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The calculus of variations (or Variational Calculus) is a field of
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s. Functionals are often expressed as definite integrals involving functions and their
derivative In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
s. Functions that maximize or minimize functionals may be found using the
Euler–Lagrange equation In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action (physics), action functional. The equations ...
of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a
straight line In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional objects, though they may exist in Two-dimensional Euclidean space, two, Three-dimensional space, three, ...
between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as ''
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path (arc (geometry), arc) between two points in a differential geometry of surfaces, surface, or more generally in a Riemannian manifold. The term also has meaning in ...
s''. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, which depends upon the material of the medium. One corresponding concept in
mechanics Mechanics (from Ancient Greek: wikt:μηχανική#Ancient_Greek, μηχανική, ''mēkhanikḗ'', "of machine, machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among Ph ...
is the principle of least/stationary action. Many important problems involve functions of several variables. Solutions of boundary value problems for the
Laplace equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \nab ...
satisfy the Dirichlet's principle. Plateau's problem requires finding a surface of minimal area that spans a given contour in space: a solution can often be found by dipping a frame in soapy water. Although such experiments are relatively easy to perform, their mathematical formulation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
.

# History

The calculus of variations may be said to begin with Newton's minimal resistance problem in 1687, followed by the
brachistochrone curve In physics and mathematics, a brachistochrone curve (), or curve of fastest descent, is the one lying on the plane between a point ''A'' and a lower point ''B'', where ''B'' is not directly below ''A'', on which a bead slides frictionlessly under ...
problem raised by
Johann Bernoulli Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educating ...
(1696). It immediately occupied the attention of Jakob Bernoulli and the Marquis de l'Hôpital, but
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
first elaborated the subject, beginning in 1733.
Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaLegendre (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima.
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
and
Gottfried Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...
also gave some early attention to the subject. To this discrimination Vincenzo Brunacci (1810),
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
(1829), Siméon Poisson (1831), Mikhail Ostrogradsky (1834), and Carl Jacobi (1837) have been among the contributors. An important general work is that of Sarrus (1842) which was condensed and improved by
Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
(1844). Other valuable treatises and memoirs have been written by Strauch (1849), Jellett (1850), Otto Hesse (1857), Alfred Clebsch (1858), and Lewis Buffett Carll (1885), but perhaps the most important work of the century is that of
Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university without a degree, ...
. His celebrated course on the theory is epoch-making, and it may be asserted that he was the first to place it on a firm and unquestionable foundation. The 20th and the 23rd Hilbert problem published in 1900 encouraged further development. In the 20th century
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
, Oskar Bolza, Gilbert Ames Bliss,
Emmy Noether Amalie Emmy NoetherEmmy (given name), Emmy is the ''Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promot ...
, Leonida Tonelli,
Henri Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are con ...
and
Jacques Hadamard Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations. Biography The son of a teac ...
among others made significant contributions. Marston Morse applied calculus of variations in what is now called Morse theory. Lev Pontryagin, Ralph Rockafellar and F. H. Clarke developed new mathematical tools for the calculus of variations in optimal control theory. The
dynamic programming Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics. I ...
of Richard Bellman is an alternative to the calculus of variations.

# Extrema

The calculus of variations is concerned with the maxima or minima (collectively called extrema) of functionals. A functional maps functions to scalars, so functionals have been described as "functions of functions." Functionals have extrema with respect to the elements $y$ of a given
function space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
defined over a given domain. A functional
sign A sign is an Physical object, object, quality (philosophy), quality, event, or Non-physical entity, entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to ...
for all $y$ in an arbitrarily small neighborhood of $f.$ The function $f$ is called an extremal function or extremal. The extremum
Euler–Lagrange equation In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action (physics), action functional. The equations ...
.

# Euler–Lagrange equation

Finding the extrema of functionals is similar to finding the maxima and minima of functions. The maxima and minima of a function may be located by finding the points where its derivative vanishes (i.e., is equal to zero). The extrema of functionals may be obtained by finding functions for which the functional derivative is equal to zero. This leads to solving the associated
Euler–Lagrange equation In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action (physics), action functional. The equations ...
. Consider the functional where *$x_1, x_2$ are constants, *$y\left(x\right)$ is twice continuously differentiable, *$y\text{'}\left(x\right) = \frac,$ *$L\left\left(x, y\left(x\right), y\text{'}\left(x\right)\right\right)$ is twice continuously differentiable with respect to its arguments $x, y,$ and $y\text{'}.$ If the functional
local minimum In mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, ...
at $f,$ and $\eta\left(x\right)$ is an arbitrary function that has at least one derivative and vanishes at the endpoints $x_1$ and $x_2,$ then for any number $\varepsilon$ close to 0, The term $\varepsilon \eta$ is called the variation of the function $f$ and is denoted by $\delta f.$ Substituting $f + \varepsilon \eta$ for $y$ in the functional the result is a function of $\varepsilon,$ Since the functional
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral (mathematics), integral of a product (mathematics), product of Function (mathematics), functions in terms o ...
on the second term. The second term on the second line vanishes because $\eta = 0$ at $x_1$ and $x_2$ by definition. Also, as previously mentioned the left side of the equation is zero so that $\int_^ \eta (x) \left(\frac - \frac\frac \right) \, dx = 0 \, .$ According to the fundamental lemma of calculus of variations, the part of the integrand in parentheses is zero, i.e. $\frac -\frac \frac=0$ which is called the Euler–Lagrange equation. The left hand side of this equation is called the functional derivative of
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable (mathematics), variable and involves the derivatives of those functions. The term ''ordinary ...
which can be solved to obtain the extremal function $f\left(x\right).$ The Euler–Lagrange equation is a necessary, but not sufficient, condition for an extremum A sufficient condition for a minimum is given in the section Variations and sufficient condition for a minimum.

## Example

In order to illustrate this process, consider the problem of finding the extremal function $y = f\left(x\right),$ which is the shortest curve that connects two points $\left\left(x_1, y_1\right\right)$ and $\left\left(x_2, y_2\right\right).$ The
arc length ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...
of the curve is given by with $y'(x) = \frac \, , \ \ y_1=f(x_1) \, , \ \ y_2=f(x_2) \, .$ Note that assuming is a function of loses generality; ideally both should be a function of some other parameter. This approach is good solely for instructive purposes. The Euler–Lagrange equation will now be used to find the extremal function $f\left(x\right)$ that minimizes the functional $\frac -\frac \frac=0$ with $L = \sqrt \, .$ Since $f$ does not appear explicitly in $L,$ the first term in the Euler–Lagrange equation vanishes for all $f\left(x\right)$ and thus, $\frac \frac = 0 \, .$ Substituting for $L$ and taking the derivative, $\frac \ \frac \ = 0 \, .$ Thus $\frac = c \, ,$ for some constant $c.$ Then $\frac = c^2 \, ,$ where $0 \le c^2<1.$ Solving, we get which implies that $f'(x)=m$ is a constant and therefore that the shortest curve that connects two points $\left\left(x_1, y_1\right\right)$ and $\left\left(x_2, y_2\right\right)$ is $f(x) = m x + b \qquad \text \ \ m = \frac \quad \text \quad b = \frac$ and we have thus found the extremal function $f\left(x\right)$ that minimizes the functional

# Beltrami's identity

In physics problems it may be the case that $\frac = 0,$ meaning the integrand is a function of $f\left(x\right)$ and $f\text{'}\left(x\right)$ but $x$ does not appear separately. In that case, the Euler–Lagrange equation can be simplified to the Beltrami identity $L - f' \frac = C \, ,$ where $C$ is a constant. The left hand side is the
Legendre transformation In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involution (mathematics), involutive List of transforms, transformation on real number, real-valued convex functions of one real variable ...
of $L$ with respect to $f\text{'}\left(x\right).$ The intuition behind this result is that, if the variable $x$ is actually time, then the statement $\frac = 0$ implies that the Lagrangian is time-independent. By Noether's theorem, there is an associated conserved quantity. In this case, this quantity is the Hamiltonian, the Legendre transform of the Lagrangian, which (often) coincides with the energy of the system. This is (minus) the constant in Beltrami's identity.

# Euler–Poisson equation

If $S$ depends on higher-derivatives of $y\left(x\right),$ that is, if $S = \int_^ f(x, y(x), y'(x), \dots, y^(x)) dx,$ then $y$ must satisfy the Euler– Poisson equation,

# Du Bois-Reymond's theorem

The discussion thus far has assumed that extremal functions possess two continuous derivatives, although the existence of the integral $J$ requires only first derivatives of trial functions. The condition that the first variation vanishes at an extremal may be regarded as a weak form of the Euler–Lagrange equation. The theorem of Du Bois-Reymond asserts that this weak form implies the strong form. If $L$ has continuous first and second derivatives with respect to all of its arguments, and if $\frac \ne 0,$ then $f$ has two continuous derivatives, and it satisfies the Euler–Lagrange equation.

# Lavrentiev phenomenon

Hilbert was the first to give good conditions for the Euler–Lagrange equations to give a stationary solution. Within a convex area and a positive thrice differentiable Lagrangian the solutions are composed of a countable collection of sections that either go along the boundary or satisfy the Euler–Lagrange equations in the interior. However Lavrentiev in 1926 showed that there are circumstances where there is no optimum solution but one can be approached arbitrarily closely by increasing numbers of sections. The Lavrentiev Phenomenon identifies a difference in the infimum of a minimization problem across different classes of admissible functions. For instance the following problem, presented by Manià in 1934: $= \.$ Clearly, $x\left(t\right) = t^$minimizes the functional, but we find any function $x \in W^$ gives a value bounded away from the infimum. Examples (in one-dimension) are traditionally manifested across $W^$ and $W^,$ but Ball and Mizel procured the first functional that displayed Lavrentiev's Phenomenon across $W^$ and $W^$ for $1 \leq p < q < \infty.$ There are several results that gives criteria under which the phenomenon does not occur - for instance 'standard growth', a Lagrangian with no dependence on the second variable, or an approximating sequence satisfying Cesari's Condition (D) - but results are often particular, and applicable to a small class of functionals. Connected with the Lavrentiev Phenomenon is the repulsion property: any functional displaying Lavrentiev's Phenomenon will display the weak repulsion property.

# Functions of several variables

For example, if $\varphi\left(x, y\right)$ denotes the displacement of a membrane above the domain $D$ in the $x,y$ plane, then its potential energy is proportional to its surface area: Plateau's problem consists of finding a function that minimizes the surface area while assuming prescribed values on the boundary of $D$; the solutions are called minimal surfaces. The Euler–Lagrange equation for this problem is nonlinear: $\varphi_(1 + \varphi_y^2) + \varphi_(1 + \varphi_x^2) - 2\varphi_x \varphi_y \varphi_ = 0.$ See Courant (1950) for details.

## Dirichlet's principle

It is often sufficient to consider only small displacements of the membrane, whose energy difference from no displacement is approximated by The functional $V$ is to be minimized among all trial functions $\varphi$ that assume prescribed values on the boundary of $D.$ If $u$ is the minimizing function and $v$ is an arbitrary smooth function that vanishes on the boundary of $D,$ then the first variation of
Peter Gustav Lejeune Dirichlet Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
. However Weierstrass gave an example of a variational problem with no solution: minimize among all functions $\varphi$ that satisfy $\varphi\left(-1\right)=-1$ and $\varphi\left(1\right)=1.$ $W$ can be made arbitrarily small by choosing piecewise linear functions that make a transition between −1 and 1 in a small neighborhood of the origin. However, there is no function that makes $W=0.$ Eventually it was shown that Dirichlet's principle is valid, but it requires a sophisticated application of the regularity theory for
elliptic partial differential equation Second-order linear differential equation, linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic partial differential equation, hyperbolic, or parabolic partial differential equation, parabolic. Any second-or ...
s; see Jost and Li–Jost (1998).

## Generalization to other boundary value problems

A more general expression for the potential energy of a membrane is This corresponds to an external force density $f\left(x,y\right)$ in $D,$ an external force $g\left(s\right)$ on the boundary $C,$ and elastic forces with modulus $\sigma\left(s\right)$acting on $C.$ The function that minimizes the potential energy with no restriction on its boundary values will be denoted by $u.$ Provided that $f$ and $g$ are continuous, regularity theory implies that the minimizing function $u$ will have two derivatives. In taking the first variation, no boundary condition need be imposed on the increment $v.$ The first variation of

# Eigenvalue problems

Both one-dimensional and multi-dimensional eigenvalue problems can be formulated as variational problems.

## Sturm–Liouville problems

The Sturm–Liouville eigenvalue problem involves a general quadratic form where $\varphi$is restricted to functions that satisfy the boundary conditions $\varphi(x_1)=0, \quad \varphi(x_2)=0.$ Let $R$ be a normalization integral The functions $p\left(x\right)$ and $r\left(x\right)$ are required to be everywhere positive and bounded away from zero. The primary variational problem is to minimize the ratio $Q/R$ among all $\varphi$ satisfying the endpoint conditions. It is shown below that the Euler–Lagrange equation for the minimizing $u$ is $-(p u')' +q u -\lambda r u = 0,$ where $\lambda$ is the quotient $\lambda = \frac.$ It can be shown (see Gelfand and Fomin 1963) that the minimizing $u$ has two derivatives and satisfies the Euler–Lagrange equation. The associated $\lambda$ will be denoted by $\lambda_1$; it is the lowest eigenvalue for this equation and boundary conditions. The associated minimizing function will be denoted by $u_1\left(x\right).$ This variational characterization of eigenvalues leads to the
Rayleigh–Ritz method The Rayleigh–Ritz method is a direct numerical method of approximating eigenvalues and eigenvectors, eigenvalues, originated in the context of solving physical Boundary value problem, boundary value problems and named after Lord Rayleigh and Walth ...
: choose an approximating $u$ as a linear combination of basis functions (for example trigonometric functions) and carry out a finite-dimensional minimization among such linear combinations. This method is often surprisingly accurate. The next smallest eigenvalue and eigenfunction can be obtained by minimizing $Q$ under the additional constraint $\int_^ r(x) u_1(x) \varphi(x) \, dx = 0.$ This procedure can be extended to obtain the complete sequence of eigenvalues and eigenfunctions for the problem. The variational problem also applies to more general boundary conditions. Instead of requiring that $\varphi$ vanish at the endpoints, we may not impose any condition at the endpoints, and set where $a_1$ and $a_2$ are arbitrary. If we set $\varphi = u + \varepsilon v$the first variation for the ratio $Q/R$ is where λ is given by the ratio

## Eigenvalue problems in several dimensions

Eigenvalue problems in higher dimensions are defined in analogy with the one-dimensional case. For example, given a domain $D$ with boundary $B$ in three dimensions we may define and Let $u$ be the function that minimizes the quotient with no condition prescribed on the boundary $B.$ The Euler–Lagrange equation satisfied by $u$ is $-\nabla \cdot (p(X) \nabla u) + q(x) u - \lambda r(x) u=0,$ where $\lambda = \frac.$ The minimizing $u$ must also satisfy the natural boundary condition $p(S) \frac + \sigma(S) u = 0,$ on the boundary $B.$ This result depends upon the regularity theory for elliptic partial differential equations; see Jost and Li–Jost (1998) for details. Many extensions, including completeness results, asymptotic properties of the eigenvalues and results concerning the nodes of the eigenfunctions are in Courant and Hilbert (1953).

# Applications

## Optics

Fermat's principle states that light takes a path that (locally) minimizes the optical length between its endpoints. If the $x$-coordinate is chosen as the parameter along the path, and $y=f\left(x\right)$ along the path, then the optical length is given by where the refractive index $n\left(x,y\right)$ depends upon the material. If we try $f\left(x\right) = f_0 \left(x\right) + \varepsilon f_1 \left(x\right)$ then the first variation of $A$ (the derivative of $A$ with respect to ε) is After integration by parts of the first term within brackets, we obtain the Euler–Lagrange equation The light rays may be determined by integrating this equation. This formalism is used in the context of Lagrangian optics and Hamiltonian optics.

### Snell's law

There is a discontinuity of the refractive index when light enters or leaves a lens. Let $n(x,y) = \begin n_ & \text \quad x<0, \\ n_ & \text \quad x>0, \end$ where $n_$ and $n_$ are constants. Then the Euler–Lagrange equation holds as before in the region where $x < 0$ or $x > 0,$ and in fact the path is a straight line there, since the refractive index is constant. At the $x = 0,$ $f$ must be continuous, but $f\text{'}$ may be discontinuous. After integration by parts in the separate regions and using the Euler–Lagrange equations, the first variation takes the form The factor multiplying $n_$ is the sine of angle of the incident ray with the $x$ axis, and the factor multiplying $n_$ is the sine of angle of the refracted ray with the $x$ axis.
Snell's law Snell's law (also known as Snell–Descartes law and ibn-Sahl law and the law of refraction) is a Mathematical formula, formula used to describe the relationship between the angle of incidence (optics), angles of incidence and refraction, when ...
for refraction requires that these terms be equal. As this calculation demonstrates, Snell's law is equivalent to vanishing of the first variation of the optical path length.

### Fermat's principle in three dimensions

It is expedient to use vector notation: let $X = \left(x_1,x_2,x_3\right),$ let $t$ be a parameter, let $X\left(t\right)$ be the parametric representation of a curve $C,$ and let $\dot X\left(t\right)$ be its tangent vector. The optical length of the curve is given by Note that this integral is invariant with respect to changes in the parametric representation of $C.$ The Euler–Lagrange equations for a minimizing curve have the symmetric form $\frac P = \sqrt \, \nabla n,$ where $P = \frac.$ It follows from the definition that $P$ satisfies $P \cdot P = n(X)^2.$ Therefore, the integral may also be written as This form suggests that if we can find a function $\psi$ whose gradient is given by $P,$ then the integral $A$ is given by the difference of $\psi$ at the endpoints of the interval of integration. Thus the problem of studying the curves that make the integral stationary can be related to the study of the level surfaces of $\psi.$In order to find such a function, we turn to the wave equation, which governs the propagation of light. This formalism is used in the context of Lagrangian optics and Hamiltonian optics.

### = Connection with the wave equation

= The
wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seism ...
for an inhomogeneous medium is $u_ = c^2 \nabla \cdot \nabla u,$ where $c$ is the velocity, which generally depends upon $X.$ Wave fronts for light are characteristic surfaces for this partial differential equation: they satisfy $\varphi_t^2 = c(X)^2 \, \nabla \varphi \cdot \nabla \varphi.$ We may look for solutions in the form $\varphi(t,X) = t - \psi(X).$ In that case, $\psi$ satisfies $\nabla \psi \cdot \nabla \psi = n^2,$ where $n=1/c.$ According to the theory of first-order partial differential equations, if $P = \nabla \psi,$ then $P$ satisfies $\frac = n \, \nabla n,$ along a system of curves (the light rays) that are given by $\frac = P.$ These equations for solution of a first-order partial differential equation are identical to the Euler–Lagrange equations if we make the identification $\frac = \frac.$ We conclude that the function $\psi$ is the value of the minimizing integral $A$ as a function of the upper end point. That is, when a family of minimizing curves is constructed, the values of the optical length satisfy the characteristic equation corresponding the wave equation. Hence, solving the associated partial differential equation of first order is equivalent to finding families of solutions of the variational problem. This is the essential content of the Hamilton–Jacobi theory, which applies to more general variational problems.

## Mechanics

In classical mechanics, the action, $S,$ is defined as the time integral of the Lagrangian, $L.$ The Lagrangian is the difference of energies, $L = T - U,$ where $T$ is the
kinetic energy In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical sci ...
of a mechanical system and $U$ its
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...
. Hamilton's principle (or the action principle) states that the motion of a conservative holonomic (integrable constraints) mechanical system is such that the action integral $S = \int_^ L(x, \dot x, t) \, dt$ is stationary with respect to variations in the path $x\left(t\right).$ The Euler–Lagrange equations for this system are known as Lagrange's equations: $\frac \frac = \frac,$ and they are equivalent to Newton's equations of motion (for such systems). The conjugate momenta $P$ are defined by $p = \frac.$ For example, if $T = \frac m \dot x^2,$ then $p = m \dot x.$
Hamiltonian mechanics Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
results if the conjugate momenta are introduced in place of $\dot x$ by a Legendre transformation of the Lagrangian $L$ into the Hamiltonian $H$ defined by $H(x, p, t) = p \,\dot x - L(x,\dot x, t).$ The Hamiltonian is the total energy of the system: $H = T + U.$ Analogy with Fermat's principle suggests that solutions of Lagrange's equations (the particle trajectories) may be described in terms of level surfaces of some function of $X.$ This function is a solution of the
Hamilton–Jacobi equation In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechan ...
: $\frac + H\left(x,\frac,t\right) = 0.$

## Further applications

Further applications of the calculus of variations include the following: * The derivation of the
catenary In physics and geometry, a catenary (, ) is the curve that an idealized hanging chain or wire rope, cable assumes under its own weight when supported only at its ends in a uniform gravitational field. The catenary curve has a U-like shape, ...
shape * Solution to Newton's minimal resistance problem * Solution to the brachistochrone problem * Solution to the tautochrone problem * Solution to isoperimetric problems * Calculating
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path (arc (geometry), arc) between two points in a differential geometry of surfaces, surface, or more generally in a Riemannian manifold. The term also has meaning in ...
s * Finding minimal surfaces and solving Plateau's problem *
Optimal control Optimal control theory is a branch of mathematical optimization that deals with finding a Control (optimal control theory), control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applica ...
*
Analytical mechanics In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the ...
, or reformulations of Newton's laws of motion, most notably Lagrangian and
Hamiltonian mechanics Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
; * Geometric optics, especially Lagrangian and Hamiltonian optics; * Variational method (quantum mechanics), one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states; * Variational Bayesian methods, a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning; * Variational methods in general relativity, a family of techniques using calculus of variations to solve problems in Einstein's general theory of relativity; *
Finite element method The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical models, mathematical modeling. Typical problem areas of interest include the traditional fields of struct ...
is a variational method for finding numerical solutions to boundary-value problems in differential equations; * Total variation denoising, an
image processing An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
method for filtering high variance or noisy signals.

# Variations and sufficient condition for a minimum

Calculus of variations is concerned with variations of functionals, which are small changes in the functional's value due to small changes in the function that is its argument. The first variation is defined as the linear part of the change in the functional, and the second variation is defined as the quadratic part. For example, if

* First variation * Isoperimetric inequality * Variational principle * Variational bicomplex * Fermat's principle * Principle of least action * Infinite-dimensional optimization *
Finite element method The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical models, mathematical modeling. Typical problem areas of interest include the traditional fields of struct ...
*
Functional analysis Functional analysis is a branch of mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, ...
* Ekeland's variational principle * Inverse problem for Lagrangian mechanics * Obstacle problem * Perturbation methods * Young measure *
Optimal control Optimal control theory is a branch of mathematical optimization that deals with finding a Control (optimal control theory), control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applica ...
* Direct method in calculus of variations * Noether's theorem * De Donder–Weyl theory * Variational Bayesian methods * Chaplygin problem * Nehari manifold * Hu–Washizu principle * Luke's variational principle * Mountain pass theorem * * Measures of central tendency as solutions to variational problems * Stampacchia Medal * Fermat Prize * Convenient vector space

# References

* Benesova, B. and Kruzik, M.
"Weak Lower Semicontinuity of Integral Functionals and Applications"
''SIAM Review'' 59(4) (2017), 703–766. * Bolza, O.
Lectures on the Calculus of Variations
Chelsea Publishing Company, 1904, available on Digital Mathematics library. 2nd edition republished in 1961, paperback in 2005, . * Cassel, Kevin W.
Variational Methods with Applications in Science and Engineering
Cambridge University Press, 2013. * Clegg, J.C.

Interscience Publishers Inc., 1968. * Courant, R.
Dirichlet's principle, conformal mapping and minimal surfaces
Interscience, 1950. * Dacorogna, Bernard:
Introduction

Introduction to the Calculus of Variations
', 3rd edition. 2014, World Scientific Publishing, . * Elsgolc, L.E.
Calculus of Variations
Pergamon Press Ltd., 1962. * Forsyth, A.R.
Calculus of Variations
Dover, 1960. * Fox, Charles

Dover Publ., 1987. * Giaquinta, Mariano; Hildebrandt, Stefan: Calculus of Variations I and II, Springer-Verlag, and * Jost, J. and X. Li-Jost
Calculus of Variations
Cambridge University Press, 1998. * Lebedev, L.P. and Cloud, M.J.
The Calculus of Variations and Functional Analysis with Optimal Control and Applications in Mechanics
World Scientific, 2003, pages 1–98. * Logan, J. David
Applied Mathematics
3rd edition. Wiley-Interscience, 2006 * * Roubicek, T.:
Calculus of variations
. Chap.17 in:
Mathematical Tools for Physicists
'. (Ed. M. Grinfeld) J. Wiley, Weinheim, 2014, , pp. 551–588. * Sagan, Hans

Dover, 1992. * Weinstock, Robert
Calculus of Variations with Applications to Physics and Engineering
Dover, 1974 (reprint of 1952 ed.).

Variational calculus
''
Encyclopedia of Mathematics The ''Encyclopedia of Mathematics'' (also ''EOM'' and formerly ''Encyclopaedia of Mathematics'') is a large reference work in mathematics. Overview The 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduat ...
''.
calculus of variations
'' PlanetMath''.
Calculus of Variations
''
MathWorld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Dig ...
''.
Calculus of variations
Example problems.
Mathematics - Calculus of Variations and Integral Equations
Lectures on
YouTube YouTube is a global online video sharing and social media platform headquartered in San Bruno, California. It was launched on February 14, 2005, by Steve Chen, Chad Hurley, and Jawed Karim. It is owned by Google, and is the second mo ...
. * Selected papers on Geodesic Fields
Part IPart II
{{Authority control Optimization in vector spaces


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Big Data Mining and Analytics 2022, 5 (3): 245-256
Published: 09 June 2022
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Negative emotion classification refers to the automatic classification of negative emotion of texts in social networks. Most existing methods are based on deep learning models, facing challenges such as complex structures and too many hyperparameters. To meet these challenges, in this paper, we propose a method for negative emotion classification utilizing a Robustly Optimized BERT Pretraining Approach (RoBERTa) and $p$-norm Broad Learning ( $p$-BL). Specifically, there are mainly three contributions in this paper. Firstly, we fine-tune the RoBERTa to adapt it to the task of negative emotion classification. Then, we employ the fine-tuned RoBERTa to extract features of original texts and generate sentence vectors. Secondly, we adopt $p$-BL to construct a classifier and then predict negative emotions of texts using the classifier. Compared with deep learning models, $p$-BL has advantages such as a simple structure that is only 3-layer and fewer parameters to be trained. Moreover, it can suppress the adverse effects of more outliers and noise in data by flexibly changing the value of $p$. Thirdly, we conduct extensive experiments on the public datasets, and the experimental results show that our proposed method outperforms the baseline methods on the tested datasets.

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# Reaction-diffusion systems treated as dynamical systems

I wonder if there is a good reference on reaction-diffusion systems on $$\mathbb{R}^N$$, that treats them as dynamical systems.

I have the books of Alain Haraux – Systèmes dynamiques dissipatifs et applications, and Joel Smoller – Shock waves and reaction–diffusion equations, but I look for something newer and more detailed.

I'm interested in the stability of a steady state of a homogenous reaction-diffusion system with two or three interacting populations in a general domain $$\Omega\subseteq\mathbb{R}^N$$ with $$N=2$$ or $$N=3$$ with Robin boundary conditions. They have the form:

$$\begin{cases} \dfrac{\partial y_1}{\partial t}(t,x)-d_1\Delta y_1=F_1(y_1,y_2) ,\ (t,x)\in (0,T)\times\Omega\\ \dfrac{\partial y_2}{\partial t}(t,x)-d_2\Delta y_2=F_2(y_1,y_2) ,\ (t,x)\in (0,T)\times\Omega \\ \dfrac{\partial y_1}{\partial \nu}(t,x)+b_1y_1(t,x)=\dfrac{\partial y_2}{\partial \nu}(t,x)+b_2y_2(t,x)=0,\ (t,x)\in (0,T)\times\partial\Omega \\ y_1(0,x)=y_{01}(x),\ y_2(0,x)=y_{02}(x)\ x\in\Omega\end{cases}$$

I heard from some authors in that field that the stationary problem associated with a reaction-diffusion system has exactly two positive solutions in general but they gave me only some intuitive proof of that thing. Maybe they utilize $$\omega$$-limits and some parabolic estimates, but I can't figure out how.

Here is the stationary system:

$$\begin{cases} -d_1\Delta Y_1=F_1(Y_1,Y_2) ,\ x\in\Omega\\ -d_2\Delta Y_2=F_2(Y_1,Y_2) ,\ x\in \Omega \\ \dfrac{\partial Y_1}{\partial \nu}(x)+b_1Y_1(x)=\dfrac{\partial Y_2}{\partial \nu}(x)+b_2Y_2(x)=0,\ x\in\partial\Omega\end{cases}$$

I can't figure this out even for a single population, that is why if:

$$\begin{cases} \dfrac{\partial y}{\partial t}(t,x)-d\Delta y=F(y) ,\ (t,x)\in (0,T)\times\Omega\\ \dfrac{\partial y}{\partial \nu}(t,x)+by(t,x)=0,\ (t,x)\in (0,T)\times\partial\Omega\\ y(0,x)=y_{0}(x),\ x\in\Omega\end{cases}$$

then the stationary problem:

$$\begin{cases} -d\Delta Y=F(Y) ,\ x\in\times\Omega\\ \dfrac{\partial Y}{\partial \nu}(x)+bY(x)=0,\ x\in \partial\Omega\end{cases}$$

has exactly two positive solutions: $$Y=0$$ and other strictly positive?

I wonder if there is a book treating this type of problems in details.

However, you will not find there general recipes for your second question and that's why. A stationary state, say $$y_{0}(x)$$, can be homogeneous (given by constant functions $$y_{0}(x) \equiv y_{0})$$ and non-homogeneous ($$y_{0}(x)$$ is not constant). For Robin conditions the only homogeneous state is $$0$$ (I assume $$b \not = 0$$ and $$F(0)=0$$). To find non-homogeneous states you have to solve a nonlinear PDE that is impossible to do analytically. So, the only way is to provide qualitative analysis of the equation that shows at least the existence (or non-existence) of non-homogeneous states (and their number if we get lucky). This highly relies on the structure of the nonlineraity $$F$$. For example, when $$F$$ is Lipschitz with a small Lipschitz constant, then the zero will be usually globally stable and there cannot be non-homogeneous states. Increasing the Lipschitz constant you may get the pitchfork bifurcation (the appearance of two non-homogeneous stationary states in a neighbourhood of zero) and so on. So, the result you mentioned (on the exsitence and number of non-homogeneous states) must use some very specifity of $$F$$.


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# nLab structure

Contents

This entry is about a general concepts of “mathematical structure” such as formalized by category theory and/or dependent type theory. This subsumes but is more general than the concept of structure in model theory.

# Contents

## Idea

It is common in informal language to speak of mathematical objects “equipped with extra structure” of some sort. The archetypical examples are algebras over a Lawvere theory in Set: these are sets equipped with the structure of certain algebraic operations. For instance a group $(G, e, {\cdot})$ is a set $G$ equipped with a binary operation ${\cdot} : G \times G \to G$, etc.

One may formalize the notion of structure using the language of category theory. This is discussed at stuff, structure, property. In that formalization objects in some category $D$ are objects in some category $C$ equipped with extra structure if there is a functor $p \colon D \to C$ such that

Depending on author and situation, more properties are required of this functor (Ehresmann 57, Ehresmann 65, Adamek-Rosicky-Vitale 09, remark 13.18):

• $p$ is an amnestic functor ($p$-vertical isomorphisms are identities),

• $p$ is an isofibration (isomorphisms can be lifted along $p$).

However, notice that these two conditions violate the principle of equivalence for categories. In the terminology of strict categories one might hence refer to these conditions as expressing “strict extra structure”.

## Notions of structure

A special class of examples of this is the notion of structure in model theory. In this case one defines a “language” $L$ that describes the constants, functions (say operations) and relations with which we want to equip sets, and then sets equipped with those operations and relations are called $L$-structures for that language. (Equivalently one might say “sets with $L$-structure”. Or one might generally say “$X$-structure” for “set with $X$-structure”.) In this case there is a faithful functor from $L$-structures to their underlying sets, and so this is a special case of the general definition.

We instead say model of a theory when we restrict to those structures which satisfy the axioms of a theory (in other words, satisfy properties specified by the axioms). In this case there is a full and faithful functor from the category of models of a theory $T$ to the category of structures of the underlying language $L(T)$, while the composition of forgetful functors

$Mod_T \to Struct_{L(T)} \to Set$

is again faithful.

###### Remarks

Thus, the English word “structure” is used in several slightly differing mathematical senses.

1. Within category theory itself, “structure” can function as a kind of mass noun, as in a phrase like “forgetting structure”. Here it refers to data comprising operations, relations, constants, and also properties borne by models of a theory or relative theory, considered abstractly (for example, the functor $Grp \to Set$ which forgets group structure, or the functor $Ring \to Ab$ which forgets multiplicative structure). On the other hand, it can also operate in the singular, where one says for example “a topological group is a topological space equipped with a group structure, such that…”

2. In model theory, however, the term structure is not a mass noun; it refers to a particular set (or “structures” for a family of sets) together with functions, relations, and elements that interpret the symbols of operations, predicates, and constants of a language. When one adds axioms to a language to make a theory, then a structure of the language where those axioms get interpreted as properties satisfied by the structure is called a model of the theory. Thus, in summary, the category theorist might refer to “the structure of a group” as consisting of a multiplication, a unit, etc., satisfying group axioms, while the model theorist would say that each particular group (like $\mathbb{Z}$) is a model of a theory of groups. For a model theorist, being a model does entail being a structure for the language of groups, but she would also say that a structure for the language of groups need not satisfy any of the axioms of a group (like associativity or unitality).

## Examples

There are gazillions of examples of objects equipped with extra structure. The most familiar is maybe

Generally the forgetful functor from a category of algebras over an algebraic theory down to the base category exhibits the equipment with the corresponding algebraic structure.

## Structures in dependent type theory

In dependent type theory the notion of “mathematical structure” and/or data structure on a base $B \,\colon\, Type$ is given by iterated dependent pairings (hence forming “type telescopes” also called “records” in Coq) with $B$-dependent types encoding operations on/with this type together with their behavioural specification.

The following shows some examples, using the notation for dependent pairs from here.

Via the extensionality principles for dependent pairs (here) and for dependent functions (here) such type theoretic structure automatically obey the structure identity principle:

In that any equivalence/identification between pairs of data of the following types are isomorphisms in the sense of bijective homomorphisms.

### Lensed data structure

To say that a given data (base) type $B \,\colon\, Set$ is

1. equipped with

1. a function $read_D \,\colon\, B \to D$ reading out $D$-data;

2. a function $write_D \,\colon\, D \times B \to B$ (over-)writing $D$-data;

2. such that this does behave as expected, namely as a well-behaved $D$-lens-structure on $B$

means to declare it to be of the following iterated dependent pair-telescope type:

### Group data structure

The dependent pair-telescope type declaration of group structure:

Further restriction to abelian groups:

Via the structure identity principle (above), the identitifications/equivalences between such group data types are indeed bijective group homomorphisms, hence group-isomorphisms:

The structure of subgroups of a given group structure:

and their underlying abstract group structure:

### Ring data structure

The dependent pair-telescope type declaration of unital ring structure:

### Module data structure

Given a unital ring type $R \colon Ring$ (as above), the type of $R$-module objects is the following dependent pair-telescope:

## Higher structures in homotopy type theory

The above examples of structures formulated in dependent type theory all had data sets as their base, which is the classical situation. But in dependent type theory with untruncated identification types, hence in homotopy type theory, we may simply drop this constraint and consider structures whose base is any higher homotopy type: This yields the notion of higher structures.

### Delooping

Given a set-based group structure (as above), delooping structure is

(…)

### Higher delooping

More generally, one may consider delooping of $n$-groups, but this is a lot of (higher) structure if spelled out in detail. Yet more generally and again more readily axiomatized there is the notion of deloopings of any pointed types (cf. Wärn (2023, §2), BCFR23), which we may write as follows:

Similarly there is iterated delooping structure:

## References

### General

On the history of the notion of “mathematical structure” with some focus on what Bourbaki did and did not contribute to the subject:

• Leo Corry, Mathematical Structures from Hilbert to Bourbaki: The Evolution of an Image of Mathematics, in: Changing Images of Mathematics in History. From the French Revolution to the new Millenium Harwood Academic Publishers (2001) 167-186 [ISBN:9780415868273, pdf]

• Leo Corry, Modern Algebra and the Rise of Mathematical Structures, Springer (2004) [doi:10.1007/978-3-0348-7917-0]

Early proposal to grasp the notion of mathematical structures via category theory (specifically via forgetful functors between groupoids):

### In dependent type theory

Discussion of mathematical structures via dependent type theory:

the general idea of representing mathematical structures as type telescopes:

and with emphasis of the notion of isomorphism between such structures:

in Coq:

in Lean:

Discussion of classes of mathematical structures in univalent homotopy type theory (univalent foundations of mathematics) which satisfy a structure identity principle:

Discussion in the context of the philosophy of structuralism:

Expressions towards the notion that, thereby the notions of mathematical structures are data structures may be understood to coincide (just along the lines of types being data types):

“An example of a use case for non-binary sum types is the definition of mathematical “data structures” like a ring or a group. These data structures are usually defined as…”

Last revised on February 4, 2023 at 13:10:46. See the history of this page for a list of all contributions to it.


text:
## Cryptology ePrint Archive: Report 2020/1385

An Alternative Approach for SIDH Arithmetic

Cyril Bouvier and Laurent Imbert

Abstract: In this paper, we present new algorithms for the field arithmetic of supersingular isogeny Diffie-Hellman; one of the fifteen remaining candidates in the NIST post-quantum standardization process. Our approach uses a polynomial representation of the field elements together with mechanisms to keep the coefficients within bounds during the arithmetic operations. We present timings and comparisons for SIKEp503 and suggest a novel 736-bit prime that offers a $1.17\times$ speedup compared to SIKEp751 for a similar level of security.

Category / Keywords: public-key cryptography / Supersingular isogeny Diffie-Hellman, Polynomial Modular Number System, Efficient arithmetic


text:
A lattice-theoretic question related to noncommutative tori

[NCG] So I'm trying to pin down a fairly well-known bit of noncommutative-geometric folklore that says that for $\Theta \in M_N(\mathbb{Q})$ skew-symmetric, the corresponding noncommutative $N$-torus $C(\mathbb{T}^N_\Theta)$ is isomorphic to $C(\mathbb{T}^N,\mathcal{M}_\Theta)$, where $\mathcal{M}_\Theta \to \mathbb{T}^N$ is a bundle of full matrix algebras. Due to the way $C(\mathbb{T}^N_\Theta)$ is constructed, such a result presumably arises from solving the following Diophantine(-ish?) problem: [/NCG]

So, let $\Theta \in M_N(\mathbb{Q})$ be skew-symmetric. For which $r \in \mathbb{Z}^N$ is $\Theta r \in \mathbb{Z}^N$?

The case when $N = 2$ is well known: we can write $$\Theta = \begin{pmatrix} 0 & -p/q \\ p/q & 0 \end{pmatrix}$$ where $q > 0$ and $p$ are coprime integers, whence $\Theta r \in \mathbb{Z}^2$ if and only if $r \in (q\mathbb{Z})^2$. The moment that $N > 2$, I'm afraid I don't even know the appropriate jargon to look up this type of (I guess) Diophantine problem myself.

Of course, if this does turn out to have been already answered here or at MathOverflow, I'd be immensely grateful just to know it is a duplicate, and of what.

EDIT: In the context I care about, $\Theta$ is invertible WLOG. Hence (thank you, Will Jagy!) what I'm really after is the lattice $$\Lambda_\Theta := \mathbb{Z}^N \cap \Theta^{-1}\mathbb{Z}^N.$$ By a basic result in lattice theory, since $(\mathbb{Z}^N)^\ast = \mathbb{Z}^N$ and $(\Theta^{-1}\mathbb{Z}^N)^\ast = \Theta \mathbb{Z}^N$ (since $\Theta$ is skew-symmetric and invertible), it follows that $$\Lambda_\Theta = ((\mathbb{Z}^N)^\ast \cup (\Theta^{-1}\mathbb{Z}^N)^\ast)^\ast = (\mathbb{Z}^N \cup \Theta \mathbb{Z}^N)^\ast;$$ since the columns of $I_N$ are a basis for $\mathbb{Z}^N$ and the columns of $\Theta$ are a basis for $\Theta \mathbb{Z}^N$, it apparently follows that a basis for $\Lambda_\Theta^\ast$ is given by the columns of the Hermite normal form of the augmented matrix $$[I_N \mid \Theta].$$ Now, are there any overall simplifications that happen with this Hermite normal form, given that $\Theta$ is skew-symmetric and invertible, or is it really just whatever it is?

• It's a lattice in any case. How does 3 by 3 work out? I know some stuff by G. L. Watson but I'm not sure it adapts to skew symmetric matrices. In case of interest, it's in Watson_Transformations_1.pdf at zakuski.math.utsa.edu/~kap/forms.html Sep 25, 2013 at 2:55
• Right. I'm this context, I can actually assume WLOG that $\Theta$ is invertible, so that I'm just computing the intersection of the lattices $\mathbb{Z}^N$ and $\Theta^{-1}\mathbb{Z}^N$, which is apparently a fairly standard procedure. Thank you! Sep 25, 2013 at 3:11


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16 Apr 2010, 11:06
Expert's post
shaselai wrote:
same here... maybe they randomly select people with different scores?

I sent an email to Josh@Knewton - waiting to hear back...
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16 Apr 2010, 11:27
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nickk wrote:
Thanks bb. I also wanted to incorporate their practice exam into my study program (GMAT is on Monday).

Good Luck!!!
Take a good rest over the weekend and prepare to fight for every single question on the exam; give it your best but don't get carried away. There is a time to move on. I spent 4 mins on one of my quant questions and never cracked it, so if you don't figure it out in 3 mins, probably not worth it. It could be an experimental question and then you wasted a whole bunch of time.
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16 Apr 2010, 11:33
bb wrote:
nickk wrote:
Thanks bb. I also wanted to incorporate their practice exam into my study program (GMAT is on Monday).

Good Luck!!!
Take a good rest over the weekend and prepare to fight for every single question on the exam; give it your best but don't get carried away. There is a time to move on. I spent 4 mins on one of my quant questions and never cracked it, so if you don't figure it out in 3 mins, probably not worth it. It could be an experimental question and then you wasted a whole bunch of time.

Thanks! I feel like I'm ready for the quant & verbal section, right now I'm going over the AWA (memorising a few templates).
I was planning to re-do GMAT prep 1 & 2 over the weekend (a full test each day). Do yo think this is a good idea? Should I completely forget about GMAT on the day before the exam?
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16 Apr 2010, 12:10
bakfed wrote:
I'm having a nervous breakdown right now (my test is tomorrow at 12:30pm).

I'm currently taking my "8-minute-break" between quant and verbal.

Anyway, I'd like to shed some light on the Knewton practice test.

I signed up through the first link that BB posted; then in a couple of days, Josh emailed me with a link. The link doesn't seem to be any special link, but simply a link that leads you to Knewton's GMAT website. From there, I signed up for an account and took my practice test. I believe that's all you need to do (though I don't know if the same will qualify if Josh did not send an email out to each individual user). I took the exam and that was it...until tomorrow - D-DAY, baby.

One question to piggy-back on this thread, does anyone seem to know the correlation between Knewton's free GMAT test and the real one? Do we have a large enough of a sample pool to know the correlation?

For me, the total score was spot on. The Verbal was lower by 6 percentage points from the real exam and the quant was higher by 5 percentage points. The total percentiles didn't match up even though the score was the same.
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16 Apr 2010, 13:12
bakfed wrote:
Thanks, nickk; I've always thought I could keep the usual calm as d-day approaches. I'll go watch Sound of Music or something

lagomez, thanks for your input. I'd love to have the same score as I had for Knewton's free practice test. I guess we shall see

Good luck. Keep us updated as to the discrepancy, if any.
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18 Apr 2010, 09:04
I too got the same message no links whatsoever
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21 Apr 2010, 12:27
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Sorry for any lag.

Here's the process.

2) I'll send you an email within 7 days of when you told me you're going to take the real GMAT.
3) Signup for the Knewton GMAT course's free trial (with the same email address you gave me when you signed up for GMAT club) and take a CAT.
4) Send me via pdf or via snail mail your official score report.
5) Every Thurs night we will verify all the score reports we got in that week, and make sure that effort was expended on our CATs.
6) We'll paypal money to your email address on Friday. If have a paypal account you can transfer to your bank account easily. If you don't have an account you can just create a free one and claim your money.

Any other questions?

Here's the snail mail address:

Knewton
c/o Josh Anish
19 Union Square West
12th floor
New York, NY 10003

Cheers,
Josh
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Free GMAT Club tests ($250 value) in addition to any other discounts or coupons when you buy the Knewton Course with KnewtonBest-GMAT-Club discount code. Use this promo code when you sign up for Knewton: KnewtonBest-GMAT-Club Current Student Status: What's your raashee? Joined: 12 Jun 2009 Posts: 1847 Location: United States (NC) Concentration: Strategy, Finance Schools: UNC (Kenan-Flagler) - Class of 2013 GMAT 1: 720 Q49 V39 WE: Programming (Computer Software) Followers: 22 Kudos [?]: 229 [0], given: 52 Re: Get Paid$100 to take a GMAT practice test conditions apply [#permalink]

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21 Apr 2010, 16:03
does this apply to someone(me) who has taken knewton already? i am certainly interested and had my score 2 saturdays ago..
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22 Apr 2010, 05:51
Lastly, former and current Knewton students are not eligible for the promotion.
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22 Apr 2010, 13:58
thanks for the clarification
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23 Apr 2010, 13:52
I took my real GMAT on the 12th. I sent my score after taking the Knewton GMAT a day later. I took a week to mail the score.
Some one from Josh's team sent me $100 via paypal. Thanks Josh for the quick turnaround. They processed my$100 in 2 days i think.

Thanks
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Re: Received $100 via paypal [#permalink] ### Show Tags 23 Apr 2010, 14:19 mbafall2011 wrote: I took my real GMAT on the 12th. I sent my score after taking the Knewton GMAT a day later. I took a week to mail the score. Some one from Josh's team sent me$100 via paypal. Thanks Josh for the quick turnaround. They processed my $100 in 2 days i think. Thanks What was the correlation between your Knewton score and the official score? Re: Received$100 via paypal   [#permalink] 23 Apr 2010, 14:19

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## Limit hulls?

In the course of trying to explain $\liminf$ and $\limsup$ “visually” I was led to this variation on the notion of convex hull. Given a set $A$ in the plane, the convex hull is constructed by considering all the half-planes $H$ that contain $A$ and taking their intersection. The requirement $A\subset H$ can be relaxed for instance by saying that $A\setminus H$ has measure zero. But what happens if we instead ask that $A\setminus H$ be bounded? Suppose for instance that $A$ is the first quadrant. Then any half-plane of the form $H_t=\{v: v\cdot (1,1)>t\}$, for $t>0$, is a candidate and we see that their intersection is empty. But it isn’t empty if we pass to the projective plane. In that case we would get an arc of the line at infinity, which could maybe be called the “limit hull” of $A$. It seems the case, but I haven’t sat down to check, that whenever $A$ is unbounded, its limit hull is non-empty.

The connection with $\liminf$ was as follow. Let $y=f(x)$ be the graph of your favorite function. First define the infimum of $f$ “visually” by saying that “we draw a horizontal line under the graph and then raise it as far as we can until it bumps into the graph of $f$. The $\liminf_{x\rightarrow+\infty} f(x)$ can be defined analogously by drawing a horizontal line under the graph and continue raising it as long as the set of $x$‘s where $f$ is under the line forms a bounded set.

## Lectures about 5, 8 and 24

These three videos of John Baez talking about his favorite numbers are a lot of fun to watch.

## Benford’s law and Banach limits

In this nice short paper (gated), R. Raimi makes a connection between Benford’s law and Banach limits.

## An inequality useful for the p-Laplacian

Fix $p>1$. Let $u,v\in{\mathbb R}^n$, then

$\displaystyle (u-v)\cdot(|u|^{p-2}u-|v|^{p-2}v)\geq 0.$

Proof: Unfold and rewrite what we want to show as:

$\displaystyle (|u|^{p-2}+|v|^{p-2})(u\cdot v)\leq |u|^{p}+|v|^{p}.$

By Cauchy-Schwarz the left hand-side is less than

$\displaystyle |u|^{p-1}|v|+|v|^{p-1}|u|,$

which can be written as

$\displaystyle |u|^{p}+|v|^{p}+(|u|-|v|)(|v|^{p-1}-|u|^{p-1}).$

But, since $p>1$,

$\displaystyle (|u|-|v|)(|v|^{p-1}-|u|^{p-1})\leq 0.$

Posted in Analysis | 2 Comments

Walking out of my Finite Math class I wondered if one can determine all possible triples of non-negative integers $(a,b,c)$ such that

$\displaystyle a!\ b!=c!$

There are some trivial solutions. Whenever $c=p!$ for some other non-negative integer $p$, then $(c-1,p,c)$ is a solution.

Are there any other? (I haven’t put much thought to this question).

## A little-known definition of Hardy spaces

In the paper by Sedleckiĭ, A. M.
An equivalent definition of the $H\sp{p}$ spaces in the half-plane, and some applications. (Russian)
Mat. Sb. (N.S.) 96(138) (1975), 75–82, 167, the following is proved: given an analytic function $f$ on the upper-half plane ${\mathbb H}=\{z: {\mathcal Im} z>0\}$ consider the following two quantities for $0:

$\displaystyle \|f\|_{H_p}^p=\sup_{y>0}\int_{-\infty}^{+\infty}|f(x+iy)|^pdx$

and

$\displaystyle \|f\|_{H_p^\star}^p=\sup_{0<\theta<\pi}\int_0^\infty |f(re^{i\theta})|^pdr.$

Then

$\displaystyle A_p\|f\|_{H_p}\leq \|f\|_{H_p^\star}\leq B_p\|f\|_{H_p}$

for some constants $0.

## Global harmonic miracle

That harmonic functions would satisfy the mean value property for infinitesimally small circles is intuitively somewhat clear. But that this property should persist on circles of arbitrary radius is quite miraculous. To recall, if $h$ is harmonic on a domain $\Omega\subset{\mathbb C}$ and one considers a disk $D(z_0,r)\subset\Omega$, then

$\displaystyle h(z_0)=\frac{1}{\pi r^2}\int_{D(z_0,r)}h(w)dA(w)$.

One might wonder whether such a global property occurs for other shapes, say ellipses. The answer is no.

Fact: Suppose $U\subset {\mathbb C}$ is open and has finite area $A$, $z_0\in U$, and

$\displaystyle h(z_0)=\frac{1}{A}\int_U h(w)dA(w)$

for every integrable harmonic function $h$ on $U$. Then $U$ is a disk centered at $z_0$.

To see why, pick a point $z_1\not\in U$ that is closest to $z_0$. Then play with the function

$\displaystyle h(z)=2{\mathcal Re}\left(\frac{z-z_0}{z-z_1}\right).$


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ACT Math : How to find compound interest

Example Questions

Example Question #1 : How To Find Compound Interest

Ashley makes a bank deposit of  at  annual interest, compounded monthly. About how many years will it take her deposit to grow to ?

years

years

year

years

years

years

Explanation:

The formula for compound interest is

where P is the principal (original) amount, r is the interest rate (in decimal form), n is the number of times per year the interest compounds, t is the time in years, and A is the final amount.

In this problem, we are solving for time, t. The given variables from the problem are:

Plugging these into the equation above, we get

This simplifies to

We can solve this by taking the natural log of both sides

Example Question #2 : How To Find Compound Interest

Alice wants to invest money such that in  years she has . The interest rate is  compounded quarterly. How much should she invest?

Explanation:

The formula for compound interest is

where P is the principal (original) amount, r is the interest rate (in decimal form), n is the number of times per year the interest compounds, t is the time in years, and A is the final amount.

In this problem, we are solving for the principal, P. The given variables from the problem are:

Plugging these into the equation above, we get

Solving for P, we get

Example Question #1 : How To Find Compound Interest

A five-year bond is opened with  in it and an interest rate of %, compounded annually. This account is allowed to compound for five years. Which of the following most closely approximates the total amount in the account after that period of time?

Explanation:

Each year, you can calculate your interest by multiplying the principle () by . For one year, this would be:

For two years, it would be:

, which is the same as

Therefore, you can solve for a five year period by doing:

Using your calculator, you can expand the  into a series of multiplications. This gives you , which is closest to

Example Question #1 : How To Find Compound Interest

If a cash deposit account is opened with  for a three year period at % interest compounded once annually, which of the following is closest to the positive difference between the interest accrued in the third year and the interest accrued in the second year?

Explanation:

It is easiest to break this down into steps. For each year, you will multiply by  to calculate the new value. Therefore, let's make a chart:

After year 1: ; Total interest:

After year 2: ; Let us round this to ; Total interest:

After year 3: ; Let us round this to ; Total interest:

Thus, the positive difference of the interest from the last period and the interest from the first period is:

Example Question #3 : How To Find Compound Interest

If an account has interest compounded annually at a rate of , what is the balance of the account after  years of compounding if the initial balance is ? Round to the nearest cent.

Explanation:

Recall that the equation for compounded interest (with annual compounding) is:

Where  is the balance,  is the rate of interest, and  is the number of years.

Thus, for our data, we need to know:

This is approximately .

Example Question #4 : How To Find Compound Interest

If an account has interest compounded annually at a rate of , what is the balance of the account after  years of compounding if the initial balance is ? Round to the nearest cent.

Explanation:

Recall that the equation for compounded interest (with annual compounding) is:

Where  is the balance,  is the rate of interest, and  is the number of years.

Thus, for our data, we need to know:

This is approximately .

Example Question #5 : How To Find Compound Interest

If an account has interest compounded quarterly at an annual rate of , what is the balance of the account after  years of compounding if the initial balance is ? Round to the nearest cent.

Explanation:

Recall that the equation for compounded interest (with quarterly compounding) is:

Where  is the balance,  is the rate of interest,  is the number of years, and  is the number of times it is compounded per year.

Thus, for our data, we need to know:

This is approximately .

Example Question #6 : How To Find Compound Interest

An account is compounded at a given rate of interest annually for  years.  What is this rate if the beginning balance for the account was  and its ending balance ? Round to the nearest hundredth of a percent.

Explanation:

Recall that the equation for compounded interest (with annual compounding) is:

Where  is the balance,  is the rate of interest, and  is the number of years.

Thus, for our data, we need to know:

Now, let's use  for . This gives us:

Using a logarithm, this can be rewritten:

This can be rewritten:

Now, you can solve for :

or

Now, finally you can rewrite this as:

Thus,

Now, round this to  and recall that

Thus,  and  or

Example Question #1 : How To Find Compound Interest

Jack has , to invest. If he invests two-thirds of it into a high-yield savings account with an annual interest rate of , compounded quarterly, and the other third in a regular savings account at  simple interest, how much does Jack earn after one year?

Explanation:

First, break the problem into two segments: the amount Jack invests in the high-yield savings, and the amount Jack invests in the simple interest account (10,000 and 5,000 respectively).

Now let's work with the high-yield savings account. $10,000 is invested at an annual rate of 8%, compounded quarterly. We can use the compound interest formula to solve: Plug in the values given: Therefore, Jack makes$824.32 off his high-yield savings account. Now let's calculate the other interest:

Add the two together, and we see that Jack makes a total of,  off of his investments.

Example Question #1 : How To Find Compound Interest

Michelle makes a bank deposit of $1,500 at 4.2% annual interest, compounded monthly. Approximately how much money will be in Michelle’s bank account in 3 years? Possible Answers:$1,697

$1,701$1,695

$1,693 Correct answer:$1,701

Explanation:

The formula to use for compounded interest is

where P is the principal (original) amount, r is the interest rate (expressed in decimal form), n is the number of times per year the interest compounds, and t is the total number of years the money is left in the bank. In this problem, P=1,500, r=0.042, n=12, and t=3.

By plugging in, we find that Michelle will have about \$1,701 at the end of three years.


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# If the earth did not have atmosphere would its average temperature be higher or lower than it is now ?

$\begin{array}{1 1} \text{with out atmosphere , the temperature would be much higher} \\ \text{with out atmosphere , there would be no green hours effect } \\ \text{with out atmosphere , the temperature would not charge} \\ \text{with out atmosphere , the earth will have hotter day colder nights} \end{array}$


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What are the finite subgroups of $SU_2(C)$?

Is there any reference which classifies the finite subgroups of $SU_2(C)$ up to conjugacy ?

What I know is that lifting the finite subgroups of $SO_3(R)$ by the map $SU_2(C) \rightarrow PSU_2(C)$ gives rise to the groups : cyclic groups of even order, dicyclic groups, binary tetrahedral/octahedral/icosahedral groups. But I dont know if they are the only one.

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I claim that if you have all cyclic groups of even order, then you also have all cyclic groups of odd order! –  Pete L. Clark May 20 '11 at 19:43

The image of a finite subgroup of $\text{SU}(2)$ in $\text{SO}(3)$ is a finite subgroup of $\text{SO}(3)$; moreover, the kernel is either trivial or $\{ \pm 1 \}$. But $-1$ is the unique element of order $2$ in $\text{SU}(2)$, so any group of even order contains it.

I claim all the finite subgroups of odd order are cyclic. This follows because the inclusion $G \to \text{SU}(2)$ cannot define an irreducible representation of $G$ (since otherwise $2 | |G|$), hence it must break up into a direct sum of dual $1$-dimensional representations.

So once you know the finite subgroups of $\text{SO}(3)$, you already know the finite subgroups of $\text{SU}(2)$.

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The diagonal matrix with entries 1, -1 is also unitary and of order 2, so your uniqueness claim is incorrect. –  i. m. soloveichik Aug 3 '12 at 0:27
Oops-sorry--ignore my comment above –  i. m. soloveichik Aug 3 '12 at 0:39

Yes, the ADE classification enumerates all possible finite subgroups of $SU(2)$. Just a trivial correction, when it comes to the Abelian cyclic groups, A, the order may be both even and odd because $U(1)$ inside $SU(2)$ has all these $Z_N$ subgroups. The odd ones aren't linked to subgroups of $SO(3)$ in the same way.

The dicyclic groups, D, and the three exceptions, E, are as you wrote.

-


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User vkrouglov - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T18:37:12Z http://mathoverflow.net/feeds/user/10847 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/125603/solution-to-the-fractional-differential-equation Solution to the fractional differential equation vkrouglov 2013-03-26T07:54:58Z 2013-03-29T21:56:47Z
What is the solution of the fractional differential equation $$f^{(\alpha-1)}(t) = tf(t)$$

where $(\alpha)$ denotes the fractional derivative of order $\alpha$

EDIT: Background behind this question.

I am interested in this equation in relation with the alpha-stable version of the Stein's lemma. Recall, that if $X \in N(0,1)$ then $$E(X g(X)) = E(g'(X))$$ for every function $g$ for which the expectations in the left and right parts exist.

The simplest way to prove this is to use the property of normal density $f'(x) = -x f(x)$ and integration by parts.

Let $\phi$ be a characteristic function of the standard symmetric stable density $S(\alpha, 1, 0)$. If I apply the usual derivative to $\phi$ and take a Fourier transform I will get the equation above (modulo signs and coefficients). A closed form solution would give an analytic presentation of the stable density (highly unlikely).

2nd question: Is it possible to obtain a differential equation for the stable density to be used in the proof of the Stein's lemma?

Thanks, Vladimir
 http://mathoverflow.net/questions/119425/one-sample-test-for-the-alpha-stable-distribution One-sample test for the alpha-stable distribution vkrouglov 2013-01-20T21:03:49Z 2013-01-20T21:03:49Z
Hi,

This is a repost of the question from http://math.stackexchange.com/questions/281920/one-sample-prametric-test-for-stable-distributions/281929#281929.

For my needs the K-S test appears to unapplicable due to a small number of observations. Maybe there are some alpha-stable specific statistics that can have better confidence interval

Thanks,
 http://mathoverflow.net/questions/119002/what-is-the-maximal-sparsity-of-a-matrix What is the maximal sparsity of a matrix? vkrouglov 2013-01-15T17:08:49Z 2013-01-15T18:08:22Z
Given a $m \times 2m$ matrix G with rank(G)=m, what is the maximal number of zeros in the product $WG$ where $W$ is an $m \times m$ nondegenerate matrix?

An obvious lower bound is $m(m-1)$ and $W$ is given by a matrix that transforms the first $m$ columns of $G$ to a standard basis in $\mathbb{R}^m$. When $m=1$ this lower bound is in fact attained. Is it possible to have more zeros when m>>1?
 http://mathoverflow.net/questions/85578/when-the-adjoint-of-a-hypoelliptic-operator-hypoelliptic When the adjoint of a hypoelliptic operator hypoelliptic vkrouglov 2012-01-13T14:00:39Z 2012-03-05T14:49:19Z
Assume, $M$ is a smooth manifold with a measure $\mu$ and let $L^2(M, \mu)$ be a space of all square-integrable functions on $M$.

Recall that $L$ is a hypoelliptic differential operator, if for every $f \in \mathcal{D}(L)$, if $Lf$ is in $C^\infty(M)$ then $f$ is also in $C^\infty(M)$.

Could anyone give a reference to the example when $L$ is hypoelliptic but its adjoint w.r.t to $\mu$ is not hypoelliptic? Could this happen to the Hormander operators, when $L$ is defined as $$L = \sum_i X_i^2 + X_0$$ and ${X_i}$'s are bracket generating?

I am mostly interested in the case when $\mu$ is induced by the Riemannian metric and $M$ is complete.

Thanks,
 http://mathoverflow.net/questions/86484/reference-request-hamiltonian-and-quantum-completeness Reference Request: Hamiltonian and quantum completeness. vkrouglov 2012-01-23T20:48:08Z 2012-02-02T16:36:40Z
Let $L$ be a differential operator in $L^2(M, dvol)$ wrt to a Riemannian volume form (say). Let us call it quantum complete if it is essentially-self-adjoint. Consider $H$ - the symbol of $L$. It is a function on a cotangent bundle of $M$, i.e. a Hamiltonian. This Hamiltonian defines a Hamiltonian flow on $T^\ast M$.

I am looking for a good reference which discusses the relation between the quantum completeness of $L$ and the completeness of the Hamiltonian flow of $H$. Some information is contained in Reed and Simon but they seem to consider only the case of $M=(0, \infty)$.

Particular questions I am interested in:

1) Sometimes the Hamiltonian flow blows up in finite time but the corresponding $L$ is complete. What is the physical meaning of this?

2) Is it possible to use the completeness of the Hamiltonian flow to prove that $L$ is ess. self.adjoint? For example, is it possible to apply this logic to prove that the Laplace-Beltrami operator is ess. self-adjoint on a geodesically complete manifold (which is a Hamiltonian flow of its symbol)?

Thanks,
 http://mathoverflow.net/questions/85726/invariant-complement-to-invariant-subspace Invariant complement to invariant subspace. vkrouglov 2012-01-15T10:14:00Z 2012-01-16T03:13:14Z
Let $G$ be a compact group and $\rho: G \to End(U)$ its linear representation in a finite dimensional vector space $U$. Fix $V \subset U$ - a subspace invariant under $\rho(G)$. Then it is well known that $V$ has some invairant complement.

What are the sufficient conditions (on $G$, $\rho$ or $V$) to ensure this complement is unique?

Stated another way, starting with any scalar product on $U$, invariant complements to $V$ can be found by averaging it w.r.t. to a Haar measure on $G$ and taking $V^\perp$. In this case my question becomes

What are the sufficient conditions that for any choice of the initial scalar product, the resulting $V^\perp$ is unique.

There are examples when the complement is unique. For example, consider $\mathbb{R}^3$ and the action of $SO(2)$ given by the rotation around the $z$-axis. Then if $V$ is a $z$-axis it has a unique complement.

Particular example I am trying to understand is the following: consider the group $U(n)$ acting on $\mathbb{R}^{2n}$ in a standard way. This action induces a representation $\rho$ in $U=Hom(\mathbb{R}^{2n} \wedge \mathbb{R}^{2n}, \mathbb{R}^{2n})$ given by $$(g S) (u\wedge v) = g^{-1}S (gu \wedge gv)$$ Consider a map $A: Hom(\mathbb{R}^{2n}, \mathfrak{u}(n)) \to Hom(\mathbb{R}^{2n} \wedge \mathbb{R}^{2n}, \mathbb{R}^{2n})$ defined by $$(AS)(u \wedge v) = S(u)v - S(v)u$$ and define $V = A(Hom(\mathbb{R}^{2n}, \mathfrak{u}(n)))$.

For what $n$, there is a unique invariant complement to $V$?

Such construction appears in the Cartan's method of equivalence. Each $U(n)$-invariant complement to $V$ corresponds to a certain $U(n)$-invariant linear connection.

Thanks,
 http://mathoverflow.net/questions/85726/invariant-complement-to-invariant-subspace/85784#85784 Comment by vkrouglov vkrouglov 2012-01-16T08:19:14Z 2012-01-16T08:19:14Z Thanks Gil, your help is really appreciated.


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# python – ValueError: invalid literal for int() with base 10: ”

## The Question :

388 people think this question is useful

I am creating a program that reads a file and if the first line of the file is not blank, it reads the next four lines. Calculations are performed on those lines and then the next line is read. If that line is not empty it continues. However, I am getting this error:

ValueError: invalid literal for int() with base 10: ''.

It is reading the first line but can’t convert it to an integer.

What can I do to fix this problem?

The code:

file_to_read = raw_input("Enter file name of tests (empty string to end program):")
try:
file_to_write = raw_input("Enter output file name (.csv will be appended to it):")
file_to_write = file_to_write + ".csv"
outfile = open(file_to_write, "w")
global count

• You should consider using with open(file_to_read, 'r') as infile: there.
• For anyone currently looking here. The error may be that one of the lines isn’t in integer form. Eg: “yes” isn’t in the correct form but “3” is. For this question the first line may not have any “1”s, “2”s, “3”s… to convert to an int.
• i got this error when input string had space between digits. this error basically means your input string is not valid for string to integer conversion. for conversion, your string should only and only contain following characters: +-.0123456789

392 people think this answer is useful

Just for the record:

>>> int('55063.000000')
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
ValueError: invalid literal for int() with base 10: '55063.000000'

Got me here…

>>> int(float('55063.000000'))
55063.0

Has to be used!

93 people think this answer is useful

The following are totally acceptable in python:

• passing a string representation of an integer into int
• passing a string representation of a float into float
• passing a string representation of an integer into float
• passing a float into int
• passing an integer into float

But you get a ValueError if you pass a string representation of a float into int, or a string representation of anything but an integer (including empty string). If you do want to pass a string representation of a float to an int, as @katyhuff points out above, you can convert to a float first, then to an integer:

>>> int('5')
5
>>> float('5.0')
5.0
>>> float('5')
5.0
>>> int(5.0)
5
>>> float(5)
5.0
>>> int('5.0')
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
ValueError: invalid literal for int() with base 10: '5.0'
>>> int(float('5.0'))
5

61 people think this answer is useful

Pythonic way of iterating over a file and converting to int:

for line in open(fname):
if line.strip():           # line contains eol character(s)
n = int(line)          # assuming single integer on each line

What you’re trying to do is slightly more complicated, but still not straight-forward:

h = open(fname)
for line in h:
if line.strip():
[int(next(h).strip()) for _ in range(4)]     # list of integers

This way it processes 5 lines at the time. Use h.next() instead of next(h) prior to Python 2.6.

The reason you had ValueError is because int cannot convert an empty string to the integer. In this case you’d need to either check the content of the string before conversion, or except an error:

try:
int('')
except ValueError:
pass      # or whatever

25 people think this answer is useful

I found a work around. Python will convert the number to a float. Simply calling float first then converting that to an int will work: output = int(float(input))

15 people think this answer is useful

The reason is that you are getting an empty string or a string as an argument into int. Check if it is empty or it contains alpha characters. If it contains characters, then simply ignore that part.

11 people think this answer is useful

The reason you are getting this error is that you are trying to convert a space character to an integer, which is totally impossible and restricted.And that’s why you are getting this error.

Check your code and correct it, it will work fine

8 people think this answer is useful

So if you have

floatInString = '5.0'

You can convert it to int with floatInInt = int(float(floatInString))

3 people think this answer is useful

You’ve got a problem with this line:

while file_to_read != " ":

This does not find an empty string. It finds a string consisting of one space. Presumably this is not what you are looking for.

Listen to everyone else’s advice. This is not very idiomatic python code, and would be much clearer if you iterate over the file directly, but I think this problem is worth noting as well.

3 people think this answer is useful

Please test this function (split()) on a simple file. I was facing the same issue and found that it was because split() was not written properly (exception handling).

1 people think this answer is useful
    readings = (infile.readline())
global count

There’s a problem with that code. readings is a new line read from the file – it’s a string. Therefore you should not compare it to 0. Further, you can’t just convert it to an integer unless you’re sure it’s indeed one. For example, empty lines will produce errors here (as you’ve surely found out).

1 people think this answer is useful

This could also happen when you have to map space separated integers to a list but you enter the integers line by line using the .input(). Like for example I was solving this problem on HackerRank Bon-Appetit, and the got the following error while compiling

So instead of giving input to the program line by line try to map the space separated integers into a list using the map() method.

1 people think this answer is useful

I recently came across a case where none of these answers worked. I encountered CSV data where there were null bytes mixed in with the data, and those null bytes did not get stripped. So, my numeric string, after stripping, consisted of bytes like this:

\x00\x31\x00\x0d\x00

To counter this, I did:

countStr = fields[3].replace('\x00', '').strip()
count = int(countStr)

…where fields is a list of csv values resulting from splitting the line.

0 people think this answer is useful

I am creating a program that reads a file and if the first line of the file is not blank, it reads the next four lines. Calculations are performed on those lines and then the next line is read.

Something like this should work:

for line in infile:
next_lines = []
if line.strip():
for i in xrange(4):
try:
next_lines.append(infile.next())
except StopIteration:
break
# Do your calculation with "4 lines" here

0 people think this answer is useful

I was getting similar errors, turns out that the dataset had blank values which python could not convert to integer.

0 people think this answer is useful

I got into the same issue when trying to use readlines() inside for loop for same file object… My suspicion is firing readling() inside readline() for same file object caused this error.

Best solution can be use seek(0) to reset file pointer or Handle condition with setting some flag then create new object for same file by checking set condition….

0 people think this answer is useful

This seems like readings is sometimes an empty string and obviously an error crops up. You can add an extra check to your while loop before the int(readings) command like:

while readings != 0 | readings != '':
global count

0 people think this answer is useful

I had hard time figuring out the actual reason, it happens when we dont read properly from file. you need to open file and read with readlines() method as below:

with open('/content/drive/pre-processed-users1.1.tsv') as f:

It corrects the formatted output

readings = int(readings)


text:
# Coleman-Mandula theorem in mathematical language

+ 5 like - 0 dislike
278 views

Every supersymmetry text starts off mentioning the Coleman-Mandula theorem. Often it is introduced using rather colloquial terminology. I was wondering if anyone knew a precise mathematical formulation of the theorem.

This post imported from StackExchange Physics at 2014-08-11 14:58 (UCT), posted by SE-user SWV
For everyone interested, here's a link to the original paper: prola.aps.org/abstract/PR/v159/i5/p1251_1

This post imported from StackExchange Physics at 2014-08-11 14:58 (UCT), posted by SE-user Vibert
Weinberg Volume 3 has a derivation of the theorem in the first (or maybe second) chapter.

This post imported from StackExchange Physics at 2014-08-11 14:58 (UCT), posted by SE-user Prahar

This post imported from StackExchange Physics at 2014-08-11 14:58 (UCT), posted by SE-user Sanath Devalapurkar
I find that this paper is particularly accessible and constitutes a nice derivation of the individual elements of the proof. ippp.dur.ac.uk/~busbridge/files/documents/CMtheorem.pdf

This post imported from StackExchange Physics at 2014-08-11 14:58 (UCT), posted by SE-user Autolatry

+ 4 like - 0 dislike

See http://arxiv.org/pdf/hep-th/9605147v1.pdf. Most of the following is verbatim from there.

Theorem: With the following assumptions: $G$ is a group of symmetries of the $S$-matrix $\mathscr{S}$$G$ contains a subgroup $\mathcal{P}^\prime_0$ that is locally isomorphic to $\mathcal{P}(r,1)$ for $r\geq3$; all particle types correspond to positive-energy time-like representations of the universal covering group of $\mathcal{P}^\prime_0$; the number of particle types is finite; at least locally, $G$ is generated by generators represented in the one-particle space $\mathcal{H}^{(1)}$ by (generalized) integral operators in momentum space, with distributions as kernels; the amplitudes for elastic scattering of two particles do not vanish identically; and the scattering amplitudes are regular functions of the momenta, and the amplitudes for scattering between two-particle states are analytic functions in some neighborhood of the physical region, we can state that $G$ is isomorphic to the direct product of $\mathcal{P}^\prime_0$ and the direct product of some internal symmetry group.

See the linked paper for the proof of the statement, if needed.

answered Sep 12, 2014 by (285 points)

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text:
1. Improper integrals:

$\int_{\frac{1}{2}}^{2} \frac{dz}{z(ln{z})^{\frac{1}{5}}}$

so:

$\lim{t\to{1^{-}}} \int_{\frac{1}{2}}^{t} {\frac{dz}{z(ln{z})^{\frac{1}{5}}} + \lim{t\to{1^{+}}}\int_{t}^{2}\frac{dz}{z(ln{z})^{\ frac{1}{5}}}$

$\lim{t\to 1^{-}} \frac{5}{4} (ln{z})^{\frac{4}{5}}\big|_{\frac{1}{2}}^{t} + \lim{t\to 1^{+}} \frac{5}{4} (ln{z})^{\frac{4}{5}}$

when i evaluate this.... the answer doesnt appear to be the sam as the back of book:
Answer at back of book : 0]

Edit:
kindly delete this thread for double posting its because of lagging

2. When I evaluated the integral in Mathematica 6 I got an imaginary solution so if the book was looking for real solutions you might have mistaken a null set (zero with a line through it) for a zero. Below is the answer mathematica gave me.. Hope this helps

1.6866 -0.548011 ä

3. Originally Posted by jandrewross
When I evaluated the integral in Mathematica 6 I got an imaginary solution so if the book was looking for real solutions you might have mistaken a null set (zero with a line through it) for a zero. Below is the answer mathematica gave me.. Hope this helps

1.6866 -0.548011 ä