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The scalar sign action of a cyclic group of order two on ${\mathbb{C}}^2$ has invariant ring $\mathbb{C}[X^2,XY,Y^2]\cong \mathbb{C}[a,b,c]/(ac-b^2)$. It is not factorial: $a,b,c$ are pairwise nonassociate irreducibles, since every nonconstant invariant has degree at least two, while $ac=b^2$ gives two distinct factorizations.

Let $\zeta =e^{\pi i/n}$ and let the binary dihedral group act on ${\mathbb{C}}^2$ by $(x,y)\mapsto (\zeta x,{\zeta }^{-1}y)$ and $(x,y)\mapsto (-y,x)$. For $n\ge 2$, the invariantspresent the invariant ring as $\mathbb{C}[U,V,W]/(W^2-V{U}^2+4V^{n+1})$. First take the cyclic invariants $A=x^{2n},B=y^{2n},C=xy$ with $AB=C^{2n}$. The residual involution interchanges $A,B$ and negates $C$, so invariant normal forms are polynomials in $A+B,C^2$ plus $C(A-B)$ times such polynomials. For $n=3$ the group has order twelve and the equation is $W^2-V{U}^2+4V^4=0$.

For three distinct times, the columns $1,t,t^2$ are linearly independent by the Vandermonde determinant. Thus a quadratic Poisson regression for the log mean can reproduce any three positive time-group means. A grouped goodness-of-fit test of that mean form has zero residual degrees of freedom. Replicated observations still allow checks of the Poisson distribution and overdispersion within each time group; saturation of the mean does not establish the response distribution.

A Poisson distribution of parameter $a$ conditioned to lie in $\{0,\dots ,C\}$. It is the occupancy law of a unit-resource loss network. Its mean is the carried load of an Erlang loss resource, $a[1-E(C,a)]$. Differentiating the mean with respect to $\log a$ gives its variance, which is positive for $a>0$ and $C\ge 1$; this proves strict monotonicity of the carried load.

For $u\ge 0$ and a positivity-preserving operator, this is the negative log-likelihood for independent Poisson observations with intensity $1+Tu$, up to constants depending only on the data and the unit background. For bounded $g\ge 0$, its scalar derivatives are $1-g/(1+s)$ and $g/(1+s{)}^2$ on $s=Tu\ge 0$. It is convex, and strictly convex in the predicted intensity when $g>0$ almost everywhere. On a unit-area domain, the Jensen inequality gives $D_g(u)\ge \Vert Tu{\Vert }_1-\Vert g{\Vert }_{\infty }\log (1+\Vert Tu{\Vert }_1)$. The shift makes $\log (1+Tu)$ integrable whenever $Tu\in L^1$; no integrability of $\log u$ is needed.

For a particle initially having semimajor axis one and encountering a circular planet at radius one, the incoming stellar-frame speed is one. A conservative encounter at planet-relative speed $u$ changes the stellar-frame velocity by at most $2u$. Since stellar escape requires speed $\sqrt{2}$, a necessary bound is $u>(\sqrt{2}-1)/2$. Retaining the outgoing relative-velocity geometry sharpens the necessary bound to $u>\sqrt{2}-1$. Neither inequality by itself guarantees a realizable deflection at a specified impact parameter.

Started at radius one, planar Brownian motion avoids the origin, and its logarithmic radius is a continuous local martingale because the logarithmic potential is harmonic away from zero. It is a strict local martingale: at deterministic $t>0$, the angular average of a logarithmic potential gives $\mathbb{E}\log |B_t|={\int }_1^{\infty }{e}^{-r^2/(2t)}dr/r>0$, whereas the initial value is zero. Absolute integrability follows from integrability of $r|\log r|$ near zero and the Gaussian tail. Its positive expectation does not contradict nonnegative-local-martingale bounds, since this process takes both signs.

For one-dimensional flux $F(p)=p|p|e^{-p^2/(2{\lambda }^2)}$, the nonlinear diffusion equation is $u_t=F^{\prime }(u_x)u_{xx}$. The sole one-dimensional member of the principal diffusion coefficients is positive for $0<|u_x|<\sqrt{2}\lambda$, negative above $\sqrt{2}\lambda$, and zero at the two endpoints. Negative diffusion gives formal sharpening with ill-posedness through high-frequency growth. The extra factor $|p|$ matters: the unweighted exponential conductance has threshold $\lambda$ instead.

Suppose the one-arm probability on the cubic lattice satisfies ${\beta }_r=\exp (-\lambda r+o(r))$ with $\lambda >0$. Define $M_n$ as the largest maximum-norm radius of any cluster rooted in $[-n,n{]}^d$. Then $M_n/\log n\to d/\lambda$ convergence in probability. An upper bound is a union bound over $O(n^d)$ roots. For a lower bound, pack $\Theta ((n/\log n{)}^d)$ disjoint radius-$O(\log n)$ boxes and use the independent first-hit connection events inside them. Clusters are finite almost surely under the positive-rate hypothesis.

For nonnegative independent identically distributed summands with polynomial upper tail $\mathbb{P}(X_1\ge x)=x^{-\beta }$ for $x\ge 1$, positivity gives $\mathbb{P}(S_n\ge na)\ge (na{)}^{-\beta }$ for fixed $a>0$ and large $n$. Since this probability is also at most one, its logarithm divided by $n$ tends to zero. No positive exponential moment exists, so the exponential-moment form of the Cramér theorem cannot be applied. The bound proves a logarithmic rate and does not assert an exact tail asymptotic.

Passing to norm limits in Artin reciprocity gives an exact sequence connecting closures of global units, local pro-$p$ units at primes over $p$, the p-ramified Iwasawa module and the unramified Iwasawa module. It explains arithmetic Iwasawa-module ranks by comparing local and global unit ranks. Roots of unity and the precise splitting conventions must be retained when seeking exact integral statements.

A language in P to which every P language has a logspace many-one reduction. The reduction strength matters: using arbitrary polynomial-time reductions instead would make completeness trivial for nontrivial P decision problems.

In exponential scale coordinates, the real affine group has matrices $\left(\begin{array}{cc}{e}^{\alpha }& \beta \\ 0& 1\end{array}\right)$. The left and right Maurer-Cartan forms are respectively $Dd\alpha +T{e}^{-\alpha }d\beta$ and $Dd\alpha +T(d\beta -\beta d\alpha )$. Their dual vector fields obey $[D_L,T_L]=T_L$ and $[D_R,T_R]=-T_R$, displaying the opposite bracket sign for right-invariant vector fields.

With independent uniform distributions on $[0,1]$ as sequential offers and a finite number of rounds, the continuation value with $j-1$ offers remaining is $v_{j-1}$. Accept the current offer when it reaches that threshold. Integrating the maximum of the offer and continuation value gives the recursion, by the Snell envelope principle.

If $x$ interchanges the new point and $\alpha$, put $H=G_{\alpha }$. Then $⟨G,x⟩$ is a one-point extension exactly when $x^2\in H$, $xH{x}^{-1}=H$, and $xgx\in GxG$ for every $g\notin H$. These conditions make $G\cup GxG$ multiplication-closed, hence a group in the finite setting, with new-point stabilizer $G$. Necessity comes from the two double cosets in the extended two-transitive action.

A continuous one-cocycle for a Galois group acting on an abelian module $M$ is a map satisfying the displayed identity. Cocycles modulo maps $\sigma \mapsto \sigma u-u$ form $H^1(K,M)$ in Galois cohomology. For an elliptic curve, choosing $mQ=P$ gives the cocycle $\sigma \mapsto \sigma Q-Q$ in $E[m]$, which represents the Kummer map of an elliptic curve.

The prenucleolus uses the same sorted excesses of a coalition but minimizes over all efficient payoff vectors, without imposing individual rationality. It can differ from the nucleolus when coalition values are not superadditive. Any computation must state which domain is used rather than silently relaxing imputation inequalities.

Every NP verifier can be compiled into a polynomial-size Boolean circuit whose inputs encode its certificate, and then into an equisatisfiable conjunctive normal form by a Tseitin transformation. This gives a polynomial-time many-one reduction from every NP language to SAT. Checking a guessed assignment proves membership, so SAT is NP-complete.

For a nonconstant polynomial over any field, use triangular substitutions $x_i=y_i+y_n^{B^i}$, $x_n=y_n$, where $B$ exceeds every exponent in its support. Base-$B$ weights distinguish all original monomials, so the transformed polynomial has a unique highest $y_n$ power with coefficient in $k^{\times }$. Rescale to make it monic. The monic polynomial quotient is finite free over $k[y_1,\dots ,y_{n-1}]$, yielding a finite flat projection. The nonlinear substitution works over finite fields as well as infinite ones; a generic linear-direction argument alone would not do so.

In an ideal linear temperature-dependent destruction model with equal warming and cooling durations, normalized temperature is $1-e^{-\alpha t}$ during warming and $(1-e^{-\alpha T})e^{-\alpha (t-T)}$ during cooling. Integrating the destruction rate gives $D(T)$, the logarithm of the surviving-count reduction factor. If the endpoint destruction rate and thermal relaxation constant stay fixed, changing the temperature gap alone leaves this integral unchanged. This is a property of the stipulated model.