For a nonnegative normalized mollifier supported in the radius-$ϵ$ ball, integrate the BV translation estimate with weight ${\rho }_{ϵ}(h)$. The full-space derivative variation controls the full-space error. Replacing it by variation on a smaller domain is invalid without a support condition: a nonconstant bump outside that domain has zero derivative measure inside but a nonzero mollification error.

Let $a\in BV(\Omega )$, $b\in BV(\Omega )\cap L^{\infty }(\Omega )$ and ${\delta }_{t}c=c\circ {\Phi }_t-c$, where ${\Phi }_t$ is the local flow of $\phi e_j$ with nonnegative $\phi \in C_c^{\infty }(\Omega )$. The formula uses common oriented BV traces on a hypersurface; the trace difference of $a$ is zero almost everywhere on $J_b\setminus J_a$. It also holds for both negative-time increments with denominator positive $t$.

The BV slicing theorem reduces the calculation to one dimension. With a right-continuous representative of $a$ and $\mu =Da$, Fubini's theorem yieldsThe inner factor tends to $\phi (s)[b](s)$ and is bounded in absolute value by $2\Vert b{\Vert }_{\infty }\Vert \phi {\Vert }_{\infty }$. Dominated convergence against $|\mu |$ leaves only its measure atoms at the countable slice jumps of $b$. That same bound times $|D{a}_y|$ permits integration over the transverse coordinates. No essential bound on $a$ is used, even if its slice values grow without bound. The formula is useful for opposite-flow fidelity identity for quadratic data with unbounded data and a bounded comparison function.

For $Y_t=y+B_t+at$, $y>0$, and $S=\inf \{t\ge 0:Y_t=0\}$,where $e^{-\lambda \infty }=0$. The diffusion generator is $\frac{1}{2}{\partial }_{yy}+a{\partial }_y$. Its bounded eigenfunction on the positive half-line with boundary value one is $u(y)=\exp [-y(a+\sqrt{a^2+2\lambda })]$, giving the formula through the discounted boundary-hitting representation. The zero-discount limit is the Hitting probability $e^{-2y\max (a,0)}$.

For a positive level $a$, this density describes the first-passage time of $B_t+ct$. The Girsanov theorem multiplies the zero-drift density by the stopped likelihood $e^{ca-c^{2}t/2}$. Its total mass is one for nonnegative drift and $e^{2ac}$ for negative drift; the latter has remaining mass at infinite hitting time.

Suppose an integrable measurable-function class can be covered by finitely many function brackets of every positive $L^1(P)$ width. For an independent sample with common law $P$, the empirical measure satisfies ${\sup }_h|P_{n}h-Ph|\to 0$ on a common probability-one event. For one finite $\epsilon$-cover, the supremum is bounded by $\epsilon$ plus the largest empirical error among its endpoints. The strong law of large numbers makes that finite maximum vanish. Taking a countable sequence of widths decreasing to zero proves the assertion. A measurable supremum can be obtained from a pointwise separable function class; otherwise the probability-one-event formulation expresses the same pathwise conclusion.

A measurable function $M$ is an integrable envelope of a function class for a class $\mathcal{H}$ under a probability law $P$ if $|h(x)|\le M(x)$ for every $h\in \mathcal{H}$ and $PM<\infty$. Such an envelope permits the dominated convergence theorem to control shrinking function brackets.

For a scalar BV space function, clipping to $[-M,M]$ partitions its coarea formula for BV functions into middle and tail levels. The clipped function has the original superlevel sets at heights in $(-M,M)$. The positive tail levels of $u-T_{M}u$ correspond to original heights above $M$, and its negative tail levels to heights below $-M$. Thus their total variation seminorms add exactly. This is stronger than the triangle inequality and is specific to scalar monotone clipping. It supports residual-preserving clipping of an ROF minimizer.

Commuting creation operators make multiparticle states invariant under exchange of their labels. With $[a,a^{†}]=1$, the normalized number states $(a^{†}{)}^r|0⟩/\sqrt{r!}$ exist for all $r\ge 0$, giving unrestricted bosonic occupation numbers. The single-mode thermal sum is $Z={\sum }_{r\ge 0}{e}^{-\beta Er}=(1-e^{-\beta E}{)}^{-1}$ and the mean occupation is $(e^{\beta E}-1{)}^{-1}$. Thus the exchange symmetry and the Bose-Einstein distribution are two expressions of the same bosonic algebra.

For the blowup of a complex manifold at a point in complex dimension $n$, the differential gives a section of $K_{\tilde{M}}\otimes ({\varphi }^{\ast }{K}_M{)}^{-1}$ whose divisor on a complex manifold is $(n-1)E$. Indeed, a local chart has $x_i=u$ and $x_j=u{t}_j$ for $j\ne i$, so its Jacobian determinant is $\pm u^{n-1}$. The holomorphic line bundle associated to a divisor then gives the formula for the canonical bundle.

Under repeated identical amplitude damping channels, all excited population eventually relaxes to the ground state for any fixed $0<p\le 1$. The Bloch vector approaches $(0,0,1)$. For $p=1$ the limit is reached in one action, while $p=0$ is the identity channel and preserves every input. Keeping the zero-damping endpoint separate prevents an incorrect universal relaxation claim.

For a nonnegative symmetric unit-integral kernel for density estimation and $f\in H^2(\mathbb{R})$, the integral Taylor remainder and Minkowski inequality give $\Vert K_h\ast f-f{\Vert }_2\le h^2{\mu }_2(K)\Vert f^{\prime \prime }{\Vert }_2/2$. Translation preserves the L2 norm. For a signed kernel the same proof requires the absolute moment $\int u^2|K(u)|du$.

For the Benjamini-Hochberg procedure, replace a true-null p-value $P_i$ by zero and call the new rejection count $R^{(i)}$. The Benjamini-Hochberg leave-one-out identity identifies rejection with $R=r$ with the event $P_i\le \alpha r/m$ and $R^{(i)}=r$. If $P_i$ is uniform and independent of all the other p-values, each possible rejection count contributes $\alpha \mathbb{P}(R^{(i)}=r)/m$ to its expected false-discovery fraction. Summation gives $\alpha /m$ per true null and the displayed exact false discovery rate. Super-uniform independent nulls yield the corresponding upper bound.

Since the chirality matrix commutes with spinor Lorentz generators, $j_5^{\mu }=\bar{\psi }{\gamma }^{\mu }{\gamma }^5\psi$ transforms as a vector under proper Lorentz transformations and as a pseudovector under parity. Contracting two copies gives the Lorentz scalar $j_{5\mu }{j}_5^{\mu }$; the extra parity signs cancel. Lorentz invariance of this interaction does not require current conservation.

For $\beta =-b{g}^3$ and $\gamma =c{g}^2$, integrate the running coupling equation and then the anomalous-dimension multiplier. The correlator correction grows as a power $c/b$ of the logarithm of energy, rather than as a fixed power of energy. Higher perturbative orders change subleading terms while retaining this leading logarithmic exponent.

A Non-Abelian gauge theory can have a weak infrared fixed point when the one-loop coefficient of its renormalization-group beta function is small and positive while the two-loop coefficient is negative. For $a=g^2/(16{\pi }^2)$ and evolution in $\log {\mu }^2$, $\beta (a)=-{\beta }_0{a}^2-{\beta }_1{a}^3$ has the displayed zero and slope $-{\beta }_0^2/{\beta }_1>0$. Fundamental-fermion flavor number just below the asymptotic-freedom boundary produces this regime.

A scalar function in the BV space has an approximate jump at a point when its mean on each of two half-balls approaches a different finite value. The half-balls are determined by a unit normal ${\nu }_u$. Reversing the normal exchanges the two BV traces on a hypersurface. Their collection is the jump set of a BV function.

The within-block ANOVA stratum contains response vectors whose coordinates sum to zero separately in every block. For $b$ blocks of size $k$ its dimension is $b(k-1)$. A shared additive block effect cancels in every vector in this subspace.

Four infinite exterior arms entering a square in alternating primal-open and dual-open order force the two primal arms to be disjoint outside the square. If they met, their finite connecting crosscut together with a boundary arc would enclose one dual arm, preventing its escape to infinity. Connecting the two primal arms through the square creates a proper doubly infinite open path, separating the two remaining dual infinite tails. finite-energy property of Bernoulli percolation can then turn this configuration into two infinite dual clusters, contradicting uniqueness. Care is needed to preserve the exterior dual tails even if the finite modification destroys their original boundary attachments.

A minimal-genus Seifert surface has a $2g_s(K)$-by-$2g_s(K)$ Seifert matrix $V$. Each entry of $tV-V^T$ has degree at most one, so its determinant has breadth of a Laurent polynomial at most $2g_s(K)$. This lower bound on Seifert genus need not be sharp: an untwisted Whitehead double can have Alexander polynomial of a knot $1$ and positive Seifert genus.

Over an algebraically closed field of characteristic different from two, a degree-two map from a smooth genus one curve to the projective line has a deck involution. After choosing an origin, all its fiber divisors on an algebraic curve are linearly equivalent and have the same group sum $s$, by the Abel-Jacobi map of a genus-one curve. The involution is consequently $p\mapsto s-p$, including its fixed ramification points. Composing two such involutions gives a translation on an elliptic curve.