Mathematics branches {analysis, mathematics} {mathematical analysis} can be about theory of real-variable functions and theory of integrals and integral equations.
purposes
Mathematical analysis studies continuous but non-differentiable functions. It studies continuous-function series whose sum is discontinuous. It studies continuous functions that are not piecewise monotonic. It studies functions with bounded derivatives that are not Riemann integrable. It studies curves that are rectifiable, but not by calculus arc-length definition. It studies non-integrable functions that are limits of integrable-function series. It studies Fourier series relations to represented functions.
point density
Number of interval points and number of subinterval points are the same.
Laplace transform
Integral from t = -infinity to t = +infinity of e^(-x * t) * g(t) * dt, where g(t) = (0.5 * i) * (integral from x = a - infinity to x = a + infinity of (e^(x * t))*(f(x)) * dx), where a is large.
integral existence
If interval points are differentiable, function can integrate over interval. Intervals have variable maximum and minimum values. Function f(x) has maximum and minimum over interval. Maximum-M limit minus minimum m times x-change dx goes to zero as dx goes to zero: (M - m) * dx.
series expansion
Functions can be equivalent to series. Function series expansions are in integral-equation theory {convergence of mean} {Lebesgue square integral} {Riesz-Fischer theorem} {moment problem} {Holder inequalities} {strong convergence} {weak convergence} {singular integral equations}.
Over intervals, functions can be almost equivalent to power-series functions {Comega-smooth function} {analytic function}. Real analytic functions {Cinfinity-smooth function} can be differentiable at domain points any number of times. Complex analytic functions are complex differentiable, typically only once, at domain points. Complex entire functions are differentiable at all complex-plane finite points. Non-analytic functions are not differentiable at some singularity point or along branch cut.
Analytic multiple-valued functions in complex plane can be discontinuous across curve {branch cut} {cut line} {slit, mathematics} {branch line}.
Sums, from n = 1 to n = infinity, of a(n) * n^-z, where n is number of terms, a(n) is complex general term, and z is complex number, are number series {Dirichlet series}.
Gradient matrices {Hessian matrix} can have first-derivative components. Hessian matrices also have basis-function-parameter second-derivative components, which are typically negligible, because they are like random measurement errors and cancel each other.
Integral equations relate to complete orthogonal-system theory {orthogonal system}. Functions can expand using orthogonal systems {Fredholm alternative theorem} {complete continuity}.
Complex function has complex conjugate {norm, function}. Two infinite complex-number sequences are orthogonal if and only if norm equals zero.
Using integral equations and initial or boundary values can solve differential equations. Methods {method of successive substitutions} {successive substitutions method} can use iterative substitutions. Methods can use complex-variable functions that are monodrome, monogenic, and holomorphic. Methods can use meromorphic functions.
Complex-variable functions can be single-valued {monodrome}.
Complex-variable functions can have only one derivative at domain values {monogenic}.
Single-valued functions {meromorphic function} can be differentiable except at singularities, where they go to infinity. Polynomials can be meromorphic at points {pole, meromorphic function} but cannot have other singularity types. Meromorphic functions are entire-function ratios.
Functions {Lebesgue integral} can have sums of lengths over intervals.
purposes
Lebesgue integrals can integrate discontinuous functions.
finite
Lebesgue integrals can be finite {summable function}. Limit from x = a to x = b of f(x) * cos(n*x) * dx equals zero. Limit from x = a to x = b of f(x) * sin(n*x) * dx equals zero. Therefore, Lebesgue integral can use Fourier series {Riemann-Lebesgue lemma}.
finite: convergence
Functions can have no bound in interval, but Lebesgue integral can converge absolutely.
extensions
Lebesgue-integral extensions include spectral theory {Lebesgue-Stieltjes integral} {ergodic theory} {harmonic analysis} {generalized Fourier analysis}.
Analytic-function sequence limits are integrals {Stieltjes integral, Riemann}. Stieltjes integrals generalize simpler integrals {Riemann-Darboux integral}.
Integrals, from x = a to x = b, of f(x) * dg(x) * dx equal limits of sums, from i = 0 to i = n, of f(e(i)) * (g(x(i + 1)) - g(x(i))), where x(i) are partition intervals and e(i) are inside intervals (x(i), x(i + 1)).
Riemann
Functions {Riemann integrable function} can have no discontinuities or have discontinuities that form measure-zero sets. Riemann integrals are Lebesgue integrable, but Lebesgue integrals can be not Riemann integrable.
Analytic-function sequence limits are integrals {Stieltjes integral}. Stieltjes integrals are generalized Riemann-Darboux integrals. Integrals, from x = a to x = b, of f(x) * dg(x) * dx equals limits of sums, from i = 0 to i = n, of f(e(i)) * (g(x(i+1)) - g(x(i))), where x(i) are partition intervals and e(i) are inside intervals (x(i), x(i+1)).
Theories {measure theory} can find discontinuous-function magnitudes {mathematical measure} {measure, mathematics}, for quantum mechanics, statistics, and probability.
process
Enclose set points in an open-set interval inside a finite or countably infinite set of non-overlapping intervals {union of non-overlapping denumerable open intervals}. To obtain lower bound {exterior measure}, sum non-overlapping intervals. Use sum to find set-point complement {interior measure}.
measure
If function has bound and is measurable, length, area, or volume is greatest lower bound {greatest exterior measure} and equals least upper bound {least interior measure}.
types
Boolean sigma-algebra can represent discontinuous-function measures. In intervals, Lebesgue generalized ordinary integrals, over discontinuous-function points, can find function values {P-measure}. If Lebesgue integrals are constant, P-measures {Lebesgue measure} are constant.
Point-set {measurable set} exterior measure can equal interior measure. If functions are greater than a number, and point sets are measurable, functions are measurable.
Non-linear least-squares parameter estimation methods {steepest descent method} {method of steepest descent} can use points, far from minimum, where first derivative is maximum.
Methods {normal equations method} {method of normal equations} can find function minimum.
Non-linear least-squares parameter estimation methods {inverse-Hessian method} can use points near minimum, where first derivative equals zero.
Non-linear least-squares parameter estimation methods {Levenberg-Marquardt method} {Marquardt method} can generalize normal-equations method to find minimum and avoid steepest-descent and inverse-Hessian extremes.
Taking gradient by differentiating eliminates equation constants and so cannot calculate equation-constant magnitude. However, Hessian-matrix components can indicate constant magnitude.
Using scale factor can transform matrix into diagonally dominant matrix. After finding minimum, set scale factor to zero, and compute estimated fitted-parameter standard-error covariance matrix.
To generalize solutions involving constants or additions, use axioms A(c*x) = c * A(x) and A(x + y) = A(x) + A(y) {continuous additive operator}.
Linear bounded Hermitean operators can be operators {Einzeltransformation} E such that E-E- = E-, E+E+ = E+, and I = E- + E+. Einzeltransformations are commutative and commute with any operator that commutes with Hermitean operator. Hermitean times E- is greater than zero, and Hermitean times E+ is greater than zero.
In equation A(x) = x, operation or transformation A leaves function or vector constant {fixed-point analysis}. If operator is continuous and operates on n-dimensional spheres, function has at least one fixed point {Brouwer fixed-point theorem}.
Subsets can project {projection operator, set} onto element sets.
Operators {adjoint operator} {transposed operator} can find function scalar products, which are linear transformations from one function to another: (A(f1), f2). Adjoint operators can have inverses {self-adjointness}: (A(f1), f2) = (f1, A(f2)). The situation is analogous to the symmetric-integral-equation kernel. (T*f, g) = (f, T*g) and ||T*|| = ||T||, where T* is matrix-T transpose.
Adjoint-operator theory can apply to operators {Riesz operator} with form I - lambda * V, where lambda is parameter, I is identity operator, and V is complex continuous L^2-space operator.
Lengths are closed intervals. The idea of length {theory of content} {content theory} can extend to open intervals. For open intervals, sum of subintervals that enclose points has greatest lower bound {outer content} and sum of polygonal regions makes least upper bound {inner content}. If outer content is less than or equal to inner content, interval has content.
length
If inner content equals outer content, inner content is interval length for one dimension.
additive
For finite number of intervals, sum of disjoint sets with content is sum of set contents {additivity property}.
For adjoint spaces, map of vector space onto line can extend to map of space including line {Hahn-Banach theorem}.
In closed intervals (a,b) that have countably infinite interval sets, if a <= x <= b, and x is inside at least one interval, x is inside at least one interval of any finite interval set {Heine-Borel theorem} (Eduard Heine) [1821 to 1881].
Closed space subsets have two unique elements, one in subset and the other orthonormal to every element in subspace {projection theorem} {Riesz representation theorem}.
Main contribution to integral is from points at which derivative equals zero {stationary phase principle} {principle of stationary phase}.
In bounded infinite point sets, a point exists in which any neighborhood has set points {Weierstrass-Bolzano theorem}.
In intervals, analysis can find function-root sets {spectral theory}. Root index is less than its multiplicity. General spectral theory can be for symmetric kernels. Function kernels can have orthonormal eigenfunctions {Hilbert-Schmidt theorem}. Eigenfunction roots {eigenvalue, spectral theory} can be point, band, or continuous spectrum.
Spectral theory can generalize Volterra's method {spectral radius theorem}.
Integral from e = a to e = x of K(x,e) * u(e) * de, where K(x,e) are differential equations {kernel, equation} and u(e) equals integral from x = a to x = b of K(x,e) * f(x) * dx, is limiting form of n linear algebraic-equations with n unknowns, as n goes to infinity.
Integrals can be from e = a to e = b for K(x,e) * u(e) * de {Fredholm's equations} {Fredholm equations}.
Integrals can be equal to zero to make homogeneous equations {Volterra's equations} {Volterra equations}.
All continuous bounded linear functionals have spaces {dual space} {adjoint space} {Banach space}. Dual-space norms are functional bounds. Duals change L^q to L^p, where q = p / (p - 1). Dual spaces are generalized Hilbert spaces.
Spaces {complex sequence space} can be equivalent to L^2 spaces of complex-valued, measurable, and square integrable functions. Complex space has inequalities {Schwarz's inequality} {Parseval's inequality}.
Spaces {function space} can have functions as points or distances. Spaces can have finite numbers of elements. Spaces can have infinite numbers of elements, with two limiting elements at interval ends {closed set, function}. Spaces can have no sequence gaps {sequentially compact}. Spaces can have only one limit element {relatively sequentially compact}.
Function spaces can use generalized Pythagorean theorem.
Function spaces can have triangle inequality, Schwarz's inequality, and other inequalities {Bessel's inequality}.
Mutually orthogonal space elements are linearly independent.
Complex abstract spaces {Hilbert space, analysis} can have infinite dimensions.
transformation
Coordinates can transform. Integral from x = a to x = b of K(x,y) * u(x) * dx can transform differentiable function u(x), where K(x,y) are differential equations.
purposes
Differential and integral equation eigenvalue theory is similar to n-dimensional-space linear transformations. Quantum-mechanics equations can use Hilbert-space spectral theory. Observed values are linear symmetric operators in Hilbert space. Linear symmetric energy-operator eigenvalues and eigenfunctions are energy levels. Eigenvalue differences show emitted-light frequencies. Lebesgue integrable functions {square summable function} make spaces similar to Hilbert spaces of sequences. Banach spaces {complete normed vector space} are generalized Hilbert spaces.
Spaces {linear vector space} can have complete orthonormal countable sets. Topology is about equivalencies during continuous motions. Linear vector space can define metric using the norm {strong topology}. Linear vector spaces can have systems of neighborhoods with weak convergence {weak topology}.
Banach space includes spaces {L^p space}, of continuous bounded measurable functions, which does not need two orthogonal elements and in which inner product does not define norm.
Spaces {metric space} can have distances. Integral from x = a to x = b of K(x,y) * u(x) * dx, where K(x,y) are differential equations, can transform distance function u(x) into another distance function.
Permutations, combinations, binomial theorem, magic squares, and partition theory can combine into one subject {combinatorial analysis}.
Continuous functions, convergences, and limits can combine into one subject {functional analysis}. Analysis includes infinite series, ordinary differential equations, partial differential equations, differential geometry, calculus, and calculus of variations. Analysis excludes plane geometry, solid geometry, and computational methods. Analysis uses arithmetic, variable, function, continuity, differentiability, integrability, limits, infinitesimals, infinite, least upper bound, uniformity, convergence, and fundamental theorem of calculus.
purposes
Functional analysis can be for generalized moment problem, statistical mechanics, fixed-point theorems, partial-differential-equation existence and uniqueness theorems, calculus of variations, and continuous compact-group representation. Linear functional analysis can study integral equations.
functional
Functions of functions {functional} map function to number. Functionals can find areas of products of two functions, over intervals. Functionals can evaluate functions at points. Functionals have generalized derivatives that are also functionals, but can have singularities.
Sum from p = 1 to p = infinity of (z(p) * Z(p))^0.5, where Z(p) is z(p) complex conjugate, can study calculus of variations {linear functional analysis}.
Analysis {standard analysis} can use limits and exhaustion method. Standard analysis and nonstandard analysis use same language and rules, but interpretation is different.
Analysis {nonstandard analysis} can use infinitesimals that can never get large. In nonstandard analysis, numbers, metrics, and spaces always have nearby values.
contradiction
Non-standard analysis can introduce contradictions, because added infinitesimals do not necessarily stay small. Infinitesimals can violate Archimedes principle {non-Archimedian}.
theorem
For proposition sets, if finite proposition subsets are true in standard analysis, whole proposition set is true in nonstandard analysis {compactness theorem}.
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Date Modified: 2022.0225