The Skorokhod space consists of real paths that are right-continuous and have a finite left limit at every positive time. Its $J_1$ topology allows small changes of time: $x_n\to x$ when there are increasing homeomorphisms ${\lambda }_n$ of $[0,1]$ such that $\Vert {\lambda }_n-\mathrm{id}{\Vert }_{\infty }\to 0$ and $\Vert x_n\circ {\lambda }_n-x{\Vert }_{\infty }\to 0$, where $\mathrm{id}$ denotes the identity time map. This topology aligns nearby jumps rather than insisting on uniform closeness at identical times. The step-path version of the Donsker invariance principle converges here to continuous Brownian motion; its polygonal version uses the ordinary supremum norm on continuous paths.

A function in extension is a function specified by its domain and the value at each input in that domain. Distinct functions in intension can have this same extension. Properties of the extension of an arbitrary program are semantic properties, to which Rice theorem applies when the property is nontrivial.

A pointwise periodic self-map is a self-map $f:X\to X$ for which every $x\in X$ satisfies $f^{n_x}(x)=x$ for some positive integer $n_x$ depending on $x$. Powers denote repeated function composition.

Such a self-map is a bijection. For surjectivity, $f^{n_x-1}(x)$ is a preimage of $x$. For injectivity, suppose $f(x)=f(y)$ and choose a common multiple $N$ of periods of $x$ and $y$; applying $f^{N-1}$ gives $x=y$. Thus $X$ decomposes into disjoint finite permutation cycles.

There need not be one positive $N$ with $f^N=\mathrm{id}$ on an infinite set. Take the disjoint union of one cycle of every positive length: any such $N$ would have to be divisible by every positive integer. On a finite set, in contrast, the least common multiple of the finitely many cycle lengths is a common period.

A finite modification of the identity on the real line is a function $f:\mathbb{R}\to \mathbb{R}$ agreeing with the identity map outside a finite set. For example, changing only $f(0)$ to $3$ gives such a function; both $0$ and $3$ then map to $3$, so a finite modification need not be a bijection.

Sorting the exceptional set gives a unique finite record of ordered pairs $(x,f(x))$. Real-number pairing by separated digits encodes every fixed-length record in $(0,1)$, and placing length-$n$ records in disjoint intervals $(n,n+1)$ gives an injection into the real line. Include the zero-length record for the unchanged identity map. Conversely, prescribing $f(0)=t$ and $f(x)=x$ for $x\ne 0$ gives distinct functions for distinct real numbers $t$. The Cantor-Schröder-Bernstein theorem therefore shows that all such finite modifications have the same cardinality as the real line.

For a partition of an integer $\lambda ⊢n$, padded to $m$ rows, let $\delta =(m-1,\dots ,0)$. If a permutation has ${\alpha }_q$ cycles of length $q$, put $p_{\alpha }={\prod }_q({\sum }_i{x}_i^q{)}^{{\alpha }_q}$. The character of the Specht module $S^{\lambda }$ is the displayed coefficient. Equivalently $p_{\alpha }{A}_{\delta }={\sum }_{\lambda }{\chi }^{\lambda }(\alpha )A_{\lambda +\delta }$, the alternant form of the Frobenius characteristic map. It gives finite coefficient computations without constructing representation matrices.

For $\alpha >0$, polar coordinates reduce the integral to $2\pi {\int }_0^{\infty }r{e}^{-(\alpha -i\lambda )r^2}dr$. The result is $\pi /(\alpha -i\lambda )$. Its damped limit for real $\lambda >0$ is $i\pi /\lambda$, which fixes the sign of the oscillatory complex Gaussian integral. The Fresnel integral convention can be compared with NIST definitions.

Let $p$ be a prime number and $L\subseteq {\mathbb{F}}_p$ have $s$ elements. A set family whose member sizes modulo $p$ avoid $L$, and whose distinct pairwise intersection sizes modulo $p$ belong to $L$, has at most ${\sum }_{j=0}^{\min (s,n)}\left(\genfrac{}{}{0ex}{}{n}{j}\right)$ members. The intersection polynomials ${\prod }_{\ell \in L}({\sum }_{i\in A}{x}_i-\ell )$ have a diagonal, nonzero evaluation matrix on the family's characteristic vectors of sets. Multilinear reduction on the Boolean cube places these linearly independent functions in the space spanned by square-free monomials of degree at most $s$.

Let $p$ be a prime number, $0\le a<p$, $1\le s<p$, and $n\ge a+s$. A multilinear polynomial over ${\mathbb{F}}_p$ of degree less than $s$ that vanishes on every Boolean vertex whose weight is not congruent to $a$ modulo $p$ vanishes everywhere. Alternating sums over intervals of $s$ free coordinates eliminate the possible nonzero weight levels one at a time. This proves independence of the auxiliary functions used for the uniform Frankl-Wilson theorem bound.

In the polynomial method in combinatorics, a set family of sets with common size residue $r$ modulo a prime number $p$ annihilates the displayed auxiliary functions. After multilinear reduction on the Boolean cube, functions with $|S|\le s-1$ have degree at most $s$. If $r$ avoids every $j=0,\dots ,s-1$ modulo $p$, these functions are linearly independent: evaluation at a set $T$ gives $(|T|-r){\sum }_{S\subseteq T}{c}_S$, and increasing $|T|$ eliminates coefficients. Combining them with diagonal intersection polynomials removes the lower-degree monomial count and strengthens the dimension bound from ${\sum }_{j\le s}\left(\genfrac{}{}{0ex}{}{n}{j}\right)$ to $\left(\genfrac{}{}{0ex}{}{n}{s}\right)$.

A shared frailty model assigns the same latent positive hazard multiplier $V_i$ to correlated event histories or group members. Gamma or lognormal frailty distributions are common examples. Modeling dependence through this latent effect yields conditional covariate effects; integrating out frailty can change the form of marginal hazards. Eligibility and event history still have to be encoded in the at-risk indicator.

For $0<t\le \pi$, split the sum at $\lfloor 1/t\rfloor$. The first part is bounded using $|\sin kt|\le kt$; summation by parts and the geometric bound on interval sine sums control the rest. The harmonic coefficient sum grows like $\log m$, but cancellations keep the whole trigonometric polynomial uniformly bounded. Modulating this polynomial lets a Fourier partial sum expose the large uncancelled half.

A Fourier extension operator forms an oscillatory integral from a density on a frequency surface with measure $\sigma$. The opposite sign in the phase merely reflects the output variable, so it gives identical Lp norms. In the usual dual formulation it is the adjoint of restricting the Fourier transform to that surface.

Assume a finite diagonal Fourier extension estimate on the unit circle. For direction-separated rectangles of dimensions ${\delta }^{-2}\times {\delta }^{-1}$, choose disjoint frequency caps and translate their transforms to the rectangles by the Fourier modulation and translation identity. Randomize their signs. Disjointness gives an input $p$th-power norm at most $CM\delta$; the Khintchine inequality converts the averaged output norm into the square function. The circle cap Fourier lower bound gives ${\delta }^p\int ({\sum }_R{1}_R{)}^{p/2}{\lesssim }_{p}M\delta$. Each rectangle has area ${\delta }^{-3}$, giving the displayed form. Input cap disjointness is compatible with arbitrary spatial rectangle overlap.

For a compact circle set with normalized Lebesgue measure at most $\exp (-8\pi M/\epsilon )$, where $M\ge 1$ and $0<\epsilon \le 1$, the displayed bounds can be realized by a trigonometric polynomial with spectrum in a positive-frequency interval arbitrarily far from zero. Take the logarithm of a positive-real-part Schwarz integral on the unit disk of a small-mean cutoff, truncate after radial dilation, and modulate its bounded imaginary part. A prefix isolates one analytic half and thus reveals the large real part. The separation between small function norm and large Fourier partial sums drives the Kahane-Katznelson divergence theorem.

If trigonometric polynomials have spectra in increasing disjoint positive-frequency intervals, the displayed summability gives a continuous uniform sum. At any cutoff its Fourier partial sum contains complete earlier blocks, at most one active prefix and no later blocks. Differences across an active prefix therefore isolate that block exactly. Fixed-size excursions occurring in arbitrarily late blocks contradict the Cauchy criterion, even when the complete blocks vanish at the point under consideration.

A continuous complex-valued function can have uniformly bounded Fourier partial sums which fail to converge at one point. Normalize harmonic sine polynomials by their harmonic coefficient sums, modulate them into disjoint high-frequency intervals and choose their degrees so that the complete norms are summable. At zero, complete blocks vanish while their midpoint prefixes have a common nonzero value. The frequency-separated Fourier block series gives a uniform bound on every partial sum as well as two distinct subsequential values.

This average of Fourier partial sums reproduces every degree-at-most-$n$ trigonometric polynomial. With ${\sigma }_r=r^{-1}{\sum }_{j=0}^{r-1}{s}_j$,The bound follows because Fejér summation is a uniform-norm contraction. For $n=0$, the term involving ${\sigma }_0$ is omitted.

For $p>-1$ this formula is interpreted away from $k=0$ as a tempered distribution or by exponential regularization. It also supplies local high-frequency cusp terms after multiplying by a smooth cutoff. Even nonnegative integer $p$ give smooth polynomials and only contact terms at $k=0$, so their localized forms have no algebraic tail. For example, the cusp expansion of $(1+|x{|}^3{)}^{-1}$ gives $-12|k{|}^{-4}+29!|k{|}^{-10}+O(|k{|}^{-16})$.

For a periodic absolutely continuous function with derivative in $L^2$, integration by parts gives $\widehat{f^{\prime }}(n)=in\hat{f}(n)$. Bessel's inequality and the Cauchy-Schwarz inequality prove the displayed bound in the normalized circle norm. Adding the constant coefficient gives an absolutely and uniformly convergent Fourier series. Merely bounding individual coefficients by $O(1/|n|)$ would not establish absolute convergence.

When two massive particles coalesce without additional outgoing radiation, four-momentum conservation makes the final four-momentum their sum. For collinear incident velocities, its rest mass is the positive square root of the displayed invariant, and its final velocity equals the center-of-momentum frame velocity $c^2{p}_{\mathrm{total}}/E_{\mathrm{total}}$. Relative kinetic energy can become internal rest energy, so the final rest mass is generally larger than the sum of the incident rest masses.