In 1968 Vizing conjectured that any independent vertex set of an edge-chromatic critical graph G contains at most half of the vertices of G, that is, α(G) [les ] ½[mid ]V(G)[mid ]. Let Δ be the maximum vertex degree in a critical graph. For each Δ, we determine c(Δ) such that α(G) [les ] c(Δ)[mid ]V)[mid ].

We consider the functional equation F∘T−F=f, where T is a measure-preserving transformation and f is a continuous function. We show that if there is an L∞ function F which satisfies this equation, then F is constrained to satisfy a number of regularity conditions, and, in particular, if T is a one-sided Bernoulli shift, then we show that there is a continuous function F satisfying this equation. We show that this is not the case for the two-sided shift.

It is shown that for each compactly generated totally disconnected locally compact group G, there is a finite number of prime numbers, p1, p2, …, pn, such that the scale function s : G → N satisfies s(x) = p1a1(x)p2a2(x)…, pnan(x), where x ∈ G.

Consider the family of exponential maps $\Ek(z)=\exp(z)+\kappa$. This paper shows that any unbounded Siegel disk $U$ of $\Ek$ contains the singular value $\kappa$ on its boundary. By a result of Herman, this implies that $\kappa\in \partial U$ if the rotation number is diophantine.

We say that a Hopf algebra is copolynomial if its dual is polynomial as an algebra. We re-derive Milnor's result that the mod 2 Steenrod algebra is copolynomial by means of a more general result that is also applicable to a number of other related Hopf algebras.

The $\ell$-adic parabolic cohomology groups attached to noncongruence subgroups of $\mathop{\rm SL_2(\mathbb{Z})$ are finite-dimensional $\ell$-adic representations of $\mathop{\rm Gal}(\overline{/mathbb{K}})$ for some number field $K$. We exhibit examples (with $K=\overline{\mathbb{Q}}$) for which the primitive parts give Galois representations whose images are open subgroups of the full group of symplectic similitudes (of arbitrary dimension). The determination of the image of the Galois group relies on Katz's classification theorem for semisimple subalgebras of $\mathfrak{sl}_n$ containing a principal nilpotent element, for which we give a short conceptual proof, suggested by I. Grojnowski.

Belyi˘'s Theorem implies that a Riemann surface X represents a curve defined over a number field if and only if it can be expressed as U/Γ, where U is simply-connected and Γ is a subgroup of finite index in a triangle group. We consider the case when X has genus 1, and ask for which curves and number fields Γ can be chosen to be a lattice. As an application, we give examples of Galois actions on Grothendieck dessins.

In this paper we consider extensions of Livsic's well-known classical results on rigidity of measurable cocycles. We consider transformations which need not be hyperbolic, and cocycles which take values in certain non-abelian non-compact groups.

A noncommutative version of the Hilbert basis theorem is used to show that certain [Rscr ]-symmetric algebras S[Rscr ](V) are Noetherian. This result applies in particular to the coordinate ring of quantum matrices A[Rscr ](V) associated with an R-matrix [Rscr ] operating on the tensor square of a vector space V, to show that, under a natural set of hypotheses on [Rscr ], the algebra A[Rscr ](V) is Noetherian and its augmentation ideal has a polynormal set of generators. As a corollary we deduce that these properties hold for the generic quantized function algebras Rq [G] over any field of characteristic zero, for G an arbitrary connected, simply connected, semisimple group over [Copf ]. That Rq[G] is Noetherian recovers a result due to Joseph [10], with a different proof.

In this note, we show that a homotopy class of [[ ]3, [ ]2] admits an [ ]1-invariant harmonic map if and only if its Hopf invariant is ±k2 for some integer k, where [ ]3 is the unit 3-sphere and [ ]2 is a 2-sphere with an arbitrary metric.

Let $K$ and $L$ be two convex bodies in ${\bb R}^n$ . The volume ratio ${\rm vr}(K, L)$ of $K$ and $L$ is defined by ${\rm vr}(K, L) = \inf(\vert K\vert/\vert T(L)\vert)^{1/n}$ , where the infimum is over all affine transformations $T$ of ${\bb R}^n$ for which $T(L) \subseteq K$ . It is shown in this paper that ${\rm vr}(K, L) \leqslant c \sqrt{n} \log n$ , where $c > 0$ is an absolute constant. This is optimal up to the logarithmic term.

In this paper, some identities in laws involving ladder processes for random walks and Lévy processes are extended and unified.

Let $K$ be a $p$-adic field, let $Z_{\Phi }(s,f)$, $s\,{\in}\,\mathbb{C}$, with Re$(s)\,{>}\,0$, be the Igusa local zeta function associated to $f(x)\,{=}\,(f_{1}(x),\ldots,f_{l}(x))\,{\in}\,[ K( x_{1},\ldots,x_{n})]^{l}$, and let $\Phi $ be a Schwartz–Bruhat function. The aim of this paper is to describe explicitly the poles of the meromorphic continuation of $Z_{\Phi }(s,f)$. Using resolution of singularities it is possible to express $Z_{\Phi }(s,f)$ as a finite sum of $p$-adic monomial integrals. These monomial integrals are computed explicitly by using techniques of toroidal geometry. In this way, an explicit list of the candidates for poles of $Z_{\Phi }(s,f)$ is obtained.

En 1914, Littlewood a montré, contre l'opinion courante à l'époque, qu'il existe des valeurs de x pour lesquelles le nombre de nombres premiers inférieurs à x, π(x), dépasse le logarithme intégral de x, li x. Plus précisément, il a établi que, pour un K>0 convenable, il existe une infinité de valeurs de x pour lesquelles

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et aussi [1] une infinité de valeurs de x pour lesquelles

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En 1937, Beurling a instauré une nouvelle manière de considérer les problèmes sur les nombres premiers, en introduisant les ‘nombres premiers généralisés’ [2]. L'idée est de partir d'une fonction croissante P(x) (x[ges ]0), nulle sur [0, 1], qui joue le rôle de π(x). On lui associe la fonction dzeta et la fonction croissante N(x) (qui joue le rôle de la partie entière de x) selon la formule

formula here

l'hypothèse est toujours que les intégrales ci-dessus existent lorque σ>1 (s = σ+it). Le but de Beurling est, partant de propriétés convenables de la fonction N(x), d'obtenir pour P(x) le ‘théorème des nombres premiers’, P(x) ∼ li x (x → ∞). Nous allons, au contraire, partir d'une hypothèse simple sur P(x) (par exemple, P(x) < li x) et en tirer des conséquences pour la fonction ζ(x), en laissant de côté la fonction N(x). Bien entendu, nous verrons en passant que l'hypothèse P(x) < li x est incompatible avec le fait que ζ(s) soit la fonction dzeta de Riemann.

Il sera commode d'associer à la fonction ζ(s) la fonction

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elle aussi définie pour σ > 1. Ainsi

formula here

Nous nous servirons seulement des deux faits suivants.

Let B=(Bt)t[ges ]0 be standard Brownian motion started at zero. We prove

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for all c>1 and all stopping times τ for B satisfying E(τr)<∞ for some r>1/2. This inequality is sharp, and equality is attained at the stopping time

τ*=inf{t>0[mid ]St [ges ]u*, Xt =1∨αSt},

where u*=1+1/ec(c−1) and α=(c−1)/c for c>1, with Xt=[mid ]Bt[mid ] and St= max0[les ]r[les ]t[mid ]Br[mid ]. Likewise, we prove

formula here

for all c>1 and all stopping times τ for B satisfying E(τr<∞ for some r>1/2. This inequality is sharp, and equality is attained at the stopping time

σ*=inf{t>0[mid ]St [ges ]v*, Xt =αSt},

where v*=c/e(c−1) and α=(c−1)/c for c>1. These results contain and refine the results on the Llog L-inequality of Gilat [6] which are obtained by analytic methods. The method of proof used here is probabilistic and is based upon solving the optimal stopping problem with the payoff

formula here

where F(x) equals either xlog+x or xlog x. This optimal stopping problem has some new interesting features, but in essence is solved by applying the principle of smooth fit and the maximality principle. The results extend to the case when B starts at any given point (as well as to all non-negative submartingales).

Let ${\mathcal P}_n$ be the collection of all polynomials of degree at most $n$ with real coefficients. A subtle Bernstein-type extremal problem is solved by establishing the inequality \[\big\|U_n^{(m)}\big\|_{L_q({\mathbb{R}})} \leq \big(c^{1+1/q}m\big)^{m/2} n^{m/2} \|U_n\|_{L_q({\mathbb{R}})}\] for all $U_n \in \widetilde{G}_n\!$, $q \in (0,\infty]$, and $m=1,2, \ldots$, where $c$ is an absolute constant and $\[ \widetilde{G}_n :=\biggl\{f: f(t) = \sum_{j=1}^N {P_{m_j}(t) e^{-(t-\lambda_j)^2}}, \; \lambda_j \in {\mathbb{R}}, \; P_{m_j}\in\, {\mathcal P}_{m_j}\!, \; \sum_{j=1}^N{(m_j+1)} \leq n \biggr\}. \]$ Some related inequalities and direct and inverse theorems about the approximation by elements of $\widetilde{G}_n$ in $L_q({\mathbb{R}})$ are also discussed.

The purpose of this note is to establish a uniform estimate for the mass function ℙ(Sm = y) of an integer-valued random walk when y → ∞ and (y − mμ)/√m → ∞, where μ is the mean of the step distribution. (The local central limit theorem provides such an estimate when (y − mμ)/√m is bounded.) The assumptions are that the mass function p of the step distribution is regularly varying at ∞ with index −κ, where κ > 3, and that [sum ]∞n=0nκ′p(−n) < ∞ for some κ′ > 2. From this result, a ratio limit theorem is derived, and this in turn is applied to yield some new information about the space–time Martin boundary of certain random walks.

A set A of real numbers is called universal (in measure) if every measurable set of positive measure necessarily contains an affine copy of A. All finite sets are universal, but no infinite universal sets are known. Here we prove some results related to a conjecture of Erdős that there is no infinite universal set. For every infinite set A, there is a set E of positive measure such that (x+tA)⊆E fails for almost all (Lebesgue) pairs (x, t). Also, the exceptional set of pairs (x, t) (for which (x+tA)⊆E) can be taken to project to a null set on the t-axis. Finally, if the set A contains large subsets whose minimum gap is large (in a scale-invariant way), then there is E⊆R of positive measure which contains no affine copy of A.

In [1] Almkvist and Meurman proved a result on the values of the Bernoulli polynomials (Theorem 5 below). Subsequently, Sury [5] and Bartz and Rutkowski [2] have given simpler proofs.

In this paper we show how this theorem can be obtained from classical results on the arithmetic of the Bernoulli numbers. The other ingredient is the remark that a polynomial with rational coefficients which is integer-valued on the integers is ℤ(p)-valued on ℤ(p). Here ℤ(p) denotes the ring of rational numbers whose denominator is not divisible by the prime p. An application is given in Section 3 to the arithmetic of generalised Bernoulli numbers.

Some years ago, Blatter [1] gave a result of the form

formula here

for any function f regular and univalent in D[ratio ][mid ]z[mid ]<1, where ρ is the hyperbolic distance between z1 and z2. Kim and Minda [5] pointed out that the multiplier on the right is incorrect. They say that Blatter's proof gives the correct multiplier, but Blatter's formula for ρ in terms of z1, z2 is wrong. Kim and Minda proved the generalized formula

formula here

where D1(f)=f′(z) (1−[mid ]z[mid ]2), valid for p[ges ]P with some P, 1<P[les ]3/2. In each case there was an appropriate equality statement. Kim and Minda made the important and easily verified remark that these problems are linearly invariant in the sense that if the result is proved for f, then it follows for f˜=UfT, where U is a linear transformation of the plane onto itself and T is a linear transformation of D onto itself. This means that we need to prove such results only in an appropriately normalized context.