For an off-axis nearly occluding sphere with radial offset $e<d$, the local minimum gap is $d(\varphi )=d+e\cos \varphi$ to leading order. Each circumferential strip has pressure-driven conductance proportional to $d(\varphi {)}^{5/2}$, and the strips act in parallel. Thus its speed at fixed load is multiplied relative to the concentric value byConvexity and the zero average of cosine make this greater than one for nonzero $ϵ$. It increases strictly with $ϵ$: pair the positive and negative cosine contributions to the derivative, obtaining factors $c[(1+ϵc{)}^{3/2}-(1-ϵc{)}^{3/2}]>0$ for $c>0$. Consequently an off-axis sphere falls faster in this narrow-gap, no-through-flow regime.

Assume $2\le k<m$ and ${\bigcup }_i{S}_i=S$. Construct a two-player game with row $i$ for $S_i$ and columns $0,1,\dots ,n$. Column $0$ pays both players one; column $j>0$ pays $(1,0)$ when $j\in S_i$, and $(0,k/(k-1))$ otherwise. There is a Nash equilibrium whose row player's support of a mixed strategy has size $k$ exactly when $k$ rows cover the universe. A uniform mixture on a cover makes column $0$ a best response. Conversely, if some element is uncovered by the support, the column player's best responses are uncovered elements and the row player's current payoff is zero. The union promise provides a profitable row deviation. Without the union promise the equivalence is false: a globally uncovered element is a best response against every row mixture.

// Target: mathematical-optimization.bigb

For a fixed coherent sheaf on a smooth projective curve, one degree bound works for all divisors on an algebraic curve, rather than just powers of one ample divisor. Choose a point $P$. The complement of $P$ is affine, so the surjective transition maps on the finite-dimensional groups $H^1(C,\mathcal{F}(nP))$ eventually kill every class. The Euler characteristic of a coherent sheaf gives $h^0(\mathcal{O}(A))\ge \mathrm{deg}A+1-g$. Every divisor of sufficiently large degree is therefore linearly equivalent to $n_{0}P+E$ with $E$ effective. The point-supported quotient for adding $E$ transfers vanishing to that divisor. Torsion is first removed so that the relevant inclusions are injections of locally free sheaves.

The coefficient of $t^{-1}dt$ gives a trace on $H^1({\mathbb{P}}^1,{\Omega }^1)$. Multiplication followed by this trace pairs the Laurent bases of the Čech cohomology of twists on the projective line perfectly: $t^a$ pairs with $t^{-a-1}$. For any vector bundle, evaluation into ${\Omega }^1$ defines the same natural pairing. Saturate a rational rank-one subspace to a line subbundle, and induct on rank using the long exact sequence in sheaf cohomology and compatibility of evaluation with connecting maps. This proves the duality without using the later splitting theorem.

For vector bundles on a smooth projective curve, a dualizing sheaf is a line bundle $D_C$ with a trace $H^1(C,D_C)\to k$ making the evaluation pairings $H^i(C,E)\times H^{1-i}(C,E^{\vee }\otimes D_C)\to k$ perfect for $i=0,1$. Choose a finite generically separable map to the projective line. The trace-dual module for this finite map, tensored with ${\Omega }_{{\mathbb{P}}^1}^1$, is ${\Omega }_C^1$ by the different–differential identity. Finite pushforward and residue duality on the projective line then prove the pairings, rather than assuming the curve-duality theorem to establish them.

In a binary logistic regression, write the signed directional margin of observation $i$ as $(2y_i-1)x_i^{T}d$. Quasi-complete separation occurs when a nonzero direction $d$ makes all these margins nonnegative, with some zero and some strictly positive. Along coefficients $\beta +td$, $t\to \infty$, each observation's log-likelihood contribution is nondecreasing, and those with positive margins strictly increase toward their limiting value. Hence the unpenalized fit cannot have a finite maximum in that direction. A group containing only outcome zeros is a simple example: sending its indicator coefficient to $-\infty$ improves that group's likelihood while leaving all other observations unchanged. Zero fitted risk is then a boundary limit, and ordinary finite-coefficient Wald confidence intervals are unavailable.

A word in positive integers is a lattice word if every initial segment has at least as many $i$'s as $(i+1)$'s for every $i$. This is the reading-word condition in the Littlewood–Richardson rule. Under good-letter matching in a tableau word, every letter of a lattice word is good.

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.