Represent inputs accurately using few strongly active coefficients, for example by minimizing $\Vert x-Az{\Vert }^2/2+\lambda {\sum }_i|z_i|$. Learning basis functions on natural images can yield localized oriented filters resembling simple-cell receptive fields. This demonstrates one possible functional account of selectivity, not a proof that cortex implements that exact objective or learning algorithm.

Construct low-dimensional coordinates representing a dissimilarity matrix. Different methods preserve different features: classical multidimensional scaling approximates a centered Gram matrix, while other methods minimize discrepancies between fitted and supplied distances or preserve distance ordering. The method and dimension determine what geometric claims the resulting display supports.

For $\Phi =U+\alpha aU/r$ and $\chi =\beta a^{3}U\cdot \nabla (1/r)$ in the Unscaled Papkovich–Neuber representation, the Newtonian fluid stress tensor gives the displayed traction at $r=a$. The $\alpha$ contribution is the Stokeslet stress; the $\beta$ contribution is twice the viscosity times the Hessian of its harmonic scalar potential. The prefactor is $6\mu /a$, not $12\mu /a$. The rigid translating-sphere limit is a direct normalization check.

For an unramified extension of local fields $L/K$, an integral element whose residue generates $k_L/k_K$ generates $L/K$: its degree is at least the residue degree $f=[L:K]$. The reduced powers $1,\bar{\alpha },\dots ,{\bar{\alpha }}^{f-1}$ are a basis of the residue field extension. If $x=\sum c_j{\alpha }^j$, factor out a coefficient of least valuation. Reduction of the remaining sum is nonzero by that residue basis, so $v_L(x)={\min }_j{v}_K(c_j)$. Consequently $x$ is integral exactly when all $c_j$ are integral.

The left multiplication representation of a C-star algebra is isometric: test $L_a$ on $a^{\ast }/\Vert a\Vert$ to get $\Vert L_a\Vert =\Vert a\Vert$. For a nonunital algebra, its closed image plus the identity operator is the algebraic unitization. For a formal $z=a+\lambda 1$ and $b\in A$, $\Vert zb{\Vert }^2=\Vert b^{\ast }{z}^{\ast }zb\Vert \le \Vert b\Vert \Vert L_{z^{\ast }z}b\Vert$. Thus $\Vert L_z{\Vert }^2\le \Vert L_{z^{\ast }z}\Vert \le \Vert L_{z^{\ast }}\Vert \Vert L_z\Vert$; applying the same estimate to $z^{\ast }$ proves the algebra involution is isometric and the C-star identity holds. This gives a unitization without first assuming an operator representation on a Hilbert space.

Vector normalization converts a nonzero vector in a normed vector space into a unit vector. Absolute homogeneity of a norm givesThe direction is unchanged by multiplying by a positive real scalar. For $c>0$, $\Vert cv\Vert =c\Vert v\Vert$ and $\widehat{cv}=\hat{v}$; for a negative real scalar, $\widehat{cv}=-\hat{v}$. Thus $(3,4)$ and $(6,8)$ normalize to the same unit vector, while $(-3,-4)$ normalizes to its negative. This decomposition is useful for specifying a direction independently of a force, velocity or displacement magnitude.

In a complex normed vector space, a nonzero complex scalar $c$ instead gives $\widehat{cv}=(c/|c|)\hat{v}$. The remaining complex argument is not removed by normalization. For a pure quantum state, vectors differing only by this overall global phase represent the same physical state; vector normalization alone does not identify them.

The zero vector cannot be normalized this way. The construction also applies in a function space: an eigenfunction of a Sturm-Liouville problem can be scaled to unit weighted norm byThe positive weight $w$ specifies the weighted inner product. Multiplication by this nonzero constant preserves the eigenvalue equation and its homogeneous boundary conditions.

The epimorphism $S\to S_{\mathrm{ab}}\cong \mathbb{Z}$ has preimage of $n\mathbb{Z}$ as the unique index-$n$ subgroup. Thus every finite-index subgroup contains the commutator subgroup. Residual finiteness makes their intersection trivial, forcing the commutator subgroup to vanish and $S\cong \mathbb{Z}$.

// Target: geometry-and-topology.bigb

Let $K$ be complete for a Non-Archimedean absolute value and let $V$ be a finite-dimensional vector space with a norm satisfying $\Vert av\Vert =|a|\Vert v\Vert$ and the ultrametric inequality. All such norms are equivalent to the maximum of coordinates in any basis. The upper bound is immediate. For the lower bound, induct on dimension. The span $W$ of all but the last basis vector is complete by induction, hence closed for the given norm. Thus $\delta ={\inf }_{w\in W}\Vert e_n-w\Vert >0$. For $v=w+a{e}_n$, $\Vert v\Vert \ge |a|\delta$; also $\Vert w\Vert \le \max (\Vert v\Vert ,|a|\Vert e_n\Vert )$. The induction bound on coordinates of $w$ now bounds all coordinates by a constant times $\Vert v\Vert$. This proof needs no local compactness.

An integral quadratic lattice that is both an even lattice and a unimodular lattice. The K3 intersection lattice has signature $(3,19)$, while the positive-definite E8 lattice has rank eight.

On a bounded domain, classical solutions continuous on the closure have difference $w=u-v$ satisfying $D_i(A_{ij}{D}_{j}w)=0$, where $A={\int }_0^1{F}_{pp}(x,Dv+tDw)dt$. The coefficients are uniformly elliptic and locally smooth. Apply the weak maximum principle on inner domains whose closures lie in the original domain, then let their boundary distances tend to zero. The common continuous boundary values make their boundary differences tend uniformly to zero, proving uniqueness without assuming global bounds on derivatives of the linearized coefficients.

For scalar minimizers of $\int F(Du)$ with $F\in C^2({\mathbb{R}}^n)$ and globally bounded uniformly positive Hessian matrix, no higher differentiability of $F$ is needed for interior gradient Hölder continuity. The weak Euler-Lagrange equation is $\mathrm{div}DF(Du)=0$. Difference quotients solve uniformly elliptic equations with coefficient matrices obtained by averaging $D^{2}F$ along gradient segments. The Caccioppoli inequality and difference quotient characterization give $u\in H_{\mathrm{loc}}^2$. The Sobolev chain rule then gives $\mathrm{div}(D^{2}F(Du)D(D_{k}u))=0$. The De Giorgi-Nash-Moser theorem makes each derivative Hölder continuous, with exponent depending only on dimension and the ellipticity ratio. This scalar conclusion should not be assumed for vector-valued elliptic systems.

For $2\le p<\infty$, the Clarkson inequality isFor $1<p\le 2$ and $q=p/(p-1)$, the other Clarkson inequality isTogether these inequalities prove uniform convexity of $L^p$ for $1<p<\infty$.

Every nonempty closed convex subset of a uniformly convex Banach space has a unique nearest point to each vector. A minimizing sequence is Cauchy: otherwise uniform convexity would put some midpoint strictly closer than the infimum. Completeness and closedness give its limiting point. In ${\ell }^p$, $p\ge 2$, the Clarkson inequality makes this quantitative: the distance between two minimizing vectors tends to zero after subtracting the midpoint's lower distance bound. Nonemptiness is required.

A fixed-mass spherical shell has acceleration $\ddot{R}=-GM/R^2+\Lambda R/3$, in units $c=1$. At a finite maximum radius $R_m$, inward acceleration requires $\eta =\Lambda R_m^3/(3GM)<1$. Equality is the unstable static radius, approached only in the limiting separatrix case rather than a finite expansion-to-collapse turnaround. The effective potential reaches its maximum there, so an initially expanding bound inner-branch shell also needs energy below that maximum.

Couple the process with independent uniform edge labels, and expose $p_{-}=(\log n-\frac{1}{4}\log \log n)/n$. A spanning-tree union bound excludes components of orders $2$ through $n/\log n$, and a cut bound excludes orders up to $n/2$. The graph has one large component and at most $\log n$ isolated vertices, with at least one isolated vertex. Before $p_{+}=(\log n+\frac{1}{4}\log \log n)/n$, the probability of any new edge between those isolated vertices is $O((\log n{)}^2\log \log n/n)=o(1)$. All have joined the large component by $p_{+}$ by the isolated-vertex threshold in the Erdős-Rényi model. Throughout this window the only obstruction to connectivity is an isolated vertex, establishing the equality of hitting times.

For initial thickness $H(1+k^2{z}_0^2)$ and constant positive force, the material-map integral is dominated by a shrinking region around the minimum thickness as $\Delta \uparrow H$. With $k\ne 0$, the integral of $1/[H-\Delta +H{k}^2{z}_0^2]$ gives the displayed inverse-square-root divergence, whose coefficient is independent of the initial half-length. If $k=0$, every material point thins equally and $L=H{L}_0/(H-\Delta )$, with the stronger inverse-time divergence $L\sim 4\mu H{L}_0/[F(t^{\ast }-t)]$. These are ideal slender-sheet predictions up to their singular limit.

Let $0\le s\le k\le n$ and form the integer matrix with rows indexed by $S\subseteq [n]$, $|S|\le s$, columns indexed by $k$-subsets $T$, and entry $M_{S,T}=1_{S\subseteq T}$. Its matrix rank over any finite field ${\mathbb{F}}_p$ is at most $\left(\genfrac{}{}{0ex}{}{n}{s}\right)$. Over the rational numbers, the row of a $j$-subset $S$ is ${\left(\genfrac{}{}{0ex}{}{k-j}{s-j}\right)}^{-1}$ times the sum of the rows of all $s$-subsets containing $S$. Thus the rows of degree exactly $s$ span, giving the bound over the rational numbers. Every larger square minor therefore has integer determinant zero, and remains zero modulo $p$. The matrix rank cannot increase under this reduction. This argument avoids dividing by an integer that might vanish in the finite field, and gives the sharp uniform-layer step in the Frankl-Wilson theorem.

The displayed solenoidal vector field has $\nabla \times \mathbf{b}=k\mathbf{b}$, ${\nabla }^2\mathbf{b}=-k^2\mathbf{b}$ and ${\mathbf{e}}_z\times \mathbf{b}=-i\mathbf{b}$. Thus the non-gradient part of its linearized Lorentz force about $\mathbf{B}=Jr{\mathbf{e}}_{\varphi }$ vanishes. Choosing $p_1=-\mathbf{B}\cdot \mathbf{b}/{\mu }_0$ cancels the magnetic-pressure gradient, and $\mathbf{u}=0$, $\omega =0$ is a neutral perturbation. It is regular at the axis and realizes the endpoint $m=2$, $\alpha =0$. A boundary condition can exclude this growing-in-radius field, but without such a condition strict positivity at the endpoint cannot be asserted.

Nonnegative weights ${\lambda }_A$ on coordinate subsets form a fractional uniform cover when the displayed equality holds for each coordinate. Rational weights can be multiplied by a common denominator to become an integer uniform cover. The uniform covers theorem then gives $\log |S|\le {\sum }_A{\lambda }_A\log |S_A|$. Exact equality of coordinate coverage matters when projection measures may be less than one.

A subset of a normed space is norm bounded if it is bounded under every continuous linear functional. Apply the Uniform boundedness principle on the complete dual space to the evaluation family $f\mapsto f(x)$. Hahn-Banach theorem identifies the evaluation norm with $\Vert x\Vert$, even when the original normed space is not complete. As a consequence, two norms with exactly the same continuous dual as sets must be equivalent: apply this criterion to either norm's unit ball in the other norm.