Search

According to Tzouvaras, a set is nontypical in the Russell sense if it belongs to a countable ordinal definable set. The class \mathbf {HNT} of all hereditarily nontypical sets satisfies all axioms of \mathbf {ZF} and the double inclusion \mathbf {HOD}\subseteq \mathbf {HNT}\subseteq \mathbf {V} holds. Several questions about the nature of such sets, recently proposed by Tzouvaras, are solved in this paper. In particular, a model of \mathbf {ZFC} is presented in which \mathbf {HOD}\subsetneqq \mathbf {HNT}\subsetneqq \mathbf {V}, and another model of \mathbf {ZFC} in which \mathbf {HNT} does not satisfy the axiom of choice. [ABSTRACT FROM AUTHOR]

Ritt's theorem of zeroes and Siedenberg's embedding theorem are classical results in differential algebra allowing to connect algebraic and model-theoretic results on nonlinear PDEs to the realm of analysis. However, the existing proofs of these results use sophisticated tools from constructive algebra (characteristic set theory) and analysis (Riquier's existence theorem). In this paper, we give new short proofs for both theorems relying only on basic facts from differential algebra and the classical Cauchy-Kovalevskaya theorem for PDEs. [ABSTRACT FROM AUTHOR]

In this paper, temporal homogeneity of an Rd-valued additive process is studied. When an additive process has the stationary independent increments property, the process is called a temporally homogeneous additive process or L'evy process. Moreover, an additive process is said to have the independent increments property in a strong sense if the process has the independent increments property at every finite stopping time (that is, its increments starting at any finite stopping time and events before the stopping time are independent). This paper shows that if an additive process has the independent increments property in a strong sense, then the process is temporally homogeneous, provided the process is immediately random. In particular, in the case of additive processes with Poisson distributed independent increments, it follows that, under some non-degeneracy conditions, the temporal homogeneity is equivalent to the independent increments property at the first jumping time and that, in the degenerate cases, whether each process has the independent increments property at the first jumping time is determined. 1. Preliminaries. [ABSTRACT FROM AUTHOR]

In this paper we show that any Bohr-zero non-periodic set B of traceless integer-valued matrices, denoted by Λ, intersects non-trivially the conjugacy class of any matrix from Λ. As a corollary, we obtain that the family of characteristic polynomials of B contains all characteristic polynomials of matrices from Λ. The main ingredient used in this paper is an equidistribution result for an SLd(ℤ) random walk on a finite-dimensional torus deduced from Bourgain-Furman-Lindenstrauss-Mozes work [J. Amer. Math. Soc. 24 (2011), 231-280]. [ABSTRACT FROM AUTHOR]

In this paper, we derive the sharp lower and upper bounds of nodal lengths of Laplacian eigenfunctions in the disc. [ABSTRACT FROM AUTHOR]

It is due to Milnor that there is no Herman ring for any rational map with two critical values. In this paper, we are concerned with if there exist Herman rings for rational maps with three critical values. We first prove that such rational maps cannot have Herman rings of period 1, 2, or 3. Then we prove all regularly ramified rational maps have no Herman rings in their Fatou sets. [ABSTRACT FROM AUTHOR]

In this paper we show the existence of weak solutions for a class of semipositone problems of the type (P) { -ΔΦu = f(u) - a in Ω, u(x) > 0 in Ω, u = 0 on ∂Ω, where Ω ⊂ ℝN, N ≥ 2, is a smooth bounded domain, f : [0,+∞) → ℝ is a continuous function with subcritical growth, a > 0, and ΔΦu stands for the Φ-Laplacian operator. By using variational methods, we prove the existence of a solution for a small enough. [ABSTRACT FROM AUTHOR]

In this paper we prove the containment I(nm) ⊂ M(n-1)mIm, for a radical ideal I of s general points in Pn, where s ≥ 2n. As a corollary we get that the Chudnovsky Conjecture holds for a very general set of at least 2n points in Pn. [ABSTRACT FROM AUTHOR]

We show that Woodin's \mathsf {AD}^+ Conjecture follows from various hypotheses extending the Continuum Hypothesis (\mathsf {CH}). These results complement Woodin's original result that the \mathsf {AD}^+ Conjecture follows from \mathsf {MM}(\mathfrak {c}). [ABSTRACT FROM AUTHOR]

This paper is concerned with homogenization of systems of linear elasticity with rapidly oscillating periodic coefficients. We establish sharp convergence rates in L² for the mixed boundary value problems with bounded measurable coefficients. [ABSTRACT FROM AUTHOR]

We consider the iteration of quasiregular maps of transcendental type from Rd to Rd. In particular we study quasi-Fatou components, which are defined as the connected components of the complement of the Julia set. Many authors have studied the components of the Fatou set of a transcendental entire function, and our goal in this paper is to generalise some of these results to quasi-Fatou components. First, we study the number of complementary components of quasi-Fatou components, generalising, and slightly strengthening, a result of Kisaka and Shishikura. Second, we study the size of quasi-Fatou components that are bounded and have a bounded complementary component. We obtain results analogous to those of Zheng, and of Bergweiler, Rippon and Stallard. These are obtained using techniques which may be of interest even in the case of transcendental entire functions. [ABSTRACT FROM AUTHOR]

In this paper, we give some properties of the zeros of special families of Brenke type polynomials. In particular, we consider an extension of Laguerre polynomials known as d-orthogonal Laguerre polynomials. For these polynomials, we prove that all the zeros are simple, positive and interlaced. A q-analogue is considered as well. [ABSTRACT FROM AUTHOR]

This paper proves explicit formulae for the number of edges, 2-sets and diagonals in the associahedron of dimension n modulo the action of the dihedral group. A generating function for the number of k-sets modulo this action, as well as a formula for the cycle index, is given. A table of values is also provided. [ABSTRACT FROM AUTHOR]

A particular case of the Hindman-Galvin-Glazer theorem states that, for every partition of an infinite abelian group G into two cells, there will be an infinite X ⊆ G such that the set of its finite sums {x1 + ⋅⋅⋅ + xn | n ∈ N ∧ x1, ⋅⋅⋅, xn ∈ X are distinct} is monochromatic. It is known that the same statement is false, in a very strong sense, if one attempts to obtain an uncountable (rather than just infinite) X. On the other hand, a recent result of Komjáth states that, for partitions into uncountably many cells, it is possible to obtain monochromatic sets of the form FS(X), for X of some prescribed finite size, when working with sufficiently large Boolean groups. In this paper, we provide a generalization of Komjáth's result, and we show that, in a sense, this generalization is the strongest possible. [ABSTRACT FROM AUTHOR]

In set theory without the Axiom of Choice (AC), we investigate the open problem of the deductive strength of Juhász's topological generalization of Neumer's Theorem from his paper On Neumer's Theorem [Proc. Amer. Math. Soc. 54 (1976), 453-454]. Among other results, we show that Juhász's Theorem is deducible from the Principle of Dependent Choices and (when restricted to the class of T1 spaces) implies the Axiom of Countable Multiple Choice, and hence implies van Douwen's Countable Choice Principle, but does not imply either the full van Douwen's Choice Principle or the axiom of choice for linearly ordered families of nonempty finite sets. Furthermore, we prove that Juhász's Theorem (for T1 spaces) implies each of the following principles: ''ℵ1 is a regular cardinal'', ''every infinite set is weakly Dedekind-infinite'', and ''every infinite linearly ordered set is Dedekind-infinite''. We also establish that Juhász's Theorem for T2 spaces is not provable in ZF. In contrast to the above results, we show that Neumer's Theorem and Juhász's Theorem for compact T1 spaces are both provable in ZF. [ABSTRACT FROM AUTHOR]

In this paper we characterize the shift spaces which have Li-Yorke chaos (an uncountable scrambled set). We focus primarily on shifts over a countably infinite alphabet. We represent them as either edge-shifts on an infinite graph (the subshift of finite type case) or as label led edge-shifts on an infinite graph (the sofic shift case). We show in the setting of a subshift of finite type on a shift over a countable alphabet that the shift space has Li-Yorke chaos if, and only if, it has a single scrambled pair, and in this case the scrambled set is closed and perfect (but not necessarily compact). We give an example of a sofic shift over an infinite alphabet which has a single scrambled pair but does not have Li-Yorke chaos. [ABSTRACT FROM AUTHOR]

In this paper, we analyze carefully the behaviour in L8(ℝ) of the square functions S and SI, originating from ergodic theory. First, we show that we can find some function f ? L8(ℝ), such that Sf equals infinity on a nonzero measurable set. Second, we can find compact supported function f ? L8(ℝ) and I such that SIf does not belong to BMO space. Finally, we show that S is bounded from L8c, the space of compactly supported L8(ℝ) functions, to BMO space. As a consequence, we solve an open question posed by Jones, Kaufman, Rosenblatt and Wierdl (2000). That is, SI are uniformly bounded in Lp(ℝ) with respect to I for 2 < p < 8. [ABSTRACT FROM AUTHOR]

We recall that a finitely presented group G is properly 3-realizable if for some finite 2-dimensional CW-complex X with π1(X) ≅ G, the universal cover X has the proper homotopy type of a 3-manifold. This purely topological property is closely related to the asymptotic behavior of the group G. We show that proper 3-realizability is also a geometric property meaning that it is a quasi-isometry invariant for finitely presented groups. In fact, in this paper we prove that (after taking wedge with a single n-sphere) any two infinite quasiisometric groups of type Fn (n ≥ 2) have universal covers whose n-skeleta are proper homotopy equivalent. Recall that a group G is of type Fn if it admits a K(G, 1)-complex with finite n-skeleton. [ABSTRACT FROM AUTHOR]

Let T be the group of smooth concordance classes of topologically slice knots, and let {0} ⊂ ··· ⊂ Tn+1 ⊂ Tn ⊂ ·· · ⊂ T0 ⊂ T be the bipolar filtration. Hedden, Kim, and Livingston showed that T contains a subgroup isomorphic to Z∞2 . In this paper, we show that a subset of the set of knots constructed by Hedden, Kim, and Livingston generates a subgroup isomorphic to Z∞2 in T0/T1. [ABSTRACT FROM AUTHOR]

Let K be a homogeneous self-similar set satisfying the strong separation condition. This paper is concerned with the quantitative recurrence properties of the natural map T : K → K induced by the shift. Let μ be the natural self-similar measure supported on K. For a positive function ϕ defined on N, we show that the μ-measure of the following set: R(ϕ) := {x ∈ K : |Tnx − x| < ϕ(n) for infinitely many n ∈ N} is null or full according to convergence or divergence of a certain series. Moreover, a similar dichotomy law holds for the general Hausdorff measure, which completes the metric theory of this set. [ABSTRACT FROM AUTHOR]

In this paper, we show that the action of a characteristically simple, non-extremely amenable (non-strongly amenable, non-amenable) group on its universal minimal (minimal proximal, minimal strongly proximal) flow is effective. We present necessary and sufficient conditions, for the action of a topological group with trivial center on its universal minimal proximal flow, to be effective. A theorem of Furstenberg about the isomorphism of the universal minimal proximal flows of a discrete group and its subgroups of finite index ([Proximal flows, Springer-Verlag, Berlin-New York, 1976]) is strengthened. Finally, for a pair of groups H < G the same method is applied in order to extend the action of H on its universal minimal proximal flow to an action of its commensurator group CommG(H). [ABSTRACT FROM AUTHOR]

In this paper we establish an upper bound on the second eigenvalue of n-dimensional spheres in the conformal class of the round sphere. This upper bound holds in all dimensions and is asymptotically sharp as the dimension increases. [ABSTRACT FROM AUTHOR]

We present a spectrum-level version of the norm map in equivariant homotopy theory based on the algebraic construction in the 1997 paper by Greenlees and May. We show that this new norm map is the same as the construction in the 2009 paper by Hill, Hopkins and Ravenel. Our comparison of the two norm maps gives a conceptual understanding of the choices inherent in the definition of the multiplicative norm map. [ABSTRACT FROM AUTHOR]

If X is a compact subset of a Banach space with X -X homogeneous (equivalently 'doubling' or with finite Assouad dimension), then X can be embedded into some R...n (with n sufficiently large) using a linear map L whose inverse is Lipschitz to within logarithmic corrections. More precisely, there exist c, α > 0 such that c ‖x - y‖/| log ‖x - y‖ |α ≤ |Lx - Ly| ≤ c‖x - y‖ for all x, y ∊ X, ‖x - y‖ < δ, for some δ sufficiently small. It is known that one must have α > 1 in the case of a general Banach space and α > 1/2 in the case of a Hilbert space. It is shown in this paper that these exponents can be achieved. While the argument in a general Banach space is relatively straightforward, the Hilbert space case relies on the fact that the maximum volume of a hyperplane slice of a k-fold product of unit volume N-balls is bounded independent of k (this provides a 'qualitative' generalisation of a result on slices of the unit cube due to Hensley (Proc.AMS 73 (1979), 95-100) and Ball (Proc.AMS 97 (1986), 465-473)). [ABSTRACT FROM AUTHOR]

In this paper we use Hofbauer towers for unimodal maps to study the collection of endpoints of the associated inverse limit spaces. It is shown that if f is a unimodal map for which the kneading map Qf (k) tends to infinity and f| ω(c) is one-to-one, then the collection of endpoints of (I, f) is precisely the set Ef = {(x0, x1, . . .) ∊ (I, f) | xi ∊ ω(c) for all i ∊ N}. [ABSTRACT FROM AUTHOR]

In this paper, we consider the partially overdetermined problem in integral equations as follows: ... where 0 < α < n, p > n/n-α, A, B, C are positive constants, Ω ⊂ Rn (n = 2) is a bounded domain with ∂Ω ∈ C¹, and Γ is a proper open set of ∂Ω. Under some assumptions on the geometry of Γ, we prove that Ω must be a ball and u is radially symmetric and monotone decreasing with respect to the radius. [ABSTRACT FROM AUTHOR]

In this paper, we proved that for generic Tonelli Lagrangian, there always exists a residual set G ⊂ H¹(M,R) such that ...(c)= Ã (c)= Ñ (c), ∀c ∈G, with ...(c) supports on a uniquely ergodic measure. [ABSTRACT FROM AUTHOR]

The Banach–Tarski paradox holds using six pieces in Zermelo–Fraenkel set theory with the Hahn–Banach theorem. [ABSTRACT FROM AUTHOR]

In this paper we consider some open questions concerning ?- convergence in complete CAT(0) metric spaces (i.e. Hadamard spaces). Suppose (X, d) is a Hadamard space such that the sets {z ? X| d(x, z) = d(z, y)} are convex for each x, y ? X. We introduce a so-called half-space topology such that convergence in this topology is equivalent to ?-convergence for any sequence in X. For a major class of Hadamard spaces, our results answer positively open questions nos. 1, 2 and 3 in [W. A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68 (2008) 3689- 3696]. Moreover, we give a new characterization of ?-convergence and a new topology that we call the weak topology via a concept of a dual metric space. The relations between these topologies and the topology which is induced by the distance function have been studied. The paper concludes with some examples. [ABSTRACT FROM AUTHOR]

In 1962, P. Gabriel showed the existence of a lattice isomorphism between the set of Serre subcategories of the category consisting of finitely generated modules and the set of specialization closed subsets of the set of prime ideals. In this paper, we consider subcategories consisting of the extensions of modules in two given Serre subcategories to find a method of constructing Serre subcategories of the category of modules. We shall give a criterion for this subcategory to be a Serre subcategory. [ABSTRACT FROM AUTHOR]

This paper deals with an example of a class of ordinary differential equations which are singular near a codimension 2 set with a homogeneous singularity of degree 0. Under some structural assumptions, we prove that for almost all initial data there exists a unique global solution. [ABSTRACT FROM AUTHOR]

According to a celebrated result by Löwner, a real-valued function f is operator monotone if and only if its Löwner matrix, which is the matrix of divided differences [Multiple line equation(s) cannot be represented in ASCII text] , is positive semidefinite for every integer N > 0 and any choice of x1, x2, … , xN. In this paper we answer a question of R. Bhatia, who asked for a characterisation of real-valued functions g defined on (0,+∞) for which the matrix of divided sums [Multiple line equation(s) cannot be represented in ASCII text], which we call its anti-Löwner matrix, is positive semide-finite for every integer N >0 and any choice of x1, x2, … , xN. ϵ? (0,+∞). Such functions, which we call anti-Löwner functions, have applications in the theory of Lyapunov-type equations. [ABSTRACT FROM AUTHOR]

In this paper, by using Gröbner-Shirshov bases for Rota-Baxter algebras, we prove that every dendriform algebra over a field of characteristic 0 can be embedded into its universal enveloping Rota-Baxter algebra. [ABSTRACT FROM AUTHOR]

We study two complexity notions of groups - the syntactic complexity of a computable Scott sentence and the m-degree of the index set of a group. Finding the exact complexity of one of them usually involves finding the complexity of the other, but this is not always the case. Knight et al. determined the complexity of index sets of various structures. In this paper, we focus on finding the complexity of computable Scott sentences and index sets of various groups. We give computable Scott sentences for various different groups, including nilpotent groups, polycyclic groups, certain solvable groups, and certain subgroups of Q. In some of these cases, we also show that the sentences we give are optimal. In the last section, we also show that d-Σ2 ⊊ Δ3 in the complexity hierarchy of pseudo-Scott sentences, contrasting the result saying d-Σ2 = Δ3 in the complexity hierarchy of Scott sentences, which is related to the boldface Borel hierarchy. [ABSTRACT FROM AUTHOR]

Let S denote the class of all functions f(z) = z + ∈n=2∞ anzn analytic and univalent in the unit disk D. For f ∈ S, Zalcman conjectured that |a2n− a2n−1| ≤ (n − 1)2 for n ≥ 3. This conjecture has been verified for only certain values of n for f ∈ S and for all n ≥ 4 for the class C of close-to-convex functions (and also for a couple of other classes). In this paper we provide bounds of the generalized Zalcman coefficient functional |λa2n−a2n−1| for functions in C and for all n ≥ 3, where λ is a positive constant. [ABSTRACT FROM AUTHOR]

In this paper, we study the intersection of an entire holomorphic mapping and a complex Monge-Ampère current with a bounded potential. By using our results, we show a sufficient condition involving order functions of an entire holomorphic mapping such that the complement of its image is a pluripolar set. [ABSTRACT FROM AUTHOR]

The main result of the paper is a flat extension theorem for positive linear functionals on *-algebras. The theorem is applied to truncated moment problems on cylinder sets, on matrices of polynomials and on enveloping algebras of Lie algebras. [ABSTRACT FROM AUTHOR]

In this short note, we build upon recent results from our earlier paper to present a precise expression for the asymptotic variance of the Euler- Poincaré characteristic for the excursion sets of Gaussian eigenfunctions on S²; this result can be written as a second-order Gaussian kinematic formula for the excursion sets of random spherical harmonics. The covariance between the Euler-Poincaré characteristics for different level sets is shown to be fully degenerate; it is also proved that the variance for the zero level excursion sets is asymptotically of smaller order. [ABSTRACT FROM AUTHOR]

An abelian arrangement is a finite set of codimension one abelian subvarieties (possibly translated) in a complex abelian variety. In this paper, we study the cohomology of the complement of an abelian arrangement. For unimodular abelian arrangements, we provide a combinatorial presentation for a differential graded algebra whose cohomology is isomorphic to the rational cohomology of the complement. Moreover, this DGA has a bi-grading that allows us to compute the mixed Hodge numbers. [ABSTRACT FROM AUTHOR]

Cameron and Erdős raised the question of how many maximal sum-free sets there are in {1, . . ., n}, giving a lower bound of 2⎣n/4⎦. In this paper we prove that there are in fact at most 2(1/4+o(1))n maximal sum-free sets in {1, . . ., n}. Our proof makes use of container and removal lemmas of Green as well as a result of Deshouillers, Freiman, Sós and Temkin on the structure of sum-free sets. [ABSTRACT FROM AUTHOR]

In this paper we present a strong local version of the Gagliardo-Nirenberg estimate that holds for elliptic systems of vector fields with smooth complex coefficients. We also consider L 1 estimates on forms analogous to those known in the case of the de Rham complex on ℝ1. [ABSTRACT FROM AUTHOR]

Given a number field K and a finite set S of places of K, the first main result of this paper shows that the quadratic rational maps ... defined over K which have good reduction at all places outside S form a Zariskidense subset of the moduli space M2 parametrizing all isomorphism classes of quadratic rational maps. We then consider quadratic rational maps with double unramified fixed-point structure, and our second main result establishes a Zariski nondensity result for the set of such maps with good reduction outside S. We also prove a variation of this result for quadratic rational maps with unramified 2-cycle structure. [ABSTRACT FROM AUTHOR]

We prove that for certain positive integers k, such as 12, a normal subgroup of a finite group which consists of kth powers is necessarily soluble. This gives rise to new solubility criteria, and solves an open problem from a 2013 paper by the authors. [ABSTRACT FROM AUTHOR]

In this paper, a formula for the number of right cosets contained in a double coset with respect to the unimodular group of invertible (2 × 2)- matrices over a Dedekind domain is developed. As applications we derive an index formula for congruence subgroups and an algorithm for the explicit calculation of products in Hecke algebras. [ABSTRACT FROM AUTHOR]

In this paper, we study effective monotonic approximations of sets and sequences of sets. We show that there is a sequence of sets which has no uniform computable monotonic approximation but has an x-computable monotonic approximation for every hyperimmune degree x. We also construct a Σ20 set which is not limitwise monotonic but is x-limitwise monotonic relative to every non-zero Δ20 degree x. We show that if a sequence of sets is uniformly limitwise monotonic in x for all except countably many degrees x, then it has to be uniformly limitwise monotonic. Finally, we apply these results to investigate degree spectra of abelian groups, equivalence relations, and N1- categorical structures. [ABSTRACT FROM AUTHOR]

Let D be an integral domain with quotient field K, * be a staroperation on D, and GV *(D) be the set of finitely generated ideals J of D such that J* = D. Then the map *w defined by I*w = {x Є K | Jx ⊆ I for some J Є GV *(D)} for all nonzero fractional ideals I of D is a finite character staroperation on D. In this paper, we study several properties of *w-Noetherian domains. In particular, we prove the Hilbert basis theorem for *w-Noetherian domains. [ABSTRACT FROM AUTHOR]

In one dimension, sofic shifts are fairly well understood and are special examples of shift spaces which must satisfy very restrictive properties. However, in multiple dimensions there are very few known conditions which guarantee nonsoficity of a shift space. In this paper, we show that for any Zd sofic shift X which satisfies a uniform mixing condition called block gluing in all directions ...2, . . ., ...d, the set of legal rows of X in the ...1-direction has a synchronizing word. This allows us to define a (new) large class of nonsofic Zd shift spaces. [ABSTRACT FROM AUTHOR]

Let SY Tn be the set of all standard Young tableaux with n cells and =weak be Melnikov's weak order on SY Tn. The aim of this paper is to introduce a conjecture on the weak order, named the inner tableau translation property. We prove the conjecture for some special cases and discuss its significance. [ABSTRACT FROM AUTHOR]

In this paper, some properties of a strictly A-coupled-expanding map in compact subsets of a metric space are studied, where A is a transition matrix. It is shown that this map has a compact invariant set on which it is topologically semi-conjugate to the subshift for A. If the subshift for A has positive topological entropy, then the map is chaotic in the sense of Li- Yorke. Moreover, in the one-dimensional case, the map is at most two-to-one conjugate to the subshift for A and chaotic in the sense of Devaney. [ABSTRACT FROM AUTHOR]

In this paper we study set expansion in finite fields. Fourier analytic proofs are given for several results recently obtained by other authors using spectral graph theory. In addition, several generalizations of these results are given. In the case that A is a subset of a prime field Fp of size less than p1/2 it is shown that |{a² + b : a, b ∊ A}| ≥ C1|A|147/146 and |{...: a, b ∊ A}| ≥ C2|A|110/109, where | · | denotes the cardinality of a set and C1 and C2 are absolute constants. [ABSTRACT FROM AUTHOR]