Codex @codex  0

A credibility estimate combines a risk's observed experience with collective or prior information, typically as $Z\overline{X}+(1-Z)m_0$. The credibility factor $Z\in [0,1]$ is the weight assigned to the individual experience: more reliable or more extensive data generally increases it. Bayesian credibility treats the risk parameter as random with a prior distribution and predicts future claims by averaging the conditional claim mean under its posterior distribution. Under squared-error loss this posterior predictive mean is optimal. It need not be an affine combination in every Bayesian model; the Poisson-gamma conjugacy calculation below yields that credibility form exactly, rather than as a best-linear approximation.

A scalar M-estimator is a selected root of $n^{-1}{\sum }_i\psi (Y_i,\hat{\theta })=0$, or equivalently a minimizer of a differentiable empirical objective whose derivative is this estimating equation. Its population statistical functional $T(F)={\theta }_F$ solves $\int \psi (x,{\theta }_F)dF(x)=0$. Assume local uniqueness, consistency of the sample root, a valid differentiable local expansion, finite ${\mathbb{E}}_F{\psi }^2$, andThese conditions include domination sufficient to differentiate under the integral and the required local uniform control of the sample derivative.

For $F_{\epsilon }=(1-\epsilon )F+\epsilon {\delta }_z$, differentiate the population estimating equation at $\epsilon =0$. Since its uncontaminated integral is zero,Thus the influence function is $\mathrm{IF}(z;T,F)=-\psi (z,{\theta }_F)/A$, with a proportionality constant independent of $z$.

Expanding the sample equation at its population root givesThe summands have mean zero. The central limit theorem and Slutsky's theorem therefore yield the scalar sandwich variance of an M-estimatorThis $V$ is the limiting variance of the $\sqrt{n}$-scaled error; the estimator's variance is approximately $V/n$. In a correctly centred location family with $\psi (x,\theta )=\psi (x-\theta )$, let $m={\mathbb{E}}_F{\psi }^{\prime }(X-\theta )$. Then $A=-m$, so the influence function is $\psi (z-\theta )/m$ and $V=\mathbb{E}{\psi }^2/m^2$. The minus sign from differentiating $x-\theta$ is essential.

The clipped Huber score in the subsequent subparts requires $0<b<\infty$. The PDF only displays an upper bound: at $b=0$ the estimating function vanishes identically, and at $b<0$ it is a constant with no root. A positive clipping threshold is the regular interpretation.

Write $a$ for the scalar intercept denoted $\mu$ in the question, to distinguish it from the individual means ${\mu }_i$. The log-likelihood of the independent Poisson observations isDifferentiating gives the score equations for an interior maximum-likelihood estimator:In particular, when $m={\sum }_i{y}_i>0$,The left side is a weighted covariate mean whose derivative is its weighted variance, so it is strictly increasing when the covariates are not all equal. An interior solution is unique if the observed count-weighted mean is strictly between the smallest and largest covariates. At an extreme it may require a limiting slope, and if all counts are zero the maximum occurs as $a\to -\infty$, rather than at finite parameters. Constant covariates make the slope and intercept unidentifiable.

Put $S_j={\sum }_i{x}_i^j{\mu }_i$. The negative Hessian matrix is deterministic at given parameters and is also the Fisher information matrix:Under the usual interior, increasing-information regularity for asymptotic normality of a maximum likelihood estimator, replace the unknown means by their fitted values and read the lower-right entry:Equivalently, Poisson slope information after eliminating an intercept is ${\sum }_i{\mu }_i(x_i-{\bar{x}}_{\mu }{)}^2$, with ${\bar{x}}_{\mu }=S_1/S_0$. This explains both the positive denominator for varying covariates and the loss of slope information when their spread is small.

Use the natural partial orderThe paper's shorthand $x\le x+y$ specifies the join order: more formally, $x\le z$ means $z=x+y$ for some $y$, which is equivalent to the displayed condition by idempotence. Associativity, commutativity and idempotence make addition the join operation. Distributivity makes multiplication monotone, while absorption gives $xy\le x$ and $xy\le y$.

Let $g_1,\dots ,g_r$ generate the incline. Distributivity writes every element as a nonempty finite join of monomialsThere is no assumed multiplicative identity: an exponent zero means omit that factor, and at least one factor remains. If $u\le v$ coordinatewise, absorption gives $m(v)\le m(u)$. If the two vectors are equal the conclusion is immediate; otherwise multiply $m(u)$ by the additional factors and apply absorption. For a polynomial with finite exponent support $A$, put $U_A=\{v:(\exists u\in A)u\le v\}$, its upward closure. If $U_A\supseteq U_B$, every monomial in $B$ is below some monomial in $A$, and consequently the join represented by $B$ is below the join represented by $A$.

We prove the reverse-inclusion well-quasi-ordering of monomial ideals, often called Maclagan's theorem, in the equivalent form that upward-closed subsets of ${\mathbb{N}}^r$ are well-quasi-ordered by reverse inclusion. Include the empty subset. For $r=1$, these are the tails $[n,\infty )$ and the empty set, corresponding to $\mathbb{N}\cup \{\infty \}$ in its usual well-order.

Induct on $r$. For an upward-closed $U\subseteq {\mathbb{N}}^r$, its slices $U_k=\{v\in {\mathbb{N}}^{r-1}:(v,k)\in U\}$ increase with $k$. Their union is upward closed and has finitely many minimal elements by the Dickson lemma; all those elements have appeared by some finite slice. Thus the slices eventually stabilize. Encode $U$ by the finite word of its pre-stabilization slices and its final stable slice. By the induction hypothesis, the slice order given by reverse inclusion is a well-quasi-ordering. The Higman lemma and finite product closure of well-quasi-orderings make these word-and-tail encodings well-quasi-ordered.

For a good pair of encodings $U,V$, an increasing subsequence map $f$ satisfies $U_k\supseteq V_{f(k)}$ at every pre-stabilization position of $U$, and the final slice of $U$ contains the final slice of $V$. Since $f(k)\ge k$ and the $V$ slices increase, $U_k\supseteq V_k$ before stabilization. Afterwards the final-slice comparison also gives $U_k\supseteq V_k$. Hence $U\supseteq V$, proving the induction. The use of Higman lemma is essential: finite-subset reverse domination does not preserve an arbitrary well-quasi-order, as the Rado order shows.

Given any infinite sequence of incline elements, choose a finite monomial support $A_i$ for each. The result just proved gives $i<j$ with $U_{A_i}\supseteq U_{A_j}$, hence $x_i\ge x_j$. Every finitely generated incline is therefore well-quasi-ordered by its reversed natural order.

We first prove invariance, before using any assertion that requires it. Write $A_n\phi (x)=n^{-1}{\sum }_{k=0}^{n-1}\phi (f^{k}x)$. Choose a point whose forward orbit has the prescribed convergence against every continuous function; the hypothesis gives a set of such points of full measure. For each continuous function $\phi$, telescoping givesbecause a continuous function on a compact metric space is bounded. Both test functions are continuous, so their averages converge to their prescribed integrals. Hencefor every continuous function $\phi$. The continuous-test criterion for an invariant measure therefore gives $f_{\ast }\mu =\mu$. The points in the hypothesis can now be called generic points for an invariant measure.

For ergodicity, let $B$ be a Borel set invariant modulo null sets, and set $h=1_B$, $p=\mu (B)$. Then $h\circ f=h$ almost everywhere. Since invariance preserves null preimages, deleting the union of all iterated preimages of the exceptional null set gives $h(f^{k}x)=h(x)$ for every $k\ge 0$ on one full-measure set. Thus $A_{n}h=h$ almost everywhere for every $n$.

Given $ϵ>0$, the continuous approximation of Borel indicators on a compact metric space, proved in the supporting steps below, supplies $\phi \in C(X)$ with $0\le \phi \le 1$ and $\Vert h-\phi {\Vert }_{L^1(\mu )}<ϵ$. By the hypothesis, $A_n\phi \to \int \phi d\mu$ almost everywhere. Since $0\le A_n\phi \le 1$, the dominated convergence theorem also gives convergence in $L^1(\mu )$. Invariance and the triangle inequality give the uniform estimateConsequently,Letting $n\to \infty$ and then $ϵ\downarrow 0$ proves $\Vert h-p{\Vert }_1=0$. Explicitly,so $p\in \{0,1\}$. Every invariant Borel set has measure zero or one, which provesThis proof controls measurable-set averages through a uniform $L^1$ estimate; convergence only against continuous test functions is not silently assumed to include indicator functions.

Let $V\subseteq U$ be open subsets, and let $s\in {\mathcal{F}}_3(V)$. The assumed flasque-kernel section-lifting lemma lifts $s$ to $\tilde{s}\in {\mathcal{F}}_2(V)$. Since ${\mathcal{F}}_2$ is a flabby sheaf, extend $\tilde{s}$ to a section on $U$, then project to ${\mathcal{F}}_3(U)$. Its restriction is $s$. Thus every restriction map of ${\mathcal{F}}_3$ is surjective, proving that the quotient of two flabby sheaves in a short exact sequence is flabby.

For the construction of sheaf cohomology, embed a sheaf of abelian groups $\mathcal{F}$ into the sheafby sending a section to all its germs. This is injective because a section with every germ zero is zero. Arbitrary families of stalk elements glue point by point, and restriction maps are projections; hence $C(\mathcal{F})$ is a flabby sheaf. Set ${\mathcal{Q}}^0=\mathcal{F}$, ${\mathcal{I}}^j=C({\mathcal{Q}}^j)$ and ${\mathcal{Q}}^{j+1}={\mathcal{I}}^j/{\mathcal{Q}}^j$. Compose the quotient map with the next embedding to obtain the flasque resolutionThe sheaf cohomology groups are the cohomology groups of the resulting cochain complex of global sections:The usual comparison of resolutions identifies this construction with the right derived functors of global sections; the construction above is the Godement resolution.

If $\mathcal{F}$ is itself a flabby sheaf, the quotient argument shows inductively that every ${\mathcal{Q}}^j$ is flabby. The assumed lifting result therefore makesan exact sequence for every $j$. The global-section complex is consequently exact in every positive degree. In particular,This establishes the vanishing from the section-lifting property rather than assuming it in advance.

For the smooth algebraic curve $X$, a divisor on an algebraic curve is a finite formal sum $D={\sum }_P{n}_{P}P$ of closed points with integer coefficients. At each closed point, the local ring of a smooth algebraic curve is a discrete valuation ring; write its normalized discrete valuation as $v_P$. For $f\in k(X{)}^{\ast }$ the principal divisor is $\mathrm{div}(f)={\sum }_P{v}_P(f)P$. Only finitely many coefficients are nonzero: zeros and poles are proper closed subsets in the Noetherian Zariski topology of a curve. The divisor class group is the abelian groupEquivalently, its elements are divisors on an algebraic curve modulo linear equivalence of divisors.

Let $\pi :E\to B$ be an oriented real vector bundle of rank $r$ over a paracompact base. A Thom class is a class $U\in H^r(E,E\setminus B;\mathbb{Z})$ restricting on every oriented fiber pair $({\mathbb{R}}^r,{\mathbb{R}}^r\setminus 0)$ to its positive generator. The Thom isomorphism theorem states that this class exists uniquely with those fiber restrictions and thatHere $B$ is identified with the zero section. Equivalently use the disk/sphere bundle pair. The compact-support version is $H_c^q(B;\mathbb{Z})\cong H_c^{q+r}(E;\mathbb{Z})$, with fiberwise compact support in the Thom construction. If the bundle is not oriented, the integral assertion requires the corresponding orientation local system.

For a closed co-oriented embedded submanifold $Y$ of codimension $r$, co-orientation means an orientation of its normal bundle. A tubular neighborhood identifies a neighborhood of $Y$ with that bundle. Its Thom class, transferred by excision, is a class in $H^r(M,M\setminus Y;\mathbb{Z})$. For compact $Y$ this maps into the direct limit defining compactly supported cohomology, giving $\boxed{{\epsilon }_Y\in H_c^r(M;\mathbb{Z})}$. The construction is independent of the tubular neighborhood and uses the chosen co-orientation. In an oriented ambient manifold it is the Poincare dual of the oriented submanifold. Here “closed submanifold” has the usual compact-without-boundary meaning; if it meant merely a closed subset and $Y$ were noncompact, the supported relative class would still exist but need not have compact support.

We next prove the fixed-point implication without imposing an unprinted orientability assumption on $M$. SetUse a finite triangulation of the closed smooth manifold. The elementary Hopf trace identity says that the alternating trace of a chain endomorphism equals its alternating trace on homology. To verify it, split each finite-dimensional chain space as boundaries, a complement representing homology, and a complement mapping isomorphically onto the preceding boundaries. The traces on these two boundary summands cancel in adjacent degrees, leaving exactly the homology traces. The cohomology traces equal the homology traces because the induced maps are dual over $\mathbb{Q}$.

If $f$ has no fixed points, compactness gives $\delta ={\min }_{x}d(x,f(x))>0$. Choose a triangulation $K$ of mesh less than $\delta /4$ and a sufficiently fine subdivision $K^{\prime }$ admitting a simplicial approximation $F:K^{\prime }\to K$ uniformly within $\delta /4$ of $f$. The subdivision chain map $s:C_{\ast }(K)\to C_{\ast }(K^{\prime })$ sends each simplex to simplices inside it. For a vertex $v$ of such a small simplex, $d(v,F(v))>3\delta /4$, so $F(v)$ cannot lie in the original coarse simplex. Consequently $F_{\#}s$ has zero diagonal coefficient on every coarse simplex. Its trace vanishes in every degree, while it induces $f_{\ast }$ on homology. The trace identity yields $L(f)=0$. Taking the contrapositive provesThis is the Lefschetz fixed-point theorem, established here rather than invoked. For oriented $M$ the same identity is the graph-diagonal formula for the Lefschetz number: $L(f)=⟨(1,f{)}^{\ast }{\epsilon }_{\Delta },[M]⟩$, which vanishes when the graph misses the diagonal. Finally, homotopic maps induce the same cohomology maps, so their Lefschetz numbers agree.

There are genuine normalization inconsistencies in this PDF. In the standard linearized chord diagram model $G_n^{(1)}$, time $n$ gives $n$ edges and total vertex degree $2n$, with self-loops counted twice. The paper calls $2n$ the number of edges. Its right-endpoint hint also uses $\sqrt{i/(2n)}$ rather than $\sqrt{i/n}$. These cannot both describe the standard model; part (c)'s exponent is consequently inconsistent as well. We give the definition, prove the alternative construction and its limits, and explicitly identify the corrected conclusion rather than infer an invalid constant from the printed hint.

Start from the empty graph at time zero, or from one vertex with one self-loop at time one. To obtain $G_t$ from $G_{t-1}$, add vertex $t$ and an edge directed from $t$ to $I_t$, whereA self-choice creates a self-loop. The old vertex degrees sum to $2(t-1)$, so these probabilities sum to one. This is exact preferential attachment, including the outward half of the new edge as weight one for the new vertex; it is not a rule normalized by $2t$ or a process adding two edges at each step. The degree of a vertex here is its total incoming plus outgoing vertex degree. The construction in part (a) has exactly one chord per vertex and makes the edge-count discrepancy explicit.

Identify the discrete cube $Q_n$ with the subsets of $[n]$, with hypercube graph adjacency given by changing one element. Write $N[A]$ for the closed graph neighbourhood and $\partial A=N[A]\setminus A$ for the external vertex boundary. The simplicial order on the discrete cube first orders by set size and then by lexicographic order, with the smallest differing coordinate belonging to the earlier set.

Harper theorem states that, for every set family $A\subseteq Q_n$,Here is the simplicial section compression proof. Induct on $n$, the one-dimensional case being immediate. Split $A$ by a coordinate into sections $A_0,A_1\subseteq Q_{n-1}$. Its neighbourhood sections areReplace $A_0,A_1$ by equally large simplicial initial segments $I_0,I_1$. By induction $|N[I_j]|\le |N[A_j]|$. Also the neighbourhood of a simplicial initial segment is itself a simplicial initial segment: at its last occupied level the extra vertices form the upper shadow of a lexicographic initial segment, again lexicographically initial. Consequently the compressed unions have sizes $\max \{|N[I_0]|,|I_1|\}$ and $\max \{|N[I_1]|,|I_0|\}$, no larger than the original unions. The compression preserves $|A|$ and cannot increase $|N[A]|$.

Repeat in every coordinate until all sections are initial. Every nontrivial compression strictly decreases the sum of the global simplicial positions, so the process terminates. To describe the terminal families for simplicial section compression, suppose an earlier absent vertex $S$ precedes a later present vertex $T$. They cannot share a coordinate value: otherwise they lie in the same compressed section, contradicting its initiality. Thus $T=[n]\setminus S$. No vertex can lie strictly between them in the order, since it would make another absent/present pair that is not complementary. Hence the terminal family is either $C(|A|)$ or that segment with its last vertex $S$ exchanged for its immediate complementary successor $T$.

For odd $n=2m+1$, that exceptional exchange crosses the middle levels: $S=\{m+2,\dots ,2m+1\}$ and $T=\{1,\dots ,m+1\}$. For $m\ge 1$, all sets of size at most $m+1$ remain in the neighbourhood after the exchange, since an $(m+1)$-set has at least two lower neighbours and only one was removed. The new $T$ can only add vertices of size $m+2$.

For even $n=2m$, the only consecutive complementary pair in one level is $S=\{1,m+2,\dots ,2m\}$ and $T=\{2,\dots ,m+1\}$. The original segment contains all sets below size $m$ and all $m$-sets containing 1. For $m\ge 2$, its neighbourhood contains all sets of size at most $m$ and all $(m+1)$-sets containing 1; every such upper set still has an included lower neighbour after the single removal. The new $T$ can only add further neighbours. The cases $n=1,2$ follow directly, with the exceptional families related to the initial ones by a graph automorphism. Thus neither exception improves the neighbourhood, completing the induction.

For the half-cube vertex-boundary extrema, if $n=2m+1$, the initial half cube is all sets of size at most $m$; its boundary is the next level, of size $\left(\genfrac{}{}{0ex}{}{2m+1}{m+1}\right)$. If $n=2m$, it contains all levels below $m$ and the $m$-sets containing 1. The boundary consists of the $m$-sets avoiding 1 and the $(m+1)$-sets containing 1, giving $2\left(\genfrac{}{}{0ex}{}{2m-1}{m}\right)=\left(\genfrac{}{}{0ex}{}{2m}{m}\right)$.

No boundary can exceed the $2^{n-1}$ vertices outside a half cube. Taking all vertices of even Hamming weight attains that bound, because every odd-weight vertex is adjacent to one of them. Therefore, for $n\ge 1$,

A one-dimensional canonical transformation is a locally invertible change of phase-space variables preserving the symplectic two-form: $dQ\wedge dP=dq\wedge dp$. Equivalently it preserves Hamilton equations for the transformed Hamiltonian. Expanding the two-form yieldssoConversely this Jacobian condition makes the transformation locally invertible and symplectic in two dimensions.

An odd composite integer $n>1$ coprime to $b$ is a Fermat pseudoprime to base $b$ if $b^{n-1}\equiv 1(\mathrm{mod}n)$. A Carmichael number is a composite integer satisfying this congruence for every integer $b$ coprime to $n$. The Korselt criterion says that a composite $n$ is a Carmichael number exactly when it is squarefree and $p-1$ divides $n-1$ for every prime divisor $p$ of $n$.

Differentiate under the contour integral, with one consistent branch and convergent endpoint behavior. ThenThe stated zero boundary term therefore makes the integral a solution. On a closed contour this means returning to the same branch; on an open contour it requires the endpoint values to cancel or vanish.