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The critical strip for the Riemann zeta function is $0<\Re s<1$.

local polynomial regression fits a low-degree polynomial by weighted least squares near each target point.

A linear estimator at $x_0$ has the form $\hat{m}(x_0)={\sum }_i{W}_i(x_0)Y_i$, where the weights depend on the design and tuning parameters but not on the observed responses.

For $C_t=k\nabla \cdot (C\nabla C)$ with total mass $2\pi M$,Its compact support has radius $r_0(t)=(32Mkt{)}^{1/4}$.

Let $B_1$ be a nondegenerate bilinear form on a finite-dimensional vector space $V$. Every bilinear form $B_2$ on $V$ has a unique representationby a linear map $\alpha :V\to V$. Thus a fixed nonsingular pairing converts another two-input form into a one-input linear operator.

For each $w$, the rule $v\mapsto B_2(v,w)$ is a linear functional. Nondegeneracy of $B_1$ identifies every such linear functional with $v\mapsto B_1(v,u)$ for a unique $u$; define $\alpha w=u$. Bilinearity of $B_2$ makes $\alpha$ linear, and uniqueness follows from the same nondegeneracy.

If $A$ and $C$ are the bilinear form matrices of $B_1$ and $B_2$ in one basis, thenFor the standard dot product on ${\mathbb{R}}^2$ and $B_2(v,w)=3v_1{w}_1+v_1{w}_2+v_2{w}_1+2v_2{w}_2$, the representing matrix is $\left(\begin{array}{cc}3& 1\\ 1& 2\end{array}\right)$. Finally, the right radical of $B_2$ is exactly $\ker \alpha$: annihilating every first input is equivalent to $\alpha w=0$.

A symplectic vector space is a vector space equipped with a nondegenerate bilinear form $\omega$ that is an alternating bilinear form: $\omega (v,v)=0$ for every vector $v$. This implies $\omega (v,w)=-\omega (w,v)$. When the characteristic is two, that latter equation alone does not imply the alternating condition. A symplectic vector space of finite dimension has even dimension: splitting off a hyperbolic pair leaves a smaller nondegenerate bilinear form, and repeating gives a decomposition into two-dimensional planes.

The intensity function $\lambda$ is the density of the expected counting measure: $\mathbb{E}[N(A)]={\int }_A\lambda (x)dx$.

The ultrametric inequality, also called the strong triangle inequality, bounds a direct distance by the larger of the two distances through an intermediate point:A metric space satisfying this rule is an ultrametric space. For a Non-Archimedean absolute value, the corresponding inequality is $|a+b|\le \max \{|a|,|b|\}$, and the associated metric is $d(x,y)=|x-y|$. The ordinary triangle inequality follows because the maximum of two nonnegative real numbers is at most their sum.

For example, the 2-adic absolute value gives the three points $0,1,8$ distancesEach distance is bounded by the maximum of the other two. The two largest distances are equal: every triangle in an ultrametric space has this property. Otherwise its unique longest side would exceed the maximum of the other two, contradicting the ultrametric inequality. The drawn lengths of a triangle on paper need not represent these distances.

The ultrametric inequality gives a useful exact rule when $|a|\ne |b|$:To see why, suppose $|a|>|b|$. The upper bound is $|a+b|\le |a|$; applying the ultrametric inequality to $a=(a+b)-b$ forces the reverse bound. Equal sizes can instead cancel: $|1{|}_2=|1{|}_2=1$, but $|1+1{|}_2=1/2$. This makes a uniquely largest term decisive in a sum, a principle used in the Newton polygon and in arguments about roots of a polynomial.

A reducible complex curve is a union of proper complex subcurves; intersecting components can create singular points.

A local obstruction must apply to neighborhoods that would have to model a Euclidean ball, rather than to an arbitrarily chosen disconnected or large neighborhood. For a nodal crossing, two complex discs meet only at their common center. Removing that center from any sufficiently small connected neighborhood leaves at least two connected components, one in each branch.

In contrast, a connected open subset of ${\mathbb{R}}^2$ remains connected after one point is removed: paths can be detoured around that point inside a small disc. Thus the nodal crossing has no neighborhood homeomorphic to a planar ball and is not a two-dimensional topological manifold there. Path avoidance in a surface gives the same local argument. Finding a disconnected punctured neighborhood without controlling its local form would not establish this obstruction.

In dimension $d$, the mass-critical sharp inequality isFor nonzero $u$, equality holds exactly on the amplitude, phase, translation, and spatial-scaling orbit of the NLS ground state $Q$; $u=0$ also gives equality. For $d=2$ this becomes $\Vert u{\Vert }_4^4\le 2\Vert Q{\Vert }_2^{-2}\Vert u{\Vert }_2^2\Vert \nabla u{\Vert }_2^2$.

A central series is a chain of normal subgroups whose successive quotients lie in the centers of the corresponding quotient groups.

A one-parameter family has a monotone likelihood ratio in a statistic $T$ when $f(x\mid {\theta }_1)/f(x\mid {\theta }_0)$ is nondecreasing in $T(x)$ whenever ${\theta }_1>{\theta }_0$.

For logits $z_1,\dots ,z_K$, the softmax probabilities are $p_k=e^{z_k}/{\sum }_j{e}^{z_j}$.

For a one-dimensional body with endpoints displaced by zero and $\Delta$, the static Navier-Cauchy equation reduces to $u^{\prime \prime }=0$ and gives the uniform strain solution $u(x)=\Delta x/L$.

The cusp--core problem is the tension between the central cusps of collisionless cold-dark-matter halo profiles and the approximately constant-density cores inferred for many dwarf and low-surface-brightness galaxies.

The characteristic density of a Navarro--Frenk--White profile is its normalization ${\rho }_s$ at fixed scale radius of a Navarro--Frenk--White profile $r_s$. It is not the density at that radius: substituting $r=r_s$ in the NFW profile gives $\rho (r_s)={\rho }_s/4$. Once $r_s$ is fixed, doubling ${\rho }_s$ doubles the enclosed mass at every radius. This separates a profile's density normalization from its radial scale.

The NFW scale radius is where the logarithmic density slope equals $-2$.

For the natural inflation inflaton potential $V(\varphi )=V_0[1+\cos (\varphi /f)]$, take $V_0>0$, $f>0$ and the rolling branch $0<{\varphi }_i<{\varphi }_f<\pi f$. Here $V^{\prime }<0$, so the slow-roll equations make $\varphi$ increase. Sincethe slow-roll e-fold count givesThe result is positive because the sine increases along this branch. Reversing these logarithms would give a negative expansion count. The upper endpoint should lie within the slow-roll approximation, or be understood as an approximate extrapolation to the end of cosmic inflation. Starting closer to the potential maximum at $\varphi =0$ increases this classical count, while an exactly stationary field there is not a monotonic rolling trajectory.

Normalize each player's positive pure-strategy payoff improvements to define a continuous self-map of the product of their mixed-strategy simplices. At a Brouwer fixed point every positive improvement must vanish, giving a Nash equilibrium.