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A set of universally quantified equality-free first-order clauses is unsatisfiable if and only if some finite set of its ground instances is propositionally unsatisfiable. Equivalently, such a set is satisfiable if and only if it has a Herbrand model. General first-order sentences can first undergo Skolemization. A fresh constant is added if needed to make the Herbrand universe nonempty; first-order equality requires its corresponding equality treatment.

For a satisfiable equality-free universal logical conjunction of Horn clauses, intersecting the sets of true ground atoms of its Herbrand models gives another Herbrand model, contained in every one of them. A ground atom is true in this least Herbrand model exactly when it holds in every Herbrand model. Positive existential sentences are true in the least Herbrand model exactly when they follow from the Horn clauses; allowing negated atoms makes this equivalence fail. For example, the least Herbrand model of $\forall x(P(x)\to Q(x))$ over a single constant $a$ makes both predicates empty and satisfies $\exists x\neg P(x)$, whereas the first-order structure making both predicates universal is a countermodel to that consequence.

A Henkin witness axiom names a witness to an existential sentence using a fresh constant symbol. Freshness is part of the consistency-preservation argument in the Henkin construction.

A full set of first-order sentences has existential witnesses among names in its own first-order language: if it proves an existential sentence, it proves some closed-term instance of the matrix. For a deductively closed set this may be stated by membership of the first-order sentences. In a function-free first-order language the witness is a constant symbol. A first-order consistent complete full first-order theory has a canonical term model.

For $0<a<1$ in the unit disk, the Möbius transformation $(z-a)/(1-az)$ takes $a$ to zero and the left semicircle to the arc centered at $-1$ with half-angle $\arctan ((1-a^2)/(2a))$. Brownian exit from the centered disk is uniform in angle, so conformal invariance of harmonic measure gives the displayed probability. Scaling and the reflection identity for Brownian exit from a half-disc give probability $4\arctan (1/r)/\pi$ of exiting the radius-$r$ circle from start one before reaching the imaginary axis.

For a Gödel sentence $G\leftrightarrow \neg \square G$ with the Hilbert-Bernays-Löb derivability conditions, fix a refutable contradiction sentence $\perp$. Since $\perp \to G$ is provable, $\square \perp \to \square G$, giving $G\to \neg \square \perp$. For the reverse direction, $\square G$ implies both $\square \square G$ and $\square \neg \square G$, hence $\square \perp$. Contraposition gives $\neg \square \perp \to \neg \square G\to G$. This is an internal equivalence of sentences, not a rule asserting the theory's own consistency.

Assuming PA is consistent, the theory $S=\mathrm{PA}+\neg {\mathrm{Con}}_{\mathrm{PA}}$ is consistent by second incompleteness. Its false axiom $G=\neg {\mathrm{Con}}_{\mathrm{PA}}$ is provable, so the standard semantic biconditional $G\leftrightarrow \neg {\mathrm{Pr}}_S(┌G┐)$ is true with both sides false. Thus a true biconditional alone does not make an arbitrary sentence a true canonical Gödel sentence. A canonical Pi-1 construction adds the necessary syntactic or sound-base information. Truth, internal equivalence and canonical construction must be distinguished.

Let $Z=\exp (X-⟨X⟩/2)$ and $dQ=Z_{T}dP$. The Girsanov theorem first makes $N=M-⟨M,X⟩$ a continuous local Q-martingale. Boundedness of $Y$ gives $⟨X{⟩}_T\le C^{2}T$ and ${\mathbb{E}}_P{Z}_T^2\le e^{C^{2}T}$: write $Z_T^2=\mathcal{E}(2X{)}_T{e}^{⟨X{⟩}_T}$ and apply Novikov's condition to $2X$.

The quadratic covariation bound gives ${\sup }_{s\le T}|⟨M,X{⟩}_s|\le C\sqrt{T}⟨M{⟩}_T^{1/2}$. For a square-integrable martingale $M$, both its maximal square moment and its expected terminal bracket are finite. Cauchy-Schwarz inequality with the density $Z_T$ therefore yields ${\mathbb{E}}_Q{\sup }_{s\le T}|N_s|<\infty$. The dominated limit of its stopped conditional identities proves the true martingale property. Mere P-integrability of $M_T$ does not suffice.

Let $W$ be a P-Brownian motion on $[0,1]$ and tilt by the bounded constant drift integrand one, so $dQ=e^{W_1-1/2}dP$. Then $W_s^Q=W_s-s$ is Q-Brownian motion, either by Girsanov theorem or direct Gaussian exponential tilting. Set $a=1/2$,The density of $W_a$ under P is ${\pi }^{-1/2}{e}^{-x^2}$, so ${\mathbb{E}}_{P}H={\pi }^{-1/2}\int (1+x^2{)}^{-1}dx<\infty$. Independence of future Brownian increments makes $M$ a continuous true P-martingale. Under Q, that density is multiplied by $e^{x-1/4}$, so ${\mathbb{E}}_{Q}H=\infty$.

For $s>a$, its compensated local martingale isThe future Q-Brownian increment is independent of the positive variable $H$ and has positive finite mean absolute value. Hence the displayed variable has infinite expected absolute value under Q and is not a true martingale. This shows why the general Girsanov theorem promises a local martingale, while a true-martingale conclusion needs an additional integrability condition such as Girsanov compensation with square-integrable input.

Let $\mathbb{P}(X=n)=p(1-p{)}^n$ for $n\ge 0$ and $Y\mid X=n\sim \mathrm{Binomial}(n,q)$, with $0<p,q<1$. Write $D=p+q-pq$. Summing the joint probability mass function with the differentiated geometric series identity ${\sum }_{n=k}^{\infty }\left(\genfrac{}{}{0ex}{}{n}{k}\right)t^{n-k}=(1-t{)}^{-k-1}$ gives $\mathbb{P}(Y=k)=(p/D)((1-p)q/D{)}^k$. By Bayes' theorem,Thus $X-k$, given $Y=k$, has the failure-count negative binomial distribution for $k+1$ successes and success parameter $D$. Its conditional expectation is $k+(k+1)(1-D)/D$ for $X$ itself. The shift by $k$, the number of retained successes, must be retained when identifying the posterior distribution.

One step of restarted GMRES searches along the current residual. In the displayed nonsingular system, the first residual minimization is ${\min }_{\alpha }[(1-\alpha {)}^2+(1+\alpha {)}^2]$, whose minimizer is zero. Restarting then repeats the same problem forever and never moves. Unrestarted GMRES instead solves exactly at step two, since $b,Ab$ span the entire space. Thus a smaller search-space cost can come at the price of stagnation; nonsingularity alone does not guarantee progress for one-step restarting.

Let $X_i=\theta +E_i$ for $n\ge 2$ independent and identically distributed observations, with $E_i$ having an exponential distribution of unknown scale $\beta >0$. Test the null hypothesis $\theta =0$ within $\theta \ge 0$. For nondegenerate data, the likelihood function isMaximizing over the scale gives $\hat{\beta }(\theta )=\overline{X}-\theta$, and the resulting profile likelihood increases with $\theta$. Thus the unrestricted maximum-likelihood estimates are $\hat{\theta }=X_{(1)}$ and $\hat{\beta }=\overline{X}-X_{(1)}$, while the null estimate is $\tilde{\beta }=\overline{X}$. The likelihood ratio is consequently

Under the null, put $A=n{X}_{(1)}$ and $D={\sum }_i(X_i-X_{(1)})$. The memoryless property gives independent gamma distributions of shapes $1$ and $n-1$ and common scale $\beta$. For a direct verification, fix which observation is the minimum and write its value as $z$ and the other observations as $z+e_j$. Their joint density is proportional to $e^{-nz/\beta }{\prod }_j{e}^{-e_j/\beta }$, which factors into an exponential density for the minimum and independent exponential densities for the excesses. The minimum's index does not affect the resulting laws of $A,D$.

By beta sampling by gamma ratios, $D/(A+D)\sim \mathrm{Beta}(n-1,1)$. ThereforeAn exact test size $\alpha$ is obtained by rejecting for $R\le {\alpha }^{n/(n-1)}$. Equivalently,Although the alternative adds one endpoint parameter, a regular one-parameter Wilks theorem calibration is incorrect: the support depends on the endpoint, and the null lies on its boundary. In particular, $-2\log R$ tends to a chi-squared distribution with two statistical degrees of freedom, not one.

Put $\kappa ={\aleph }_{\omega }$. ZFC gives ${\kappa }^{\kappa }=2^{\kappa }$. König's theorem and the shift of product coordinates give $\kappa <{\prod }_n{\aleph }_n\le {\kappa }^{{\aleph }_0}\le 2^{\kappa }$. The single instance $2^{\kappa }={\kappa }^{+}$ of the generalized continuum hypothesis therefore fixes all three displayed values. The strict lower bound explains why countable exponentiation cannot leave this singular cardinal unchanged.

Let $G=⟨x,y\mid y^2=1⟩$. For each $n\ge 2$, a free basis of a subgroup of rank $n$ isA covering-space proof uses $r=n-1$ spheres double-covering the real projective plane, with antipodal points $a_j,b_j$ over the wedge point. Add lifted $x$ edges in a cyclic order through the $a_j$, and an $x$ loop at each $b_j$. This is a connected $2r$-sheeted cover of a circle wedged with a projective plane. After collapsing one connecting arc in each simply connected sphere, its fundamental group is that of a graph with $r$ vertices and $2r$ edges, hence free of rank $r+1$.

A spanning tree along the first $r-1$ cyclic edges gives the displayed words: the closing cycle is $x^r$, and the loop reached through $a_j,b_j$ is $x^{j}yxy{x}^{-j}$. Injectivity of the induced fundamental-group map of a covering map proves they form a free basis in $G$.

For $I(u)={\int }_a^{b}f(x,u,u^{\prime })dx$ on $W^{1,\infty }$, continuity of $f,f_u,f_v$ on $[a,b]\times {\mathbb{R}}^2$ gives a bounded Fréchet derivative in the displayed form. Fixing $u$ and restricting $\Vert w{\Vert }_{W^{1,\infty }}$ to a bounded neighborhood confines all intermediate pairs $(u+tw,u^{\prime }+t{w}^{\prime })$ to a compact range up to null sets. Uniform continuity of the partial derivatives bounds the integrated first-order remainder by $(b-a)\omega (\Vert w\Vert )(\Vert w{\Vert }_{\infty }+\Vert w^{\prime }{\Vert }_{\infty })$, where $\omega (s)\to 0$. This is little-oh of the Sobolev norm. The result needs no continuous derivative representative for $u$; essential boundedness suffices.

For a positive integer $m$, the negative matrix power is differentiable on the open general linear group. The Neumann series gives $(A+H{)}^{-1}=A^{-1}-A^{-1}H{A}^{-1}+O(\Vert H{\Vert }^2)$ locally around any invertible matrix. Applying the product rule to the $m$ factors of this inverse gives the displayed Fréchet derivative. Each summand chooses one inverse factor to differentiate while keeping the others in order.

The factor order matters. Only when $A$ and $H$ commute does the sum simplify to $-m{A}^{-(m+1)}H$. This is a useful example of why scalar differentiation rules for powers cannot be transferred to matrices without accounting for noncommutativity.

Let $c(u)$ be the Black-Scholes formula value divided by stock price for an at-the-money call of duration $u$. On a discrete set of possible entry dates, homogeneous function and the discounted stock martingale reduce the Snell envelope to $b_i=\max (c(T-T_i),b_{i+1})$, with zero terminal value. Thus the displayed maximum is exact. When $r>0$ and volatility is positive, $c$ increases with maturity, so immediate entry is optimal and the initial value equals the ordinary at-the-money European call option. The maximum recursion remains useful if a different parameter regime loses that monotonicity.

For adequate arithmetic provability predicates, the arithmetized deduction theorem identifies inconsistency of $S+\neg {\mathrm{Con}}_S$ with $S$-provability of ${\mathrm{Con}}_S$. Formalized second incompleteness gives ${\square }_S{\mathrm{Con}}_S\to {\square }_S\perp$. Contraposition yields the displayed conditional. This is an internal theorem and does not license removal of its consistency premise.

Take Lebesgue decompositions relative to a fixed finite positive measure $\kappa$. The finite union $S$ of null supports of their singular components is still $\kappa$-null. On $S$ each original measure equals its singular component; on $S^c$ it equals its absolutely continuous component. Restriction commutes with the partition formula defining the finite supremum of positive measures. Thus the supremum restricts to ${\bigvee }_i{\mu }_i^s$ on $S$ and to ${\bigvee }_i{\mu }_i^a$ off $S$. The latter is absolutely continuous with respect to $\kappa$ because it is bounded by the sum of the absolutely continuous components. Uniqueness of the Lebesgue decomposition proves the displayed identities, and the Radon-Nikodym derivative of the absolutely continuous component is the pointwise maximum of the individual derivatives.

Attempt scalar multiplication on an elliptic curve modulo a composite integer. An addition denominator that is not invertible yields a greatest common divisor with the modulus; a value strictly between one and the modulus is a factor. The method exploits different orders of the reduced point at the unknown prime factors. For $y^2=x^3+5x-5$ modulo $221$, doubling $(1,1)$ gives $(14,-53)$; adding the original point requires division by thirteen, revealing $221=13\cdot 17$.