For adjacent letters $i,i+1$, pair entries in the same column and leave those pairs fixed. In each row, interchange the numbers of unpaired $i$ and $i+1$ entries, preserving their weak order. Strict columns are preserved: a letter that could obstruct a change from $i$ to $i+1$, or conversely, would already have been paired. Applying the operation twice restores the tableau. The paired letters contribute equal amounts to the two contents, so the involution swaps their total multiplicities.

If $Q\ll P$ and $X$ is a semimartingale under both measures, the two integrals of any locally bounded predictable $H$ against $X$ agree up to $Q$-indistinguishability. Elementary integrals are the same increment sums. Bounded pointwise convergence of integrands, stochastic dominated convergence, and transfer of probability convergence from $P$ to $Q$ extend equality by the monotone-class theorem. Localize to remove boundedness.

// Target: probability-and-statistics.bigb

For all interval origins $a\ge 0$, the number of multiples of any $d$ in $(a,a+x]$ is $x/d+O(1)$ with an absolute error bound. The Selberg sieve diagonalization minimizes ${\sum }_{d,e\le z}{\lambda }_d{\lambda }_e/[d,e]$, subject to ${\lambda }_1=1$, at $1/G(z)$. The optimal Selberg weights have modulus at most one, so the total remainder is $O(z^2)$. Restoring the at most $z$ small primes gives $\pi (a+x)-\pi (a)\le z+x/G(z)+O(z^2)$. Take $z=\lfloor \sqrt{x}/(\log x{)}^2\rfloor$ and use the Selberg sieve denominator asymptotic. The resulting $o(1)$ is independent of $a$.

The two-layer cell seismic layer transfer matrix has reciprocal eigenvalues because its determinant is one. Writing them as $e^{\pm iKd}$ gives the displayed half-trace relation. At a real frequency, an absolute half-trace exceeding one forces an exponentially growing/decaying pair instead of propagating cell modes. A long finite stack then suppresses transmission in this stop band, without dissipating energy: the missing transmitted flux is reflected. Quarter-wave layers of unequal seismic impedance give half-trace $-\frac{1}{2}(q_1/q_2+q_2/q_1)<-1$.

Up to multiplication by a unit $\pm t^n$, the Alexander polynomial isIt satisfies ${\Delta }_K(t^{-1})\doteq {\Delta }_K(t)$ and ${\Delta }_K(1)=\pm 1$.

Cut a knot exterior along a Seifert surface and stack copies to form its infinite cyclic cover. Linking-dual generators for the homology of the cut-open piece give the positive and negative inclusion matrices $V$ and $V^T$ in row convention. The gluing relations therefore have matrix $tV-V^T$. The H0 gluing map is multiplication by $t-1$ and is injective, so these relations present the entire Alexander module of a knot. Its order is the determinant, up to a Laurent unit.

A closed symplectic manifold of real dimension four with $b_2^{+}>1$ has a nonzero Seiberg–Witten invariant of a four-manifold in its canonical structure, with value $\pm 1$. The positive index of the intersection form hypothesis avoids chamber dependence. This provides the symplectic connected-sum obstruction.

If a closed oriented smooth four-manifold splits as a connected sum $X_1\#X_2$ with both $b_2^{+}(X_i)>0$, its ordinary Seiberg–Witten invariants of a four-manifold vanish. Combined with Taubes nonvanishing theorem, this obstructs a symplectic structure in that orientation.

On a complete connected Riemannian manifold of nonpositive sectional curvature, the exponential map at every point is a covering map. If the manifold is simply connected, each exponential map is a diffeomorphism from a tangent space onto the manifold.

A Riemannian manifold is flat when every sectional curvature is zero. A complete simply connected flat $n$-manifold is isometric to Euclidean space ${\mathbb{R}}^n$.

The origin is a saddle point of a scalar function for every real parameter. For $\alpha >0$ four further critical points have $x^2=y^2=1/(2\alpha )$. Equal-sign coordinates give global maxima and opposite-sign coordinates give global minima, with values $\pm 1/(2e\alpha )$. The Hessian matrix there is $-4\alpha xy{e}^{-1}I$. Their distance from the origin is ${\alpha }^{-1/2}$, so as $\alpha \downarrow 0$ they escape every compact region. At nonpositive $\alpha$ the origin is the only critical point. This illustrates why the number of critical points on an unbounded domain can change without a finite degenerate critical point appearing.

A Tangent-disk Schottky group uses side-pairing disks with disjoint interiors but permits boundaries to meet. This convention must be distinguished from the strictly separated, purely loxodromic Schottky group convention. For example, the maps $A(z)=z+2$ and $B(z)=z/(iz+1)$ pair the half-plane disks $\Re z<-1,\Re z>1$ and the disks $|z-i|<1,|z+i|<1$. The ping-pong lemma gives a free discrete group. With $C=AB{A}^{-1}{B}^{-1}$, the subgroup generated by $B,C$ is conjugate by $M(z)=iz/(2-z)$ to the principal level-two modular group, generated by $\left(\begin{array}{cc}1& 0\\ 2& 1\end{array}\right)$ and $\left(\begin{array}{cc}1& 2\\ 0& 1\end{array}\right)$. Thus its Kleinian limit set contains the circle $|z-1|=1$ and its translate $|z-3|=1$. Its Hausdorff dimension is at least one and it cannot be homeomorphic to a single circle.

For nonnegative partition-normalized order-$k$ B-splines, let $Vf(t)={\sum }_{i}f({\tau }_i)N_i(t)$ with each sample point inside $[t_i,t_{i+k}]$. If $N_i(t)$ is nonzero, both $t$ and ${\tau }_i$ lie in an interval of length at most $k|\Delta |$. Their function values differ by at most the modulus of continuity at that length. Summing with nonnegative weights of total one proves the displayed bound. With fixed order and vanishing maximum knot gap, uniform continuity gives uniform convergence. The same argument works for bounded piecewise-continuous outputs if full interior knot multiplicities are allowed; their individual continuity should not then be asserted.

For $\sqrt{p}|000⟩+\sqrt{1-p}|111⟩$, a projective measurement of the number of ones in one party's $n$ local qubits gives a common type class of size $d_k=\left(\genfrac{}{}{0ex}{}{n}{k}\right)$. The conditional pure state is $d_k^{-1/2}{\sum }_{|x|=k}|x,x,x⟩$. It can be converted exactly and deterministically to a uniform common-label state of any dimension $m\le d_k$: one party measures an $m$-element subset $S$ with Kraus operator ${\Pi }_S/\sqrt{\left(\genfrac{}{}{0ex}{}{d_k-1}{m-1}\right)}$, and all parties relabel their common retained labels after receiving $S$. Choosing $m=2^f$ gives $f$ independent GHZ states. Stirling's formula and the weak law of large numbers give $f/n\to h_2(p)$ in probability. Measuring every local bit separately would instead destroy the coherence and leave a product state.

For $x>0$, the decimal significand is the unique $S_{10}(x)\in [1,10)$ such that $x=10^k{S}_{10}(x)$ for an integer $k$. Explicitly, $k=\lfloor {\log }_{10}x\rfloor$ and $S_{10}(x)=10^{\{{\log }_{10}x\}}$, using the floor function and the fractional part. For $0.0042$, $4.2$ and $4200$, the decimal significand is $4.2$ and the first nonzero digit is $\lfloor S_{10}(x)\rfloor =4$. In the classical terminology of logarithm tables, the logarithmic mantissa is $\{{\log }_{10}x\}$, which differs from the decimal significand $10^{\{{\log }_{10}x\}}$.

At $\omega =-3/2$, the combined scalar equation of Brans-Dicke theory becomes the matter consistency condition $T=0$. It does not determine $\square \Phi$ by division, since the wave coefficient has vanished. The original metric tensor and scalar Euler-Lagrange equations must be retained. Introducing scalar-dependent matter or a scalar potential changes the consistency condition.

Choose a rational basis of the dual of a vector bundle and an effective divisor $D$ bounding all its poles. Evaluation gives a generically injective map $E\to {\mathcal{O}}_C(D{)}^{\oplus r}$. A line subbundle maps nontrivially to at least one summand, so ${\mathcal{O}}_C(D)\otimes L^{-1}$ has a nonzero regular section. Consequently $\mathrm{deg}L\le \mathrm{deg}D$. This bound depends on $E$ and the chosen rational frame, and does not require $E$ to be stable.

For a rank-two bundle $E$, the sufficiently positive twist $E(m{p}_0)$ is globally generated. The nowhere-vanishing section of a globally generated bundle on a curve gives a subbundle ${\mathcal{O}}_C\subset E(m{p}_0)$. Twisting back produces a genuine line subbundle of degree $-m$, with locally free quotient. Thus line-subbundle degrees are bounded above but not below. Merely inserting a negative divisor on an algebraic curve into a fixed subline would allow a torsion quotient and would not establish this stronger conclusion.

A sample correlation matrix rescales a sample covariance matrix $S$ to unit diagonal, for variables with positive sample variance. Its off-diagonal entries are sample correlation coefficients. Strong correlations between predictors can indicate multicollinearity, but pairwise correlations alone do not exclude higher-order linear dependence.

If a density has polynomially decaying tails on both half-lines, its moment-generating function is infinite at every nonzero real parameter: exponential growth defeats polynomial decay on the positive tail for a positive parameter, and on the negative tail for a negative parameter. Thus no open real tilting domain exists for a saddlepoint density approximation, even when the distribution has finite mean and variance and a central limit theorem applies.