Showing papers in "Israel Journal of Mathematics in 2013"

The proximal point algorithm in metric spaces

Abstract: The proximal point algorithm, which is a well-known tool for finding minima of convex functions, is generalized from the classical Hilbert space framework into a nonlinear setting, namely, geodesic metric spaces of non-positive curvature. We prove that the sequence generated by the proximal point algorithm weakly converges to a minimizer, and also discuss a related question: convergence of the gradient flow.

Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces

Abstract: A new moment inequality is obtained for the eigenfunctions ϕ of the flat tours Πn. More specifically, it is known that the \(p = \frac{{2n}} {{n - 1}} \) is almost uniformaly bounded. Simillar results are established for the periodiec Schrodinger group.

On Ulam stability

Abstract: We study $\epsilon$-representations of discrete groups by unitary operators on a Hilbert space. We define the notion of Ulam stability of a group which loosely means that finite-dimensional $\epsilon$-represendations are uniformly close to unitary representations. One of our main results is that certain lattices in connected semi-simple Lie groups of higher rank are Ulam stable. For infinite-dimensional $\epsilon$-representations, the similarly defined notion of strong Ulam stability is defined and it is shown that groups with free subgroups are not strongly Ulam stable. We also study deformation rigidity of unitary representations and show that groups containing a free subgroup are not deformation rigid.

Non-localization of eigenfunctions on large regular graphs

Abstract: We give a delocalization estimate for eigenfunctions of the discrete Laplacian on large (d+1)-regular graphs, showing that any subset of the graph supporting e of the L 2 mass of an eigenfunction must be large. For graphs satisfying a mild girth-like condition, this bound will be exponential in the size of the graph.

Externally definable sets and dependent pairs

Abstract: We prove that externally definable sets in first order NIP theories have honest definitions, giving a new proof of Shelah’s expansion theorem. Also we discuss a weak notion of stable embeddedness true in this context. Those results are then used to prove a general theorem on dependent pairs, which in particular answers a question of Baldwin and Benedikt on naming an indiscernible sequence.

The graded structure of Leavitt path algebras

Abstract: A Leavitt path algebra associates to a directed graph a ℤ-graded algebra and in its simplest form it recovers the Leavitt algebra L(1, k). In this note, we first study this ℤ-grading and characterize the (ℤ-graded) structure of Leavitt path algebras, associated to finite acyclic graphs, C n -comet, multi-headed graphs and a mixture of these graphs (i.e., polycephaly graphs). The last two types are examples of graphs whose Leavitt path algebras are strongly graded. We give a criterion when a Leavitt path algebra is strongly graded and in particular characterize unital Leavitt path algebras which are strongly graded completely, along the way obtaining classes of algebras which are group rings or crossed-products. In an attempt to generalize the grading, we introduce weighted Leavitt path algebras associated to directed weighted graphs which have natural ⊕ℤ-grading and in their simplest form recover the Leavitt algebras L(n, k). We then show that the basic properties of Leavitt path algebras can be naturally carried over to weighted Leavitt path algebras.

A proof of uniqueness of the Gurariĭ space

Abstract: We present a short and elementary proof of isometric uniqueness of the Gurariĭ space.

Semi-tilting complexes

Abstract: We introduce the notion of semi-tilting complexes, which is a small generalization of tilting complexes. Interesting examples include APR-semitilting complexes, etc. Note that non-trivial semi-tilting complexes exist for any non-semisimple non-local artin algebras, while tilting complexes may not. We extend interesting results in the tilting theory to semi-tilting complexes. As corollaries, we also obtain some new characterizations of tilting complexes.

Right coideal subalgebras of Nichols algebras and the Duflo order on the Weyl groupoid

Abstract: We study graded right coideal subalgebras of Nichols algebras of semisimple Yetter-Drinfeld modules. Assuming that the Yetter-Drinfeld module admits all reflections and the Nichols algebra is decomposable, we construct an injective order preserving and order reflecting map between morphisms of the Weyl groupoid and graded right coideal subalgebras of the Nichols algebra. Here morphisms are ordered with respect to right Duflo order and right coideal subalgebras are ordered with respect to inclusion. If the Weyl groupoid is finite, then we prove that the Nichols algebra is decomposable and the above map is bijective. In the special case of the Borel part of quantized enveloping algebras our result implies a conjecture of Kharchenko.

On theories of random variables

Abstract: We study theories of spaces of random variables: first, we consider random variables with values in the interval [0, 1], then with values in an arbitrary metric structure, generalising Keisler’s randomisation of classical structures. We prove preservation and non-preservation results for model theoretic properties under this construction: (i) The randomisation of a stable structure is stable. (ii) The randomisation of a simple unstable structure is not simple.

The polynomial multidimensional Szemerédi Theorem along shifted primes

Abstract: If $\vec q_1 ,...,\vec q_m $ : ℤ → ℤ l are polynomials with zero constant terms and E ⊂ ℤ l has positive upper Banach density, then we show that the set E ∩ (E − $\vec q_1 $ (p − 1)) ∩ … ∩ (E − $\vec q_m $ (p − 1)) is nonempty for some prime p. We also prove mean convergence for the associated averages along the prime numbers, conditional to analogous convergence results along the full integers. This generalizes earlier results of the authors, of Wooley and Ziegler, and of Bergelson, Leibman and Ziegler.

The vanishing Euler characteristic of an isolated determinantal singularity

Abstract: Let (X, 0) be a complex analytic isolated determinantal singularity. We will define the vanishing Euler characteristic of (X, 0) and the Milnor number of a holomorphic function germ with an isolated singularity on X, f: (X, 0) → ℂ.

Prescribing the binary digits of primes

Abstract: We present a new result on counting primes p < N = 2 n for which r (arbitrarily placed) digits in the binary expansion of p are specified. Compared with earlier work of Harman and Katai, the restriction on r is relaxed to r < c(n/log n)4/7. This condition results from the estimates of Gallagher and Iwaniec on zero-free regions of L-functions with ‘powerful’ conductor.

The probability of generating a finite simple group

Abstract: We study the probability of generating a finite simple group, together with its generalisation P G,socG (d), the conditional probability of generating an almost simple finite group G by d elements, given that these elements generate G/socG. We prove that P G,socG (2) ⩽ 53/90, with equality if and only if G is A6 or S6, and establish a similar result for P G,socG (3). Positive answers to longstanding questions of Wiegold on direct products, and of Mel’nikov on profinite groups, follow easily from our results.

The Bishop-Phelps-Bollobás Theorem for operators from c 0 to uniformly convex spaces

Abstract: We show that the pair of Banach spaces (c0, Y) has the Bishop-Phelps-Bollobas property when Y is uniformly convex. Further, when Y is strictly convex, if (c0, Y) has the Bishop-Phelps-Bollobas property then Y is uniformly convex for the case of real Banach spaces. As a corollary, we show that the Bishop-Phelps-Bollobas theorem holds for bilinear forms on c0 × lp (1 < p < ∞).

Legendre polynomials and Ramanujan-type series for 1/ π

Abstract: We resolve a family of recently observed identities involving 1/π using the theory of modular forms and hypergeometric series. In particular, we resort to a formula of Brafman which relates a generating function of the Legendre polynomials to a product of two Gaussian hypergeometric functions. Using our methods, we also derive some new Ramanujan-type series.

A discrete Gauss-Green identity for unbounded Laplace operators, and the transience of random walks

Abstract: On a finite network (connected weighted undirected graph), the relationship between the natural Dirichlet form E and the discrete Laplace operator Δ is given by $$\varepsilon (u,v) = {\langle u,\Delta v\rangle _{{\ell ^2}}}$$ , where the latter is the usual l 2 inner product. This formula is not generally true for infinite networks; earlier authors have given various conditions under which this formula remains valid. Instead, we extend this formula to arbitrary infinite networks (including the case when Δ is unbounded) by including a new (boundary) term, in parallel with the classical Gauss-Green identity. This tool allows for detailed study of the boundary of the network. We construct a reproducing kernel for the space of functions of finite energy which allows us to specify a dense domain for Δ and give several criteria for the transience of the random walk on the network. The extended Gauss-Green identity and the reproducing kernel also yield a boundary integral representation for harmonic functions of finite energy. The boundary representation is developed further in [24].

Turán numbers for K s,t -free graphs: Topological obstructions and algebraic constructions

Abstract: We show that every hypersurface in ℝ s × ℝ s contains a large grid, i.e., the set of the form S × T, with S, T ⊂ ℝ s . We use this to deduce that the known constructions of extremal K 2,2-free and K 3,3-free graphs cannot be generalized to a similar construction of K s,s -free graphs for any s ≥ 4. We also give new constructions of extremal K s,t -free graphs for large t.

Clique complexes and graph powers

Abstract: We study the behaviour of clique complexes of graphs under the operation of taking graph powers. As an example we compute the clique complexes of powers of cycles, or, in other words, the independence complexes of circular complete graphs.

Splitting fields of characteristic polynomials of random elements in arithmetic groups

Abstract: We discuss rather systematically the principle, implicit in earlier works, that for a “random” element in an arithmetic subgroup of a (split, say) reductive algebraic group over a number field, the splitting field of the characteristic polynomial, computed using any faitfhful representation, has Galois group isomorphic to the Weyl group of the underlying algebraic group. Besides tools such as the large sieve, which we had already used, we introduce some probabilistic ideas (large deviation estimates for finite Markov chains) and the general case involves a more precise understanding of the way Frobenius conjugacy classes are computed for such splitting fields (which is related to a map between regular elements of a finite group of Lie type and conjugacy classes in the Weyl group which had been considered earlier by Carter and Fulman for other purposes; we show in particular that the values of this map are equidistributed).

On a uniform estimate for the quaternionic Calabi problem

Abstract: We establish a C 0 a priori bound on the solutions of the quaternionic Calabi-Yau equation (of Monge-Ampere type) on compact HKT manifolds with a locally flat hypercomplex structure. As an intermediate step, we prove a quaternionic version of the Gauduchon theorem.

Generic representations of abelian groups and extreme amenability

Abstract: If G is a Polish group and Γ is a countable group, denote by Hom(Γ, G) the space of all homomorphisms Γ → G. We study properties of the group $$\overline {\pi (\Gamma )} $$ for the generic π ∈ Hom(Γ, G), when Γ is abelian and G is one of the following three groups: the unitary group of an infinite-dimensional Hilbert space, the automorphism group of a standard probability space, and the isometry group of the Urysohn metric space. Under mild assumptions on Γ, we prove that in the first case, there is (up to isomorphism of topological groups) a unique generic $$\overline {\pi (\Gamma )} $$ ; in the other two, we show that the generic $$\overline {\pi (\Gamma )} $$ is extremely amenable. We also show that if Γ is torsionfree, the centralizer of the generic π is as small as possible, extending a result of Chacon and Schwartzbauer from ergodic theory.

The average degree of an irreducible character of a finite group

Abstract: Given a finite group G, we write acd(G) to denote the average of the degrees of the irreducible characters of G. We show that if acd(G) ≤ 3, then G is solvable. Also, if acd(G) < 3/2, then G is supersolvable, and if acd(G) < 4/3, then G is nilpotent.

On dynamics of Out(Fn) on PSL2(ℂ) characters

Abstract: We introduce and study an open set of PSL2(ℂ) characters of a nonabelian free group, on which the action of the outer automorphism group is properly discontinuous, and which is strictly larger than the set of discrete, faithful convex-cocompact (i.e. Schottky) characters. This implies, in particular, that the outer automorphism group does not act ergodically on the set of characters with dense image. Hence there is a difference between the geometric (discrete vs. dense) decomposition of the characters, and a natural dynamical decomposition.

A partial analog of the integrability theorem for distributions on p-adic spaces and applications

Abstract: Let X be a smooth real algebraic variety. Let ξ be a distribution on it. One can define the singular support of ξ to be the singular support of the DX-module generated by ξ (sometimes it is also called the characteristic variety). A powerful property of the singular support is that it is a coisotropic subvariety of T*X. This is the integrability theorem (see [KKS, Mal, Gab]). This theorem turned out to be useful in representation theory of real reductive groups (see, e.g., [AG4, AS, Say]).

Some new properties of composition operators associated with lens maps

Abstract: We give examples of results on composition operators connected with lens maps. The first two concern the approximation numbers of those operators acting on the usual Hardy space H 2 . The last ones are connected with Hardy- Orlicz and Bergman-Orlicz spaces H and B , and provide a negative answer to the question of knowing if all composition operators which are weakly compact on a non-reflexive space are norm-compact. Mathematics Subject Classification. Primary: 47 B 33 - Secondary: 28 B 99 ; 28 E 99 ; 46 E 30

Some irreducibility theorems of parabolic induction on the metaplectic group via the Langlands-Shahidi method

Abstract: Let \(\overline {S{p_{2n}}({\rm{ F }})} \) be the metaplectic double cover of F where F is a local field of characteristic 0. We use the Uniqueness of Whittaker model to define a metaplectic analog to Shahidi local coefficients and we use these coefficients to define gamma factors. We show that these gamma factors are multiplicative and satisfy the crude global functional equation. Then, we compute these factors in various cases and obtain explicit formulas for Plancherel measures. These computations are then used to prove some irreducibility theorems for parabolic induction on the metaplectic group over p-adic fields. In particular, we show that all principal series representations induced from unitary characters are irreducible. We also prove that parabolic induction from unitary supercuspidal representation of the Siegel parabolic sub group is irreducible if and only if a certain parabolic induction on F is irreducible.

The inverse Fueter mapping theorem in integral form using spherical monogenics

Abstract: In this paper we prove an integral representation formula for the inverse Fueter mapping theorem for monogenic functions defined on axially symmetric open sets U ⊆ ℝ n+1, i.e. on open sets U invariant under the action of SO(n), where n is an odd number. Every monogenic function on such an open set U can be written as a series of axially monogenic functions of degree k, i.e. functions of type $$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{f} _k (x) = \left[ {A\left( {x_{0,\rho } } \right) + \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega } {\rm B}\left( {x_{0,\rho } } \right)} \right]\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )$$ , where A(x 0, ρ) and B(x 0, ρ) satisfy a suitable Vekua-type system and $$\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )$$ is a homogeneous monogenic polynomial of degree k. The Fueter mapping theorem says that given a holomorphic function f of a paravector variable defined on U, then the function $$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{f} _k (x)\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )$$ given by $$\Delta ^{k + \tfrac{{n - 1}} {2}} \left( {f(x)\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )} \right) = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{f} (x)\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )$$ is a monogenic function. The aim of this paper is to invert the Fueter mapping theorem by determining a holomorphic function f of a paravector variable in terms of $$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{f} _k (x)\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )$$ . This result allows one to invert the Fueter mapping theorem for any monogenic function defined on an axially symmetric open set.

Entropy-based bounds on dimension reduction in L1

Abstract: We show that for every large enough integer N, there exists an N-point subset of L1 such that for every D > 1, embedding it into l1d with distortion D requires dimension d at least \({N^{\Omega (1/{D^2})}}\), and that for every ɛ > 0 and large enough integer N, there exists an N-point subset of L1 such that embedding it into l1d with distortion 1 + ɛ requires dimension d at least \({N^{\Omega (1/{D^2})}}\)). These results were previously proven by Brinkman and Charikar [JACM, 2005] and by Andoni, Charikar, Neiman and Nguyen [FOCS 2011]. We provide an alternative and arguably more intuitive proof based on an entropy argument.

On the spectral theory of trees with finite cone type

Abstract: We study basic spectral features of graph Laplacians associated with a class of rooted trees which contains all regular trees. Trees in this class can be generated by substitution processes. Their spectra are shown to be purely absolutely continuous and to consist of finitely many bands. The main result gives stability of the absolutely continuous spectrum under sufficiently small radially label symmetric perturbations for non-regular trees in this class. In sharp contrast, the absolutely continuous spectrum can be completely destroyed by arbitrary small radially label symmetric perturbations for regular trees in this class.