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# Fourier transform

The Fourier transform (FT) decomposes a function of time (a signal) into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies (or pitches) of its constituent notes. [1]

## Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

## Absolute continuity

In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity.

## Absolute convergence

In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite.

## Absolute value

In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.

## Airy disk

In optics, the Airy disk (or Airy disc) and Airy pattern are descriptions of the best focused spot of light that a perfect lens with a circular aperture can make, limited by the diffraction of light.

## Almost everywhere

In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities.

## American Mathematical Society

The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.

## Amplitude

The amplitude of a periodic variable is a measure of its change over a single period (such as time or spatial period).

## Analog signal processing

Analog signal processing is a type of signal processing conducted on continuous analog signals by some analog means (as opposed to the discrete Digital Signal Processing where the signal processing is carried out by a digital process).

## Analysis of variance

Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among group means in a sample.

## Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series.

## Angular frequency

In physics, angular frequency ω (also referred to by the terms angular speed, radial frequency, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate.

## Anticausal system

An anticausal system is a hypothetical system with outputs and internal states that depend solely on future input values.

## Argument (complex analysis)

In mathematics, the argument is a multi-valued function operating on the nonzero complex numbers.

## Autocorrelation

Autocorrelation, also known as serial correlation, is the correlation of a signal with a delayed copy of itself as a function of delay.

## Automorphic form

In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group G to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of the topological group.

## Ba space

In mathematics, the ba space ba(\Sigma) of an algebra of sets \Sigma is the Banach space consisting of all bounded and finitely additive signed measures on \Sigma.

## Banach algebra

In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, i.e. a normed space and complete in the metric induced by the norm.

## Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.

## Beevers–Lipson strip

Beevers–Lipson strips were a computational aid for early crystallographers in calculating Fourier transforms to determine the structure of crystals from crystallographic data, enabling the creation of models for complex molecules.

## Bessel function

Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are the canonical solutions of Bessel's differential equation for an arbitrary complex number, the order of the Bessel function.

## Bochner's theorem

In mathematics, Bochner's theorem (named for Salomon Bochner) characterizes the Fourier transform of a positive finite Borel measure on the real line.

## Borel measure

In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets).

## Bounded operator

In functional analysis, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded above by the same number, over all non-zero vectors v in X. In other words, there exists some M\ge 0 such that for all v in X The smallest such M is called the operator norm \|L\|_ \, of L. A bounded linear operator is generally not a bounded function; the latter would require that the norm of L(v) be bounded for all v, which is not possible unless L(v).

## Bulletin of the American Mathematical Society

The Bulletin of the American Mathematical Society is a quarterly mathematical journal published by the American Mathematical Society.

## C*-algebra

C∗-algebras (pronounced "C-star") are an area of research in functional analysis, a branch of mathematics.

## Cambridge University Press

Cambridge University Press (CUP) is the publishing business of the University of Cambridge.

## Cauchy distribution

The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution.

## Cauchy principal value

In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.

## Cauchy's integral theorem

In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in the complex plane.

## Character group

In mathematics, a character group is the group of representations of a group by complex-valued functions.

## Characteristic function (probability theory)

In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution.

## Charles Fefferman

Charles Louis Fefferman (born April 18, 1949) is an American mathematician at Princeton University.

## Chebyshev polynomials

In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively.

## Chirplet transform

In signal processing, the chirplet transform is an inner product of an input signal with a family of analysis primitives called chirplets.

## Chord (music)

A chord, in music, is any harmonic set of pitches consisting of two or more (usually three or more) notes (also called "pitches") that are heard as if sounding simultaneously.

## Circle group

In mathematics, the circle group, denoted by T, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers The circle group forms a subgroup of C×, the multiplicative group of all nonzero complex numbers.

## Circumflex

The circumflex is a diacritic in the Latin, Greek and Cyrillic scripts that is used in the written forms of many languages and in various romanization and transcription schemes.

## Closed-form expression

In mathematics, a closed-form expression is a mathematical expression that can be evaluated in a finite number of operations.

## Compact space

In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).

## Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers.

## Complex conjugate

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis.

## Conjugate variables

Conjugate variables are pairs of variables mathematically defined in such a way that they become Fourier transform duals, or more generally are related through Pontryagin duality.

## Constant (mathematics)

In mathematics, the adjective constant means non-varying.

## Continuous wavelet transform

In mathematics, a continuous wavelet transform (CWT) is used to divide a continuous-time function into wavelets.

## Convolution

In mathematics (and, in particular, functional analysis) convolution is a mathematical operation on two functions (f and g) to produce a third function, that is typically viewed as a modified version of one of the original functions, giving the integral of the pointwise multiplication of the two functions as a function of the amount that one of the original functions is translated.

## Convolution theorem

In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution is the pointwise product of Fourier transforms.

## CRC Press

The CRC Press, LLC is a publishing group based in the United States that specializes in producing technical books.

## Critical point (mathematics)

In mathematics, a critical point or stationary point of a differentiable function of a real or complex variable is any value in its domain where its derivative is 0.

## Cross-correlation

In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other.

## Cyclic group

In algebra, a cyclic group or monogenous group is a group that is generated by a single element.

## Derivative

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

## DFT matrix

In applied mathematics, a DFT matrix is an expression of a discrete Fourier transform (DFT) as a transformation matrix, which can be applied to a signal through matrix multiplication.

## Differential entropy

Differential entropy (also referred to as continuous entropy) is a concept in information theory that began as an attempt by Shannon to extend the idea of (Shannon) entropy, a measure of average surprisal of a random variable, to continuous probability distributions.

## Differential equation

A differential equation is a mathematical equation that relates some function with its derivatives.

## Diffusion

Diffusion is the net movement of molecules or atoms from a region of high concentration (or high chemical potential) to a region of low concentration (or low chemical potential) as a result of random motion of the molecules or atoms.

## Dirac comb

In mathematics, a Dirac comb (also known as an impulse train and sampling function in electrical engineering) is a periodic tempered distribution constructed from Dirac delta functions for some given period T. The symbol \operatorname(t), where the period is omitted, represents a Dirac comb of unit period.

## Dirac delta function

In mathematics, the Dirac delta function (function) is a generalized function or distribution introduced by the physicist Paul Dirac.

## Discrete Fourier transform

In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency.

## Discrete-time Fourier transform

In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to the uniformly-spaced samples of a continuous function.

## Distribution (mathematics)

Distributions (or generalized functions) are objects that generalize the classical notion of functions in mathematical analysis.

## Dot product

In mathematics, the dot product or scalar productThe term scalar product is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space.

## Dover Publications

Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward Cirker and his wife, Blanche.

## Edward Condon

Edward Uhler Condon (March 2, 1902 – March 26, 1974) was a distinguished American nuclear physicist, a pioneer in quantum mechanics, and a participant in the development of radar and nuclear weapons during World War II as part of the Manhattan Project.

## Eigenfunction

In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue.

## Eigenvalues and eigenvectors

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.

## Elias M. Stein

Elias Menachem Stein (born January 13, 1931) is a mathematician.

## Entire function

In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic at all finite points over the whole complex plane.

## Entropic uncertainty

In quantum mechanics, information theory, and Fourier analysis, the entropic uncertainty or Hirschman uncertainty is defined as the sum of the temporal and spectral Shannon entropies.

## Envelope (waves)

In physics and engineering, the envelope of an oscillating signal is a smooth curve outlining its extremes.

## Equivariant map

In mathematics, equivariance is a form of symmetry for functions from one symmetric space to another.

## Euler's formula

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function.

## Euler–Mascheroni constant

The Euler–Mascheroni constant (also called Euler's constant) is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter gamma.

## Even and odd functions

In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses.

## Fast Fourier transform

A fast Fourier transform (FFT) is an algorithm that samples a signal over a period of time (or space) and divides it into its frequency components.

## Filter (mathematics)

In mathematics, a filter is a special subset of a partially ordered set.

## Filter (signal processing)

In signal processing, a filter is a device or process that removes some unwanted components or features from a signal.

## Four-momentum

In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional spacetime.

## Fourier integral operator

In mathematical analysis, Fourier integral operators have become an important tool in the theory of partial differential equations.

## Fourier inversion theorem

In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform.

## Fourier series

In mathematics, a Fourier series is a way to represent a function as the sum of simple sine waves.

## Fourier-transform infrared spectroscopy

Fourier-transform infrared spectroscopy (FTIR) is a technique used to obtain an infrared spectrum of absorption or emission of a solid, liquid or gas.

## Fourier–Deligne transform

In algebraic geometry, the Fourier–Deligne transform, or ℓ-adic Fourier transform, or geometric Fourier transform, is an operation on objects of the derived category of ℓ-adic sheaves over the affine line.

## Fourier–Mukai transform

In algebraic geometry, a Fourier–Mukai transform ΦK is a functor between derived categories of coherent sheaves D(X) → D(Y) for schemes X and Y, which is, in a sense, an integral transform along a kernel object K ∈ D(X×Y).

## Fractional Fourier transform

In mathematics, in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform.

## Frequency

Frequency is the number of occurrences of a repeating event per unit of time.

## Frequency response

Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system.

## Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

## Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.

## Gaussian function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the form: for arbitrary real constants, and.

## Gelfand representation

In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) has two related meanings.

## Generalized function

In mathematics, generalized functions, or distributions, are objects extending the notion of functions.

## Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

## Haar measure

In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.

## Hankel transform

In mathematics, the Hankel transform expresses any given function f(r) as the weighted sum of an infinite number of Bessel functions of the first kind.

## Harmonic analysis

Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis).

## Harmonic function

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U → R where U is an open subset of Rn that satisfies Laplace's equation, i.e. everywhere on U. This is usually written as or.

## Harmonic series (mathematics)

In mathematics, the harmonic series is the divergent infinite series: Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are,,, etc., of the string's fundamental wavelength.

## Hartley transform

In mathematics, the Hartley transform (HT) is an integral transform closely related to the Fourier transform (FT), but which transforms real-valued functions to real-valued functions.

## Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods.

## Hausdorff–Young inequality

In mathematics, the Hausdorff−Young inequality bounds the ''L''''q''-norm of the Fourier coefficients of a periodic function for q ≥ 2.

## Heat equation

The heat equation is a parabolic partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time.

## Heat transfer

Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems.

## Heaviside step function

The Heaviside step function, or the unit step function, usually denoted by or (but sometimes, or), is a discontinuous function named after Oliver Heaviside (1850–1925), whose value is zero for negative argument and one for positive argument.

## Heisenberg group

In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form \end under the operation of matrix multiplication.

## Hermite polynomials

In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.

## Hermitian function

In mathematical analysis, a Hermitian function is a complex function with the property that its complex conjugate is equal to the original function with the variable changed in sign: (where the ^* indicates the complex conjugate) for all x in the domain of f. This definition extends also to functions of two or more variables, e.g., in the case that f is a function of two variables it is Hermitian if for all pairs (x_1, x_2) in the domain of f. From this definition it follows immediately that: f is a Hermitian function if and only if.

## Hertz

The hertz (symbol: Hz) is the derived unit of frequency in the International System of Units (SI) and is defined as one cycle per second.

## Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.

## Hilbert transform

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).

## Holomorphic function

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.

## Homogeneous distribution

In mathematics, a homogeneous distribution is a distribution S on Euclidean space Rn or that is homogeneous in the sense that, roughly speaking, for all t > 0.

## Homogeneous polynomial

In mathematics, a homogeneous polynomial is a polynomial whose nonzero terms all have the same degree.

## Hyperbolic function

In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions.

## If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

## Imaginary number

An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit,j is usually used in Engineering contexts where i has other meanings (such as electrical current) which is defined by its property.

## Imaginary unit

The imaginary unit or unit imaginary number is a solution to the quadratic equation.

## Improper integral

In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number, \infty, -\infty, or in some instances as both endpoints approach limits.

## Impulse response

In signal processing, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse.

## Indicator function

In mathematics, an indicator function or a characteristic function is a function defined on a set X that indicates membership of an element in a subset A of X, having the value 1 for all elements of A and the value 0 for all elements of X not in A. It is usually denoted by a symbol 1 or I, sometimes in boldface or blackboard boldface, with a subscript specifying the subset.

## Indirect Fourier transform

In a Fourier transform (FT), the Fourier transformed function \hat f(s) is obtained from f(t) by: where i is defined as i^2.

## Injective function

In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## Integral

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

## Integral transform

In mathematics, an integral transform maps an equation from its original domain into another domain where it might be manipulated and solved much more easily than in the original domain.

## Inverse Laplace transform

In mathematics, the inverse Laplace transform of a function F(s) is the piecewise-continuous and exponentially-restricted real function f(t) which has the property: where \mathcal denotes the Laplace transform.

## Involution (mathematics)

In mathematics, an involution, or an involutory function, is a function that is its own inverse, for all in the domain of.

## John Wiley & Sons

John Wiley & Sons, Inc., also referred to as Wiley, is a global publishing company that specializes in academic publishing.

## Joseph Fourier

Jean-Baptiste Joseph Fourier (21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations.

## Journal of Computational Physics

The Journal of Computational Physics is a bimonthly scientific journal covering computational physics that was established in 1966 and is published by Elsevier.

## Laplace distribution

In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace.

## Laplace transform

In mathematics, the Laplace transform is an integral transform named after its discoverer Pierre-Simon Laplace.

## Lebesgue integration

In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis.

## Lebesgue measure

In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space.

## Lie group

In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

## Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.

## Linear canonical transformation

In Hamiltonian mechanics, the linear canonical transformation (LCT) is a family of integral transforms that generalizes many classical transforms.

## Linear form

In linear algebra, a linear functional or linear form (also called a one-form or covector) is a linear map from a vector space to its field of scalars.

## Linear time-invariant theory

Linear time-invariant theory, commonly known as LTI system theory, comes from applied mathematics and has direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas.

## Locally compact space

In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.

## Low-pass filter

A low-pass filter (LPF) is a filter that passes signals with a frequency lower than a certain cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency.

## Lp space

In mathematics, the Lp spaces are function spaces defined using a natural generalization of the ''p''-norm for finite-dimensional vector spaces.

## Magnetic resonance imaging

Magnetic resonance imaging (MRI) is a medical imaging technique used in radiology to form pictures of the anatomy and the physiological processes of the body in both health and disease.

## Mass spectrometry

Mass spectrometry (MS) is an analytical technique that ionizes chemical species and sorts the ions based on their mass-to-charge ratio.

## MATLAB

MATLAB (matrix laboratory) is a multi-paradigm numerical computing environment and proprietary programming language developed by MathWorks.

## Mellin transform

In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform.

## Modular form

In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition.

## Moment (mathematics)

In mathematics, a moment is a specific quantitative measure, used in both mechanics and statistics, of the shape of a set of points.

## Momentum

In Newtonian mechanics, linear momentum, translational momentum, or simply momentum (pl. momenta) is the product of the mass and velocity of an object.

## Multidimensional transform

In mathematical analysis and applications, multidimensional transforms are used to analyze the frequency content of signals in a domain of two or more dimensions.

## Multiplier (Fourier analysis)

In Fourier analysis, a multiplier operator is a type of linear operator, or transformation of functions.

## Multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution or multivariate Gaussian distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions.

## Natural number

In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").

## NGC 4622

NGC 4622 is a face-on unbarred spiral galaxy with a very prominent ring structure located in the constellation Centaurus.

## Noncommutative geometry

Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions (possibly in some generalized sense).

## Noncommutative harmonic analysis

In mathematics, noncommutative harmonic analysis is the field in which results from Fourier analysis are extended to topological groups that are not commutative.

## Norbert Wiener

Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician and philosopher.

## Normal distribution

In probability theory, the normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a very common continuous probability distribution.

## Nuclear magnetic resonance

Nuclear magnetic resonance (NMR) is a physical phenomenon in which nuclei in a magnetic field absorb and re-emit electromagnetic radiation.

## Number theory

Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.

## Numerical integration

In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations.

## Omega

Omega (capital: Ω, lowercase: ω; Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the 24th and last letter of the Greek alphabet.

## Operator norm

In mathematics, the operator norm is a means to measure the "size" of certain linear operators.

## Ordinary differential equation

In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and its derivatives.

## Orthonormality

In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal and unit vectors.

## Oxford University Press

Oxford University Press (OUP) is the largest university press in the world, and the second oldest after Cambridge University Press.

## Paley–Wiener theorem

In mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform.

## Partial differential equation

In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.

## Periodic function

In mathematics, a periodic function is a function that repeats its values in regular intervals or periods.

## Periodic summation

In signal processing, any periodic function, s_P(t) with period P, can be represented by a summation of an infinite number of instances of an aperiodic function, s(t), that are offset by integer multiples of P. This representation is called periodic summation: When s_P(t) is alternatively represented as a complex Fourier series, the Fourier coefficients are proportional to the values (or "samples") of the continuous Fourier transform, S(f) \ \stackrel \ \mathcal\, at intervals of 1/P.

## Peter–Weyl theorem

In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian.

## Phase (waves)

Phase is the position of a point in time (an instant) on a waveform cycle.

## Phase angle

In the context of phasors, phase angle refers to the angular component of the complex number representation of the function.

## Physics

Physics (from knowledge of nature, from φύσις phýsis "nature") is the natural science that studies matterAt the start of The Feynman Lectures on Physics, Richard Feynman offers the atomic hypothesis as the single most prolific scientific concept: "If, in some cataclysm, all scientific knowledge were to be destroyed one sentence what statement would contain the most information in the fewest words? I believe it is that all things are made up of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another..." and its motion and behavior through space and time and that studies the related entities of energy and force."Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves."Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of the human intellect in its quest to understand our world and ourselves."Physics is an experimental science. Physicists observe the phenomena of nature and try to find patterns that relate these phenomena.""Physics is the study of your world and the world and universe around you." Physics is one of the oldest academic disciplines and, through its inclusion of astronomy, perhaps the oldest. Over the last two millennia, physics, chemistry, biology, and certain branches of mathematics were a part of natural philosophy, but during the scientific revolution in the 17th century, these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics often enable advances in new technologies. For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.

## Plancherel theorem

In mathematics, the Plancherel theorem is a result in harmonic analysis, proven by Michel Plancherel in 1910.

## Planck constant

The Planck constant (denoted, also called Planck's constant) is a physical constant that is the quantum of action, central in quantum mechanics.

## Polar coordinate system

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## Real number

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## Rectangular function

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## Relativistic quantum mechanics

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## Representation theory

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## Riemann integral

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## Riemann sum

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## Symbolic integration

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## Symplectic vector space

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## Tannaka–Krein duality

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## Time–frequency analysis

In signal processing, time–frequency analysis comprises those techniques that study a signal in both the time and frequency domains simultaneously, using various time–frequency representations.

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## Topological group

In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology.

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## Triangular function

A triangular function (also known as a triangle function, hat function, or tent function) is a function whose graph takes the shape of a triangle.

## Trigonometric functions

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## Two-sided Laplace transform

In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform equivalent to probability's moment generating function.

## Uncertainty principle

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## Uniform continuity

In mathematics, a function f is uniformly continuous if, roughly speaking, it is possible to guarantee that f(x) and f(y) be as close to each other as we please by requiring only that x and y are sufficiently close to each other; unlike ordinary continuity, the maximum distance between f(x) and f(y) cannot depend on x and y themselves.

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## Unitary transformation

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## Vector (mathematics and physics)

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## Wave function

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## Wavelet transform

In mathematics, a wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated by a wavelet.

## Wolfram Alpha

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# Endomorphisms of free modules and extension of scalars

Let $B$ be a commutative ring with $1$, let $A$ be a subring such that any unit of $B$ which belongs to $A$ is a unit of $A$, and let $\phi:F\to F$ be an endomorphism of a free $A$-modules $F$ such that $B\otimes_A\phi:B\otimes_AF\to B\otimes_AF$ is an automorphism.

Does this imply that $\phi$ is an automorphism?

The answer is of course yes if $F$ has finite rank, by determinant theory.

The answer is also yes if $B$ is faithfully flat over $A$.

Julian Rosen made the following interesting comment on Mathematics Stackexchange:

"Choose bases for $F,F'$, and write $[\phi]$ (resp. $[\phi^{-1}]$) for the matrix of $\phi$ (resp. $(B\otimes\phi)^{-1}$), which is a column-finite (possibly) infinite with entries in $A$ (resp. $B$). WLOG we may assume $B$ is generated as an $A$-algebra by the entries of $[\phi^{-1}]$. If two ring homomorphisms $f,g:B\to C$ agree on $A$, both $f([\phi^{-1}])$ and $g([\phi^{-1}])$ are inverses of $f([\phi])=g([\phi])$, so $f([\phi^{-1}])=g([\phi^{-1}])$ and hence $f=g$ (here functions act on matrices entry-wise). This means $A\to B$ is an epimorphism of rings. An example of an injective ring epimorphism introducing no new units is $k[x^2,x(x^2-1)]\subset k[x,(x-1)^{-1}]$ (see this MO answer), so if a counterexample exists this might be a place to start."

Here $k$ is a field of characteristic $\ne2$. (For an elementary proof of the fact that the inclusion $k[x^2,x(x^2-1)]\subset k[x,(x-1)^{-1}]$ is an epimorphism, see this text.)

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# American Institute of Mathematical Sciences

January  2017, 13(1): 155-185. doi: 10.3934/jimo.2016010

## Asymptotics for ruin probabilities of a non-standard renewal risk model with dependence structures and exponential Lévy process investment returns

 1 School of Mathematical Sciences, School of Management and Economics, University of Electronic Science and Technology of China, Chengdu, 611731, China 2 School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China

* Corresponding author: Dingcheng Wang

Received  February 2015 Revised  December 2015 Published  March 2016

Fund Project: The research of Jiangyan Peng is supported by the National Natural Science Foundation of China (project no: 71501025) and China Postdoctoral Science Foundation (project no: 2015M572467). The research of Dingcheng Wang is supported by the National Natural Science Foundation of China (project no: 71271042).

Consider a non-standard renewal risk model with dependence structures, where claim sizes follow a one-sided linear process with independent and identically distributed step sizes, the step sizes and inter-arrival times respectively form a sequence of independent and identically distributed random pairs, with each pair obeying a dependence structure. An insurance company is allowed to make risk-free and risky investments, where the price process of the investment portfolio follows an exponential Lévy process. When the step-size distribution is dominatedly-varying-tailed, some asymptotic estimates for the finite-and infinite-time ruin probabilities are obtained.

Citation: Jiangyan Peng, Dingcheng Wang. Asymptotics for ruin probabilities of a non-standard renewal risk model with dependence structures and exponential Lévy process investment returns. Journal of Industrial & Management Optimization, 2017, 13 (1) : 155-185. doi: 10.3934/jimo.2016010
##### References:

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GCSE Maths Algebra Law Of Indices

Dividing Indices

Dividing Indices

Here we will learn about dividing indices including how to use the laws of indices to divide indices and how to divide indices that have different bases.

There are also laws of indices worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is dividing indices?

Dividing indices is where we divide terms that involve indices or powers.

There are two methods we can use to divide terms involving indices.

1 When the bases are the same:

E.g

$a^{5} \div a^{3}=a^{5-3}=a^{2}$

These questions usually ask you to ‘simplify’ the calculation.

2 When the bases are different:

E.g

$2^{5} \div 4^{2}=32 \div 16=2$

These questions usually ask you to ‘evaluate’ (work out) the calculation

How to divide indices when the bases are the same

When dividing indices with the same base, subtract the powers.

$a^{m} \div a^{n}=a^{m-n}$

1. Subtract the indices
2. Divide any coefficients of the base number or letter

In order to divide indices when the bases are the same we can use one of the laws of indices.

E.g.

To simplify the following expression:

$8^{7} \div 8^{4}$

We can write out each power in its expanded form:

$8^{7} \div 8^{4}=\frac{8 \times 8 \times 8 \times 8 \times 8 \times 8 \times 8}{8 \times 8 \times 8 \times 8}$

We know that 8 ÷ 8 = 1, so we can keep canceling an 8 on the top with an 8 on bottom until we are left with:

$8 \times 8 \times 8=8^{3}$

We can simplify this process by simplifying 87 ÷ 84 to 87-4 = 83:

$8^{7} \div 8^{4}=8^{7-4}=8^{3}$

The base has stayed the same and we have subtracted the indices.

This is the division law of indices.

Related lessons on laws of indices

Dividing indices is part of our series of lessons to support revision on laws of indices. You may find it helpful to start with the main laws of indices lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

Dividing indices examples (with the same base)

Example 1: no coefficient in front of base

$a^{5} \div a^{2}$

1. Subtract the indices 5 and 2.

$5 – 2 = 3$

So,

$a^{5} \div a^{2}=a^{3}$

Example 2: with a coefficient in front of base

$21 a^{9} \div 7 a^{2}$

$9 – 2 = 7$

$21 ÷ 7 =3$

So,

$21 a^{9} \div 7 a^{2}=3a^{7}$

Example 3: dividing indices (negative Indices)

$18 a^{-6} \div 9 a^{-4}$

$-6 -(-4) = -6+4=-2$

$18 ÷ 9 = 2$

So,

$18 a^{-6} \div 9 a^{-4}=2 a^{-2}$

Example 4: algebraic expressions

$100 a^{6 b} \div-10 a^{-7 b}$

$6b -(- 7b) = 6b+7b=13b$

$100 \div-10=-10$

So,

$100 a^{6 b} \div-10 a^{-7 b}=-10 a^{13 b}$

Example 5: with fractional powers

$x^{\frac{4}{5}} \div x^{\frac{2}{3}}$

$\frac{4}{5}-\frac{2}{3}$

Remember, to subtract fractions we need to find the lowest common denominator.

Multiples of 5           Multiples of 3

$5, 10, 15, 20 \quad \quad \quad 3, 6, 9, 12, 15, 18$

We can see that 15 is the lowest common denominator.

To change the denominator 5 into 15 we need to multiply by 3, so we multiply the numerator by 3 too.

To change the denominator 3 into 15 we need to multiply by 5, so we multiply the numerator by 5 too.

$\frac{12}{15}-\frac{10}{15}$

Now that we have a common denominator we can subtract the numerators:

$\frac{12}{15}-\frac{10}{15}=\frac{12-10}{15}=\frac{2}{15}$

So,

$x^{\frac{4}{5}} \div x^{\frac{2}{3}}=x^{\frac{2}{15}}$

Practice dividing indices questions (with the same base)

x^{9} \div x^{2}

x^{11}

x^{4.5}

7x

x^{7}

9-2=7 so

x^{9} \div x^{2} = x^{7}

2. Simplify

12 b^{13} \div 4 b^{7}

3b^{6}

8b^{6}

3b^{\frac{13}{7}}

3^{6}

13-7=6

12 \div 4 = 3

12 b^{13} \div 4 b^{7}=3b^{6}

3. Simplify

63^{-2} \div 7 a^{-3}

9a^{\frac{2}{3}}

56a

9a

9a^{-5}

-2 – (-3) = -2+3 = 1

63 \div 7=9

63a^{-2} \div 7 a^{-3} = 9a

(When the power is 1 we do not need to write it)

4. Simplify

\frac{24c^{6}d^{3}}{4c^{2}d^{2}}

6cd^{5}

20c^{4}d

6c^{4}

6c^{4}d

For c: 6-2=4

For d: 3-2=1

24 \div 4 = 6

\frac{24c^{6}d^{3}}{4c^{2}d^{2}} = 6c^{4}d

5. Simplify

-100 a^{8 x+2 b} \div 10 a^{3 b}

10 a^{8 x-b}

-10 a^{8 x-b}

10 a^{8 x+5b}

90 a^{8 x-b}

8x+2b-3b=8x-b

-100 \div 10 = -10

-100 a^{8 x+2 b} \div 10 a^{3 b}=-10 a^{8 x-b}

6. Simplify

4 x^{\frac{3}{4}} \div 2 x^{\frac{1}{3}}

4 x^{\frac{5}{12}}

2 x^{\frac{5}{12}}

0.5 x^{\frac{5}{12}}

2 x^{\frac{9}{4}}

\begin{aligned} \frac{3}{4} – \frac{1}{3} &= \frac{9}{12} – \frac{4}{12}\\\\ &= \frac{5}{12} \end{aligned}

4 \div 2 = 2

4 x^{\frac{3}{4}} \div 2 x^{\frac{1}{3}} = 2 x^{\frac{5}{12}}

How to divide indices when the bases are different

In order to divide indices when the bases are different we need to write out each term and calculate the answer.

We cannot simplify them using the laws of indices as the bases are not the same.

1 Write out each term without the indices.

2 Work out the calculation and simplify.

E.g.

To evaluate the following expression:

$2^{3} \div 3^{2}$

We need to write each term of the calculation without using index notation:

$2^{3}=2 \times 2 \times 2=8$ $3^{2}=3 \times 3=9$

So we have:

$2^{3} \div 3^{2}=\frac{2 \times 2 \times 2}{3 \times 3}$

Now we need to work out the calculation:

\begin{aligned} 2^{3} \div 3^{2} &=\frac{2 \times 2 \times 2}{3 \times 3} \\ &=\frac{8}{9} \end{aligned}

Dividing indices examples (with different bases)

Example 1: with positive indices

Evaluate:

$4^{2} \div 2^{3}$

1. Write out each term without the indices.

$4^{2} \div 2^{3}=\frac{4 \times 4}{2 \times 2 \times 2}$

2 Work out the calculation and simplify.

\begin{aligned} 4^{2} \div 2^{3} &=\frac{4 \times 4}{2 \times 2 \times 2} \\\\ &=\frac{16}{8} \\\\ &=2 \end{aligned}

Example 2: with three terms

Evaluate:

$8^{2} \div 2^{4} \div 3^{2}$

$8^{2}=8 \times 8=64$ $2^{4}=2 \times 2 \times 2 \times 2=16$ $3^{2}=3 \times 3=9$

\begin{aligned} 8^{2} \div 2^{4} \div 3^{2} &=64 \div 16 \div 9 \\\\ &=\frac{4}{9} \end{aligned}

Example 3: with negative powers and fractional powers

This example uses the Negative and Fractional indices. It is a good idea to check our Laws of Indices page for more information before attempting this question.

Evaluate:

$8^{\frac{1}{3}} \div 2^{-2}$

$8^{\frac{1}{3}}=(\sqrt[3]{8})^{1}=2$ $2^{-2}=\frac{1}{2^{2}}=\frac{1}{4}$

\begin{aligned} 8^{\frac{1}{3}} \div 2^{-2} &=2 \div \frac{1}{4} \\\\ &=2 \times 4 \\\\ &=8 \end{aligned}

Practice dividing indices questions (with different bases)

1. Evaluate

5^{2} \div 3^{3}

\frac{25}{27}

2^{-1}

\frac{25}{81}

(\frac{5}{3})^{-1}
\begin{aligned} 5^{2} \div 3^{3} &= \frac{5 \times 5}{3 \times 3 \times 3}\\\\ &=\frac{25}{27} \end{aligned}

2. Evaluate

2^{6} \div 4^{2} \div 3^{3}

\frac{5}{30}

\frac{2}{27}

\frac{4}{27}

\frac{8}{27}
\begin{aligned} 2^{6} &= 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64\\\\ 4^{2} &= 4 \times 4 = 16\\\\ 3^{3} &= 3 \times 3 \times 3 = 27 \end{aligned}

\begin{aligned} 2^{6} \div 4^{2} \div 3^{3} &= 64 \div 16 \div 27\\\\ &=\frac{4}{27} \end{aligned}

3. Evaluate

3^{4} \div 2^{-2}

20.25

108

\frac{81}{\sqrt{2}}

324
\begin{aligned} 3^{4}&=3 \times 3 \times 3 \times 3=81\\\\ 2^{-2}&=\frac{1}{-2}=\frac{1}{4} \end{aligned}

\begin{aligned} 3^{4} \div 2^{-2} &= 81 \div \frac{1}{4}\\\\ &=81 \times 4\\\\ &=324 \end{aligned}

4. Evaluate

27^{\frac{2}{3}} \div 2^{-2} \div 3^{2}

4

9

\frac{1}{4}

\frac{9}{4}
\begin{aligned} 27^{\frac{2}{3}}&=9\\\\ 2^{-2}&=\frac{1}{4}\\\\ 3^{2}&=9 \end{aligned}

\begin{aligned} 27^{\frac{2}{3}} \div 2^{-2}\div 3^{2} & = 9 \div \frac{1}{4} \div 9\\\\ &=4 \end{aligned}

Dividing indices GCSE questions

1. Simplify

x^{2} \div x^{3}

(1 mark)

x^{-1}

(1)

2. Simplify

8 h^{3} m^{6} \div 2 h^{4} m^{2}

(2 marks)

h^{-1} \text { or } m^{4} seen (evidence of subtracting powers)

(1)

4 h^{-1} m^{4}

(1)

3. Simplify

\frac{30 x^{3} y^{-4}}{5 x^{-2} y^{-3}}

(2 marks)

x^{5} \text { or } y^{-1} seen (evidence of subtracting powers)

(1)

6 x^{5} y^{-1}

(1)

Common misconceptions

• The dividing indices law can only be used for terms with the same base

E.g. We cannot simplify

$a^{4} \div b^{3}$

as the bases are different

• Confusing integer and fractional powers

Raising a term to the power of 2 means we square it

E.g.

$a^{2}=a \times a$

Raising a term to the power of ½ means we find the square root of it

E.g.

$a^{\frac{1}{2}}= \sqrt{a}$

Raising a term to the power of 3 means we cube it

E.g.

$a^{3}=a \times a \times a$

Raising a term to the power of ⅓ means we find the cube root of it

E.g.

$a^{\frac{1}{3}}=\sqrt[3]{a}$

• Indices, powers or exponents

Indices can also be called powers or exponents.

Learning checklist

You have now learned how to:

• Simplify expressions involving the laws of indices
• Calculate with roots, and with integer and fractional indices

Still stuck?

Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.

Find out more about our GCSE maths revision programme.

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FREE GCSE Maths Practice Papers - 2022 Topics

Practice paper packs based on the advanced information for the Summer 2022 exam series from Edexcel, AQA and OCR.

Designed to help your GCSE students revise some of the topics that will come up in the Summer exams.


text:
# Break axes on listcontourplot

I would like to insert a break on both of the axes of a ListContourPlot near the origin, to include a datapoint which has axis values quite far away from the others. The question is similar in nature to this one:

Generating a broken or snipped axis in ListPlot

Although, this is for a single axis on a simple Listplot, and I don't think it would be usable in this case.

To be clear, I am looking for a break in BOTH of the axes on a ListContourPlot, which would allow me to include the origin (0,0) on this graph below:

Although it looks like a single break on the x axis would work, I actually beed a double break, as between 0.01 and 0 on the y is not plotable.

Any help would be most appreciated!

Best,

Ben

• But what you want to do with the contours around the breaks? They will look rather ugly! – Dr. belisarius Dec 12 '14 at 16:23
• @belisarius Thanks for your reply. The whole graph would simply be shifted towards top right. I only want to label a point, having whitespace between there and the rest of the contour plot. – ben18785 Dec 12 '14 at 16:41
• Ok, but that isn't "Breaking axes on ListContourplot[] since ListContourplot` doesn't do that kind of things. – Dr. belisarius Dec 12 '14 at 16:50


text:
# Academic year 2022-2023 : Number Theory and Algebraic Geometry.

Number theory and algebraic geometry are two domains of mathematics linked by many fructuous interactions. For example, one can not progress in the study of the solutions with integral coordinates of polynomial equations without understanding the geometry of the varieties defined by these equations.

The proposed courses form an introduction to these wide and very active fields in Mathematics, with minimal prerequisite essentially limited to basic notions in algebra.

## Introductory course: Algebraic number theory (S. Checcoli)

### Description

Algebraic number theory studies algebraic numbers (i.e., complex numbers that are roots of polynomials with integer coefficients) using techniques and tools from algebra.

Its origins probably go back to the work of Gauss, Eisenstein, Jacobi and its development over the centuries, under the impulse of Dirichlet, Dedkind, Hilbert and many others, is motivated by the study of some important problems in number theory, such as generalizations of the quadratic reciprocity law or the famous Fermat’s last theorem.

Algebraic number theory has now become one of the essential areas of number theory, with applications ranging from diophantine geometry, factorization of integers, primality tests, and transcendence theory, but it has also enriched on its way many other fields of mathematics. For example, the search for general formulas for roots of polynomials has been a powerful source for the development of group theory, thanks to the work of Ruffini, Abel, up to the magnificent Galois theory.

This course aims to study number fields which are extensions of finite degree of the field of rational numbers, the structure of their ring of integers and its prime ideals (ramification theory) and their completions with respect to archimedean and non-archimedean valuations (the theory of p-adic fields). We will start by recalling several tools from Galois theory (for finite and infinite extensions) and, if time allows, we will end the course with an introduction to the theory of heights.

### Program

• Algebraic extensions : reminders on algebraic and finite extensions of fields, existence and uniqueness of an algebraic closure, extensions of homomorphisms with values in an algebraically closed fields, primitive element theorem.
• Galois theory: reminders on Galois extensions, Galois correspondence, inverse Galois problem and examples, solvability by radicals and solvable groups, infinite Galois extensions.
• p-adic fields: Ostrowski’s theorem, local degrees, local norms, Krasner’s lemma, classification of unramified and tamely ramified extensions of a p-adic field, wild ramification, Galois extensions of p-adic fields and their structure, ramification groups: structure and jumps.
• Complements: Chebotarev density theorem and applications. Shafarevich theorem on the “local” realisability of p-groups. Introduction to the Weil height.

### Prerequisites

Theory of finite groups; theory of field extensions; number fields: structure, ring of integers and prime ideals, valuation theory over a number fields; basic notions in Galois theory.

### Bibliography

1. J.W.S. Cassels and A. Frohlich (eds.), Algebraic Number Theory, Academic Press 1967.
2. S. Lang, Algebra, 3rd Edition, Addison Wesley, 1993.
3. S. Lang, Algebraic Number Theory, 2nd Edition, Springer Verlag 1994.
4. W. Narkiewicz, Elementary and analytic theory of algebraic numbers, Springer Monographs in Mathematics (3 ed.), Berlin: Springer-Verlag, 2004.
5. J. Neukirch, Class Field Theory, Springer Verlag, 1986.
6. J-P. Serre, Topics in Galois Theory, Jones and Bartlett Publishers, 1992.
7. J-P. Serre, Local fields, Springer Verlag 1979.

## Introductory course: Arithmetics under the influence of Geometry (E. Peyre)

### Description

Exhibiting strange relations between integers is as old as the notion of numbers. The fermat equation $X^p+Y^p=Z^p$ has fascinated many would-be mathematicians for 300 years. With computers, it is possible to get quite surprising relations like $95800^4 + 217519^4 + 414560^4 = 422481^4$ or $2682440^4 + 15365639^4 + 18796760^4 = 20615673^4$ with which Noam Elkies disproved a long standing conjecture of Euler.

Therefore one would like to have criteria to determine from an explicit system of polynomial equations with integer coefficients:

• Whether the system has solutions with integer coordinates;
• Whether it has infinitely many solutions;
• How many such solution one may find with coordinates smaller than a real number B.

The first question is known as Hilbert’s tenth problem. It was proven to be not solvable by Matiyasevich: no algorithm can determine in finite time whether a system of polynomial equations is solvable over the integers.

On the bright side, it turned out that the set of solutions was under the influence of the geometry of the complex variety defined by the same equations. One of the most famous result in that direction is a theorem of Faltings: A complex curve may be seen as an oriented real surface which is classified up to homeomorphism by a topological invariant: its genus. If the genus of a curve is bigger or equal to two, there exists only a finite numbers of points with rational coordinates on the curve. So the nature of the set of solutions is ruled by purely geometric invariants.

In higher dimensions, the situation is more complicated but similar phenomena are expected to be true.

The objective of these lectures is to present the tools of arithmetic geometry which provide the link between the arithmetic setting and the geometry, including projective varieties, line bundles, heights,\dots We shall illustrate these notions on various examples explaining also techniques to produce solutions and to count them.

As an example, the figure below displays points with rational coordinates on the surface given by the equation $Y^2+Z^2=X^3-X.$

### Program

• Classical examples;
• Projective setting;
• Heights;
• Northcott’s theorem;
• Line bundles; Picard group;
• Conics and cubics;
• Law group on elliptic curves;
• Mordell-Weil’s theorem;
• Statement of Falting’s theorem;
• Higher dimension, Lang’s conjectures;
• Varieties with many points.

### References

1. J.-P. Serre, Lectures on the Mordell-Weil theorem, 3rd edition, Aspects of Math, Springer Fachmedien Wiesbaden (1997).
2. G. Cornell, J. H. Silverman, Arithmetic Geometry, Springer Verlag (1986).
3. S. Lang, Number theory III: Diophantine Geometry, Encyclopaedia of Math. Sciences \textbf{60}, Springer Verlag (1991).

### Prerequisite

Notions of algebra (groups, rings, algebras, polynomials in several variables) topology (topological space, compacity) and differential geometry (partial derivatives, jacobian matrix, tangent space).

## Introductory course: An introduction to algebraic geometry (J. Fasel)

### Description

Algebraic geometry is roughly the study of objects obtained as solutions of system of polynomials, combining geometric intuition and algebraic methods. It is one of the richest and most interesting field of mathematics, having connections with diverse branches such as topology and number theory. Its successes are manifold, including among other the proof of Fermat s conjecture by Wiles or the proof of Weil's conjectures by Deligne and Grothendieck.

The goal of this course is to provide solid ground for the more advanced notions in algebraic geometry. In the first part, we will focus on affine schemes. We will introduce the Zariski topology and study its behaviour under ring homomorphisms, focusing on localizations and quotient by ideals. We will then briefly discuss the notion of sheaf on a topological space before turning to the notions of scheme and morphism of schemes. In the second part, we will study more specific properties of morphisms and useful constructions such as the fiber product. We will also spend a fair amount of time on the notion of point. Time permitting, we will introduce the Chow groups associated to a scheme.

### Program

1. Affine schemes, Zariski topology.
2. Schemes and their properties, points.
3. Functorial behaviour.

### References

1. R. Hartshorne, Algebraic geometry, Graduate texts in mathematics, Springer-Verlag, Berlin, New York (1977).
2. R. Vakil, Foundations of algebraic geometry, available online.

### Prerequisite

Basic notions in commutative algebra.

## Advanced Course - Diophantine Approximation for values of special functions (T. Rivoal)

### Description,

This course will be devoted to the Diophantine properties of the values taken at algebraic points by certain special functions. The question is to decide whether or not such a function takes an irrational value (or more generally a transcendental value) at a rational point or an algebraic point. Roughly speaking, a special function is a power series solution of a linear differential equation with coefficients in Q(z), or more generally a solution of some linear functional equation. The most classical examples of special functions are the hypergeometric series, amongst which we find exp(z) et log(1-z).

In the first part of the course, we will first review a few basic results and constructions of Diophantine Approximation. We will then be interested in certain “explicit” constructions based on Hermite-Padé approximants. These constructions enable to show that exp(α) is a transcendental number (ie, not an algebraic number) for every non zero algebraic number α, from which we deduce the transcendence log(α) for every algebraic number α different of 0 and 1. We will also apply this type of constructions to the polylogarithmic functions $$\sum_{n=1}^\infty z^n/n^s$$ that generalize log(1-z). Important corollaries will be obtained concerning the Riemann zeta function, such as the irrationality of ζ(3).

In the second part of the course, we will generalize the previous theorems to larger classes of functions, including E-functions (that generalize $\exp(z)$) and G-functions (that generalize log(1+z)); the underlying constructions, based on Siegel’s lemma, are now “inexplicit”. The differential equations satisfied by these functions are crucial and the results are fundamentaly different from one class of functions to the other.

At last, depending on time available, we will study the Diophantine properties of Mahlerian functions. These functions are solutions of a functional equation of the form $$\sum_{j=0}^d a_{j+1}(z)F(z^{k^j})+a_0(z)=0$$ with $$a_j(z)\in \mathbf Q(z)$$. A classical example is $$F(z)=\sum_{n=0}^\infty z^{k^n}$$ (k⩾2 an integer) solution of $$F(z^k)=F(z)-z$$, and it will be shown that it takes transcendental values at all algebraic points z such that $$0<|z|<1.$$. More generally, we will show a result due to Mahler, on the transcendence of the values of these functions, where fonctionnal equations play a fundamental role similar to differential equations for E-and G-functions.

### Prerequisite

It will be useful to attend the introductory courses in number theory proposed during the first term by Sara Checcoli and Emmanuel Peyre.

### Program

1. Generalities on Diophantine Approximation
2. Transcendence at algebraic points of exp(z) and log(z).
3. Linear independence of values of polylogarithms, and applications to the values of the Riemann zeta function.
4. Gel'fond-Schneider's Theorem.
5. Siegel-Shidlovkii's Theorem for E-functions.
6. Chudnovsky's Theorem for G-functions.
7. Mahler's Theorem for Mahlerian functions.

### References

1. A. Baker, Transcendental number theory, Cambridge University Press, coll. Cambridge Mathematical Library (1990).
2. A. Shidlovskii, Transcendental Numbers, Studies in Math 12, de Gruyter (1987).
3. K. Nishioka, Mahler Functions and Transcendence, Lecture Notes in Mathematics 1631 (1996).

## Advanced course: Grassmannians and Enumerative Geometry (J. Guéré)

### Description

The projective space classifies all lines in the affine space. It admits a generalization, called the Grassmannian Gr(k,n), which classifies all subvector spaces of dimension k inside a given vector space of dimension n>k. We prove it is a variety of dimension $$\binom{n}{k}$$ and it plays a crucial role in algebraic geometry.

Enumerative geometry focuses on counting problems for some objects satisfying special conditions. For instance, in the Grassmannian Gr(k,n), if we look at all subvector spaces containing a given line, we get a subvariety of Gr(k,n). When sufficiently many constraints of this sort are imposed, the resulting subvariety can be zero-dimensional. Hence, it means it is a collection of points and we can count them. For instance, the number of lines in the plane meeting at two given points is just 1, so is the number of quadrics meeting at five general points.

In this course, we aim at counting how many lines there are on a smooth cubic complex surface. For this reason, we will investigate the geometry of the Grassmannian, compute its Chow ring, and work out what is now called Schubert calculus.

### Program

1. Preliminaries on Chow rings and the example of the projective space
2. Introduction to Grassmannians and Schubert calculus
3. The cubic surface and its lines

### References

1. D. Eisenbud and J. Harris 3264 and All That, Intersection Theory in Algebraic Geometry, Cambridge University Press (2016). The whole book is available online

### Prerequisite

An introduction to algebraic geometry


text:
Without using Lagrange's theorem, show that an abelian group of odd order cannot have an element of even order.

Let $G$ be an abelian group of odd order, say $n$. Then, by fundamental theorem of finite abelian groups, $G$ is a direct product of groups of prime power order. Let $$G=\mathbb Z_{{p_1}^{n_1}} \oplus \mathbb Z_{{p_2}^{n_2}}\oplus \cdots\oplus\mathbb Z_{{p_k}^{n_k}}.$$ Then $\vert G\vert=n={p_1}^{n_1}{p_2}^{n_2}\cdots{p_k}^{n_k}$. Now let $(g_1,g_2,...,g_k)\in G$. Then $\vert (g_1,g_2,...,g_k )\vert=\mathrm{lcm}(\vert g_1 \vert,\vert g_2 \vert,...,\vert g_k \vert)$.

From here how to move, i don't know but it is given in the hint to make use of fundamental theorem of cyclic groups (but i'm not getting how to apply it here).

• How do you get the decomposition into a direct sum of cyclic groups without Lagrange theorem? – egreg Oct 1 '16 at 17:23
• @egreg:Since G is abelian,then by fundamental theorem of finite abelian groups(which states that-every finite abelian group is the direct product of cyclic groups of prime power order) it can be represented as the direct product of cyclic groups. – P.Styles Oct 1 '16 at 17:36
• The point is that it does not make any sense to avoid using Lagrange's Theorem if you are going to use a more advanced theorem like the fundamental theorem of abelian groups. – Derek Holt Oct 1 '16 at 20:37

Assuming you can prove the fundamental theorem without using Lagrange (which I don't believe), a consequence of the theorem is that, for every $x\in G$, $x^n=1$, because $n$ is a multiple of the order of every cyclic component and, clearly, in a cyclic group $C$ of order $k$, we have $g^k=1$, for every $c\in C$.

The map $x\mapsto x^2$ is an endomorphism of $G$ and, since $n=2k+1$ is odd, we have $$x=(x^{k+1})^2$$ so the map is surjective. Thus the map is also injective, therefore $x^2=1$ implies $x=1$ and there's no element of order $2$.

• :I did'nt noticed this fact that proof of FTFAG makes use of Lagrange's theorem.But this exercise is from Gallian's algebra text.Whatever i mentioned in my post is directed through the hint in the very text.May be i'm wrong in understanding the hint.Hints are given as-use 1.Fundamental theorem of finite abelian groups.2.Order of elements in a direct product.3.Fundamental theorem of cyclic groups . – P.Styles Oct 1 '16 at 18:19
• @PKStyles That's what I used – egreg Oct 1 '16 at 19:20
• :How does $x=(x^{k+1})^2$ happen? – P.Styles Oct 1 '16 at 20:53
• @PKStyles Since $n=2k+1$, $(x^{k+1})^2=x^{2k+2}=x^{n+1}=x^nx=x$ – egreg Oct 1 '16 at 21:13
• :thanks for the response – P.Styles Oct 2 '16 at 4:41

Suppose $g\in G$ has even order, say $2m$, then $g^m$ has order $2$ so assume $g$ has order $2$, then we can partition $G$ into pairs, $\{h,k\}$ where $h=gk$ and $k=gh$.

This is of course impossible since $G$ has odd order, so we have the result by contradiction.

• Nice. This works even if $G$ is not abelian. – Arnaud D. Oct 2 '16 at 9:54

I agree with egreg that using the FTFGAG seems very much like overkill; and Robert Chamberlain's proof is very nice, but seems essentially to be re-proving Lagrange's theorem in this case. I wonder whether what Gallian has in mind isn't just to use the fact that multiplication by $g^n$ fixes the product of the elements of $G$, hence is trivial; so, if $g$ had even order $m$, then we would have that there existed integers $a$ and $b$ so that $g^{\gcd(m, n)} = g^{a m + b n} = (g^m)^a(g^n)^b = 1$. This is a contradiction, since $\gcd(m, n)$ is positive but strictly less than $m$ (it divides $m$, and is odd so it divides $n$, so it doesn't equal $m$).


text:
Rating changes for the last round are temporarily rolled back. They will be returned soon. ×

### thienlongtpct's blog

By thienlongtpct, history, 2 years ago, ,

### Overview

This is just a subproblem that I have learned before. However, I don't have the testcases or the place to submit this.

### Describe

Give 2 * n colors with the their frequency (Each of them have to appear at least once). Sum of all their frequency is m * n. Make a table with size m * n (m rows, n column) that minimize the max numbers of different color in each column (And the frequency of color is same above) ?

#### Input

The first line of input contains two integers m and n (m, n <  = 50). The second line contains 2 * n integers — the frequency of each color. The sum values of them equal exactly m * n.

4 4
1 1 2 2 2 4 1 3

#### Output

Output the table that satisfied the statement above.

1 7 4 6
2 3 4 6
6 3 5 8
6 8 5 8

### Explain

• Color 1 has to appear 1 times;
• Color 2 has to appear 1 times;
• Color 3 has to appear 2 times;
• Color 4 has to appear 2 times;
• Color 5 has to appear 2 times;
• Color 6 has to appear 4 times;
• Color 7 has to appear 1 times;
• Color 7 has to appear 3 times;

Sum of them is: 1 + 1 + 2 + 2 + 2 + 4 + 1 + 3 = 16 = m * n.

• Column 1 has 3 different colors.
• Column 2 has 3 different colors.
• Column 3 has 2 different colors.
• Column 4 has 2 different colors.

So the maximum different colors in each column is 3, which is the best answer in this case.

### My approach

My teacher said that the answer is only equal 2 or 3.

• In case the answer is 2, we just sort the frequency of colors and merge the most frequency and the least frequency color together in the first column, do the same thing with (n - 1) other column.
• However, when the answer is 3, he just said the the greedy approach could do this.

I have tried some greedy approach, like merge this first least and the second least frequency color (we can prove that sum of them always  <  = m) with the most frequency color. But I already found a conner case for this.

I have ask many people (include my teacher — who always said is just a easy greedy problem -_-) but I couldn't found a correct approach. So now I post this, thank you Codeforces community.

• +18

By thienlongtpct, history, 3 years ago, ,

UPD 1: Add the application to automatic draw the rectangle.

#### Overview

This problem is a varient of the classic problem, "Packing rectangles with the smallest area rectangle.". This is also an optimization with you may not find the most optimal solution but the more rectangles you can place, the more score you get.

In this problem, given the size of the main rectangle and the size of others rectangles, find the maximum number of rectangle could be place in the main rectangle without overlapping the other rectangle ( note: these rectangles must be inside the main rectangle — but it can be touch in the borders of the main rectangle).

Input

The first line contains three integers n, m and k (1n, m100, 1k1000) — the height, the width of the fixed rectangle and the number of rectangles to be placed.

The next k lines have two integers xi and yi (1xin, 1yim) — the height and the width of the rectangle, remember that we can't rotate these rectangles.

Output

The first line of the ouput print k.

The next k line print two number xj and yj is the top — left point where the rectangle j placed. In case we don't place this rectangle, print "0 0" (without quotes).

Sample input

8 7 10
4 5
4 2
2 3
1 1
1 1
1 2
1 5
5 1
6 1
2 1

Sample output

10
1 1
5 1
5 3
1 6
7 3
7 4
8 3
1 7
2 6
6 7

Illustration

Picture for sample test case

Illustration application

Using the application

To use the application, you run your code to make a new ".exe" file (it must be in same folder with the input test). After that, upload the test input by clicking the button "Load input file...". Finally, click the button "Run external solution ..." and chose your ".exe" file.

P/s: Hope you to help with problem in case we want to maximize the area be covered. I just think of the brute force solution and some of the trival greedy approachs.

• +14

By thienlongtpct, history, 3 years ago, ,

As you see, my new rating is 1608, however, it dislay as 1689 (old rating before the Codeforces Round #226 (Div. 2)). The rating changes in the contest and the graph have show may exactly my new rating — except the dislay in my home page — sorry I don't know how to call it.

I have to see some other user and I have the same bug, I just see their new color but still old rating.

Also, when I open the Problemset Status with friend only, I just see the submission before the contest, although after the contest I have make a submission in 745B - Коровоконг решает пазл — and it change to green in the Problemset but I can't see the submisson in the Status.

So I ask my friend but he couldn't find anybug. I think maybe the problem is about my account, so I log out and see the rating again, but the bug still there :(.


text:
# megahertz to degrees per second conversion

Conversion number between megahertz [MHz] and degrees per second [deg/s] is 360000000. This means, that megahertz is bigger unit than degrees per second.

### Contents [show][hide]

Switch to reverse conversion:
from degrees per second to megahertz conversion

### Enter the number in megahertz:

Decimal Fraction Exponential Expression
 [MHz]
eg.: 10.12345 or 1.123e5

Result in degrees per second

?
 precision 0 1 2 3 4 5 6 7 8 9 [info] Decimal: Exponential:

### Calculation process of conversion value

• 1 megahertz = (exactly) (1000000) / ((1/360)) = 360000000 degrees per second
• 1 degrees per second = (exactly) ((1/360)) / (1000000) = 2.7777777777778 × 10-9 megahertz
• ? megahertz × (1000000  ("Hz"/"megahertz")) / ((1/360)  ("Hz"/"degrees per second")) = ? degrees per second

### High precision conversion

If conversion between megahertz to hertz and hertz to degrees per second is exactly definied, high precision conversion from megahertz to degrees per second is enabled.

Decimal places: (0-800)

megahertz
Result in degrees per second:
?

### megahertz to degrees per second conversion chart

 Start value: [megahertz] Step size [megahertz] How many lines? (max 100)

visual:
megahertzdegrees per second
00
103600000000
207200000000
3010800000000
4014400000000
5018000000000
6021600000000
7025200000000
8028800000000
9032400000000
10036000000000
11039600000000
Copy to Excel

## Multiple conversion

Enter numbers in megahertz and click convert button.
One number per line.

Converted numbers in degrees per second:
Click to select all

## Details about megahertz and degrees per second units:

Convert Megahertz to other unit:

### megahertz

Definition of megahertz unit: = 106 Hz. Millions of Hertz. How many cycles take place over one millionth of a second. Event per 0.000001 sec.

Convert Degrees per second to other unit:

### degrees per second

Definition of degrees per second unit: = 1/360 Hz. Degrees per second is a variant of the angular velocity unit. One hertz frequency is equal to turning a full circle (360°) in one second , 1 Hz = 360 deg / s.

← Back to Frequency units