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This paper addresses the sharpness of a weighted $L^{2}$-estimate for the Fourier extension operator associated to the circle, obtained by J. Bennett, A. Carbery, F. Soria and A. Vargas in 2006. A point left open in their paper was the necessity of a certain $\log R$-factor in the bound. Here, I show that the factor is necessary for all $1/2$-Ahlfors-David regular weights on the circle, but it can be removed for $s$-Ahlfors-David regular weights with $s \neq 1/2$., 12 pages

In this paper we deal with the hyponormality of Toeplitz operators with matrix-valued symbols. The aim of this paper is to provide a tractable criterion for the hyponormality of bounded-type Toeplitz operators (i.e., the symbol is a matrix-valued function such that Φ and are of bounded type). In particular, we get a much simpler criterion for the hyponormality of when the co-analytic part of the symbol Φ is a left divisor of the analytic part.

The Erdos-Pósa theorem (1965) states that in each graph G which contains at most k disjoint cycles, there is a 'blocking' set B of at most f(k) vertices such that the graph G - B is acyclic. Robertson and Seymour (1986) give an extension concerning any minor-closed class \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document} of graphs, as long as \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document} does not contain all planar graphs: in each graph G which contains at most k disjoint excluded minors for \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document}, there is a set B of at most g(k) vertices such that G - B is in \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document}. In an earlier paper (Kurauskas and McDiarmid, Combin, Probab Comput 20 (2011) 763-775), we showed that, amongst all graphs on vertex set [n] = {1,...,n} which contain at most k disjoint cycles, all but an exponentially small proportion contain a blocking set of just k vertices. In the present paper we build on the previous work, and give an extension concerning any minor-closed graph class \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document} with 2-connected excluded minors, as long as \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document} does not contain all fans (here a 'fan' is a graph consisting of a path together with a vertex joined to each vertex on the path). We show that amongst all graphs G on [n] which contain at most k disjoint excluded minors for \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document}, all but an exponentially small proportion contain a set B of k vertices such that G - B is in \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document}. (This is not the case when \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document} contains all fans.) For a random graph R sampled uniformly from the graphs on [n] with at most k disjoint excluded minors for \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document}, we consider also vertex degrees and the uniqueness of small blockers, the clique number and chromatic number, and the probability of being connected. © 2012 Wiley Periodicals, Inc.

For a flnite group G let i(G) denote the graph deflned on the non- identity elements of G in such a way that two distinct vertices are connected by an edge if and only if they generate G. In this paper it is shown that the graph i(G) contains a Hamiltonian cycle for many flnite groups G. In the literature many deep results about flnite simple groups G can equivalently be stated as theorems about i(G). Three examples are given. Guralnick and Shalev (10) showed that for su-ciently large G the graph i(G) has diameter at most 2. Guralnick and Kantor (9) showed that there is no isolated vertex in i(G). Finally, Breuer, Guralnick, Kantor (4) showed that the diameter of i(G) is at most 2 for all G. In this paper those flnite groups G are considered for which i(G) contains a Hamiltonian cycle. The following proposition reduces the investigations to those non-solvable groups G for which G=N is cyclic for any non-trivial normal subgroup N of G. Proposition 1.1. Let G be a flnite solvable group that has at least 4 elements. Then the graph i(G) contains a Hamiltonian cycle if and only if G=N is cyclic for all non-trivial normal subgroups N of G. The three main results of this paper are Theorems 1.2, 1.3, and 1.4.

This article presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(u(x, t)) in the quasi-linear parabolic equation u t (x, t) = (k(u(x, t))u x (x, t)) x , with Dirichlet boundary conditions u(0, t) = ψ 0 , u(1,t) = ψ 1 . The main purpose of this paper is to investigate the distinguishability of the input-output mappings Φ[·]: K →C 1 [0, T], Ψ[·]: Κ → C 1 [0, T] via semigroup theory. In this paper, it is shown that if the null space of the semigroup T(t) consists of only zero function, then the input-output mappings Φ[·] and Ψ[·] have the distinguishability property. It is also shown that the types of the boundary conditions and the region on which the problem is defined play an important role in the distinguishability property of these mappings. Moreover, under the light of measured output data (boundary observations) f(t):=k(u(0, t))u x (0, t) or/and h(t):=k (u(1,t))u x (1, t), the values k(ψ 0 ) and k(ψ 1 ) of the unknown diffusion coefficient k(u(x,t)) at (x, t) = (0,0) and (x,t) = (1,0), respectively, can be determined explicitly. In addition to these, the values k u (ψ 0 ) and k u (ψ 1 ) of the unknown coefficient k(u(x, t)) at (x,t)=(0,0) and (x, t) = (1, 0), respectively, are also determined via the input data. Furthermore, it is shown that measured output data f(t) and h(t) can be determined analytically by an integral representation. Hence the input-output mappings Φ[·]:Κ→ C 1 [0, T], Ψ[·]:K→ C 1 [0, T] are given explicitly in terms of the semigroup.

In previous papers the authors showed that almost all d-regular graphs for d≤3 are hamiltonian. In the present paper this result is generalized so that a set of j oriented root edges have been randomly specified for the cycle to contain. The Hamilton cycle must be orientable to agree with all of the orientations on the j root edges. It is shown that the requisite Hamilton cycle almost surely exists if and the limiting probability distribution at the threshold is determined when d=3. It is a corollary (in view of results elsewhere) that almost all claw-free cubic graphs are hamiltonian. There is a variation in which an additional cyclic ordering on the root edges is imposed which must also agree with their ordering on the Hamilton cycle. In this case, the required Hamilton cycle almost surely exists if j=o(n2/5). The method of analysis is small subgraph conditioning. This gives results on contiguity and the distribution of the number of Hamilton cycles which imply the facts above. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 128–147, 2001

This paper concerns a Markov operator T on a space L I , and a Markov process P, which defines a Markov operator on a space M of finite signed measures. For T, the paper presents necessary and suf ic ient conditions for: (a) the existence of invariant probability densities (IPDs) (b) existence of strictly positive IPDs, and (c) existence and uniqueness of IPDs. Similar results on invariant probability measures for P are presented. The basic approach is to pose a fixed-point problem as the problem of solving a certain linear equation in a suitable Banach space, and then obtain necessary and sufficient conditions for this equation to have a solution.

In fact, if X is any vector space on which the primitive Banach algebra A acts faithfully and irreducibly, then X can be converted in a Banach space in such a way that the requirements in the theorem are satisfied and even the inclusion A ↪→ BL(X) is contractive. Roughly speaking, the aim of this paper is to prove the appropriate Jordan variant of the above theorem. The notion of primitiveness for Jordan algebras was introduced and developed in 1981 by L. Hogben and K. McCrimmon [10]. Primitive Jordan algebras are relevant particular types of prime nondegenerate Jordan algebras but, although the celebrated Zel’manov prime theorem ([19], 1983) gave a precise description of these last algebras, it has happened only very recently that the appropriate variant of Zel’manov’s theorem for primitive Jordan algebras has been obtained (see [3] and [17]). Also very recently several particular normed versions of Zel’manov’s theorem have been provided (see [8], [6], [16], and [7]). Nevertheless, to obtain a Zel’manov type theorem for primitive Jordan-Banach algebras has remained an open problem in the last years [15]. In fact we have been able to prove such a theorem but only passing through a general normed version of the Zel’manov prime theorem (see Theorem 1) which is in our opinion one of the most important novelties in the paper. Since Theorem 1 will probably have applications different from that in the paper, we have included in its statement and proof some details not strictly needed for our main purpose. The same comment should be made concerning Theorem 2, which is nothing but a fine improvement of Theorem 1 under the additional assumption of completeness. From Theorem 2 and the main results in [3], [18], and [5], the desired Jordan variant of Theorem 0 (Theorem 3) follows easily. Again roughly speaking, it asserts that primitive complex Jordan-Banach algebras, different from the simple exceptional 27-dimensional one and the simple Jordan algebras of a continuous symmetric bilinear form on a complex Banach space, can be continuously regarded as Jordan algebras of bounded linear operators ”acting irreducibly” on a suitable complex Banach space.

We consider digital expansions with respect to complex integer bases. We derive precise information about the length of these expansions and the corresponding sum-of-digits function. Furthermore we give an asymptotic formula for the sum-of-digits function in large circles and prove that this function is uniformly distributed with respect to the argument. Finally the summatory function of the sum-of-digits function along the real axis is analyzed. where F q is a continuous, 1-periodic, nowhere dierentiable function with known Fourier expansion. Several more sophisticated digital functions have been studied since then and the fractal behaviour of the summatory functions appeared in many of these cases (cf. (5, 23)). Various methods were used to derive such summation formulae: an early one was developed by Delange (2) and is based on reinterpretation of the occurring sums as real integrals. In (23) and (5) it is observed that the classical technique of Dirichlet generating functions can be used to derive Delange's formula. In this paper we shall generalize some results about 'ordinary' digital expansions to positional number systems of the Gaussian integers, which were introduced by Knuth (20) and extensively studied by Gilbert in a series of papers (7-15). Again fractal structures are involved, but it requires some additional ideas to use the techniques mentioned above in the case of complex bases. In the introductory Section 2 we shall present a number of auxiliary results about digital expansions of complex integers (some of which recall results of Gilbert using a dierent approach). We also exhibit automata, which describe addition of 1 (respectively, other fixed Gaussian integers) in these positional number systems. Furthermore formulae for the sum-of-digits function with respect to complex bases are derived and we analyze the length of the expansion asymptotically.

The paper is concerned with completely positive maps on the algebra of unbounded operatore L+(D) and on its completion L(D, D+). A decomposition theorem for continuous positive functionals is proved in [Tim. Loef.), and [Scholz 91] contains a generalization to maps into operator algebra on finite dimensional Hilbert spaces H0. The aim of the present paper is to construct an analogous decomposition without the assumption that H0 is finite dimensional. Moreover, the Kraus - theorem [Kraus] is proved for normal completely positive mappings on L(D, D+). The paper is organized as follows. Section 1 contains the necessary definitions and notations. In Section 2 we prove the decomposition theorem. Section 3 deal with the structure of the normal completely positive mappings.

Let Gn, p denote a random graph on n vertices. It is an interesting problem when small cliques arise and what distributions of the number of small cliques may occur. Matroids are natural generalization of graphs; therefore, we can try to investigate maximal flats of a small rank in random matroids. The most studied and most interesting seem to be “random projective geometries” introduced by Kelly and Oxley. Many of the theorems in our paper are based on the results published in a long paper of Barbour, Janson, Karoski, and Ruciski. However, proofs for projective geometries generally are more complicated. © 1995 Wiley Periodicals, Inc.

In this paper we give sucient conditions for weak isomorphism of Lebesgue measure-preserving expanding endomorphisms of S 1 : rst author gave necessary and sucient conditions for two real an- alytic Lebesgue measure-preserving expanding endomorphisms of the circle to be isomorphic upto a phase factor. This was a partial answer to the problem of nding complete measure theoretic invariants for isomorphisms posed by Shub and Sullivan in (5). In this paper it is shown that the condition given in (2) is sucient for weak-isomorphism. For i = 1; 2 let fi be endomorphisms of the Lebesgue spaces (Xi;Bi; i): We say that the two systems (X1;B1; 1;f1) and (X2;B2; 2;f2) are isomorphic if there are sets of measure zero A1 X1;A2 X2 and a one-to-one onto map : X1nA1! X2nA2 such that f 1 = f2 on X1nA1 and 1( 1 E) = 2(E) for all measurable E X2nA2: The classication

The standard additive coalescent starting with n particles is a Markov process which owns several combinatorial representations, one by Pitman as a process of coalescent forests, and one by Chassaing & Louchard as the block sizes in a parking scheme. In the coalescent forest representation, some edges are added successively between a random node and a random root. In this paper, we investigate an alternative construction by adding edges between the roots. This construction induces the same process at the level of cluster sizes, but allows one to make numerous connections with some combinatorial and probabilistic models that were not known to be connected with additive coalescent. The variety of the combinatorial objects involved here – size biased percolation, parking scheme in a tree, increasing trees, random cuts of trees – justifies our interests in this Acknowledgement : The research has been supported by ANR-14-CE25-0014 (ANR GRAAL).

In this paper, we introduce different kinds of growth orders for the set of entire solutions to the most general framework of higher-dimensional polynomial Cauchy-Riemann equations ∏i=1p(D−λi)kif=0, where D:=∂f∂x0+∑i=1nei∂f∂xi is the hypercomplex Cauchy-Riemann operator, λi are arbitrarily chosen nonzero complex constants, and ki are arbitrarily chosen positive integers. The core ingredient is a projection formula that establishes a relation to the ki-monogenic component functions, which are null-solutions to iterates of the Cauchy-Riemann operator that we studied in earlier works. Furthermore, we briefly outline the analogies of the Lindelof-Pringsheim theorem in this context.

In this paper, we study the topology of real analytic map-germs with isolated critical value f:(Rm,0)→(Rn,0), with 1

We continue our study of the full set of translation-invariant splitting Gibbs measures TISGMs, translation-invariant tree-indexed Markov chains for the q-state Potts model on a Cayley tree. In our previous work Kulske et al., J Stat Phys 156 2014, 189-200 we gave a full description of the TISGMs, and showed in particular that at sufficiently low temperatures their number is 2q-1. In this paper we find some regions for the temperature parameter ensuring that a given TISGM is non-extreme in the set of all Gibbs measures. In particular we show the existence of a temperature interval for which there are at least 2q-1+q extremal TISGMs. For the Cayley tree of order two we give explicit formulae and some numerical values. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 636-678, 2017

In this paper we prove an existence result for the following singular elliptic system {z > 0 in Omega, z is an element of W-0(iota,p)(Omega) : -Delta(p)z = a(x)z(q-iota)u(theta) , u > 0 in Omega, u is an element of W-0(iota,p)(Omega) : -Delta(p)u = b(x)z(q)u(theta-iota) , where Omega is a bounded open set in R-N (N >= 2), -Delta(p) is the p-laplacian operator, a(x) and b(x) are suitable Lebesgue functions and q > 0, 0 1 are positive parameters satisfying suitable assumptions. (C) 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

We investigate suitable conditions for a C0-semigroup (T(t))t≥0 of Hilbert space contractions to be unitarily equivalent to the restriction of the adjoint shift semigroup (S∗γ(t))t≥0 to an invariant subspace of the standard weighted Bergman space Aγ−2(C+,K). It turns out that (T(t))t≥0 admits a model by (S∗γ(t))t≥0 if and only if its cogenerator is γ-hypercontractive and limt→0T(t)=0 in strong operator topology. We then discuss how such semigroups can be characterized without involving the cogenerator. A sufficient condition is that, for each t>0, the operator T(t) is γ-hypercontractive. Surprisingly, this condition is necessary if and only if γ is integer. The paper is concluded with a conjecture that would imply a more symmetric characterization. (Less)

This paper deals with dimension-controllable (tractable) embeddings of Besov spaces on n-dimensional torus into small Lebesgue spaces. Our techniques rely on the approximation structure of Besov spaces, extrapolation properties of small Lebesgue spaces and interpolation.

Let f be an edge ordering of Kn: a bijection . For an edge , we call f(e) the label of e. An increasing path in Kn is a simple path (visiting each vertex at most once) such that the label on each edge is greater than the label on the previous edge. We let S(f) be the number of edges in the longest increasing path. Chvatal and Komlos raised the question of estimating m(n): the minimum value of S(f) over all orderings f of Kn. The best known bounds on m(n) are , due respectively to Graham and Kleitman, and to Calderbank, Chung, and Sturtevant. Although the problem is natural, it has seen essentially no progress for three decades. In this paper, we consider the average case, when the ordering is chosen uniformly at random. We discover the surprising result that in the random setting, S(f) often takes its maximum possible value of n – 1 (visiting all of the vertices with an increasing Hamiltonian path). We prove that this occurs with probability at least about 1/ e. We also prove that with probability 1- o(1), there is an increasing path of length at least 0.85 n, suggesting that this Hamiltonian (or near-Hamiltonian) phenomenon may hold asymptotically almost surely. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015

ii(t)+Au(t)+p(t) B(u(t)) =O. Here A is a densely defined positive selfadjoint linear operator on a real Hilbert space H with inner product ( , ), B is a locally Lipschitz map from D(A”2) (endowed with the graph norm) into H, and p(t) is a real-valued control. The finite-dimensional stabilization problem H = IW” has been considered in the recent papers of Jurdjevic and Quinn [20] and Slemrod [24]. In the case when (1.1) is bilinear, i.e., B linear, these papers give simple criteria for feedback stabilization. Specifically, it is a consequence of both [20] and [24] that if the only solution of the uncontrolled system

for a certain c. In fact I shall prove that if n is highly composite, then the next highly composite number is less than n+n(log y&)-C ; and the result just stated follows immediately from this. At, present I cannot, decide whether the number of highly composite numbers not exceeding z is greater than (logx)” for every k. The principal tool in the proof will be Ingham’s improvement,$ on Hoheisel’s theorem. This asserts that if x is sufficiently large, then the number of primes in the int,erval (x, x+&) is asymptot.ic to cxg(logz)-1. First we state three lemmas, which will be proved at the end of the paper. They are contained subst’antially in the paper of Ramanujan, but we prove Ohem here for completeness. Let n = 22 3~ . . . p+ be a sufficiently large highly composite number. Plainly

It was proved by Roth in a recent paper that if α is any real algebraic number, and if K > 2, then the inequalityhas only a finite number of solutions in relatively prime integers p, q (q > 0) The object of the present paper is to prove that the lower bound for κ can be reduced if conditions are imposed on p and q. The result obtained is as follows.

In this paper, we present some results on Fredholm and upper semi-Fredholm operators involving demicompact operators. Our results generalize many known ones in the literature, in particular those obtained by Petryshyn in [27] and Jeribi et al. in [1], [22]. They are used to establish a fine description of the Schechter essential spectrum of closed densely defined operators, and to investigate the essential spectrum of the sum of two bounded linear operators defined on a Banach space by means of the essential spectrum of each of the two operators.

The Hartley transform is an integral transformation that maps a real valued function into a real valued frequency function via the Hartley kernel, thereby avoiding complex arithmetic as opposed to the Fourier transform. Approximation of the Hartley integral by the trapezoidal quadrature results in the discrete Hartley transform, which has proven a contender to the discrete Fourier transform because of its involutory nature. In this paper, a discrete transform is proposed as a real transform with a convolution property and is an alternative to the discrete Hartley transform. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper we obtain several extensions to the quaternionic setting of someresultsconcerningtheapproximation bypolynomials of functionscontinuous onacompact set and holomorphic in its interior. The results include approximationon compact starlike sets and compact axially symmetric sets. The cases of someconcrete particular sets are described in details, including quantitative estimatestoo. AMS 2010 Mathematics Subject Classification: Primary 30G35; Secondary 30E10,41A25.Keywords and phrases: Mergelyan’s theorem, quaternions, Riemann mapping, axiallysymmetric sets, approximation by polynomials, convolution operators, Cassini pseudo-metric, Cassini cell, order of approximation, slice regular functions. 1 Introduction It is well-known the fact that the Mergelyan’s approximation theorem is the ultimatedevelopment and generalization of the Weierstrass approximation theorem and Runge’stheorem in the complex plane. It can be stated as follows (see [13]):Theorem 1.1. Let K be a compact subset of the complex plane C such that C\ K isconnected. Then, every continuous function on K, f : K → C, which is holomorphic inthe interior of K, can be approximated uniformly on K by polynomials.Notice that all the known proofs of this result, based on the methods in complex analysisuse the Riemann mapping theorem.1

The aim of the paper is to study the relation between ultra-differentiable classes of functions defined in terms of estimates on derivatives on one hand and in terms of growth properties of Fourier transforms of suitably localized functions in the class on the other hand. We establish this relation for the ultra-differentiable classes introduced in [6], [16], and show that the classes of [6], [16], can be regarded as inhomogeneous Gevrey classes in the sense of [22]. We also discuss a number of properties of the weight functions used to define the respective classes and of their Young conjugates.

Repeated convolution of a probability measure on leads to the central limit theorem and other limit theorems. This paper investigates what kinds of results remain without positivity. It reviews theorems due to Schoenberg, Greville, and Thomee which are motivated by applications to data smoothing (Schoenberg and Greville) and finite difference schemes (Thomee). Using Fourier transform arguments, we prove detailed decay bounds for convolution powers of finitely supported complex functions on . If M is an hermitian contraction, an estimate for the off-diagonal entries of the powers of is obtained. This generalizes the Carne–Varopoulos Markov chain estimate.

In this paper, we present a heuristic algorithm for solving exact, as well as approximate, shortest vector and closest vector problems on lattices. The algorithm can be seen as a modified sieving algorithm for which the vectors of the intermediate sets lie in overlattices or translated cosets of overlattices. The key idea is hence no longer to work with a single lattice but to move the problems around in a tower of related lattices. We initiate the algorithm by sampling very short vectors in an overlattice of the original lattice that admits a quasi-orthonormal basis and hence an efficient enumeration of vectors of bounded norm. Taking sums of vectors in the sample, we construct short vectors in the next lattice. Finally, we obtain solution vector(s) in the initial lattice as a sum of vectors of an overlattice. The complexity analysis relies on the Gaussian heuristic. This heuristic is backed by experiments in low and high dimensions that closely reflect these estimates when solving hard lattice problems in the average case.This new approach allows us to solve not only shortest vector problems, but also closest vector problems, in lattices of dimension$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}n$in time$2^{0.3774\, n}$using memory$2^{0.2925\, n}$. Moreover, the algorithm is straightforward to parallelize on most computer architectures.

In this paper a general theory of semi-classical d-orthogonal polynomials is developed. We define the semi-classical linear functionals by means of a distributional equation , where Φ and Ψ are matrix polynomials. Several characterizations for these semi-classical functionals are given in terms of the corresponding d-orthogonal polynomials sequence. They involve a quasi-orthogonality property for their derivatives and some finite-type relations.

Let (C1,C′(*)),(C2,C′(*)),…,(C m,C′(*)) be a sequence of ordered pairs of 2CNF clauses chosen uniformly at random (with replacement) from the set of all 4 clauses on n variables. Choosing exactly one clause from each pair defines a probability distribution over 2CNF formulas. The choice at each step must be made on-line, without backtracking, but may depend on the clauses chosen previously. We show that there exists an on-line choice algorithm in the above process which results whp in a satisfiable 2CNF formula as long as m/n ≤ (1000/999)1/4. This contrasts with the well-known fact that a random m -clause formula constructed without the choice of two clauses at each step is unsatisfiable whp whenever m/n > 1. Thus the choice algorithm is able to delay satisfiability of a random 2CNF formula beyond the classical satisfiability threshold. Choice processes of this kind in random structures are known as “Achlioptas processes.” This paper joins a series of previous results studying Achlioptas processes in different settings, such as delaying the appearance of a giant component or a Hamilton cycle in a random graph. In addition to the on-line setting above, we also consider an off-line version in which all m clause-pairs are presented in advance, and the algorithm chooses one clause from each pair with knowledge of all pairs. For the off-line setting, we show that the two-choice satisfiability threshold for k -SAT for any fixed k coincides with the standard satisfiability threshold for random 2k -SAT.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013

In this paper we prove the Bohr Theorem for slice regular functions. Following the historical path that led to the proof of the classical Bohr Theorem, we also extend the Borel-Carath\'eodory Theorem to the new setting.

Let be a sequence included in a compact subset of the unit disk ; we consider the rational Fourier series that are obtained by orthogonalization of the Blaschke product sequence {B0(eix) = 1,B1(eix), ⋯ ,Bn(eix), ⋯ }, where In order to study the point-wise convergence of this rational Fourier series, let the partial sums of f ∈ L1[ − π,π] be defined as where for . In this paper, we will show that the conditions for the point-wise convergence of f(x) − Sn(f)(x) is the same as that of the Fourier series. More precisely, one is the Dirichlet–Dini criterion, and the other is the Jordan test. Copyright © 2012 John Wiley & Sons, Ltd.

Let X be a complex Banach space and Y be a JB*-triple. Let G be a bounded balanced domain in X and BY be the unit ball in Y. Let f : G BY be a holomorphic mapping. In this paper, we obtain some generalization of Bohr's theorem that if a = f(0), then we have for z ∈ (1 / 3)G, where φa ∈ Aut(BY) such that φa(a) = 0. Moreover, we show that the constant 1 / 3 is best possible. This result generalizes Bohr's theorem for the open unit disc Δ in the complex plane . Copyright © 2012 John Wiley & Sons, Ltd.

In this paper we construct a resolution function that will provide an algorithm of resolution of singularities for binomial ideals, over a field of arbitrary characteristic. For us, a binomial ideal means an ideal generated by binomial equations without any restriction, including monomials and p-th powers, where p is the characteristic of the base field. This resolution function is based in a modified order function, called E-order. The E-order of a binomial ideal is the order of the ideal along a normal crossing divisor E. The resolution function allows us to construct an algorithm of E-resolution of binomial basic objects, that will be a subroutine of the main resolution algorithm.

The purpose of this paper is to introduce five martingale Orlicz-Hardy spaces and to establish the atomic decomposition theorem. As applications we show the relation among five martingale Orlicz-Hardy spaces and the duality, namely, the dual of martingale Orlicz-Hardy spaces are generalized martingale Campanato spaces. Further, we prove a John-Nirenberg type inequality for generalized martingale Campanato spaces when the stochastic basis is regular.

The four colour theorem states that the vertices of every planar graph can be coloured with at most four colours so that no two adjacent vertices receive the same colour. This theorem is famous for many reasons, including the fact that its original 1977 proof includes a non-trivial computer verification. Recently, a formal proof of the theorem was obtained with the equational logic program Coq [G. Gonthier, ‘Formal proof–the four color theorem’,Notices of Amer. Math. Soc.55 (2008) no. 11, 1382–1393]. In this paper we describe an implementation of the computational method introduced by C. S. Calude and co-workers [Evaluating the complexity of mathematical problems. Part 1’,Complex Systems18 (2009) 267–285; A new measure of the difficulty of problems’,J. Mult. Valued Logic Soft Comput.12 (2006) 285–307] to evaluate the complexity of the four colour theorem. Our method uses a Diophantine equational representation of the theorem. We show that the four colour theorem is in the complexity class ℭU,4. For comparison, the Riemann hypothesis is in class ℭU,3while Fermat’s last theorem is in class ℭU,1.

Let L be a number field and let OL be its ring of integers. It is a very difficult problem to decide whether OL has a power basis. Moreover, if a power basis exists, it is hard to find all the generators of OL over Z. In this paper, we show that if a is a generator of the ring of integers of an abelian imaginary field whose conductor is relatively prime to 6, then either a is an integer translate of a root of unity, or a + a is an odd integer. From this result and other remarks it follows that if â is a generator of the ring of integers of an abelian imaginary field with conductor relatively prime to 6 and â is not an integer translate of a root of unity, then ââ is a generator of the ring of integers of the maximal real field contained in Q(â). Finally, we use a result of Gras to prove that if d > 1 is an integer relatively prime to 6, then all but finitely many imaginary extensions of Q of degree 2d have a ring of integers that does not have a power basis.

In this paper an abstract condition is given yielding universal series defined by sequences a = {a(j)}infinity j=1 in boolean AND(p > 1)l(p) but not in l(1). We obtain a unification of some known results related to approximation by translates of specific functions including the Riemann zeta-function, or a fundamental solution of a given elliptic operator in R-nu with constant coefficients or an approximate identity as, for example, the normal distribution. Another application gives universal trigonometric series in R-nu simultaneously with respect to all sigma-finite Borel measures in R-nu. Stronger results are obtained by using universal Dirichlet series.

Our aim in this paper is to deal with integrability of maximal functions for generalized Lebesgue spaces with variable exponent. Our exponent approaches 1 on some part of the domain, and hence the integrability depends on the shape of that part and the speed of the exponent approaching 1. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Generalized Orlicz–Lorentz sequence spaces λφ generated by Musielak-Orlicz functions φ satisfying some growth and regularity conditions (see [28] and [33]) are investigated. A regularity condition δλ2 for φ is defined in such a way that it guarantees many positive topological and geometric properties of λφ. The problems of the Fatou property, the order continuity and the Kadec–Klee property with respect to the uniform convergence of the space λφ are considered. Moreover, some embeddings between λφ and their two subspaces are established and strict monotonicity as well as lower and upper local uniform monotonicities are characterized. Finally, necessary and sufficient conditions for rotundity of λφ, their subspaces of order continuous elements and finite dimensional subspaces are presented. This paper generalizes the results from [19], [4] and [17]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Some boundaries about the solution of the linear Volterra integral equations of the form f(t)=1−K*f were obtained as |f(t)|⩽1, |f(t)|⩽2 and |f(t)|⩽4 in (J. Math. Anal. Appl. 1978; 64:381–397; Int. J. Math. Math. Sci. 1982; 5(1):123–131). The boundary of the solution function of an equation in this type was found as |f(t)|⩽2n in (Integr. Equ. Oper. Theory 2002; 43:466–479), where t∈[0, ∞) and n is a natural number such that n⩾2. In (Math. Comp. 2006; 75:1175–1199), it is shown that the boundary of the solution function of an equation in the same form can also be derived as that of (Integr. Equ. Oper. Theory 2002; 43:466–479) under different conditions than those of (Integr. Equ. Oper. Theory 2002; 43:466–479). In the present paper, the sufficient conditions for the boundedness of functions f, f′, f′′, …, f(n+3), (n∈ℕ) defined on the infinite interval [0, ∞) are given by our method, where f is the solution of the equation f(t)=1−K*f. Copyright © 2007 John Wiley & Sons, Ltd.

Let E_G be a family of hyperelliptic curves over F2^(alg cl) with general Weierstrass equation given over a very small field F. We describe in this paper an algorithm to compute the zeta function of E_g for g in a degree n extension field of F, which has as time complexity O(n^3) and memory requirements O(n^2). With a slightly different algorithm we can get time O(n^2,667) and memory O(n^2,5), and the computation of O(n) curves of the family can be done in time and space O(n^3). All these algorithms are polynomial in the genus., Comment: 23 pages

Let K be an algebraically closed field, X a K-scheme, and X(K) the set of closed points in X. A constructible set C in X(K) is a finite union of subsets Y(K) for finite type subschemes Y in X. A constructible function f : X(K) --> Q has f(X(K)) finite and f^{-1}(c) constructible for all nonzero c. Write CF(X) for the Q-vector space of constructible functions on X. Let phi : X --> Y and psi : Y --> Z be morphisms of C-varieties. MacPherson defined a Q-linear "pushforward" CF(phi) : CF(X) --> CF(Y) by "integration" w.r.t. the topological Euler characteristic. It is functorial, that is, CF(psi o phi)=CF(psi) o CF(phi). This was extended to K of characteristic zero by Kennedy. This paper generalizes these results to K-schemes and Artin K-stacks with affine stabilizers. We define notions of Euler characteristic for constructible sets in K-schemes and K-stacks, and pushforwards and pullbacks of constructible functions, with functorial behaviour. Pushforwards and pullbacks commute in Cartesian squares. We also define "pseudomorphisms", a generalization of morphisms well suited to constructible functions problems., 25 pages, LaTeX. (v7) shorter: material moved to math.AG/0509722

In this paper, we characterize the complex sequences (zn)n which satisfy the following condition: For each complex sequence (an)n, there exists a function f such that the functions tznf(t) are Lebesgue integrable and an = ∫ tznf(t)(dt) for all n∫. In this case, we give for every sequence (an)n infinitely many C∫ functions f satisfying some growth conditions in x = 0 and x = + ∫, and such that an = ∫ tznf(t)dt. Finally, we extend this result for Banach space valued functions.

The purpose of this paper is to present characterizations of the inhomogeneous Hardy-Bessel potential spaces Fpα (K) over the 2-series field K defined by Littlewood-Paley type function, where Δ0(x) = 1(|x|≥1), = 0(other), Δj(x) = 2j ≥, = 0(other) (j ≥ 1). These characterizations are given by difference of functions, ball means of difference and atoms. As applications of these results we shall determine when Fpα(K) is a multiplication algebra, and prove the lower majorant property, the uniform localization property and the equivalence of Fourier multipliers.

In this article we show that the set of Dirichlet regular boundary points of a bounded domain of dimension up to 4, definable in an arbitrary o-minimal structure on the field ℝ, is definable in the same structure. Moreover we give estimates for the dimension of the set of non-regular boundary points, depending on whether the structure is polynomially bounded or not. This paper extends the results from the author's Ph.D. thesis [6, 7] where the problem was solved for polynomially bounded o-minimal structures expanding the real field. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In this paper, we give a lower bound of the p-part of the reduced Euler characteristic of the order complex of the centric p-radical subgroups by studying vertices of indecomposable summands of the reduced Lefschetz module. This bound is in fact best possible for at least some groups, and also provides a uniform explanation of the observed phenomenon on the reduced Euler characteristic for some sporadic simple groups.

It is known that whenever E1, … , En are infinite dimensional L∞-spaces and F is any infinite dimensional Banach space, there exists a bounded n-linear mapping from E1 × … × En into F that fails to be absolutely (1; 2)-summing. In this paper we generalize a theorem of S. Kwapien and obtain a sufficient condition in order to assure that a given n-linear mapping T from infinite dimensional L∞-spaces into an infinite dimensional Hilbert space is absolutely (1; 2)-summing. Besides, we also give a sufficient condition in order to obtain a fully (1; 1)-summing multilinear mapping from l1 ×…× l1 × l2 into an infinite dimensional Hilbert space. In the last section we introduce the concept of fully summing holomorphic mappings and give the first examples of this kind of map. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

This paper presents a new probabilistic algorithm to compute modular polynomials modulo a prime. Modular polynomials parameterize pairs of isogenous elliptic curves, and are useful in many aspects of computational number theory and cryptography. The algorithm presented here has the distinguishing feature that it does not involve the computation of Fourier coefficients of modular forms. The need to compute the exponentially large integral coefficients is avoided by working directly modulo a prime, and computing isogenies between elliptic curves via Vélu's formulas.