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Nonzero compactly supported distributions with pairwise disjoint supports are linearly independent: a smooth cutoff function isolating one support isolates its coefficient. Injectivity transfers this independence to their Fourier transforms. Exponential-type bounds and the Translation property of the Fourier transform can prove the needed support separation before the distributions themselves are known explicitly.

The Bradley-Terry model assigns positive abilities ${\theta }_i$ and the displayed pairwise winning probabilities. Multiplying all abilities by the same positive scalar leaves the probabilities unchanged, so identifiability requires a normalization. In log abilities ${\beta }_i=\log {\theta }_i$, the model has comparison probabilities obtained from the logistic function. Either parametrization induces the same ordering of players.

For each unordered pair, $g=({\Theta }_{jk},{\Theta }_{kj})$ and $c=(X_j^{\mathsf{T}}{r}_k/n,X_k^{\mathsf{T}}{r}_j/n)$ with centred columns and regression residuals. The subdifferential of the Euclidean norm is the unit ball at zero and $g/\Vert g{\Vert }_2$ otherwise. These KKT conditions are necessary and sufficient for the convex paired objective and explain joint edge deletion.

For a finitely generated group, let $G^p$ denote the subgroup generated by all $p$th powers. Its p-rank isThis is the dimension of the largest quotient that is an elementary abelian group of exponent $p$. In a group presentation with $d$ generators and $s$ relators that are not $p$th powers, reduction of the exponent sums modulo $p$ gives at most $s$ linear constraints. Hence $d_p(G)\ge d-s\ge {\mathrm{def}}_p(\mathcal{P})$. In particular a presentation of p-deficiency at least one yields a surjection onto $C_p$.

If $H$ is a normal subgroup of index $p$ in a group presented by $\mathcal{P}$, the Reidemeister–Schreier theorem produces a presentation $\mathcal{Q}$ withThe preimage of $H$ in the free group has rank $1+p(d-1)$. A relator $w^{p^k}$ produces at most $p$ conjugate relators of weight $p^{-k}$ if its root lies in that preimage. If the root lies outside, $k\ge 1$, and the coset conjugates are redundant up to conjugation; one relator of weight at most $p^{-(k-1)}$ suffices. Thus the total weighted relator cost grows by at most $p$. Convergent infinite sums are handled term by term.

If a group presentation has unshifted p-deficiency at least one, the p-rank of a group is at least one and gives a normal subgroup of index $p$. The index-p rewriting bound for p-deficiency gives this subgroup another presentation with p-deficiency at least one. Iterating produces subgroups of index $p^j$ for every $j$, so the group is infinite. This criterion applies even with infinitely many relators, provided the weighted sum in the definition converges.

Enumerate all nonidentity words $w_i$ in the rank-two free group and impose relations $w_i^{p^{i+2}}=1$. The resulting two-generated group hasIt is infinite by p-deficiency at least one implies infinitude. Each element has order a power of $p$, so it is a torsion group. The infinitely many relators are essential to this particular construction; finite generation does not imply a finite group presentation.

For odd $p$, a p-adic unit is a square exactly when its residue is a square, by the Hensel lemma. Its unit square-class group is $C_2$. For $p=2$, a unit is a square exactly when it is congruent to one modulo eight, by the strong form of Hensel lemma. The four square classes are represented by $1,3,5,7$, giving $C_2\times C_2$.

At fixed orientation, opposite travelling amplitudes can obey ${\dot{Z}}_{\pm }=(r+i{\omega }_0)Z_{\pm }-(g|Z_{\pm }{|}^2+h|Z_{\mp }{|}^2)Z_{\pm }$. For $r>0$ and $g_r=\mathrm{Re}g>0$, a travelling branch has one nonzero intensity $r/g_r$ and is stable to its counter-propagating amplitude when $h_r>g_r$. A standing branch has both intensities $r/(g_r+h_r)$ and is stable within the pair when $g_r>h_r$ and $g_r+h_r>0$. These follow by linearizing the real intensity equations. Spatial modulation and other orientations remain separate stability tests.

Integration by parts against $1$, $v_i$ and $|v{|}^2/2$ gives these equations for mass, unnormalized momentum and kinetic energy. With $M_0\ne 0$, the mean is $u(t)=e^{-t}{u}_0$, and $E(t)=d{M}_0/2+(E_0-d{M}_0/2)e^{-2t}$. Normalizing by mass is invalid if that mass is zero.

For a sufficiently regular unit-mass solution, the Ornstein-Uhlenbeck Fokker-Planck equation is ${\partial }_{t}f=\nabla \cdot (f\nabla \log (f/\gamma ))$. Integration by parts proves the displayed identity. Consequently relative entropy decreases, and its dissipation is relative Fisher information.

For a connected graph, the Moore-Penrose inverse of its oriented incidence matrix satisfies $D^{†}D=I-{\Pi }_1$, because this product is the orthogonal projection onto $(\ker D{)}^{\perp }$. Thus every signal decomposes into its constant mean and $D^{†}D\theta$. Writing $L={\max }_e\Vert (D^{†}{)}_{\cdot ,e}{\Vert }_2$, a sub-Gaussian random vector noise gives a simultaneous bound on $\Vert (D^{†}{)}^{\top }z{\Vert }_{\infty }$ with scale $L\sqrt{\log (m/\delta )}$.

A family $Y\subseteq [\mathbb{N}{]}^{\omega }$ is $\tau$-Baire when $Y△O$ is meagre for some ordinarily open family $O$. This is weaker than the corresponding regularity in the star topology. A meagre set meeting every Ramsey cone in both colours has the ordinary Baire property while failing to be a Ramsey set of infinite subsets.

The family $D=\{\{a_1<a_2<\cdots \}:a_{n+1}>2a_n\text{ for all }n\}$ is ordinarily closed and nowhere dense. A failed gap is witnessed by a finite prefix, and any cylinder can be refined by adjoining consecutive large integers that fail the gap. Nevertheless $D$ intersects every infinite-subset cone in continuum many sets: a binary tree chooses two sufficiently large next points inside its reservoir at each step.

The ordinal ${\epsilon }_0$ is the least fixed point of $\alpha \mapsto {\omega }^{\alpha }$. It is the supremum of $1,\omega ,{\omega }^{\omega },{\omega }^{{\omega }^{\omega }},\dots$, and is countable.

If $\theta$ has order at most one, set $c=\theta (1)$ and $\delta =\theta -m_c$. Then $\delta (1)=0$ and $[\delta ,m_a]=m_{\delta (a)}$, which is the Leibniz rule. Thus $\delta$ is a derivation of an algebra. Conversely any derivation satisfies this commutator identity. The sum is direct because a derivation that is multiplication must vanish at $1$.

An orbital element of the Kepler orbit having the same instantaneous position and velocity as a perturbed trajectory. Its elements vary as the perturbing force acts. Their interpretation depends on the central gravitational parameter used, for example whether static radiation pressure is included in the unperturbed attraction.

The plane containing a central-force Kepler orbit, perpendicular to its conserved specific angular momentum. Its orientation can be specified by an orbital inclination and an orbital node. Perturbing forces can change that orientation.

For an algebraic group action, the orbit dimension is the dimension of the group minus that of its stabilizer. For a finite-dimensional representation, the stabilizer is an automorphism group, open in its endomorphism ring, so their dimensions agree. In a quiver representation space, the Ringel form turns the orbit codimension into the dimension of the self-extension group.

The general linear group over a finite field acting diagonally on projective triples has five orbits of a group action: all equal, three different positions for the repeated coordinate when exactly two are equal, and all distinct. The corresponding representative stabilizer subgroups are upper triangular, diagonal, and scalar matrices. Their orders are $p(p-1{)}^2$, $(p-1{)}^2$, and $p-1$, respectively; the orbit sizes are $p+1$, $p(p+1)$, and $p(p+1)(p-1)$. Transitivity follows from the action being sharply three-transitive on a projective line.