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A fixed-aperture approach region at a boundary point of the unit disc. Its points approach the boundary point without becoming tangent to the circle. Convergence inside every such fixed region is a nontangential limit; the corresponding supremum defines a non-tangential maximal function.

Put a flex at $[0:1:0]$ with tangent line $Z=0$. The tangent section has a triple zero there, so $F(X,Y,0)=a{X}^3$ with $a\ne 0$. Smoothness gives $b\ne 0$. In characteristic different from two, replacing $Y$ by $Y+(cX+eZ)/(2b)$ completes the square and produces a Weierstrass equation of an elliptic curve using only projective linear coordinate changes.

For two independent samples of sizes $m,n$, rank their pooled observations and let $W_X$ be the sum of the ranks of the first sample. With continuous observations, the null hypothesis of a common distribution makes every allocation of $m$ ranks to that sample equally probable. A one-sided alternative that the first sample is stochastically smaller rejects for small $W_X$, equivalently small $U_X$. In a common-shape location model this tests a negative location difference. Without the common-shape assumption it is not simply a test of medians. The shifted statistic $U_X$ counts cross-sample pairs for which the first observation exceeds the second. Ties use average ranks and a conditional permutation test or a tie-adjusted normal approximation.

For independent paired observations, form differences $D_i=U_i-V_i$. Assume the differences are identically distributed, continuous and symmetric about a location $\delta$. Under the zero-location null hypothesis, the signs are independent fair signs conditional on their absolute values. Rank the absolute differences and add ranks with positive signs. A negative-location alternative rejects for small $W_{+}$. Its exact conditional distribution is that of a sum of an independently selected subset of ranks, each of the $2^n$ subsets equally likely. Thus its mean is $n(n+1)/4$ and its variance is $n(n+1)(2n+1)/24$. Symmetry is essential: a zero median alone does not justify this signed-rank null law. Zero differences are omitted in the usual version, and tied absolute differences receive average ranks.

A rank test is a hypothesis test whose statistic is a function of the ranks of observations, possibly together with fixed sample labels. Under independent observations from a common continuous probability distribution, exchangeable rank allocations give its exact null law without specifying that distribution. The Wilcoxon rank sum test is an example. Invariance under common increasing transformations follows because ranks do not change.

A sum of squares polynomial is a real polynomial expressible as $p={\sum }_i{q}_i^2$ for real polynomials $q_i$. In the homogeneous degree-$2m$ case the summands may be taken homogeneous of degree $m$. Every such polynomial is nonnegative on real inputs; the converse fails for multivariate polynomials in general.

A real polynomial $p$ is a sum of squares polynomial if $p={\sum }_{\ell }{q}_{\ell }^2$ for real polynomials $q_{\ell }$. Such a representation certifies global nonnegativity. If $m(z)$ is the vector of relevant monomials, the identity $p(z)=m(z{)}^{T}Gm(z)$ with a positive semidefinite matrix $G$ encodes a sum of squares through a Gram matrix. Equating coefficients is linear in $G$, so finding a certificate is a semidefinite program.

The nonlinear Schrödinger equation adds an amplitude-dependent interaction to Schrödinger-type wave evolution. Its Laplacian term disperses the complex wave amplitude, while the sign and form of the nonlinear term determine focusing or defocusing behavior. A one-dimensional cubic specialization is the focusing nonlinear Schrodinger equation; members of this broader family need not be integrable systems.

For unknown-location Pauli operators of weight at most $t$, there are $N(t)={\sum }_{j=0}^t{3}^j\left(\genfrac{}{}{0ex}{}{n}{j}\right)$ physically distinct errors modulo phase, including the identity. A nondegenerate quantum code assigns them mutually orthogonal error-image subspaces, each of dimension $2^k$, within the $2^n$-dimensional physical Hilbert space. Hence $N(t)2^k\le 2^n$. The hypothesis of nondegeneracy is essential to this dimension count.

For a complex scalar field with global $U(1)$ invariance and Lagrangian $\mathcal{L}=\frac{1}{2}|{\partial }_t\varphi {|}^2-\frac{1}{2}|\nabla \varphi {|}^2-U(|\varphi |)$, a Q-ball is a localized phase-rotating configuration with conserved Noether charge $Q=\int \mathrm{Im}(\bar{\varphi }{\varphi }_t)$. At fixed charge its real profile can minimizeSuch solutions require a suitable potential and charge range; the complex nature of a field alone does not ensure existence. A common condition is ${\inf }_{f>0}2U(f)/f^2<U^{\prime \prime }(0)$.

For integrable boundary data $g$ on the unit circle, this follows from the Poisson kernel bound $C_A\delta /({\delta }^2+d^2)$ in a fixed Stolz region. The arc of radius $\delta$ and successive dyadic annuli contribute at most $C_A{2}^{-k}Mg(\xi )$, which sum to the displayed estimate. This links harmonic extension to the Hardy-Littlewood maximal function and its strong $L^p$ bounds for $p>1$.

Shear banding is spatial separation into regions with different local shear rates. In simple inertialess planar shear, equal shear stress can coexist with different rates when the homogeneous constitutive curve is nonmonotonic. Gradient terms and boundary conditions can determine the interface and selected stress; a local multivalued flow curve alone does not provide that selection. Shear banding need not be confused with the ordinary smooth shear-rate variation in a pressure-driven channel.

A simple fluid has local stress determined by the history of deformation of a material element, rather than by spatial derivatives of that history. An isotropic incompressible simple fluid has an arbitrary isotropic pressure and an objective constitutive functional of the relative deformation gradients. Internal-variable equations can represent such a functional when the initial material state is fixed by the past history.

For a surjective module endomorphism of a Noetherian module, the chain $\ker f\subseteq \ker f^2\subseteq \cdots$ stabilizes at some $N$. If $f(x)=0$, choose $y$ with $f^N(y)=x$. Then $y\in \ker f^{N+1}=\ker f^N$, giving $x=0$. Thus the endomorphism is also injective and is an isomorphism.

For a short exact sequence $0\to N\to M\to M/N\to 0$, $M$ is a Noetherian module exactly when both the other terms are Noetherian modules. For the extension direction, an ascending chain $M_i$ has eventually constant intersections $M_i\cap N$ and images $(M_i+N)/N$. Equality of both forces equality of consecutive $M_i$: subtract an element with the same image and then use the common intersection.

For a finite integral inclusion $A=k[t_1,\dots ,t_r]\subset B$ of domains, a nonzero $b\in B$ has a monic minimal polynomial over the fraction field with nonzero constant term $a_0\in A$. Its equation gives $a_0\in bB\cap A$. Consequently the image of $V_B(b)$ under the finite morphism to ${\mathbb{A}}^r$ is contained in the proper Zariski-closed set $V_A(a_0)$. Finite morphisms preserve the dimension of a closed subvariety and its image, so this proves dimension drops for proper closed subsets after Noether normalization. A power of $a_0$, up to sign, is the field norm of $b$.

For $x^{\prime \mu }=x^{\mu }+\epsilon {\xi }^{\mu }$ and ${\varphi }_a^{\prime }(x^{\prime })={\varphi }_a(x)+\epsilon {\Delta }_a$, the fixed-coordinate field variation is $Q_a={\Delta }_a-{\xi }^{\nu }{\partial }_{\nu }{\varphi }_a$. Suppose action invariance is expressed as ${\delta }_0\mathcal{L}/\epsilon +{\partial }_{\mu }(\mathcal{L}{\xi }^{\mu })={\partial }_{\mu }{K}^{\mu }$. Writing ${\Pi }_a^{\mu }=\partial \mathcal{L}/\partial ({\partial }_{\mu }{\varphi }_a)$ and $E_a={\mathcal{L}}_{{\varphi }_a}-{\partial }_{\mu }{\Pi }_a^{\mu }$, the product rule gives ${\partial }_{\mu }{j}^{\mu }=-{\sum }_a{E}_a{Q}_a$. Thus $j^{\mu }$ is a conserved current on shell. A vanishing spatial boundary flux is additionally required for its Noether charge to be conserved. This formula includes internal transformations, translations and Lorentz transformations.

A virial current $V^{\mu }$ represents the trace of a conserved stress-energy tensor as a divergence. Then the dilation Noether current $D^{\mu }=x_{\nu }{T}^{\mu \nu }-V^{\mu }$ is conserved, since its divergence is $T^{\mu }{}_{\mu }-{\partial }_{\mu }{V}^{\mu }$. This establishes scale invariance; a traceless stress-energy tensor improvement is the extra condition needed for conformal currents.

For a symmetric conserved stress-energy tensor $\Theta$ and a Conformal Killing vector field $\omega$ in dimension $d$, the displayed conformal current has divergence $(\partial \cdot \omega ){\Theta }^{\mu }{}_{\mu }/d$. This follows by contracting the Conformal Killing equation with $\Theta$. A traceless tensor therefore gives a conserved current for every conformal generator.

The no-scale supergravity identity cancels the negative gravitino contribution, leaving $V=e^{|C{|}^2}|3C^2+\bar{C}(C^3+B){|}^2/(T+\bar{T}{)}^3$. Zero-energy minima occur at $C=0$ and, for $B\ne 0$, three phase-related nonzero solutions with $|B|=|C|(3+|C{|}^2)$. Their $T$ supergravity auxiliary field is nonzero if $B\ne 0$, while both real components of $T$ are flat on each vacuum branch. For $B=0$, the $C=0$ branch is instead supersymmetric and has zero gravitino mass from a superpotential.

For $s=S+\bar{S}>0$, $t=T+\bar{T}-|C{|}^2>0$, $K=-\log s-3\log t$ and a holomorphic superpotential independent of $T$, the exact F-term potential is $V=e^K[s^2|D_{S}W{|}^2+(t/3)|W_C{|}^2]$. Inverting the mixed $T,C$ metric gives $K^{T\bar{T}}=t(t+|C{|}^2)/3$, $K^{T\bar{C}}=t\bar{C}/3$, $K^{C\bar{T}}=tC/3$, and $K^{C\bar{C}}=t/3$. Substitution of $D_{T}W=-3W/t$ and $D_{C}W=W_C+3\bar{C}W/t$ cancels all mixed terms, leaving $3|W{|}^2+(t/3)|W_C{|}^2$; the first term cancels the supergravity negative term.