Ideas from calculus Flashcards
This quantity’s transfer rate divided by the product of area and the driving force concentration difference equals this quantity’s “transfer coefficient,” denoted k sub c. Choked flow imposes a limit on this quantity’s time derivative. The equation (read slowly) “d rho d t equals negative rho times the divergence of u” describes the conservation of this quantity in the Euler equations of fluid dynamics. The logarithm of the ratio of this quantity’s
mass
An analog of a solution to an equation named for this scientist is illustrated by the Kac ring model. A limit named for Harold Grad and this scientist is used to derive an equation named for this scientist for hard sphere potentials. That equation named for this scientist is derived by assuming that the velocities of colliding particles are uncorrelated and independent of position. This scientist’s use of the molecular chaos hypothesis led to criticism that irreversible dynamics should not be derivable from time-symmetric dynamics. Loschmidt’s paradox was formulated after this scientist published his H-theorem. The logarithm of the number of microstates is proportional to entropy according to an equation engraved on this scientist’s gravestone. For 10 points, name this Austrian physicist whose namesake constant is denoted k-sub-B.
Ludwig Boltzmann
This value is often multiplied by a “chicken factor” in a method developed by Erwin Fehlberg. A “velocity” parameter multiplies a ratio of two examples of these values to define the Courant number. Time goes to infinity while this value is fixed when evaluating systems for A-stability. A limit as this value approaches zero can be computed by Richardson extrapolation. Equations whose solutions are unstable, except when this value is extremely small, are called
step size
Ives and Garland modified a technique based on this function to test for phylogenetic signal. Max Welling and Yee Whye applied an algorithm based on Langevin dynamics to a model named for this function. This function is the basis of the cross-entropy loss function. In a method named for this function, the Firth correction uses Jeffreys invariant prior to penalize the likelihood. The softmax function generalizes this function to multinomial classification. The probit model is an alternative to this function’s regression method for binary dependent variables. Richards’s generalized version of this function includes in the Gompertz curve. The Hill equation is an example of this function used to model dose-response curves. The Fermi–Dirac distribution is described by this function, whose inverse is the log-odds. For 10 points, exponential growth with a limit is modeled by what S-shaped curve?
logistic function
This scientist is the alphabetically prior namesake of a function equal to the axisymmetric eigenfunctions of the curl operator for a force-free magnetic field. For homogenous, plane-parallel atmospheres with finite thicknesses, this scientist names two X and Y functions that approximate radiative transfer. With Mario Schonberg, this scientist names a limit below which main-sequence
Subrahmanyan Chandrasekhar
This quantity is negligible in a limit defined by a Henry function equal to 1.5 in the Smoluchowski (“smol-u-KOFF-skee”) approximation. For a short-range Yukawa potential whose magnitude is proportional to [read slowly] “q, divided by four-pi times permeability times r,” this quantity appears in the [emphasize] denominator of the exponent of such a potential. Mean-field theories based on the Poisson-Nernst-Planck equation are valid when this quantity is much larger than a similar quantity named for Bjerrum (“BYUR-um”). In D·L·V·O theory, the inverse of this quantity is designated
Debye length
Elements of Skorokhod spaces have this property “on the right” and are càdlàg (“cod-log”). The Picard–Lindelöf theorem applies when the derivative of a solution to an O·D·E has this property in t and has a form of this property in y that is defined in terms of the Lipschitz constant. The Cantor staircase function has this property, but not its “uniform” type. The
continuity
This design names a method in which the violation of the assumption that epsilon equals one results in an inflation in the degrees of freedom that is addressed by the Greenhouse–Geisser correction. That method named for this design fails when Mauchly’s W is large. The assumption of equality among the variances of the pairwise differences between levels of each within-subject factor is called sphericity and appears in a form of ANOVA named for this design that has greater power than
repeated measures
The product of the deformation gradient tensor and its transpose is the “left” deformation tensor named for this physicist and George Green . A statement partially named for this physicist is used to derive the Schrodinger uncertainty relation. Absent external forces, the material derivative of the flow velocity equals the divergence of an object named for this physicist divided by the density, according to an equation named for this physicist. Coefficients describing refraction and dispersion are included in an empirical expression for the
Augustin-Louis Cauchy
A “principal value” named for this scientist arises often when the Lebesgue (“Leh-bayg”) integral does not exist and is defined for an isolated singularity as the limit of the sum of two integrals. This mathematician names a test for convergence where the absolute value of the sum of the coefficients is bounded by some epsilon. A metric space is said to be
Augustin-Louis Cauchy
For a subset A of a Lorentzian manifold, this mathematician names the set of points such that every past inextensible curve through that point meets A. Boundary conditions named for this mathematician specify both a function value and the normal derivative. Any sequence whose terms become arbitrarily close together is named for this mathematician. This mathematician is the alphabetically first namesake of a set of equations that form a necessary condition for complex differentiability, which they name with Riemann. For 10 points, name this mathematician, the alphabetically first to name an inequality that compares inner products to norms along with Schwarz.
Augustin-Louis Cauchy
A form of this technique that works on objects for which the well-founded relation holds is named for Emmy Noether. Augustin-Louis Cauchy used both the standard “forward” form of this technique and its alternative “backwards” form to prove the AM-GM inequality. This technique, which relies on the last of the Peano axioms to hold, is often used to show that “n times quantity n plus 1 all divided by 2” is a formula for the
proof by induction
An equation named for this quantity sets the differential of the flow velocity field equal to the curl of the stress tensor, divided by density. That equation is named for Augustin-Louis Cauchy and is a restatement of this quantity’s conservation. This quantity is the conjugate variable of position. Kinetic energy can be given as this quantity
momentum
The Schwarz–Christoffel mapping is applied on sets of these numbers, as are all conformal mappings. Functions of these numbers are often visualized using color-wheel graphs. Several theorems about these numbers can be proven using the maximum modulus principle. If a function whose values are numbers of this type is differentiable at every point of its domain, then it is known as a holomorphic function. A field of
complex numbers
The distribution of ratios of numbers selected from a normal distribution is named after this mathematician. If a prime p divides the order of a group, then according to a theorem due to him, that group has an element of order p. A formula due to this mathematician gives the value of a holomorphic function at each point inside a circle in terms of the values of the function on the circle. If two paths have the same initial and terminal paths, then the integral over them for a holomorphic function in that region is equal for the two paths according to an “integral theorem” due to him. This man was the first to discover an inequality stating that two vectors’s dot product is less than the product of their magnitudes. For 10 points, identify this French mathematician who names a notable inequality with Hermann Schwarz.
Augustin-Louis Cauchy
This mathematician names a surface in spacetime which intersects every non-spacelike curve in that spacetime exactly once, and a value named for him appears in the Kramers–Kronig relations. A weaker form of Sylow’s first theorem is his theorem which states that if a prime number p divides the order of G, then G has a subgroup of order p. He names a theorem stating that the integral of a function on a simply connected region is equal to the sum of the function’s a sub negative one Laurent coefficients at each pole in the region. Any complex differentiable function will satisfy the pair of differential equations named for him and Riemann, and he is the first namesake of an inequality on vector spaces. For 10 points, name this French mathematician whose namesake sequence converges in a complete space.
Augustin-Louis Cauchy
The distribution of ratios of numbers selected from a normal distribution is named after this mathematician. If a prime p divides the order of a group, then according to a theorem due to him, that group has an element of order p. A formula due to this mathematician gives the value of a holomorphic function at each point inside a circle in terms of the values of the function on the circle. If two paths have the same initial and terminal paths, then the integral over them for a holomorphic function in that region is equal for the two paths according to an “integral theorem” due to him. This man was the first to discover an inequality stating that two vectors’s dot product is less than the product of their magnitudes. For 10 points, identify this French mathematician who names a notable inequality with Hermann Schwarz.
Augustin-Louis Cauchy
A result of Bela Bollobás [BAY-la BOL-lo-bash] and Andrew Thomason named for covers described by this adjective generalises the Loomis–Whitney inequality. Entourages are constructs used to define spaces described by this adjective, which generalise topological groups and metric spaces. A corollary of the Baire category theorem which deduces this kind of boundedness from pointwise boundedness was proved by Banach and Steinhaus. The Box–Muller transform produces two Gaussian variables from two of this kind of variable. The supremum norm metrises this type of
uniform
A monotonic function can lack this property only at countably many points. If a function with this property maps a compact metric space to a metric space, then it also has the uniform type of this property. This property is often defined using the epsilon-delta criterion. A function with this property must
continuity
Under one definition, a function from a set X to a set Y has this property if open subsets of Y are only mapped from open subsets of X. That form of this property is possessed by all homeomorphisms and generalizes the epsilon-delta definition of this property to non-metrizable
continuity
A measure mu (“mew”) is said to have the absolute form of this property relative to a measure nu (“new”) if, for all measurable sets A, “nu of A equals zero” implies “mu of A equals zero.” The Cantor function has this property, but not the absolute form of this property. A function f between topological spaces has this property if, for all open subsets of the codomain, the inverse image of the subset under f is open. A function has this property at a point a if f-of-x
continuity
Cantor’s “Devil’s staircase” function has the uniform version of this property, but not its absolute form, and the popcorn function has this property at exactly every irrational. For a real-valued function f, this property exists at a point c if the limit of f as x approaches c is equal to f of c. The Weierstrass function has this property but not
continuity
If a function with this property preserves all directed suprema, it has the Scott version of it. The Riesz-Markov-Kakutani representation theorem concerns linear functionals over the space of functions with this property on a compact Hausdorff (“HOWZ-dorf”) space; that space of functions with this property forms a prototypical Banach (“ba-nawk”) algebra. If all subsequences of a sequence of real-valued functions are convergent, than the sequence is both bounded and has a stronger form of this property by the
continuous
According to a namesake principle of functional analysis, pointwise-bounded linear operators on a Banach (“BAH-nuck”) space have this type of boundedness. A sequence of functions converges in the infinity norm if the sequence has this type of convergence, which is defined by the existence of a natural number capital N that depends only on epsilon, not on a point x. This type of continuity implies pointwise continuity. The CDF of a distribution named for this word is
uniform
This property is exhibited by the irrational arguments of Thomae’s function, but not by the rational arguments. A function does not have this property if it has a non-zero oscillation. Bijective functions are homeomorphic if they have this property. The Weierstrass function has this property
continuity
A Hamel basis can be used to construct an infinite set of solutions to an equation of these objects named after Cauchy. For complex numbers, one class of these objects which can be extended with a namesake “continuation” can be locally represented with convergent power series. That analytic class of these objects is equivalent to the class of holomorphic ones. The
functions
Any map from a discrete topological space X must have this property, since every subset in X is open. Roots of functions satisfying this property can be found using Bolzano’s theorem, and another theorem states that functions satisfying this property map connected spaces to connected spaces. That theorem is the intermediate value theorem, which was once part of the definition of this property, although now it is recognised as a consequence of it. A function with this property is equal to its limit at all points, and these functions are sometimes called C0 [C-nought] smooth. For 10 points, what is this property informally possessed by functions that one can draw without taking pen off paper?
continuity
Poset functions with the Scott form of this property preserve directed suprema (soo-PREE-mah). A function has this property if its oscillation at each point is zero. In topology, a function has this property if its inverse takes open sets to open sets. The uniform limit theorem concerns a sequence of functions with this property and is proven using an epsilon-over-three argument. The topologist’s sine curve lacks this property when x is zero, and the Weierstrass (VYE-er-shtross) function has this property everywhere despite being nowhere differentiable. A function f has this property at a if the limit, as x approaches a, of f-of-x, equals f-of-a. For 10 points, identify this property of functions that lack jumps, holes, and vertical asymptotes.
continuity
If the distance between two outputs of a function are less than K times the distance between the points evaluated, then the function has the Lipschitz version of this property. Rigorously, given an epsilon positive, there exists a delta such that if the distance between p and x is less than delta, the distance between f of p and f of x is less than
continuity
This principle can be strengthened by a Cauchy continuity equation that can be used to derive the Navier–Stokes equations. Any system which is invariant under translation in space will follow this principle, according to Noether’s theorem. This principle implies that a system’s center of mass will move with
conservation of momentum
A function has the uniform version of this property if one can bound the distance between two function values by bounding the distance between the corresponding arguments. The extreme and intermediate value theorems require that a function have this property. A function has this property at a point if its
being continuous
A metric q will have a form of this property if there exists some constant b such that for all pair of points c and d, the metric of “q of c d” is less than “b times q of c d”; that is the Lifshitz form of this property. A function with this property within a closed and bounded interval will attain a minimum and maximum at least once by the extreme value theorem. Though every
continuity
By the Heine-Cantor theorem, every function of this kind between a compact metric space and a metric space possesses the uniform version of this property. These functions on a closed interval can be uniformly approximated by polynomials by the
continuous
A function with this property that maps a compact convex set onto itself has a fixed point according to the Brouwer fixed point theorem. A mapping of this type from space X to space Y has this property if for any open subset V of Y, the inverse image of V is an open subset of X. A bijective mapping between topological spaces is a homeomorphism if the mapping and the inverse mapping are open and have this property. If for any epsilon there exists a delta such that the distance between f(x) and f(y) is less than epsilon when the distance between x and y is less than delta, the function is
continuity
A homotopy equivalence between two functions from spaces X to Y is the existence of an invertible mapping of this type between the two functions. The Lipschitz form of this quantity is not possessed by the square root and exponential functions because the slope becomes arbitrarily steep. One definition of this property is that each
continuous
One type of this property used in measure theory is denoted by a double-less-than sign, and means that whenever the mu of a set is zero, the nu of that set must also be zero. If a function with this property has compact domain, then it has the uniform variety of this property as well. Probability distributions with this property have a zero chance of producing any specific result. Despite having this property everywhere, the Weierstrass function is differentiable nowhere. A function f has this property if the limit of f-of-x as x approaches the point c is f-of-c. For 10 points, name this mathematical property possessed by functions whose graphs have no holes or breaks.
continuity
One proof of this theorem takes the normal closure of a finite extension and then considers a 2-Sylow subgroup of that extension’s Galois group. This theorem follows from continuity and Picard’s little theorem. Another proof considers the function 1-over-f, which is entire and bounded, and thus must be constant by Liouville’s theorem. Since algebraic extensions have finite subextensions, this theorem is equivalent to there being no
fundamental theorem of algebra
Functions with a special type of this property make up Holder spaces. The fundamental theorem of Lebesgue integral calculus relates the existence of a derivative-like function with one type of this property. Preimages of open sets on functions with this property are open. This propertyis not fulfilled by the
continuity
Any function from a set with the discrete topology will have this property, as the discrete topology includes all elements and thus all subsets are open. A function has this property if the preimage of all open sets in the range are also open. Any function with this property on a closed interval will also meet the criterion for the stronger
continuity
With an Italian, this mathematician names the result that the image of a subset of a complex plane under a function with an essential singularity in that subset will be dense in the complex plane. This non-American mathematician partially names a theorem which states that every continuous function on a closed interval can be uniformly approximated, arbitrarily closely, by a polynomial. He is the second namesake of a theorem stating that a bounded and closed subset of R-n is sequentially compact. He used the concept of uniform convergence to create the rigorous definition of continuity and revised Cauchy’s imprecise epsilon-delta definition of a limit. For 10 points, identify this German who names a theorem with Bolzano and discovered a function that is everywhere continuous but nowhere differentiable.
Karl Weierstrass
A conjecture named for this object implies that any endomorphism of a Weyl (“vile”) algebra is an automorphism, and states that a polynomial mapping over the complex numbers is an automorphism if and only if this object is nonzero. Stiff O·D·E solvers usually require this object as an input. The left pseudo·inverse of this object multiplies the previous output to iterate the Gauss–Newton algorithm. In three dimensions, this object has the value “rho-squared times sine-of-phi” when transforming from spherical to
Jacobian
Taking the auto·correlation of this phenomenon as a time series gives the Dirac delta function, since its auto·correlation at nonzero lag is zero. The integral of this phenomenon gives a Wiener process, or equivalently, a generalized time derivative of Brownian motion gives this phenomenon. Adding the constraint that samples of this phenomenon have zero mean and are normally distributed creates its “additive Gaussian” type, which is used as a channel model for
white noise
The Frenet–Serret formulas calculate derivatives taken with respect to this quantity. Elliptic integrals were developed to calculate this quantity for ellipses. This quantity is approximated as the sum of the distances between adjacent values in a partition of the domain and may equal the supremum of all such rectifications. A function “parameterized
arc length
A constant multiple of this function of the integral of P-of-x dx gives the integrating factor of a first-order O་D་E. This function of “negative s་t” appears in the definition of the Laplace [luh-PLAHSS] transform. The odd and even parts of this function are the hyperbolic sine and cosine, respectively. The nth term of this function’s Taylor series has a coefficient of “one over
exponential
The Cholesky (“shoh-LESS-kee”) decomposition can be used on one of these things instead of the slower LU (“L-U”) decomposition if it is positive-definite and Hermitian (“air-mee-shun”), meaning it is its own adjoint. Second-order partial derivatives make up one of these things known as the Hessian (“HESS-ee-un”). The roots of these things’ characteristic polynomial are known as
matrix
By the strong duality theorem, if the primal and dual problems of this type have solutions, they are equivalent. One algorithm for this task “walks” along the edges of a convex polytope. An iterative algorithm for this task became widely used in machine learning after the development of backpropagation by Geoffrey Hinton. The simplex algorithm performs this task, which is performed locally for a loss function during gradient
optimization
A mathematical object partly named for this quantity is related to a harmonic function in Bochner’s formula. A type of this quantity is related to a Jacobi field inequality in the Rauch comparison theorem. Covariant derivatives of types of this quantity are related in the contracted Bianchi identities. The Killing-Hopf theorem relates objects with a constant value of this quantity to a quotient of a space form by a group. One type of this quantity is invariant under local isometry by the
curvature
Description acceptable. One of these functions is the fixed point of a sequence of Picard iterations. Given a Lipschitz continuous function, one of these functions is unique by the Picard–Lindelöf theorem. One of these functions exists as a consequence of the Arzela–Ascoli theorem according to a theorem by Peano. These functions are approximated with total error bounded by the fourth power of step size in the standard Runge–Kutta (“ROON-guh KOOT-ah”) method. Finding these functions is the goal of initial value problems and boundary value problems. Families of these functions are plotted on slope fields. In simple cases, these functions can be determined by separation of variables. For 10 points, name these functions that satisfy equations relating them to their derivatives.
solutions to ordinary differential equations
James Wilkinson’s discovery that the results of this task are highly sensitive to small perturbations of a “perfidious” input was the “most traumatic experience [of his] career.” Polishing the results of this task when using forward or backward deflation minimizes the impact of increasing errors. The basins of convergence for an algorithm for this task form a fractal, as shown by interpreting the algorithm as a meromorphic function and looking at its Julia set. A superlinear algorithm for this task has an order of convergence equal to the golden ratio. Ridders’s method for this task makes the false position method more robust. Bracketing methods for this task rely on the intermediate value theorem. An algorithm for this task subtracts the input function over its derivative at every iteration. For 10 points, the Newton–Raphson method performs what task of determining where a function crosses the x-axis?
finding the roots of a function
If an object described by this adjective is trivial, then the corresponding manifold is called parallelizable. A Riemannian (“ree-MAHN-ee-in”) metric is a collection of inner products that each act on a space described by this adjective. For a point p on a differentiable manifold, equivalence classes of curves that pass through p define a space described by this adjective, which is isomorphic to Euclidean space. The integral of the norm of a vector described by this adjective gives the
tangent
A holomorphic function is a complex function that can have this operation performed on it. A Weierstrass function is continuous but unable to have this function applied to it at any point. One method of performing this operation takes the natural logarithm of both sides. Notation named for Gottfried Leibniz can be used to perform this operation
derivative
In complex analysis, the viability of performing this operation determines the property of holomorphism. Performing this operation on each component of a square matrix gives a Jacobian matrix. For functions satisfying the conditions of Rolle’s theorem, this operation results in zero somewhere in an open interval. This operation cannot be performed at a
differentiation
Successive versions of this type of function are known as jerk, snap, crackle, and pop. This function is typically defined using the limit of a difference quotient, and when working with polynomials, the power rule and
derivative
This function’s output is 1 minus theta squared over 2 in its small angle approximation. The reciprocal of this function is known as the secant function. This function’s derivative is negative sine, and it represents the
cosine
In a Wronskian [RAHN-skee-un] matrix, after the first row each entry is made by applying this operation to the entry above it. “Partial” versions of this operation are used to compute each entry in a Jacobian [yah-KOH-bee-un] matrix. Divergence, curl, and gradient are versions of this operation for vector-related functions. A rhyme ending “draw the line and square below” is used to remember how to perform the quotient rule for this operation. This operation can also be done using the product rule and the chain rule. For 10 points, name this operation used in calculus to find the slope of a tangent line to a function’s graph.
derivatives
One theory behind the development of this unit of measure combines the use of equilateral triangles and a sexagesimal [“sex”-uh-JESS-ih-mull] numeric system. Though it is common to use this unit of measure for many applications, this unit is generally avoided in calculus because trigonometric derivatives are not simple with this unit. Though this unit does not measure time, it can be divided into minutes and seconds. 90 of these units equals 100 gradians. Approximately 57.3 of these units is equal to one radian; the exact conversion is 180 over pi. For 10 points, name this unit used for angles that divides a circle into 360 parts.
degrees
The displacement field is split into two independent components in a hybrid method of doing this task named for Trefftz. The order of the central objects of this task can be reduced by breaking it into irreducible components via the Loewy decomposition. If the object central to this task is in self-adjoint form, then this task can be reduced to problems in Sturm–Liouville theory. Robin, Neumann, and Dirichlet name the different kinds of
solving differential equations
Many proofs attempt to avoid the Arzela-Ascoli theorem when showing that one of these things exists by Peano’s theorem. An indicial polynomial determines coefficients when expressing these things as infinite series in the method of Frobenius. A ratio of Wronskian determinants is used in variation of parameters to produce one of these functions, which can be made particular using the
solutions to ordinary differential equations
Applying this operation to the gradient and a normalized vector on a scalar field gives the directional derivative. This operation, which appears in the line integral expression over a vector field, is used in Gram-Schmidt orthonormalization to eliminate non-orthogonal components of a vector. This operation is equivalent to the product of the magnitudes of two vectors times the
dot product
L’Hôpital’s rule can be used to easily solve for one value in this type of mathematics through substitution. The area under a curve and the slope of a line tangent to a function at a specific point are given by operations in this type of math. Gottfried Leibnitz and Isaac
calculus
The harmonic mean of a set of numbers equals n divided by the sum of this function applied to each number. This function equals the derivative of the natural log. The graph of this function forms a hyperbola in the first and third quadrants, and has
reciprocal
The topologist’s sine curve equals the sine of this function of x. The sum of this function, applied to each natural number, appears in the definition of the Euler-Mascheroni Constant. The harmonic mean of a set of numbers equals n divided by the sum of this function, applied to each number. This function equals the derivative of the
reciprocal
In De Moivre’s (duh MWAH-vruh’s) theorem, this function is multiplied by i. This function is used in the Fourier (FOOR-yay) transforms for odd functions. The hyperbolic form of this function equals “e to the x minus e to the negative x over two.” The negative reciprocal of this function squared is the derivative of
sine
The quantity e to the x plus e to the negative x all over 2 is an analogue of this function used in catenary curves. The alternating series of x to the 2n over 2n factorial converges to this function, which is the real part of Euler’s formula. The dot product of two vectors is proportional to this function of the angle between them. The
cosine
The Laplace expansion uses minor examples of these constructs to determine a specific value and is also called cofactor expansion. The dimension of the vector space spanned is used to determine the rank of these constructs. When these constructs are comprised of first-order or second-order partial derivatives they are known, respectively, as Jacobian or Hessian
matrices
The Laplace transform of “f-of-t” equals the integral from zero to infinity of “f-of-t” times this function of “negative-s-t”. The limit as “n approaches infinity” of “one plus one over n all to the n power” equals this function. This function is the solution to the simple differential equation “d-y over d-x equals
exponential function
For a compact manifold M, groups whose members are co·homology classes of these objects have finite dimension. Moreover, co·homology groups for these objects are Poincaré dual to those for ones with compact support. On symplectic manifolds, there is always a local Darboux (“dar-BOO”) chart in which one of these objects has a canonical structure. These objects live in the spaces Zk(X) (“Z-K-of-X”) and Bk(X) (“B-K-of-X”), the quotient of which is the k-th (“K’th”)
differential forms
This mathematician names the following lemma: if A is a finitely-generated commutative k-algebra, and B is a finitely-generated module over a polynomial ring over k, then A and B are isomorphic. Masayoshi Nagata found an object named for this mathematician that pathologically has infinite Krull dimension. This mathematician generalized and co-names Emanuel Lasker’s theorem on the primary decomposition of ideals. Emil Artin and this mathematician name dual properties on the non-existence of infinite
Emmy Noether
This operation can be performed infinitely many times on functions in the class C infinity. A complex function is holomorphic at a given point if the complex form of this operation can be performed in a neighborhood of that point. By design, this operation cannot be performed on the Weierstrass (“vire-strass”) function. The entries in Jacobian (“ja-koh-bian”) and Hessian matrices are the results of this operation. This operation is defined as the
differentiation
This construct is iteratively updated by truncating a Taylor expansion in the SR1 method. An approximation of this construct is multiplied by the step direction and set equal to del-f in each step of the BFGS algorithm. In optimization problems, the definiteness of this construct can be used to identify solutions among the stationary points of the Lagrange multiplier. The
Hessian matrix
By Osgood’s lemma, if this property holds for each variable separately, then it holds for a function of several variables. Morera’s theorem gives sufficient conditions for a function to have this property on an open set. One can prove the fundamental theorem of algebra like so: if p(z) (“p-of-z”) has no roots, then 1 / p(z) (“one over p-of-z”) is bounded and has this property everywhere, and therefore must be constant. Contour integrals can be computed by summing
differentiable
Johann Bernoulli solved the brachistochrone (“bruh-KISS-tuh-crone”) problem using the fact that this function of theta is inversely proportional to the square root of y for a cycloid. The Fourier decompositions of odd functions are sums of this function. The hyperbolic form of this function equals one half times the quantity e-to-the-x minus e-to-the-negative-x. This function’s Taylor series expansion is the
sine
The Runge-Kutta methods are a family of algorithms that perform this operation implicitly and explicitly. The double type of this operation over a planar region is related to its line type around a curve, according to Green’s theorem. Terms like “u” and “du” are used in a method for performing this operation
integration
A two-sided Laplace transform produces one of these expressions that computes moments for a discrete probability distribution. One of these expressions is constructed around an ordinary point to solve a differential equation in the method of Frobenius. When terms of sequences are encoded into these expressions, they are called generating functions. These expressions evaluate as infinite outside of their
power series
This task can be simplified by removing variables via Fourier-Motzkin elimination. Grids called tableaux are used to track pivot operations in one algorithm that performs this task by first defining slack variables. The domain in which this task is possible is called the feasible region and is represented by a convex polytope whose corners are found in
optimization
An operation analogous to this operation characterizes how well a function can be approximated by a Mobius transformation and is called the Schwarzian. This operation returns a nonzero value everywhere for a conformal map. This operation is performed successively on each row in the matrix used to calculate the Wronskian. When using vectors, the
derivative
This operation on a constant function always gives zero. The chain, product, and quotient rules are used to simplify computing this operation, which is rigorously defined using limits. Common notation for this operation includes
differentiation
These mathematical objects represent the elements of the general linear group. A process that factors these objects is called LU decomposition. One of these objects that contains a function’s first-order partial derivatives is named for Jacobi. Cramer’s rule can be used to solve a system of equations represented by one of these objects. These objects can be reduced into
matrix
For powers of a prime p, the von Mangoldt function outputs this function of p. The first Chebyshev function is equal to the sum of this function applied to all the primes less than a given number. The limit of this function minus an expression that asymptotically approaches it was conjectured by Legendre (“luh-ZHOND-ruh”) to be slightly greater than one. The integral of one over this function is often offset using a lower bound of two and is denoted Li (“L-I”). According to the earliest form of the prime number theorem, [read slowly] “N over the prime counting function of N” is asymptotic to this function. The difference between this function and the harmonic series approaches the Euler–Mascheroni constant. The derivative of this function is equal to one over x. For 10 points, name this inverse of the exponential function.
Natural logarithm
Johann Bernoulli solved the brachistochrone (“bruh-KISS-tuh-crone”) problem using the fact that this function of theta is inversely proportional to the square root of y for a cycloid. The Fourier decompositions of odd functions are sums of this function. The hyperbolic form of this function equals one half times the quantity e-to-the-x minus e-to-the-negative-x. This function’s Taylor series expansion is the
sine
The Runge-Kutta methods are a family of algorithms that perform this operation implicitly and explicitly. The double type of this operation over a planar region is related to its line type around a curve, according to Green’s theorem. Terms like “u” and “du” are used in a method for performing this operation
integration
A two-sided Laplace transform produces one of these expressions that computes moments for a discrete probability distribution. One of these expressions is constructed around an ordinary point to solve a differential equation in the method of Frobenius. When terms of sequences are encoded into these expressions, they are called generating functions. These expressions evaluate as infinite outside of their
power series
This task can be simplified by removing variables via Fourier-Motzkin elimination. Grids called tableaux are used to track pivot operations in one algorithm that performs this task by first defining slack variables. The domain in which this task is possible is called the feasible region and is represented by a convex polytope whose corners are found in
optimization
An operation analogous to this operation characterizes how well a function can be approximated by a Mobius transformation and is called the Schwarzian. This operation returns a nonzero value everywhere for a conformal map. This operation is performed successively on each row in the matrix used to calculate the Wronskian. When using vectors, the
derivative
This operation on a constant function always gives zero. The chain, product, and quotient rules are used to simplify computing this operation, which is rigorously defined using limits. Common notation for this operation includes
differentiation
These mathematical objects represent the elements of the general linear group. A process that factors these objects is called LU decomposition. One of these objects that contains a function’s first-order partial derivatives is named for Jacobi. Cramer’s rule can be used to solve a system of equations represented by one of these objects. These objects can be reduced into
matrix
Nambu’s formalism introduces multiple of these functions. In one dimension, this function can be written as solely depending on an adiabatic invariant ‘capital I’ using action-angle coordinates. Either capital omega or J denotes a 2n by 2n matrix encoding the symplectic structure of a formalism based on this function. A quantity is conserved if its Poisson (pwuh-SAHN) bracket with this function is zero. The time derivative of a canonical coordinate q equals the derivative of this function with respect to the conjugate momentum. This quantity is related to its namesake’s “principal function” by an equation partially named for Jacobi. The Legendre (luh-JON-druh) transform of the Lagrangian is, for 10 points, what function corresponding to total energy of a system and denoted H?
Hamiltonian
An equivalent expression to the result of this function on the sum of two matrices is given by the Lie-Trotter (Lee-Trotter) formula. Lie (Lee) algebras are sent to Lie groups by a mapping named for this function. A gamma distribution with a shape parameter of 1 is a distribution named for this operation, which models the time between events for a Poisson process. This function of quantity, negative s times t, end quantity, multiplies the input function in the integrand of the Laplace (luh-PLOSS) transform. The coefficient of the n-th (ENTH) term in this function’s Taylor series is one over n factorial. Up to a constant factor, this function solves the differential equation y prime equals y, meaning it is its own derivative. For 10 points, name this function whose inverse is the natural logarithm.
exponential
A group named for this property consists of all 2 by 2 matrices with determinant 1. First-order ODEs with this property can be solved via multiplication by an integrating factor. The simplex algorithm can only be applied when both the system of constraints and objective function have this property. Non-pathological solutions to Cauchy’s functional equation have this property. A function with this property is constructed to
linearity
A tensor described by this word is given by halving the antisymmetrization of the two lower indices of a connection. For Abelian groups, the structure theorem for finitely generated modules reduces to the Chinese remainder theorem because Abelian groups are modules described by this word over the integers. This word describes group elements with finite order, and Tullio Levi-Civita names the unique metric-compatible connection for which a quantity described by this word vanishes. In the
torsion
A rule named for this mathematician states that the derivative of an integral is equal to that integral of the partial derivative of the original integrand. He’s not Cauchy, but this man created a criterion for determining alternating series’ convergence based on whether the absolute values of their terms decrease monotonically to 0. Another rule named for this man states that for the nth derivative of f
Leibniz
In 2018, Market Track and InfoScout announced a merger between the two companies to form a company which shares its name with these numbers. In a Farey sequence of order 2n-1, the first n of these numbers (excluding 0) are all 1. In Taylor series expansions, the nth of these numbers is given by
numerators
An estimator of this quantity can be adjusted to be unbiased by Bessel’s correction. This quantity equals the second derivative of the moment generating function at zero. For a random variable x, this quantity equals the expected value of x squared, minus the square of the expected value of x; or the expected value of the square of
variance
A theorem named for a “uniform” type of this operation does not necessarily hold true for a set of pointwise convergent functions. Convergence is equivalent to the inferior and superior forms of this operation being equal. Taking this operation on two distinct functions can be used to find its value for a third by the
limit
This quantity times u is constant in the Rankine-Hugoniot conditions. When this quantity is higher above an interface than below it, a Rayleigh-Taylor instability maycan form. If this quantity is constant for a fluid, then the divergence of its velocity is equal to zero. The derivative of this quantity with respect to time is zero in an
density
The inverse of x times this function of x is the Lambert W function. The integrand is multiplied by this function of negative s t in a Laplace transform. One-half of the sum of this function of x and this function of negative x equals the hyperbolic cosine. The coefficient of the nth term of this function’s Taylor series is equal to one over n
exponential function
Cauchy’s residue theorem can be used to evaluate this operation over certain closed curves. In three dimensions, this operation of the curl of a vector field over a surface is equal to this operation on that vector field over its boundary. In statistics, a CDF can be found by performing this operation on a subset of the PDF. This operation has “contour” and
integration
Philippe Flajolet pioneered an “era” of these functions in combinatorics, during which they were used to define the distribution under a Brownian excursion and the Tracy–Widom distribution. One of these functions’ value at the origin is given by: one over, quantity, three to the two-thirds power times the gamma function of two-thirds, end quantity. These functions exhibit jumps in their asymptotic behavior as their argument crosses their three anti-Stokes lines and three Stokes lines. These functions are the eigenfunctions of the Schrödinger equation for the triangular potential well used in semiconductor physics. The equation
Airy functions
The subsets of R-to-the-n with this property form a Helly family of order n-plus-one, by Helly’s theorem. Kakutani’s fixed-point theorem applies to functions which send points to sets with this property. A vector space with a topology induced by a family of seminorms is said to locally have this property. The existence of supporting hyperplanes for this kind of set can be used to show that the
convexity
A 1944 paper by Kakutani shows how to apply a general one of these statements to a certain problem using stochastic analysis. These statements can be approximated probabilistically using the Monte-Carlo ‘walk-on-spheres’ method, and can be applied at large distances by using a ‘shooting method’. A linear combination of two simple instances of them gives the
boundary conditions
This mathematician proved that if a trigonometric series converges to zero for every real input, then its coefficients are all zero. Generalising that result, this mathematician defined a ‘derivative’ named for them and Bendixson. In this person’s ‘normal form’, an ordinal is written as a finite sum of powers of omega, the first of this mathematician’s ‘transfinite’ ordinals. This person’s theorem states that no set
Cantor
Curvature is given by “the magnitude of this operation performed on the unit tangent vector” divided by “the magnitude of this operation performed on the position vector.” The directional type of this operation is maximized when taken perpendicular to contour lines. This operation is performed at successively closer points to the
derivative
Epsilon squared is defined to be this value when working with dual numbers, which can be used for automatic differentiation. In multivariable calculus, this value is the divergence of the curl, and it is also the value of any closed-path line integral over a conservative vector field. The integral from -a to a of an
zero
An inner product must be linear, bisymmetric, and [this property]-definite, meaning the inner product of a nonzero vector with itself always has this property. If a linear transformation’s determinant has this property, it preserves orientation. The norm of a nonzero vector always has this property. If a function’s second derivative has this property, the function is called
positive
According to Marden’s theorem, a function with roots at the vertices of one of these objects has zero derivative at the foci [“FOH-sy”] of that object’s Steiner inellipse [“in-ellipse”]. With their edges and vertices, these objects form a structure named for Boris Delaunay that is dual to a Voronoi diagram. Thales’s theorem states that if one of these objects is inscribed in a circle with one side as a
triangles
The Legendre symbol returns -1 or 1 based on whether an integer is this type of residue or non-residue modulo an odd prime p. An extension of Euler’s criterion is known as the law of [this adjective] reciprocity and was proven by Carl Friedrich Gauss. The third derivative of a univariate function of this type is zero. A common technique to solve this type of equation is called
quadratic
For powers of a prime p, the von Mangoldt function outputs this function of p. The first Chebyshev function is equal to the sum of this function applied to all the primes less than a given number. The limit of this function minus an expression that asymptotically approaches it was conjectured by Legendre (“luh-ZHOND-ruh”) to be slightly greater than one. The integral of one over this function is often offset using a lower bound of two and is denoted Li (“L-I”). According to the earliest form of the prime number theorem, [read slowly] “N over the prime counting function of N” is asymptotic to this function. The difference between this function and the harmonic series approaches the Euler–Mascheroni constant. The derivative of this function is equal to one over x. For 10 points, name this inverse of the exponential function.
natural logarithm
A numerical method named for this type of set can be classified as “direct” or “indirect,” based on how it converts the problem to an integral equation. The trace operator on Sobolev spaces is used to define functions on one of these sets. Unlike a fundamental solution, a Green’s function has conditions on one of these sets. The Cauchy (“koh-SHEE”), Neumann, and Dirichlet (“DEER-ih-klet”) conditions all apply to this type of set. These sets are typically notated by prefixing with a curved lowercase d. In differential equations, initial value problems are contrasted with problems named for this type of set, which impose conditions on a function and its derivatives on one of these sets. For a ball, this set is the sphere. For 10 points, name this type of set that, for a region enclosed by a polygon, traces out its perimeter.
boundary
This operation appears multiplying the homoclinic orbit function in the integrand of the Poincare-Melnikov integral. This operation is real bilinear, skew-symmetric, and satisfies Jacobi’s identity, with the consequence that smooth functions on a manifold form a Lie (“lee”) algebra under this operation, which is identical to a Lie (“lee”) directional derivative. The time derivative of any function on a symplectic manifold can be rewritten as the sum of the time partial of the function and this operation applied to the function and the Hamiltonian. This operation on two functions, f and g, of canonical position p and momentum q is given by the sum over the coordinates i of quantity partial-f-partial-q-sub-i times partial-g-partial-p-sub-i minus partial-f-partial-p-sub-i times partial-g-partial-q-sub-i. For 10 points, identify this operation from mechanics named for a French mathematician.
Poisson bracket
Poisson’s (“pwa-sohn’s”) equation is a partial differential equation of this order. Rings are defined with this many operations. Each element of the Hessian matrix is a partial derivative of this order. This is the smallest positive number such that taking the derivative of sinh (“sinch”) x this many times yields sinh x. It takes this many coordinates to uniquely specify the position of a point on the surface of a unit sphere. The sign of a function’s derivative of this order determines whether it is concave up or down, and the derivative of x squared equals this number times x. For 10 points, squaring a number is equivalent to raising it to what power?
two
