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Assume the radial moment integral exists. Rotation through $\varphi$ about the third axis multiplies $x_1+i{x}_2$ by $e^{i\varphi }$ and preserves the region, volume element and radial weight. The integral equals $e^{in\varphi }$ times itself for every angle, so it vanishes. For degree two and four this also follows by contracting the isotropic sphere moments with the complex null vector $(1,i,0)$, whose bilinear square is zero. Odd moments vanish under inversion.

Away from the origin the Hessian matrix of $r^{-1}$ is $(3x_k{x}_l-r^2{\delta }_{kl})/r^5$. Multiplication by $x_i{x}_j$ makes the singularity locally integrable. Rotational invariance and the symmetries in each index pair reduce the integral to $A{\delta }_{ij}{\delta }_{kl}+B({\delta }_{ik}{\delta }_{jl}+{\delta }_{il}{\delta }_{jk})$. The Laplacian trace gives $3A+2B=0$, while contracting the two pairs as $k=i,l=j$ gives $3A+12B={\int }_{r<a}2/rdV=4\pi a^2$. Solving these two equations yields the displayed expression.

Writing an isophote's radial residual about its fitted ellipse as $\delta r=a_4\cos 4\theta +\cdots$, positive $a_4$ extends the major and minor axes and gives a pointed, disky shape; negative $a_4$ extends diagonal directions and gives a boxy shape. The sign stated here is for radial residuals, not an unconverted brightness residual. Massive slowly rotating elliptical galaxies commonly have boxy isophotes, while embedded rotating disks commonly make lower-mass elliptical galaxies disky.

The area of a surface of revolution about the horizontal axis is $2\pi \int |y|\sqrt{1+y^{\prime 2}}dx$. For a curve below the axis with endpoints $(\pm a,0)$ and fixed arc length, its stationarity is equivalent to stationarity of $\int (y+\lambda )\sqrt{1+y^{\prime 2}}dx$. The Beltrami identity gives $(y+\lambda )/\sqrt{1+y^{\prime 2}}=C$. Choosing $C=1/k$, imposing the endpoints and integrating the length gives the displayed curve. The function $\sinh s/s$ increases strictly from $1$ to infinity on $s>0$, so every $l>2a$ determines exactly one positive $k$. Stationarity alone does not establish a global minimum.

Zeros and poles describe the positive and negative local orders of a meromorphic function. Locally $f(z)=(z-a{)}^{m}g(z)$ with $g$ holomorphic and nonzero at $a$: $m>0$ is a zero, $m<0$ is a pole, and $m=0$ is a regular nonzero value. A simple pole has $m=-1$, and the order of vanishing records $m$.

Let $H(\sigma )={\sum }_{\{x,y\}}{J}_{xy}{\sigma }_x{\sigma }_y+{\sum }_x{h}_x{\sigma }_x$ on a finite graph, with each undirected edge counted once, $J_{xy}\ge 0$, and Ising spins ${\sigma }_x\in \{-1,1\}$. Here $H$ is a dimensionless log-weight: the probability mass function is $\pi (\sigma )=e^{H(\sigma )}/Z$. If $E$ denotes physical energy, the convention is $H=-\beta E$, not $H=E$.

Order configurations coordinatewise, so $(\sigma \vee \tau {)}_x=\max ({\sigma }_x,{\tau }_x)$ and $(\sigma \wedge \tau {)}_x=\min ({\sigma }_x,{\tau }_x)$. Define $U=\{x:{\sigma }_x=1,{\tau }_x=-1\}$ and $W=\{x:{\sigma }_x=-1,{\tau }_x=1\}$. An edge contributes zero to the displayed difference unless it joins $U$ to $W$. On such an edge the joined and met configurations each have spin product $+1$, whereas both original configurations have product $-1$. Its contribution is therefore $4J_{xy}$. Each field term cancels because the coordinatewise maximum and minimum preserve the sum of the two spins. Consequently,Exponentiating gives $\pi (\sigma \vee \tau )\pi (\sigma \wedge \tau )\ge \pi (\sigma )\pi (\tau )$, the FKG lattice condition. The corresponding FKG inequality implies positive association of random variables: increasing observables have nonnegative covariance. The sign convention matters: using a positive ferromagnetic log-weight as though it were physical energy in $e^{-\beta E}$ would reverse which couplings favor alignment.

With the convention $H=J{\sum }_{⟨ij⟩}{\sigma }_i{\sigma }_j-h{\sum }_i{\sigma }_i$, positive $J$ favors opposite neighboring Ising spins. On a bipartite graph, the Néel order parameter measures opposite sublattice magnetizations; it is distinct from the uniform magnetization. Changing the signs of spins on one sublattice maps the zero-field model to the ferromagnetic Ising model.

A fixed Ising boundary condition pins the Ising spins on a specified boundary, leaving the interior distributed with the original interaction weights. The boundary can be the inner graph vertex boundary of a finite box, or an exterior neighboring set if that convention is explicitly chosen. Conditioning on pinned Ising spins preserves the FKG lattice condition of a ferromagnetic Ising model. The FKG inequality then makes interior expectations of increasing observables nondecreasing in the pinned boundary Ising spins.

For the infinite one-dimensional coupling matrix $J_{ij}=J{e}^{-\kappa |i-j|}$ with $J,\kappa >0$, its Fourier symbol is $J\sinh \kappa /(\cosh \kappa -\cos q)$. Its inverse is nearest-neighbor even though the original interaction is not. The inverse quadratic form equals $[2J\sinh \kappa {]}^{-1}{\sum }_j(m_j-m_{j+1}{)}^2+[\tanh (\kappa /2)/J]{\sum }_j{m}_j^2$. Finite open chains add endpoint corrections; a periodized coupling gives the same translation-invariant formula.

A $\pi$ rotation about the x-axis on one qubit conjugates $Z\otimes Z$ to its negative. With ideal instantaneous pulses, two equal free-evolution intervals separated by this pulse and its inverse cancel an Ising coupling exactly. Finite-duration pulses require accounting for the coupling during the pulses; the error is small only when $|J|$ times the pulse duration is small, or when the control is explicitly designed to compensate it.

A real group representation is absolutely irreducible if its complexification is irreducible. For a finite group this is equivalent to having only scalar real endomorphisms commuting with the representation. Independent coordinate sign changes force a commuting matrix to be diagonal; a transitive cyclic permutation of the coordinates then forces its entries equal. Alternatively, the coordinate projections and the cyclic permutation directly prove irreducibility over either field.

A group representation over a field is absolutely irreducible if it remains irreducible after every field extension. For a finite-dimensional representation it is enough to check an algebraic closure. Irreducibility over the original field alone is weaker: the two-dimensional real rotation representation of the cyclic group of order three splits into two lines over the complex numbers.

A steady incompressible jet at atmospheric pressure conserves Bernoulli's equation and volume flux. Its radius contracts as $R=a[u/u_0{]}^{-1/2}$. After impact, a thin nearly horizontal axisymmetric film has $2\pi rhv=\pi a^2{u}_0$. Evaluating Bernoulli's equation on its free surface gives $v^2=u_0^2+2g(H-h)$, hence $a^4/(4r^2{h}^2)=1+2g(H-h)/u_0^2$. This inviscid thin-film description applies away from the impact region and does not describe a dissipative hydraulic jump.

The expected value of a random financial return $R$. For asset return vector $R$ with mean $\mu$, a portfolio of weights $w$ has expected return $w^T\mu$ by linearity of expectation. This measures average payoff growth rather than a guaranteed realized gain.

Selling a borrowed asset and later buying it back to return it. A negative portfolio weight represents a short position. The position benefits from falling prices and loses from rising prices; its sale proceeds may be invested in the bank account. Unrestricted short positions are a modelling assumption, not a guarantee of unlimited practical borrowing.

Spreading a portfolio across assets can reduce risk that is not shared by them. For centered asset risks with covariance matrix $\Gamma$, the variance of the weighted risk is $w^T\Gamma w$. Equal weights across $n$ independent risks with common variance $s^2$ give variance $s^2/n$, whereas equal weights across risks that are all the same random variable leave variance $s^2$. Thus small individual weights do not by themselves remove common risk; the covariance criterion for diversification of factor residuals identifies sufficient dependence and weight conditions.

Define $h(x)=|x|\log (e\pi /|x|)$ on $[-\pi ,\pi ]$, set $h(0)=0$, and extend periodically. Concavity and monotonicity on $[0,\pi ]$ give $\omega (h,\delta )=\delta \log (e\pi /\delta )$ for $0<\delta \le \pi$. Its cosine Fourier coefficients areOne integration by parts gives the formula; the remaining integral is $\log j+O(1)$ by the Dirichlet test. All these coefficients are nonpositive. The de la Vallée Poussin sum $V_n=2{\sigma }_{2n}-{\sigma }_n$ reproduces degree-at-most-$n$ trigonometric polynomials and has operator norm at most three. Hence $L_n(f)=f(0)-V_{n}f(0)$ has norm at most four and annihilates those polynomials. Its weights on coefficients are zero up to $n$, nonnegative thereafter, and one from $2n$ onwards, so $|L_n(h)|\ge {\sum }_{j\ge 2n}|a_j|\gtrsim \log n/n$. Thus $E_n(h)\ge |L_n(h)|/4\gtrsim \log n/n$; the first Jackson theorem for periodic approximation proves the matching upper bound. This cusp and the critical Weierstrass modulus have the same first-modulus order but different best-approximation rates.

The Born approximation for scalar wave scattering relates an incident plane wave and the transverse Fourier transform of its scattered field to samples of the three-dimensional Fourier transform of a scattering potential on a shifted Ewald sphere. One incident direction at one frequency samples a surface, rather than an entire three-dimensional frequency region. Additional illuminations, frequencies or structural constraints are needed for a full reconstruction.

For an outgoing field outside a sphere of radius $R$, write $u_s=k{\sum }_{n,m}{i}^{n+1}{a}_n^m{h}_n^{(1)}(kr)Y_n^m$. Its far-field pattern is $\sum a_n^m{Y}_n^m$. If its trace on that sphere lies in L2 space, Parseval's identity gives $k^2\sum |h_n^{(1)}(kR){|}^2|a_n^m{|}^2<\infty$. The fixed-argument growth of spherical Hankel functions yields the displayed equivalent range condition, with the low-order weights defined finitely. Square summability of the far-field coefficients alone does not ensure continuation to the chosen sphere.

Fix ordered branch positions and conjugacy classes $C_i$ in a finite group $G$. The generating product-one tuples $(g_1,\dots ,g_r)\in \prod C_i$ form the inner Nielsen set. The type is rigid when this set is nonempty and a single simultaneous-conjugation orbit. Over $\mathbb{C}(T)$ this fixes the connected Galois cover up to isomorphism: tuples prescribe the monodromy epimorphism of the punctured sphere, and conjugate epimorphisms give isomorphic covers; the local completions at punctures are uniquely determined by their cyclic monodromy.