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This paper studies the virtual $\chi_{-y}$-genera of Grothendieck's Quot schemes on surfaces, thus refining the calculations of the virtual Euler characteristics by Oprea-Pandharipande. We first prove a structural result expressing the equivariant virtual $\chi_{-y}$-genera of Quot schemes universally in terms of the Seiberg-Witten invariants. The formula is simpler for curve classes of Seiberg-Witten length $N$, which are defined in the paper. By way of application, we give complete answers in the following cases: (i) arbitrary surfaces for the zero curve class, (ii) relatively minimal elliptic surfaces for rational multiples of the fiber class, (iii) minimal surfaces of general type with $p_g>0$ for any curve classes. Furthermore, a blow up formula is obtained for curve classes of Seiberg-Witten length $N$. As a result of these calculations, we prove that the generating series of the virtual $\chi_{-y}$-genera are given by rational functions for all surfaces with $p_g>0$, addressing a conjecture of Oprea-Pandharipande. In addition, we study the reduced $\chi_{-y}$-genera for $K3$ surfaces and primitive curve classes with connections to the Kawai-Yoshioka formula., Comment: 48 pages. Reformulation of Theorem 1 for readability. Extra explanation for the mixed terms in section 2.2.3. Reference updates

This paper continues the work started in [E. Kissin, V. S. Shulman and Yu. V. Turovskii, �Banach Lie algebras with Lie subalgebras of finite codimension: their invariant subspaces and Lie ideals�, J. Funct. Anal. 256 (2009) 323�351.] and is devoted to the study of reducibility of an infinite-dimensional Lie algebra of operators on a Banach space when its Lie subalgebra of finite codimension has an invariant subspace of finite codimension. In addition to the tools developed in the above paper; filtrations of Banach spaces with respect to Lie algebras of operators and related systems of operators on graded Banach spaces, the present paper introduces and studies some new concepts and techniques: the theory of Lie quasi-ideals and properties of Lie nilpotent finite-dimensional subspaces of Banach associative algebras. The application of these techniques to an operator Lie algebra L shows that, under some mild additional assumptions, L is reducible if its Lie subalgebra of finite codimension has an invariant subspace of finite codimension. This, in turn, leads to the main result of the paper: if a Banach Lie algebra L has a closed Lie subalgebra of finite codimension, then it has a proper closed Lie ideal of finite codimension. Moreover, if L is non-commutative, then it has a characteristic Lie ideal of finite codimension, that is, a proper closed Lie ideal of L invariant for all bounded derivations of L.

We prove, correct and extend several results of an earlier paper of ours (using and recalling several of our later papers) about the derived functors of projective limit in abelian categories. In particular we prove that if C is an abelian category satisfying the Grothendieck axioms AB3 and AB4* and having a set of generators then the first derived functor of projective limit vanishes on so-called Mittag-Leffler sequences in C. The recent examples given by Deligne and Neeman show that the condition that the category has a set of generators is necessary. The condition AB4* is also necessary, and indeed we give for each integer $m \geq 1$ an example of a Grothendieck category Cm and a Mittag-Leffler sequence in Cm for which the derived functors of its projective limit vanish in all positive degrees except m. This leads to a systematic study of derived functors of infinite products in Grothendieck categories. Several explicit examples of the applications of these functors are also studied.

The paper defines a new type of product of association schemes (and of the related objects, permutation groups and orthogonal block structures), which generalizes the direct and wreath products (which are referred to as 'crossing' and 'nesting' in the statistical literature). Given two association schemes for , each having an inherent partition (that is, a partition whose equivalence relation is a union of adjacency relations in the association scheme), a product of the two schemes is defined, which reduces to the direct product if or , and to the wreath product if and , where and are the relation of equality and the universal relation on . The character table of the crested product is calculated, and it is shown that, if the two schemes and have formal duals, then so does their crested product (and a simple description of this dual is given). An analogous definition for permutation groups with intransitive normal subgroups is created, and it is shown that the constructions for association schemes and permutation groups are related in a natural way. The definition can be generalized to association schemes with families of inherent partitions, or permutation groups with families of intransitive normal subgroups. This time the correspondence is not so straightforward, and it works as expected only if the inherent partitions (or orbit partitions) form a distributive lattice. The paper concludes with some open problems.

Inspired by the work of Bloch and Kato in [ 2 ], David Burns constructed several ‘equivariant Tamagawa invariants’ associated to motives of number fields. These invariants lie in relative $K$ -groups of group-rings of Galois groups, and in [ 3 ] Burns gave several conjectures (see Conjecture 3.1) about their values. In this paper I shall verify Burns' conjecture concerning the invariant $T\Omega^{\rm loc}( N/{\bf Q},1)$ for some families of quaternion extensions $N/{\bf Q}$ . Using the results of [ 9 ] I intend in a subsequent paper to verify Burns' conjecture for those families of quaternion fields which are not covered here.

Time-variantfilters based on Calderon and Gabor reproducingformulas are important tools in time­ frequencyanalysis.The paper studiesthe behaviorof the eigenvaluesof these filters.Optimal two-sided estimates of the number of eigenvaluescontainedin the interval (151,02), where 0 < 01 < 152 < 1, arc obtained.The estimatescoverlarge classesof localizationdomainsand generatingfunctions. 1. Introduction and statements of the results Calderon- Toeplitz and Gabor- Toeplitz operators arise naturally in two contexts: (i) Toephtz operators on Fock and Bergman spaces of holomorphic functions; (ii) time-variant filters based on Calderon and Gabor reproducing formulas. This paper is concerned with the eigenvalues of a subclass of Calderon- Toeplitz and Gabor- Toeplitz operators which have characteristic functions of bounded domains as symbols. Operators of this class are called time-frequency localiza­ tion operators. The basic idea of functional calculus is that the operators resemble the main algebraic features of their symbols. We consider symbols that are idem­ potent with respect to pointwise multiplication, so it is natural to expect that the corresponding operators are at least approximately idempotent. It is easy to verify that time-frequency localization operators are compact, self-adjoint and bounded by 1. In view of these facts and the above-mentioned correspondence principle, one is inclined to think that localization operators should resemble finite dimensional orthogonal projections. We show that this expectation is correct for Gabor- Toeplitz operators and that it is false for Calderon- Toeplitz operators. We identify the basic geometric features responsible for these two different behaviors. Our principal results are two-sided estimates of the number of eigenvalues inside the plunge region corresponding to 61, (52, where 0 < 61 < (52 < 1. The plunge region consists of the set of indices of the eigenvalues contained inside the open interval (61,62). The eigen­ val ues are ordered non-increasingly. Our work generalizes and improves previ ous results of Daubechies, Paul, Ramanathan and Topiwala [6, 8, 21].

The paper gives a formula for the probability that a randomly chosen elliptic curve over a finite field has a prime number of points. Two heuristic arguments in support of the formula are given as well as experimental evidence. The paper also gives a formula for the probability that a randomly chosen elliptic curve over a finite field has kq points where k is a small number and q is a prime.

In [6] S. Shelah showed that in the endomorphism semi-group of an infinitely generated algebra which is free in a variety one can interpret some set theory. It follows from his results that, for an algebra Fℵ which is free of infinite rank ℵ in a variety of algebras in a language L, if ℵ > |L|, then the first-order theory of the endomorphism semi-group of Fℵ, Th(End(Fℵ)), syntactically interprets Th(ℵ,L2), the second-order theory of the cardinal ℵ. This means that for any second-order sentence χ of empty language there exists χ*, a first-order sentence of semi-group language, such that for any infinite cardinal ℵ > |L|,formula hereIn his paper Shelah notes that it is natural to study a similar problem for automorphism groups instead of endomorphism semi-groups; a priori the expressive power of the first-order logic for automorphism groups is less than the one for endomorphism semi-groups. For instance, according to Shelah's results on permutation groups [4, 5], one cannot interpret set theory by means of first-order logic in the permutation group of an infinite set, the automorphism group of an algebra in empty language. On the other hand, one can do this in the endomorphism semi-group of such an algebra.In [7, 8] the author found a solution for the case of the variety of vector spaces over a fixed field. If V is a vector space of an infinite dimension ℵ over a division ring D, then the theory Th(ℵ, L2) is interpretable in the first-order theory of GL(V), the automorphism group of V. When a field D is countable and definable up to isomorphism by a second-order sentence, then the theories Th(GL(V)) and Th(ℵ, L2) are mutually syntactically interpretable. In the general case, the formulation is a bit more complicated.The main result of this paper states that a similar result holds for the variety of all groups.

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.

In this paper, we consider a basic question in commutative algebra: if I and J are ideals of a commutative ring S , when does IJ = I ∩ J ? More precisely, our setting will be in a polynomial ring k [ x 0 , …, x n ], and the ideals I and J will define subschemes of the projective space ℙ n k over k . In this situation, we are able to relate the equality of product and intersection to the behavior of the cohomology modules of the subschemes defined by I and J . By doing this, we are able to prove several general algebraic results about the defining ideals of certain subschemes of projective space. Our main technique in this paper is a study of the deficiency modules of a subscheme V of ℙ n . These modules are important algebraic invariants of V , and reflect many of the properties of V , both geometric and algebraic. For instance, when V is equidimensional and dim V [ges ]1, the deficiency modules of V are invariant (up to a shift in grading) along the even liaison class of V [ 14, 11, 15, 7 ], although they do not in general completely determine the even liaison class, except in the case of curves in ℙ 3 [ 14 ]. On the algebraic side, at least for curves in ℙ 3 , the deficiency modules have been shown to have connections to the number and degrees of generators of the saturated ideal defining V [ 12 ]. One of our main goals in this paper is to extend these results to subschemes of arbitrary codimension in any projective space ℙ n . We now describe the contents of this paper more precisely. In the first section, we set up our notation and give the basic definitions which we will use. Then we prove our main technical result: if I and J define subschemes V and Y , respectively, of ℙ n , we relate the quotient module ( I ∩ J )/ IJ to the cohomology of V , at least when V and Y meet properly. We are then able to give a different proof of a general statement due to Serre about when there is an equality of intersection and product. In the second section, we give an extension of Dubreil's Theorem on the number of generators of ideals in a polynomial ring. Specifically, our generalization works for an ideal I defining a locally Cohen–Macaulay, equidimensional subscheme V of any codimension in ℙ n , and relates the number of generators of the defining ideal to the length of certain Koszul homologies of the cohomology of V . The results in this section depend crucially on the identification done in Section 1 of the intersection modulo the product. Finally, in Section 3, we give an extension of a surprising result of Amasaki [ 1 ] showing a lower bound for the least degree of a minimal generator of the ideal of a Buchsbaum subscheme. Originally, Amasaki gave a bound in the case of Buchsbaum curves in ℙ 3 (and later gave a natural extension to Buchsbaum codimension 2 subschemes of ℙ n [ 2 ]). Easier proofs were subsequently given by Geramita and Migliore in [ 6 ], based on a determination of the free resolution of the ideal from a resolution of its deficiency module. For Buchsbaum codimension 2 subschemes of ℙ n whose intermediate cohomology vanishes, we are able to extend these considerations.

The main result of this paper is the establishment of the "full Muntz Theorem" in C[0,1]. This characterizes the sequences $\{\lambda_i\}^\infty_{i=1}$ of distinct, positive real numbers for which $$\text{\rm span}\{1, x^{\lambda_1},x^{\lambda_2}, \ldots \}$$ is dense in C[0,1]. The novelty of this result is the treatment of the most difficult case when $\inf_i{\lambda_i} = 0$ while $\sup_i{\lambda_i}=\infty$. The paper settles the $L_\infty$ and $L_1$ cases of the following. Conjecture (Full Muntz Theorem in L_p[0,1]) Let p \in [1,\infty]. Suppose $\{\lambda_i\}^\infty_{i=0}$ is a sequence of distinct real numbers greater than $-1/p$. Then $$\text{\rm span}\{x^{\lambda_0}, x^{\lambda_1}, \ldots \}$$ is dense in L_p[0,1] if and only if $$\sum^{\infty}_{i=0}{\frac{\lambda_i+1/p}{(\lambda_i+1/p)^2+1}} = \infty.$$

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

Given an iterated function system (IFS) of contractive similitudes, the theory of Gromov hyperbolic graph on the IFS has been established recently. In the paper, we introduce a notion of simple augmented tree which is a Gromov hyperbolic graph. By generalizing a combinatorial device of rearrangeable matrix, we show that there exists a near-isometry between the simple augmented tree and the symbolic space of the IFS, so that their hyperbolic boundaries are Lipschitz equivalent. We then apply this to consider the Lipschitz equivalence of self-similar sets with or without assuming the open set condition. Moreover, we also provide a criterion for a self-similar set to be a Cantor-type set which completely answers an open question raised in \cite{LaLu13}. Our study extends the previous works.

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)

If f is in the Eremenko–Lyubich class B (transcendental entire functions with bounded singular set), then Ω = {z : |f (z)| >R } and f|Ω must satisfy certain simple topological conditions when R is sufficiently large. A model (Ω ,F ) is an open set Ω and a holomorphic function F on Ω that satisfy these same conditions. We show that any model can be approximated by an Eremenko– Lyubich function in a precise sense. In many cases, this allows the construction of functions in B with a desired property to be reduced to the construction of a model with that property, and this is often much easier to do. The singular set of a entire function f is the closure of its critical values and finite asymptotic values and is denoted by S(f ). The Eremenko–Lyubich class B consists of functions such that S(f ) is a bounded set (such functions are also called bounded type). The Speiser class S⊂ B (also called finite type) is those functions for which S(f ) is a finite set. In [9], Eremenko and Lyubich showed that if S(f ) ⊂ DR = {z : |z| R } is a disjoint union of analytic, unbounded, Jordan domains, and that f acts as a covering map f :Ω j →{ x : |z| >R } on each component Ωj of Ω. Building examples where Ω has a certain geometry is important for applications to dynamics. We would like to start with a model, that is, a choice of Ω and a covering map f :Ω →{ |z| > 1} and ask whether f can be approximated by an entire function F in B or S. In this paper, we deal with approximation by functions in B. It turns out that if Ω satisfies some obviously necessary topological conditions, then the approximation by Eremenko–Lyubich functions is always possible in a sense strong enough to imply that the functions f and F have the same dynamical behavior on their Julia sets. This allows us to build entire functions in B with certain behaviors by simply exhibiting a model with that behavior (this is often much easier to do). In [4], we deal with the analogous question for the Speiser class; again the approximation is always possible, but in a slightly weaker sense (dynamically, given any model we can build a function in the Speiser class that has the model’s dynamics on some subset of its Julia set). In the next few paragraphs, we introduce some notation to make these remarks more precise. Suppose that Ω = � j Ωj is a disjoint union of unbounded simply connected domains

Let g be a finite-dimensional semi-simple Lie algebra over an algebraically closed field k of characteristic zero and letg = n+rj + n~ be a triangular decomposition of g. Denote by U = U(Q) the enveloping algebra of g. For Xe\)*, denote by Dk the primitive factor ring £//t/ ^W> where M(X) is the Verma module of highest weight X—p (where p is the half-sum of the positive roots). The main aim of this paper is to prove Theorem 3.9, which states that if A is regular, then gldim Dx < dim . n + -In{X), where n{X) is a non-negative integer less than or equal to dimfcn+. In [12] Levasseur computed the injective dimension of D^ in terms of the Gelfand-Kirillov dimension of L{X), the unique simple quotient of M(X). For X regular, Theorem 3.9 implies that the global dimension of D^ is finite and hence must be equal to the injective dimension of Dk. It can be shown that this figure coincides with the bound given in Theorem 3.9 (the authors would like to thank Levasseur for pointing this out). On the other hand, if X is not regular, Joseph and Stafford [11] have shown that gldim D^= oo. Special cases of this result have already appeared in the literature. In particular Stafford [17] computed the global dimension of Dk for all A el)* in the case when g = si (2, C). On the other hand Roos [15] computed gldim D^ for general g when X satisfies some transcendental but generic conditions. This paper is divided into two distinct parts. In Section 2 we prove a result analogous to a theorem of Roos relating the weak global dimension of a ring R to that of certain sets of torsion-theoretic localisations. If R is a commutative ring and {Si)tei i a s e t °f localisations of R such that © S{ is faithfully flat as an /^-module, then it is well known that wgldim R = sup{wgldim St}. To what extent this result is true for non-commutative rings is unknown. Roos and others have proved various results under the assumption that wgldim R is finite (see for instance [3, 7, 13,14]). We prove here a slightly more general sort of result than Roos's which does not require this assumption. However, it is necessary to impose certain additional assumptions on the localisations St. The special case of this result needed for the second part is the following. Let R be a prime Noetherian ring and let St, i = 1, ..., n be a finite collection of rings lying between R and its quotient ring. If © St is a faithfully flat right /^-module and St ®R S} = S} ®R St as /?-/?-bimodules for all pairs (i,j), then

Let $R$ be a commutative ring with identity and δ be a derivation of $R$. Then the set, $R$δ, of all derivations of $R$ of the form $r$δ : x → $r$δ(x), $r$ ∈ $R$, is a Lie subring of the Lie ring $D$(R) of derivations of $R$. In [2] the authors studied the structure of $D$(R) and found that $R$δ played an analogous role to that played by the Lie ring $I$(S) of inner derivations of a non-commutative ring S in the study of $D$(S). Furthermore it was shown in [2] that the properties of $R$δ closely resemble the known properties of $I$(S). In particular it was shown that the following results hold in the case where $R$ is 2-torsion-free: (i) if $R$ is prime or if $R$ is δ-prime noetherian then $R$δ is a prime Lie ring; (ii) if $R$ is δ-simple noetherian then $R$δ is a simple Lie ring. The purpose of this paper is to continue the study of the structure of the Lie ring $R$δ and of certain of its Lie subrings. In this paper we shall not view $R$δ as a Lie subring of $D$(R) but rather as a Lie ring whose elements are the elements of $R$ with the product [r, s] = rδ(s)—sδ(r) for all r, s ∈ $R$δ. The two approaches coincide in the case where the annihilator of δ($R$) is zero. It is perhaps worth noting that $R$δ is isomorphic to a Lie subring of the Lie ring arising from a certain non-commutative ring, namely the Ore extension $S$ = $R$[x, δ]. The set of those elements of $S$ of the form rx, r ∈ $R$, is closed under the Lie product in $S$ (that is, [s, t] = st — ts for all s, t∈S) and forms a Lie ring which is isomorphic to $R$δ.

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

M. D. ATKINSONIn a series of papers [3, 4 and 5] on insoluble (transitive) permutation groupsof degree p = 2q +1, where p and q are primes, N. Ito has shown that, apart from asmall number of exceptions, such a group must be at least quadruply transitive.One of the results which he uses is that an insoluble 2q grou +1 p of degree p =which is not doubly primitive must be isomorphi (3, 2)c wit to PSh p =L 7. Thisresult is due to H. Wielandt, and ltd gives a proof in [3]. It is quite easy to extendthis proof to give the following result: a doubly transitive group of degree 2q + l,where q is prime, which is not doubly primitive, is either sharply doubly transitiveor a group of automorphisms of a bloc A = 1 ank desigd k = 3n wit. Ouh rnotation for the parameters of a block design, v, b, X, k, i r,s standard; see [9].In this paper we shall prove two results about doubly transitive but not doublyprimitive groups which resemble the two results mentioned above.

A well known theorem of Shuzo Izumi, strengthened by David Rees, asserts that all the divisorial valuations centered in an analytically irreducible local noetherian ring are linearly comparable to each other. In the present paper we generalize this theorem to the case of Abhyankar valuations with archimedian value semigroup. Indeed, we prove that in a certain sense linear equivalence of topologies characterizes Abhyankar valuations with archimedian semigroups, centered in analytically irreducible local noetherian rings. Then we show that some of the classical results on equivalence of topologies in noetherian rings can be strengthened to include linear equivalence of topologies. We also prove a new comparison result between the Krull topology and the topology defined by the symbolic powers of an arbitrary ideal.

We provide a full and rigorous proof of a theorem attributed to \.Zuk, stating that random groups in the Gromov density model for d > 1/3 have property (T) with high probability. The original paper had numerous gaps, in particular, crucial steps involving passing between different models of random groups were not described. We fix the gaps using combinatorial arguments and a recent result concerning perfect matchings in random hypergraphs. We also provide an alternative proof, avoiding combinatorial difficulties and relying solely on spectral properties of random graphs in G(n, p) model., Comment: v2: minor corrections

International audience; We build a coding of the trajectories of billiards in regular 2n-gons, similar but different from the one in [16], by applying the self-dual induction [9] to the underlying one-parameter family of n-interval exchange transformations. This allows us to show that, in that family, for n = 3 non-periodicity is enough to guarantee weak mixing, and in some cases minimal self-joinings, and for every n we can build examples of n-interval exchange transformations with weak mixing, which are the first known explicitly for n > 6. In [16], see also [15], John Smillie and Corinna Ulcigrai develop a rich and original theory of billiards in the regular octagons, and more generally of billiards in the regular 2n-gons, first studied by Veech [17]: their aim is to build explicitly the symbolic trajectories, which generalize the famous Sturmian sequences (see for example [1] among a huge literature), and they achieve it through a new renormalization process which generalizes the usual continued fraction algorithm. In the present shorter paper, we show that similar results, with new consequences, can be obtained by using an existing, though recent, theory, the self-dual induction on interval exchange transformations. As in [16], we define a trajectory of a billiard in a regular 2n-gon as a path which starts in the interior of the polygon, and moves with constant velocity until it hits the boundary, then it re-enters the polygon at the corresponding point of the parallel side, and continues travelling with the same velocity; we label each pair of parallel sides with a letter of the alphabet (A 1 , ...A n), and read the labels of the pairs of parallel sides crossed by the trajectory as time increases; studying these trajectories is known to be equivalent to studying the trajectories of a one-parameter family of n-interval exchange transformations, and to this family we apply a slightly modified version of the self-dual induction defined in [9]. Now, the self-dual induction is in general not easy to manipulate, as its states are a family of graphs, and its typical itineraries, or paths in the so-called graph of graphs, are quite complicated to describe; but in our main Theorem 7 below, we show that for any non-periodic n-interval exchange in this particular family, after at most 2n − 2 steps our self-dual induction goes back, up to small modifications, to the initial state of another member of the family. This gives us a renormalization process, which differs from the one in [16] essentially because it is applied to lengths of intervals instead of angles, and allows us to compute the whole itinerary of the original interval exchange transformation under the self-dual induction in function of a single sequence of integers between 1 and 2n − 1, which act as the partial quotients of a continued fraction algorithm applied to initial lengths of subintervals.

The aim of this paper is to study some classes of second-order divergence-form partial differential operators L of sub-Riemannian type. Our main assumption is the C^infinity-hypoellipticity of L, together with the existence of a well-behaved fundamental solution Gamma(x, y) for L. We consider the mean-integral operator Mr naturally associated to the mean-value theorem for the L-harmonic functions and we investigate the following topics: the positivity set of the kernel associated to M_r; the role of M_r in solving the homogeneous Dirichlet problem related to L in the Perron–Wiener–Brelot sense; the existence of an inverse mean-value theorem characterizing the sub-Riemannian ‘balls’ Omega_r(x), superlevel sets of Gamma(x, ·). This last result extends a previous theorem by Kuran [Bull. London Math. Soc. 1972]. As side-results, we provide a short proof of the Strong Maximum Principle for L using M_r, a Poisson–Jensen formula for the L-subharmonic functions and several results concerning the geometry of the sets Omega_r(x).

In this paper we show how the p-adic Rankin–Selberg product construction of Hida can be combined with freeness results of Hecke modules of Wiles to establish interesting congruences between particular special values of L-functions of elliptic curves. These congruences are part of some deep conjectural congruences that follow from the work of Kato on the non-commutative Iwasawa theory of the false Tate curve extension. In the appendix by Vladimir Dokchitser it is shown that these congruences, combined with results from Iwasawa theory for elliptic curves, give interesting results for the arithmetic of elliptic curves over non-abelian extensions.

Buchi's problem asked whether there exists an integer $M$ such that the surface defined by a system of equations of the form $$x_{n}^2+x_{n-2}^2=2x_{n-1}^2+2,\quad n=2,\dotsc, M-1,$$ has no integer points other than those that satisfy $\pm x_n=\pm x_0+n$ (the $\pm$ signs are independent). If answered positively, it would imply that there is no algorithm which decides, given an arbitrary system $Q=(q_1,\dotsc,q_r)$ of integral quadratic forms and an arbitrary $r$-tuple $B=(b_1,\dotsc,b_r)$ of integers, whether $Q$ represents $B$ (see T. Pheidas and X. Vidaux, Fund. Math. 185 (2005) 171-194). Thus it would imply the following strengthening of the negative answer to Hilbert's tenth problem: the positive-existential theory of the rational integers in the language of addition and a predicate for the property '$x$ is a square' would be undecidable. Despite some progress, including a conditional positive answer (depending on conjectures of Lang), Buchi's problem remains open. In this paper we prove the following: an analogue of Buchi's problem in rings of polynomials of characteristic either 0 or $p\geq17$ and for fields of rational functions of characteristic 0; and an analogue of Buchi's problem in fields of rational functions of characteristic $p\geq19$, but only for sequences that satisfy a certain additional hypothesis. As a consequence we prove the following result in logic. Let $F$ be a field of characteristic either 0 or at least 17 and let $t$ be a variable. Let $L_{t}$ be the first order language which contains symbols for 0 and 1, a symbol for addition, a symbol for the property '$x$ is a square' and symbols for multiplication by each element of the image of $\mathbb{Z}[t]$ in $F[t]$. Let $R$ be a subring of $F(t)$, containing the natural image of $\mathbb{Z}[t]$ in $F(t)$. Assume that one of the following is true: $R\subset F[t]$; the characteristic of $F$ is either 0 or $p\geq19$. Then multiplication is positive-existentially definable over the ring $R$, in the language $L_t$. Hence the positive-existential theory of $R$ in $L_{t}$ is decidable if and only if the positive-existential ring-theory of $R$ in the language of rings, augmented by a constant-symbol for $t$, is decidable

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.

Let (An)n1 be the sequence of Apery numbers with a general term given by . In this paper, we prove that both the inequalities (An) > c0 log log log n and P(An) > c0 (log n log log n)1/2 hold for a set of positive integers n of asymptotic density 1. Here, (m) is the number of distinct prime factors of m, P(m) is the largest prime factor of m and c0 > 0 is an absolute constant. The method applies to more general sequences satisfying both a linear recurrence of order 2 with polynomial coefficients and certain Lucas-type congruences

We introduce a linearly ordered set Z and use it to prove a necessity condition for the existence of a G\^ateaux smooth norm on C(T), where T is a tree. This criterion is directly analogous to the corresponding equivalent condition for Fr\'echet smooth norms. In addition, we prove that if C(T) admits a G\^ateaux smooth lattice norm then it also admits a lattice norm with strictly convex dual norm., Comment: A different version of this paper is to appear in J. London Math. Soc

Suppose k is a field. Let A and B be two finite dimensional k-algebras such that there is a stable equivalence of Morita type between A and B. In this paper, we prove that (1) if A and B are representation-finite then their Auslander algebras are stably equivalent of Morita type; (2) The n-th Hochschild homology groups of A and B are isomorphic for all n≥1. A new proof is also provided for Hochschild cohomology groups of self-injective algebras under a stable equivalence of Morita type.

A normed space X is said to be strictly convex if x = y whenever � (x + y)/2� = � x� = � y� , in other words, when the unit sphere of X does not contain non-trivial segments. Our aim in this paper is the study of those normed spaces which admit an equivalent strictly convex norm. We present a characterization in linear topological terms of the normed spaces which are strictly convex renormable. We consider the class of all solid Banach lattices made up of bounded real functions on some set Γ. This class contains the Mercourakis space c1(Σ × Γ) and all duals of Banach spaces with unconditional uncountable bases. We characterize the elements of this class which admit a pointwise strictly convex renorming.

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 construct new obstructions for the surjectivity of the Johnson homomorphism of the automorphism group of a free group. We also determine the structure of the cokernel of the Johnson homomorphism for degrees 2 and 3.

We revise the concept of compact tripotent in the bidual space of a JB*-triple. This concept was introduced by Edwards and Ruttimann generalizing the ideas developed by Akemann for compact projections in the bidual of a C*-algebra. We also obtain some characterizations of weak compactness in the dual space of a JC*-triple, showing that a bounded subset in the dual space of a JC*-triple is relatively weakly compact if and only if its restriction to any abelian maximal subtriple C is relatively weakly compact in the dual of C. This generalizes a very useful result by Pfitzner in the setting of C*-algebras. As a consequence we obtain a Dieudonnet heorem for JC*-triples which generalizes the one obtained by Brooks, Saito and Wright for C*-algebras. One of the most celebrated and useful results characterizing weakly compact subsets in the dual space of a C*-algebra is due to Pfitzner, who established that weak com- pactness in the dual space of a C*-algebra is commutatively determined (see (30)). More concretely, Pfitzner shows, in a 'tour de force', that if K is a bounded subset in the dual space of a C*-algebra A ,t henK is relatively weakly compact if and only if the restriction of K to each maximal abelian subalgebra of A is relatively weakly compact. This result has many important consequences, one of the most interesting being that every C*-algebra satisfies property (V) of Pelczynski. Pfitzner's result is the latest advance in the study of weak compactness in the dual space of a C*-algebra developed by Takesaki (33), Akemann (1), Akemann, Dodds and Gamlen (3), Saito( 32) and Jarchow (22, 23). In the more general setting of dual spaces of JB*-triples the study of weak compactness has been developed by Chu and Iochum (12) and by Peralta and Rodr´ iguez-Palacios (28, 29). However, all the results concerning weak compactness in the dual space of a JB*-triple give characterizations in terms of the abelian subtriples of its bidual instead of the abelian subtriples of the JB*-triple itself. The question clearly is whether a bounded subset in the dual space of a JB*-triple, E, is relatively weakly compact whenever its restriction to any abelian subtriple of E is. In the main result of this paper we show that weak compactness in the dual space of a JC*-triple is commutatively determined, by showing that a bounded subset K in the dual space of a JC*-triple E is relatively weakly compact if and only if the restriction of K to each separable abelian subtriple of E also is relatively weakly compact (see Theorem 3.5).

We give a corrected and strengthened statement and proof of the �p-adic analytic arc lemma� in a paper of the author (J. London Math. Soc. (2) 73 (2006) 367�379). We show that the analytic arc is guaranteed to exist for p 5 and give a counterexample showing that this sometimes cannot be done when p = 2.

The paper is an addendum to D. Andrica and L. Funar, 'On smooth maps with finitely many critical points', J. London Math. Soc. (2) 69 (2004) 783-800.

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.

The analytic concepts of martingale type and cotype of A Banach space have an intimate relation with the geometric concepts of -concavity and -convexity of the space under consideration, as shown by pisier. In particular, for a banach space , having martingale type for some implies that has martingale cotype for some . The generalisation of these concepts to linear operators was studied by the author, and it turns out that the duality above only holds in a weaker form. An example is constructed showing that this duality result is best possible. So-called random martingale unconditionality estimates, introduced by Garling as a decoupling of the unconditional martingale differences (UMD) inequality, are also examined. It is shown that the random martingale unconditionality constant of for martingales of length asymptotically behaves like . This improves previous estimates by Geiss, who needed martingales of length to show this asymptotic. At the same time the order in the paper is the best that can be expected.

Explicit bounds are given for the residues at of the Dedekind zeta functions of number fields. As a consequence, a simple proof of the Brauer-Siegel theorem and explicit lower bounds for class numbers of number fields are obtained. Compared with Stark's original approach, the paper is explicit and more satisfactory for number fields containing quadratic subfields. Examples are given of fully explicit lower bounds for class numbers of various types of number fields, for example normal and non-normal number fields of odd degree, with an emphasis on cubic fields, real cyclic quartic number fields, and non-normal quartic number fields containing an imaginary quadratic subfield.

Hall's theorem for bipartite graphs gives a necessary and sufficient condition for the existence of a matching in a given bipartite graph. Aharoni and Ziv have generalized the notion of matchability to a pair of possibly infinite matroids on the same set and given a condition that is sufficient for the matchability of a given pair of finitary matroids, where the matroid is SCF (a sum of countably many matroids of finite rank). The condition of Aharoni and Ziv is not necessary for matchability. The paper gives a condition that is necessary for the existence of a matching for any pair of matroids (not necessarily finitary) and is sufficient for any pair of finitary matroids, where the matroid is SCF.

The paper investigates the vectorial Dirichlet problem defined by Here is an open bounded subset of with boundary , and () denote the singular values of the gradient . The existence of solutions is established under one of the following assumptions: is continuous on and locally contractive on , or is contractive on . This extends a result due to Dacorogna and Marcellini. The approach is based on the Baire category method developed earlier by the authors.

Consider the Sobolev embedding operator from the space of functions in W-1,W-p(I) with average zero into L-p, where I is a finite interval and p > 1. This operator plays an important role in recent work. The operator norm and its approximation numbers in closed form are calculated. The closed form of the norm and approximation numbers of several similar Sobolev embedding operators on a finite interval have recently been found. It is proved in the paper that most of these operator norms and approximation numbers on a finite interval are the same.

The theory of higher-dimensional shifts of finite type is still largely an open area of investigation. Recent years have seen much activity, but fundamental questions remain unanswered. In this paper we consider the following basic question. Given a shift of finite type (SFT), under what topological mixing conditions are we guaranteed the existence of Bernoulli (or even $K$ , mixing, or weakly mixing) invariant measures?

The main goals of the paper are to establish a priori bounds for the prime power divisors of the discriminants of spinor regular positive definite primitive integral ternary quadratic lattices, and to describe a procedure for determining all such lattices.

The $(C,F)$ -construction from a previous paper of the first author is applied to produce a number of funny rank one infinite measure preserving actions of discrete countable Abelian groups $G$ with ‘unusual’ multiple recurrence properties. In particular, the following are constructed for each $p\in\Bbb N\cup\{\infty\}$ : (i) a $p$ -recurrent action $T=(T_g)_{g\in G}$ such that (if $p\ne\infty$ ) no one transformation $T_g$ is $(p+1)$ -recurrent for every element $g$ of infinite order; (ii) an action $T=(T_g)_{g\in G}$ such that for every finite sequence $g_1,\dots,g_r\in G$ without torsion the transformation $T_{g_1}\times\cdots\times T_{g_r}$ is ergodic, $p$ -recurrent but (if $p\ne\infty$ ) not $(p+1)$ -recurrent; (iii) a $p$ -polynomially recurrent $(C,F)$ -transformation which (if $p\ne\infty$ ) is not $(p+1)$ -recurrent. $\infty$ -recurrence here means multiple recurrence. Moreover, it is shown that there exists a $(C,F)$ -transformation which is rigid (and hence multiply recurrent) but not polynomially recurrent. Nevertheless, the subset of polynomially recurrent transformations is generic in the group of infinite measure preserving transformations endowed with the weak topology.

As a generalization of the dynamics of a single polynomial, Hinkkanen and Martin studied the dynamics of polynomial semigroups of finite type. Among the issues addressed were possible generalizations of the filled-in Julia set of a single polynomial and the basin of attraction for infinity. In the paper, these concepts are further refined, and the connections between the dynamics of a single function and the dynamics of semigroups are strengthened.