For one-dimensional flux $F(p)=p|p|e^{-p^2/(2{\lambda }^2)}$, the nonlinear diffusion equation is $u_t=F^{\prime }(u_x)u_{xx}$. The sole one-dimensional member of the principal diffusion coefficients is positive for $0<|u_x|<\sqrt{2}\lambda$, negative above $\sqrt{2}\lambda$, and zero at the two endpoints. Negative diffusion gives formal sharpening with ill-posedness through high-frequency growth. The extra factor $|p|$ matters: the unweighted exponential conductance has threshold $\lambda$ instead.

Suppose the one-arm probability on the cubic lattice satisfies ${\beta }_r=\exp (-\lambda r+o(r))$ with $\lambda >0$. Define $M_n$ as the largest maximum-norm radius of any cluster rooted in $[-n,n{]}^d$. Then $M_n/\log n\to d/\lambda$ convergence in probability. An upper bound is a union bound over $O(n^d)$ roots. For a lower bound, pack $\Theta ((n/\log n{)}^d)$ disjoint radius-$O(\log n)$ boxes and use the independent first-hit connection events inside them. Clusters are finite almost surely under the positive-rate hypothesis.

Let $S_n={\sum }_{i=1}^n{X}_i$, where the summands are independent and identically distributed with Pareto distribution $\mathrm{Pareto}(\alpha ,\beta )$. Their positivity implies $\{X_1\ge na\}\subseteq \{S_n\ge na\}$, so for fixed $a>0$ and $na\ge \beta$,Both outer expressions tend to zero. A single large summand already prevents a strictly negative exponential rate. No positive exponential moment exists, so the exponential-moment form of the Cramér theorem cannot be applied. The bound proves this logarithmic rate and does not assert an exact tail asymptotic. In particular, it remains true when $a$ lies below a finite mean, where the event need not even be rare. The stronger fixed-summand-count asymptotic in a heavy-tailed subexponential distribution is a separate statement.

Passing to norm limits in Artin reciprocity gives an exact sequence connecting closures of global units, local pro-$p$ units at primes over $p$, the p-ramified Iwasawa module and the unramified Iwasawa module. It explains arithmetic Iwasawa-module ranks by comparing local and global unit ranks. Roots of unity and the precise splitting conventions must be retained when seeking exact integral statements.

A language in P to which every P language has a logspace many-one reduction. The reduction strength matters: using arbitrary polynomial-time reductions instead would make completeness trivial for nontrivial P decision problems.

In exponential scale coordinates, the real affine group has matrices $\left(\begin{array}{cc}{e}^{\alpha }& \beta \\ 0& 1\end{array}\right)$. The left and right Maurer-Cartan forms are respectively $Dd\alpha +T{e}^{-\alpha }d\beta$ and $Dd\alpha +T(d\beta -\beta d\alpha )$. Their dual vector fields obey $[D_L,T_L]=T_L$ and $[D_R,T_R]=-T_R$, displaying the opposite bracket sign for right-invariant vector fields.

With independent uniform distributions on $[0,1]$ as sequential offers and a finite number of rounds, the continuation value with $j-1$ offers remaining is $v_{j-1}$. Accept the current offer when it reaches that threshold. Integrating the maximum of the offer and continuation value gives the recursion, by the Snell envelope principle.

If $x$ interchanges the new point and $\alpha$, put $H=G_{\alpha }$. Then $⟨G,x⟩$ is a one-point extension exactly when $x^2\in H$, $xH{x}^{-1}=H$, and $xgx\in GxG$ for every $g\notin H$. These conditions make $G\cup GxG$ multiplication-closed, hence a group in the finite setting, with new-point stabilizer $G$. Necessity comes from the two double cosets in the extended two-transitive action.

A continuous one-cocycle for a Galois group acting on an abelian module $M$ is a map satisfying the displayed identity. Cocycles modulo maps $\sigma \mapsto \sigma u-u$ form $H^1(K,M)$ in Galois cohomology. For an elliptic curve, choosing $mQ=P$ gives the cocycle $\sigma \mapsto \sigma Q-Q$ in $E[m]$, which represents the Kummer map of an elliptic curve.

The prenucleolus uses the same sorted excesses of a coalition but minimizes over all efficient payoff vectors, without imposing individual rationality. It can differ from the nucleolus when coalition values are not superadditive. Any computation must state which domain is used rather than silently relaxing imputation inequalities.

Every NP verifier can be compiled into a polynomial-size Boolean circuit whose inputs encode its certificate, and then into an equisatisfiable conjunctive normal form by a Tseitin transformation. This gives a polynomial-time many-one reduction from every NP language to SAT. Checking a guessed assignment proves membership, so SAT is NP-complete.

For a nonconstant polynomial over any field, use triangular substitutions $x_i=y_i+y_n^{B^i}$, $x_n=y_n$, where $B$ exceeds every exponent in its support. Base-$B$ weights distinguish all original monomials, so the transformed polynomial has a unique highest $y_n$ power with coefficient in $k^{\times }$. Rescale to make it monic. The monic polynomial quotient is finite free over $k[y_1,\dots ,y_{n-1}]$, yielding a finite flat projection. The nonlinear substitution works over finite fields as well as infinite ones; a generic linear-direction argument alone would not do so.

In an ideal linear temperature-dependent destruction model with equal warming and cooling durations, normalized temperature is $1-e^{-\alpha t}$ during warming and $(1-e^{-\alpha T})e^{-\alpha (t-T)}$ during cooling. Integrating the destruction rate gives $D(T)$, the logarithm of the surviving-count reduction factor. If the endpoint destruction rate and thermal relaxation constant stay fixed, changing the temperature gap alone leaves this integral unchanged. This is a property of the stipulated model.

In a Hermitian neutral-kaon mass matrix, independently choosing square-root signs of $z$ and $z^{\ast }$ can interchange the eigenvalue labels. Choose $z=r{e}^{i\phi }$, $a=\sqrt{r}{e}^{i\phi /2}$ and $b=\sqrt{r}{e}^{-i\phi /2}$, so $ab=r>0$, continuously from the desired CP-conserving limit. Then $(a,-b)$ is the lower-mass eigenvector. At a degenerate off-diagonal zero the mass eigenbasis is not fixed by diagonalization alone.

Positive nearest- and next-nearest-neighbour exchanges give one-magnon energy $2S[J_1(1-\cos k)+J_2(1-\cos 2k)]$. At small momentum this is $S(J_1+4J_2)k^2$. The quadratic-boson result describes the harmonic many-magnon spectrum; higher-order spin-wave interactions remain in the full Quantum Hamiltonian.

The vector of category counts is the sum of independent categorical indicator vectors. Their means are $p$ and covariance is $\mathrm{diag}p-p{p}^T$, so the multivariate central limit theorem gives the displayed limit. At uniform probabilities on $k$ categories, Pearson-standardized deviations have covariance equal to the orthogonal projection onto the $(k-1)$-dimensional subspace of zero-sum vectors. Their squared norm has an asymptotic chi-squared distribution with $k-1$ degrees of freedom.

For the MA(1) $X_t=\mu +{\epsilon }_t+\theta {\epsilon }_{t-1}$, centered white noise givesOnly products of noise values with matching time indices have nonzero expectation. At lag zero there are two squared weights, $1$ and ${\theta }^2$; at lag one there is one shared noise term with weight product $\theta$; at higher lags there are none. The calculation needs zero covariance between distinct noise values, not independence. This tridiagonal finite covariance structure makes the finite-sample innovations of an MA(1) process particularly simple.

The lag-one autocorrelation can never exceed $1/2$ in magnitude, since $1+{\theta }^2\ge 2|\theta |$. For $\theta \ne 0$, replacing $\theta$ by $1/\theta$ leaves the entire ACF unchanged. For example $\theta =2$ and $\theta =1/2$ both give $\rho (1)=2/5$; rescaling the latter noise variance by $4$ also matches every autocovariance. Moving-average root reflection explains this ambiguity and why a stable invertibility convention helps identify a representation. Equality of second-order laws need not imply equality of all joint probability distributions unless the processes are Gaussian processes. See Bartlett, lecture 6: equivalent representations and covariance of ARMA processes.

Two closed symplectic forms agreeing as ambient bilinear forms along a compact embedded submanifold are equivalent near it by a diffeomorphism fixing it pointwise. Shrink until their affine interpolation is nondegenerate. A relative homotopy primitive $\eta$ for their difference vanishes as an ambient covector on the submanifold. Solving ${\iota }_{Z_t}{\omega }_t=-\eta$ gives a vector field vanishing there; its local flow fixes the submanifold and makes ${\omega }_t$ constant under pullback.

A monotone sequence of real numbers has a finite limit of a sequence exactly when it is a bounded sequence. In the increasing case, only an upper bound needs to be supplied, because the first term already gives a lower bound; in the decreasing case, only a lower bound needs to be supplied. The limit of a sequence is the exact edge of the set of terms:The upper bound $M$ or lower bound $m$ need not itself be the limit of a sequence. A supremum or infimum identifies the sharp bound. ^[ref]

For example, $a_n=1-1/n$ is increasing and bounded above by one, and $b_n=1+1/n$ is decreasing and bounded below by one. Both have limit of a sequence one, although neither ever equals one. Their absolute errors are exactly $1/n$. They also bracket any sequence $x_n$ satisfying $a_n\le x_n\le b_n$: the squeeze theorem gives $x_n\to 1$, even if $(x_n)$ itself is not a monotone sequence.

The theorem is useful when a recurrence relation gives successive improvements and a bound, before the answer has been identified. It proves existence of a limit of a sequence; further algebra, often using continuity, identifies the answer. A finite plot illustrates the formulas but does not establish the infinite claim. Error bounds for bounded monotone sequences require additional quantitative information.

A real increasing sequence bounded above converges to its supremum; a decreasing sequence bounded below converges to its infimum. For the increasing case, the definition of supremum provides a term above $L-\epsilon$, and every later term remains between that term and $L$. Negation reduces the decreasing case to the increasing case. This elementary result is distinct from the measure-theoretic monotone convergence theorem for integrals.

The existence of the real supremum in this short proof uses the least-upper-bound property, equivalently completeness of the real numbers. The approximation property of a supremum supplies $a_N>L-\epsilon$, and monotonicity keeps every later term within the same tolerance:The corresponding decreasing statement follows from the approximation property of an infimum. The converse uses the fact that every convergent sequence is a bounded sequence. ^[ref]

The kernel of $x\mapsto t^3$, $y\mapsto t^4$, $z\mapsto t^5$ is generated by $y^2-xz$, $yz-x^3$, $z^2-x^{2}y$. Reducing these three relations leaves a unique normal form $A_0(x)+A_1(x)y+A_2(x)z$: its images have distinct exponents modulo three. The curve ring is therefore finite free of rank three over $k[x]$ and has dimension one. In a three-variable polynomial ring its ideal has height two, but its three independent quadratic initial forms force at least three generators. This is a concrete failure of generation by codimension many equations.