For a linearly independent partition-normalized B-spline basis, equivalence of norms gives a lower coefficient-stability constant. The upper bound follows from the subpartition of unity for B-splines. A knot-independent stability bound can be chosen depending on spline order. This norm comparison transfers matrix estimates to the B-spline interpolation operator norm.

The auxiliary-metric brane action is a quadratic embedding action with an independent worldvolume metric. Its embedding equation is ${\partial }_{\mu }(\sqrt{-\gamma }{\gamma }^{\mu \nu }{\partial }_{\nu }{X}^a)=0$. Its metric equation is $h_{\mu \nu }-{\gamma }_{\mu \nu }({\gamma }^{\rho \sigma }{h}_{\rho \sigma }-(p-1))/2=0$. For $p\ne 1$, tracing determines ${\gamma }^{\rho \sigma }{h}_{\rho \sigma }=p+1$ and hence $\gamma =h$; eliminating the auxiliary field gives twice the invariant worldvolume volume, before including the overall brane tension. The Weyl-invariance exception for the string among branes prevents unique elimination at $p=1$.

In a cyclic group of arbitrary order $q$, a Bohr set with $d$ frequencies, defined by $\Vert rx/q\Vert \le 1/16$, contains an arithmetic progression of distinct residues of length at least $q^{1/(d+1)}/32$. Partition the $d$-dimensional cube into $Q^d$ boxes, with $Q=\lfloor q^{1/(d+1)}\rfloor$. The pigeonhole principle supplies $1\le t\le Q^d$ with $\Vert rt/q\Vert \le 1/Q$ simultaneously. The residues $jt$, $0\le j\le \lfloor Q/16\rfloor$, remain in the Bohr set and do not wrap around the cyclic group. The weaker exponent permits composite moduli without assuming that every nonzero step has large order.

Loss of summer sea ice need not be irreversible at fixed external forcing. Open water can release excess heat in autumn, and thin ice grows faster than thick ice during winter because it offers less resistance to thermal conduction. Continued climate change can nevertheless shift the long-term ice cover downward. A one-summer perturbation and persistent forcing are different physical experiments.

A monomial alternant is the determinant formed from powers with an integer exponent tuple $\ell$. It is a polynomial when the exponents are nonnegative, and otherwise a Laurent polynomial. Equal exponents give equal columns and zero determinant; interchanging exponents changes its sign. Ordered strictly decreasing nonnegative exponents give a basis of alternating polynomials, by grouping monomials into permutation orbits.

For the constant gas-radiation ratio vertical disc structure, the integrated alpha prescription $\bar{\nu }\Sigma =\int \mu dz=(\alpha /\Omega )\int pdz$ gives the displayed mean kinematic viscosity and $\lambda /(1+\lambda )=\alpha H\Omega \kappa \Sigma /(8c)$. Uniform dynamic viscosity and nonuniform pressure are compatible because this prescription is an integrated equality. Imposing the corresponding pointwise equality would not preserve the exact vertical solution.

Over a field $k$, an affine curve parametrized by positive integer powers of one parameter. Its coordinate ring is the subalgebra generated by those powers, and its defining prime ideal is the kernel of $k[x_1,\dots ,x_n]\to k[t]$ sending $x_i$ to $t^{a_i}$. Equal exponent sums yield binomial relations. Studying those relations connects an additive semigroup of exponents to the geometry of the curve.

Mixing two lotteries with the same third lottery and the same positive weight preserves their strict preference order. For affine utility on probability measures, the difference of the two mixed utilities is the original difference multiplied by that weight. At weight zero the strict preference disappears, so positivity is essential.

For three strictly ordered affine-utility values $u>v>w$, the middle lottery is indifferent to the mixture of the extreme lotteries with weight $p=(v-w)/(u-w)$ on the best one. This weight lies strictly between zero and one. It is the exact mixture-indifference property underlying cardinal utility comparisons.

Two real affine functionals on a convex set of lotteries that represent the same strict preferences differ by $V=aU+b$ with $a>0$. Equal utility values must first be shown to give equal values of the other functional. The induced function on the real interval of utility values preserves mixtures; anchor interpolation, including exterior points represented through an anchor, makes it linear throughout. Constant representations are handled separately and still admit a positive slope.

Set $U(x)=U_0({\delta }_x)$. On a finite measurable space, points in the same atom of a sigma-algebra give the same Dirac measure, so $U$ is a measurable function and constant on atoms. Every probability measure is a finite mixture of representative Dirac measures with its atom masses as weights. Affineness then proves the expected-utility representation, even when the sigma-algebra is smaller than the power set.

The map $A$ associated with an affine map $x\mapsto Ax+b$; it is independent of the translation $b$. An affine Euclidean isometry has an orthogonal linear part, and an isometry between flat tori has a lift whose linear part carries one Euclidean lattice onto the other.

The affine variety $X=V(xy-zw)\subset {\mathbb{A}}^4$ has dimension three and a unique singular point of an algebraic variety at the origin. The planes $V_X(x,z)$ and $V_X(y,w)$ have dimension two but intersect only at that origin. This is a basic failure of intersection dimension estimates at singular points.

For a finite-dimensional vector space $V$, the affine group consists of invertible affine maps $x\mapsto Ax+b$, where $A$ belongs to the general linear group and $b\in V$. Its product is $(A,b)(A^{\prime },b^{\prime })=(A{A}^{\prime },b+A{b}^{\prime })$, making it the semidirect product of the translation vector space and its general linear group. On the real line the positive-dilation subgroup is the real affine group used in the adjacent article.

The half-line solution of $C_t+U{C}_x=D{C}_{xx}$ with initially zero concentration, unit inlet concentration for positive time, and decay at infinity at fixed time. Both complementary error functions are needed to satisfy the inlet exactly. A Laplace transform gives $\tilde{C}=p^{-1}{e}^{(U-\sqrt{U^2+4Dp})x/(2D)}$. A finite layer requires an additional downstream boundary condition.

For a passive scalar on a half-line, impose constant total solute flux at its inlet, with initially zero concentration and constant positive advection speed $U$ and mass diffusivity $\mathcal{D}$. A Laplace transform in time givesWith $\xi =Ux/\mathcal{D}$, $\vartheta =U^{2}t/\mathcal{D}$ and $z_{\pm }=(\xi \pm \vartheta )/(2\sqrt{\vartheta })$, its inverse isThe complementary error function describes the smeared front. This is a flux boundary condition, not the constant-concentration inlet solution. At large axial Péclet number, both have the leading advancing front $\frac{1}{2}\mathrm{erfc}(z_{-})$.

Suppose $|f|\le 1$ on a finite cyclic group, $H$ has density $\beta$, and for each $h\in H$ the multiplicative derivative has a Fourier coefficient on a finite abelian group of magnitude at least $\eta$ at $\theta (h)$. Align the correlation phases, apply Cauchy-Schwarz, group pairs by their shift and frequency differences, and apply Parseval identity. This proves the displayed lower bound on additive energy of a frequency graph without replacing exact frequency equalities by approximate ones.

Near a regular Lagrangian torus, choose a one-form $\lambda$ with $d\lambda =-\omega$ and a basis of fibre cycles ${\gamma }_j$. The action $I_j=(2\pi {)}^{-1}{\int }_{{\gamma }_j}\lambda$ has differential of a smooth map $d{I}_j=(2\pi {)}^{-1}{\sum }_i{t}_i^{(j)}d{F}_i$, where $t^{(j)}$ is the corresponding joint-flow period. Nonsingularity of the period matrix makes these actions local coordinates transverse to the tori.

A sheet of mass per area $m$ and elastic-sheet tension $T$ backed by an identical outgoing fluid half-space presents the stated surface acoustic impedance to the other half-space, using upward sheet velocity $i\omega \eta =-V$. The fluid term is its normal acoustic impedance; the remaining term is structural. Infinite elastic-sheet tension suppresses every fixed nonzero spatial wavenumber, but not $k=0$, whereas infinite mass suppresses even spatially uniform motion. A massless untensioned sheet between identical fluids is transparent.

A harmonic line force localized at one point of a two-dimensional sheet gives $\hat{\eta }=F/D$ and $\Delta =\gamma D$ in the Fourier transform. The outgoing upper pressure is the displayed amplitude. Dividing by $c_0^2$ converts it to acoustic density perturbation. The method of steepest descent gives cylindrical sound from the saddle point, while crossed zeros of the dispersion relation give guided or leaky modal residues. Retaining $k_0=\omega /c_0$ is essential for dimensionally consistent spreading and prefactors.