A stationary axisymmetric galactic distribution function depending on the specific orbital energy and axial specific angular momentum is a two-integral distribution. These quantities are conserved in a time-independent axisymmetric Newtonian gravitational potential, so the Collisionless Boltzmann equation holds by the chain rule. Even dependence on $L_z$ removes net azimuthal streaming velocity; an odd part can supply rotation without changing the mass density.

With continuous cash settlement and a money-market risk-neutral measure, discounted futures gains have zero drift; under the necessary true-martingale integrability, the futures quote is a martingale ending at the terminal spot price. The forward contract discount-weighted expectation is a different valuation expression.

For small diagrams with complete codomain, choose each objectwise categorical limit. For $u:d\to d^{\prime }$, its projections uniquely define the arrow $L(u)$ through $p_j(d^{\prime })L(u)=H_j(u)p_j(d)$. The universal property proves identity and composition laws. The same projection calculation makes every categorical cone factorization a natural transformation. Hence these are genuine categorical limits in the functor category, and the objectwise forgetful functor is a limit-creating functor.

A natural transformation between functors to the Category of sets is a monomorphism in the functor category exactly when every component is an injective function. Componentwise cancellation proves sufficiency. For necessity, the Yoneda lemma converts two elements with equal component images into two transformations from a covariant representable functor; monicity cancels these transformations, forcing the elements to coincide. The argument remains valid for a locally small category in an ambient universe where the functor category is formed.

The equation $x^2{y}^{\prime \prime }-2x{y}^{\prime }+(2-x^2)y=0$ has a regular singular point at zero. Factoring $y=xu$ cancels the Euler derivative terms and leaves $u^{\prime \prime }-u=0$ away from zero. Its analytic solutions $x\cosh x$ and $x\sinh x$ extend across zero. Their coefficient recurrence $(n-1)(n-2)c_n=c_{n-2}$ has independent free coefficients $c_1,c_2$, giving two entire solutions without a logarithmic branch despite integer-separated indicial roots.

A real simple pole is not locally integrable in the ordinary improper sense. Its symmetric Cauchy principal value defines a tempered distribution. For the $e^{-ikx}$ transform convention, contour indentation or sine integration gives the displayed nondecaying oscillatory contribution. Upper and lower boundary-value prescriptions instead change it by $\mp i\pi \delta (x-a)$ before transforming. A pole prescription must therefore be stated before claiming a Fourier asymptotic expansion.

For a real symmetric positive-definite three-dimensional matrix, the reciprocal quadratic form is locally integrable and defines a regular tempered distribution. With the Fourier kernel $e^{-ix\cdot \xi }$, radial Abel regularization gives $\mathcal{F}(|x{|}^{-2})=2{\pi }^2/|\xi |$. The linear change of variables $y=G^{1/2}x$ gives the displayed formula. No principal value or origin-supported correction is required. The result is homogeneous of degree $-1$ on the frequency side.

Unit-mass solutions of the isotropic Maxwell molecule collision operator with matching first moments obey this comparison. The Bobylev identity, the Fourier bound by one, and $|{\xi }^{+}{|}^2+|{\xi }^{-}{|}^2=|\xi {|}^2$ give a scalar damped differential inequality. Its integral form and the Gronwall inequality yield nonexpansion in the Fourier distance of order two. This does not alone establish strict contraction or equilibrium convergence.

For a canonically quantized real scalar field, the quadratic Hamiltonian contains $\frac{1}{2}{\sum }_{\mathbf{p}}{E}_{\mathbf{p}}$ in a regulated box. Normal ordering subtracts this zero-point energy, leavingThe vacuum momentum is zero with an inversion-symmetric regulator. The canonical commutation relation then gives $[P^{\mu },a_{\mathbf{p}}^{†}]=p^{\mu }{a}_{\mathbf{p}}^{†}$, identifying the momentum and energy added by one particle. The un-subtracted energy differs by a constant and is not literally the same operator.

Two equal parallel axial point stresslets with positive strength $S$ and separation $\ell$ have relative velocity $S/(2\pi \mu {\ell }^2)$. Identical self-propulsion cancels in their relative motion. The solution is the displayed cubic separation law while the far-field approximation is valid.

For a two-coordinate fluid model with $\dot{s}=g(n/s)$, $\dot{n}=f(n/s)$ and $s>0$, set $\kappa =n/s$ and $du/dt=1/s$. Then $d\kappa /du=f(\kappa )-\kappa g(\kappa )$ and $d\log s/du=g(\kappa )$. A bounded ratio eventually confined where $g\le -\eta <0$ makes $s$ decay exponentially in the new time. Since $dt/du=s$, the original fluid time has a finite terminal value and both coordinates tend to zero. This is a useful finite-time draining of a fluid model argument; the singular origin is treated by a stopped trajectory.

A flexural-gravity wave bends a floating elastic plate while displacing the underlying water. For a uniform plate with areal mass $m$ and bending stiffness $D$ above inviscid deep water of mass density ${\rho }_w$, put $\alpha =m/{\rho }_w$ and $\beta =D/{\rho }_w$. Combining the plate equation with the water's kinematic boundary condition gives the displayed dispersion relation. Both gravitational acceleration and plate bending restore displacement; water and plate inertia resist acceleration. It assumes small amplitude, no prestress and negligible shear deformation of the plate.

Let $x_i=i/n$, $1\le i\le n$, and let $N_x$ count sites within distance $h/2$ of $x\in [0,1]$. If $nh\ge 2$, a nearest site lies in the closed window, so $N_x\ge 1$. A clipped window has length at least $\min (h,1)/2$, and a grid count gives $N_x\ge n\min (h,1)/2-1$. Combining these two bounds yields $N_x\ge n\min (h,1)/4$. For $0<h\le 2$, the stronger clipped-length bound $h/2$ directly gives $N_x\ge \lfloor nh/2\rfloor \ge nh/4$. These estimates include both endpoints of the domain.

For two independent binomial counts, form the Fisher exact test rejection region from their conditional null hypergeometric distribution. Summing alternative joint binomial probabilities over that region gives its exact power. For equal group sizes, the null allocation is symmetric, so the two-sided probability-ordering test agrees with doubled lower-tail probabilities. Conservative test discreteness can make a normal-approximation sample size underpowered even when both samples are large but event counts are low.

A smooth scalar Landau free energy can end a two-phase phase coexistence curve by merging two minima and their intervening maximum. At a generic stable endpoint, its first three derivatives vanish and its fourth derivative is positive. A field shift and two mixed control parameters reduce the local form to a quartic Landau free energy; the order-parameter jump then vanishes and the magnetic susceptibility diverges. This conclusion assumes there is no competing third phase and that the endpoint lies inside the control-parameter domain. It is not a general theorem about every first-order line: a sextic potential with fixed negative quartic coefficient has a symmetry-related ordered coexistence line that ends at a triple point with a finite jump.

A finitely generated cone is a closed convex cone. For a convergent sequence of represented vectors, use linearly independent active generators and pass to a subsequence with the same active set. Its coefficients converge through a fixed left inverse and retain nonnegativity. General linear images of closed cones need not be closed; finiteness of the generators supplies the missing argument.

Under the finite-valued assumptions, subgradients of an infimal convolution are obtained as its attained dual optimizers. Equivalently $z\in \partial (h^{\ast }+k^{\ast })(p)$. Splitting the latter subdifferential into two separate ones needs a subdifferential sum rule qualification; solving the aggregate dual objective avoids that extra assumption.

For real monic orthogonal polynomials $p_j$ with squared weighted norms $h_j$ under a positive weight $w$, the functions $p_j\sqrt{w/h_j}$ form an orthonormal set for the unweighted reference measure. Their finite-rank projection kernel turns the squared weighted Vandermonde determinant into a kernel determinant. Normalization on the full labelled configuration space is $1/m!$.

For any group $G$, evaluation at a chosen generator of a group $x$ of $C_n$ gives a bijection between group homomorphisms $C_n\to G$ and elements satisfying $g^n=1$. A group homomorphism must send $x^j$ to $g^j$. Conversely, this formula is well-defined because exponents differing by a multiple of $n$ give equal powers, and it respects multiplication. The displayed correspondence is a bijection of sets; it is not claimed to be a group isomorphism for a nonabelian target. In a symmetric group, the order of a group element is the least common multiple of its disjoint cycle lengths, making this correspondence easy to enumerate.

Thickness noise is the sound generated by displacement of fluid by a moving impermeable body, represented by the surface acoustic monopole term. For a rigid body, ${\int }_S{v}_{n}dS=d\mathcal{V}/dt=0$, so its leading acoustic compact-source approximation has no net-volume monopole. Its local source need not vanish: a moving finite-volume body can radiate through higher spatial moments. A thin-body or negligible-volume approximation is needed when all such radiation is to be omitted.