetic plastics such as polystyrene to natural biopolymers such as DNA and proteins that are fundamental to biological structure and function. Polymers, both natural and synthetic, are created via polymerization of many small molecules, known as monomers. Their consequently large molecular mass relative to small molecule compounds produces unique physical properties, including toughness, viscoelasticity, and a tendency to form glasses and semicrystalline structures rather than crystals. The terms polymer and resin are often synonymous with plastic.

It is given by the equation

where is the complex dielectric constant, and are the "static" and "infinite frequency" dielectric constants, is the angular frequency and is a time constant.

In physics, **angular frequency***ω* is a scalar measure of rotation rate. It refers to the angular displacement per unit time or the rate of change of the phase of a sinusoidal waveform, or as the rate of change of the argument of the sine function.

The exponent parameter , which takes a value between 0 and 1, allows to describe different spectral shapes. When , the Cole-Cole model reduces to the Debye model. When , the relaxation is *stretched*, i.e. it extends over a wider range on a logarithmic scale than Debye relaxation.

The separation of the complex dielectric constant was reported in the original paper by Cole and Cole^{ [1] } as follows:

Upon introduction of hyperbolic functions, the above expressions reduce to:

Here .

These equations reduce to the Debye expression when .

Cole–Cole relaxation constitutes a special case of Havriliak–Negami relaxation when the symmetry parameter (β) is equal to 1, that is, when the relaxation peaks are symmetric. Another special case of Havriliak–Negami relaxation (β<1, α=1) is known as Cole–Davidson relaxation. For an abridged and updated review of anomalous dielectric relaxation in disordered systems, see Kalmykov.

The **Havriliak–Negami relaxation** is an empirical modification of the Debye relaxation model in electromagnetism. Unlike the Debye model, the Havriliak–Negami relaxation accounts for the asymmetry and broadness of the dielectric dispersion curve. The model was first used to describe the dielectric relaxation of some polymers, by adding two exponential parameters to the Debye equation:

**Bessel functions**, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are the canonical solutions *y*(*x*) of Bessel's differential equation

In mathematics, the **Dirac delta function** is a generalized function or distribution introduced by the physicist Paul Dirac. It is used to model the density of an idealized point mass or point charge as a function equal to zero everywhere except for zero and whose integral over the entire real line is equal to one. As there is no function that has these properties, the computations made by the theoretical physicists appeared to mathematicians as nonsense until the introduction of distributions by Laurent Schwartz to formalize and validate the computations. As a distribution, the Dirac delta function is a linear functional that maps every function to its value at zero. The Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1, is a discrete analog of the Dirac delta function.

A **dielectric** is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material as they do in an electrical conductor but only slightly shift from their average equilibrium positions causing **dielectric polarization**. Because of dielectric polarization, positive charges are displaced in the direction of the field and negative charges shift in the opposite direction. This creates an internal electric field that reduces the overall field within the dielectric itself. If a dielectric is composed of weakly bonded molecules, those molecules not only become polarized, but also reorient so that their symmetry axes align to the field.

**Synchrotron radiation** is the electromagnetic radiation emitted when charged particles are accelerated radially, i.e., when they are subject to an acceleration perpendicular to their velocity. It is produced, for example, in synchrotrons using bending magnets, undulators and/or wigglers. If the particle is non-relativistic, then the emission is called cyclotron emission. If, on the other hand, the particles are relativistic, sometimes referred to as ultrarelativistic, the emission is called synchrotron emission. Synchrotron radiation may be achieved artificially in synchrotrons or storage rings, or naturally by fast electrons moving through magnetic fields. The radiation produced in this way has a characteristic polarization and the frequencies generated can range over the entire electromagnetic spectrum which is also called continuum radiation.

**Dynamic mechanical analysis** is a technique used to study and characterize materials. It is most useful for studying the viscoelastic behavior of polymers. A sinusoidal stress is applied and the strain in the material is measured, allowing one to determine the complex modulus. The temperature of the sample or the frequency of the stress are often varied, leading to variations in the complex modulus; this approach can be used to locate the glass transition temperature of the material, as well as to identify transitions corresponding to other molecular motions.

**Chebyshev filters** are analog or digital filters having a steeper roll-off and more passband ripple or stopband ripple than Butterworth filters. Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the range of the filter, but with ripples in the passband. This type of filter is named after Pafnuty Chebyshev because its mathematical characteristics are derived from Chebyshev polynomials. The type I Chebyshev filters are called usually as just "Chebyshev filters", the type II ones are usually called as "inverse Chebyshev filters".

In mathematics and in signal processing, the **Hilbert transform** is a specific linear operator that takes a function, *u*(*t*) of a real variable and produces another function of a real variable *H*(*u*)(*t*). This linear operator is given by convolution with the function :

In mathematics, there are several integrals known as the **Dirichlet integral**, after the German mathematician Peter Gustav Lejeune Dirichlet.

In calculus, **Leibniz's rule** for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form

**Instantaneous phase** and **instantaneous frequency** are important concepts in signal processing that occur in the context of the representation and analysis of time-varying functions. The instantaneous phase of a *complex-valued* function *s*(*t*), is the real-valued function:

In many-body theory, the term **Green's function** is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators.

**Kenneth Stewart Cole** was an American biophysicist described by his peers as "a pioneer in the application of physical science to biology". Cole was awarded the National Medal of Science in 1967.

A **biarc** is a smooth curve formed from two circular arcs. In order to make the biarc smooth, the two arcs should have the same tangent at the connecting point where they meet.

A **Sommerfeld expansion** is an approximation method developed by Arnold Sommerfeld for a certain class of integrals which are common in condensed matter and statistical physics. Physically, the integrals represent statistical averages using the Fermi–Dirac distribution.

The **narrow escape problem** is a ubiquitous problem in biology, biophysics and cellular biology.

The **non-radiative dielectric (NRD) waveguide ** has been introduced by Yoneyama in 1981. In Fig. 1 the cross section of NRD guide is shown: it consists of a dielectric rectangular slab of height a and width b, which is placed between two metallic parallel plates of suitable width. The structure is practically the same as the H waveguide, proposed by Tischer in 1953. Due to the dielectric slab, the electromagnetic field is confined in the vicinity of the dielectric region, whereas in the outside region, for suitable frequencies, the electromagnetic field decays exponentially. Therefore, if the metallic plates are sufficiently extended, the field is practically negligible at the end of the plates and therefore the situation does not greatly differ from the ideal case in which the plates are infinitely extended. The polarization of the electric field in the required mode is mainly parallel to the conductive walls. As it is known, if the electric field is parallel to the walls, the conduction losses decrease in the metallic walls at the increasing frequency, whereas, if the field is perpendicular to the walls, losses increase at the increasing frequency. Since the NRD waveguide has been devised for its implementation at millimeter waves, the selected polarization minimizes the ohmic losses in the metallic walls.

**Bending of plates**, or **plate bending**, refers to the deflection of a plate perpendicular to the plane of the plate under the action of external forces and moments. The amount of deflection can be determined by solving the differential equations of an appropriate plate theory. The stresses in the plate can be calculated from these deflections. Once the stresses are known, failure theories can be used to determine whether a plate will fail under a given load.

The **table of chords**, created by the astronomer, geometer, and geographer Ptolemy in Egypt during the 2nd century AD, is a trigonometric table in Book I, chapter 11 of Ptolemy's *Almagest*, a treatise on mathematical astronomy. It is essentially equivalent to a table of values of the sine function. It was the earliest trigonometric table extensive enough for many practical purposes, including those of astronomy. Centuries passed before more extensive trigonometric tables were created. One such table is the *Canon Sinuum* created at the end of the 16th century.

In thermal quantum field theory, the **Matsubara frequency** summation is the summation over discrete imaginary frequencies. It takes the following form

- ↑ Cole, Kenneth S, Robert H (1941). "Dispersion and Absorption in Dielectrics: I - Alternating Current Characteristics".
*Journal of Chemical Physics*.**9**(4): 341–351. Bibcode:1941JChPh...9..341C. doi:10.1063/1.1750906.

Cole, K.S.; Cole, R.H. (1941). "Dispersion and Absorption in Dielectrics - I Alternating Current Characteristics". *J. Chem. Phys*. **9** (4): 341–352. Bibcode:1941JChPh...9..341C. doi:10.1063/1.1750906.

The **bibcode** is a compact identifier used by several astronomical data systems to uniquely specify literature references.

In computing, a **Digital Object Identifier** or **DOI** is a persistent identifier or handle used to uniquely identify objects, standardized by the International Organization for Standardization (ISO). An implementation of the Handle System, DOIs are in wide use mainly to identify academic, professional, and government information, such as journal articles, research reports and data sets, and official publications though they also have been used to identify other types of information resources, such as commercial videos.

Cole, K.S.; Cole, R.H. (1942). "Dispersion and Absorption in Dielectrics - II Direct Current Characteristics". *Journal of Chemical Physics*. **10** (2): 98–105. Bibcode:1942JChPh..10...98C. doi:10.1063/1.1723677.

Kalmykov, Y.P.; Coffey, W.T.; Crothers, D.S.F.; Titov, S.V. (2004). "Microscopic Models for Dielectric Relaxation in Disordered Systems". *Physical Review E*. **70** (4): 041103. Bibcode:2004PhRvE..70d1103K. doi:10.1103/PhysRevE.70.041103. PMID 15600393.

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