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The Riemann hypothesis asserts that every Nontrivial zero of the Riemann zeta function lies on the critical line $\Re s=1/2$.

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 $r_0(t)=(32Mkt{)}^{1/4}$.

If $B_1$ is a nondegenerate bilinear form on a finite-dimensional vector space and $B_2$ is any bilinear form, there is a unique linear map $\alpha$ such thatThe right radical of $B_2$ is $\ker \alpha$.

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.

If a punctured neighborhood has a different number of connected components from a punctured Euclidean ball, the point is not a manifold point.

In dimension $d$, the mass-critical sharp inequality isand equality holds exactly on the phase, translation, and scaling orbit of the NLS ground state $Q$. 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 NFW characteristic density fixes the normalization of the halo profile at a given scale radius.

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

For the cosine potential,