On the size of maximal chains and the number of pairwise disjoint maximal antichains

For each integer k≥3, we find all maximal intervals I_k of natural numbers with the following property: whenever the number of elements in every maximal chain in a finite partially ordered set P lies in I_k, then P contains k pairwise disjoint maximal antichains. All such I_k are of the form

     

Sequential definitions of compactness

A subset F of a topological space is sequentially compact if any sequence of points in F has a convergent subsequence whose limit is in F. We say that a subset F of a topological group X is G-sequentially compact if any sequence of points in F has a convergent subsequence such that where G is an additive function from a subgroup of the group of all sequences of points in X. We investigate the impact of changing the definition of convergence of sequences on the structure of sequentially compactness of sets in the sense of G-sequential compactness. Sequential compactness is a special case of this generalization when G=lim.     

On the division of space by topological hyperplanes

A topological hyperplane is a subspace of (or a homeomorph of it) that is topologically equivalent to an ordinary straight hyperplane. An arrangement of topological hyperplanes in is a finite set such that for any nonvoid intersection Y of topological hyperplanes in and any that intersects but does not contain Y, the intersection is a topological hyperplane in Y. (We also assume a technical condition on pairwise intersections.) If every two intersecting topological hyperplanes cross each other, the arrangement is said to be transsective. The number of regions formed by an arrangement of topological hyperplanes has the same formula as for arrangements of ordinary affine hyperplanes, provided that every region is a cell. Hoping to explain this geometrically, we ask whether parts of the topological hyperplanes in any arrangement can be reassembled into a transsective arrangement of topological hyperplanes with the same regions. That is always possible if the dimension is two but not in higher dimensions. We also ask whether all transsective topological hyperplane arrangements correspond to oriented matroids; they need not (because parallelism may not be an equivalence relation), but we can characterize those that do if the dimension is two. In higher dimensions this problem is open. Another open question is to characterize the intersection semilattices of topological hyperplane arrangements; a third is to prove that the regions of an arrangement of topological hyperplanes are necessarily cells; a fourth is whether the technical pairwise condition is necessary.     

Optimal embeddings in quaternion algebras

Let A be a quaternion algebra over a number field k and assume that A satisfies the Eichler condition so that some infinite place of k is unramified in A. Let L be a quadratic extension of k which embeds in A. Let R_k denote the ring of integers of k and let B be an R_k-order in L. Suppose that is an Eichler order of A of square-free level S. In this paper, we determine when there exists an embedding σ:L→A over k which gives an optimal embedding of B into in the sense that . This generalises previous work of Eichler [M. Eichler, Zur Zahlentheorie der Quaternionenalgebren, J. Reine Angew. Math. 195 (1955) 127–155] and Chinburg and Friedman [T. Chinburg, E. Friedman, An embedding theorem for quaternion algebras, J. London Math. Soc. 60 (1999) 33–44].     

Boundedness and unboundedness results for some maximal operators on functions of bounded variation

We characterize the space BV(I) of functions of bounded variation on an arbitrary interval I⊂R, in terms of a uniform boundedness condition satisfied by the local uncentered maximal operator MR from BV(I) into the Sobolev space W1,1(I). By restriction, the corresponding characterization holds for W1,1(I). We also show that if U is open in Rd, d>1, then boundedness from BV(U) into W1,1(U) fails for the local directional maximal operator , the local strong maximal operator , and the iterated local directional maximal operator . Nevertheless, if U satisfies a cone condition, then boundedly, and the same happens with , , and MR.     

On the dynamics of composition of commuting interval maps

Let be two commuting continuous maps. We establish some results on the topological dynamic shared by both maps and state some conditions to get that the topological entropy of the composition f○g will be positive.     

The maximal range problem for a bounded domain

Let be a bounded domain such that 0Ω. Denote by , the set of all complex polynomials of degree at most n. Let

where . We relate the maximal polynomial range

to the geometry of Ω.     

Higher order log-concavity in Euler’s difference table

For 0≤k≤n, let be the entries in Euler’s difference table and let . Dumont and Randrianarivony showed equals the number of permutations on [n] whose fixed points are contained in {1,2,…,k}. Rakotondrajao found a combinatorial interpretation of the number in terms of k-fixed-points-permutations of [n]. We show that for any n≥1, the sequence is essentially 2-log-concave and reverse ultra log-concave.     

Pseudorandom numbers and entropy conditions

We investigate measures of pseudorandomness of finite sequences (x_n) of real numbers. Mauduit and Sárközy introduced the “well-distribution measure”, depending on the behavior of the sequence (x_n) along arithmetic subsequences (x_ak+b). We extend this definition by replacing the class of arithmetic progressions by an arbitrary class of sequences of positive integers and show that the so obtained measure is closely related to the metric entropy of the class . Using standard probabilistic techniques, this fact enables us to give precise bounds for the pseudorandomness measure of classical constructions. In particular, we will be interested in “truly” random sequences and sequences of the form {n_kω}, where {·} denotes fractional part, (n_k) is a given sequence of integers and ω[0,1).     

On maximal functions for Mikhlin-Hörmander multipliers

Given Mikhlin-Hörmander multipliers , with uniform estimates we prove an optimal bound in L^p for the maximal function and related bounds for maximal functions generated by dilations. These improve the results in [M. Christ, L. Grafakos, P. Honzík, A. Seeger, Maximal functions associated with multipliers of Mikhlin-Hörmander type, Math. Z. 249 (2005) 223-240].     

Ore-type and Dirac-type theorems for matroids

Let M be a connected binary matroid having no -minor. Let be a collection of cocircuits of M. We prove there is a circuit intersecting all cocircuits of if either one of two things hold:

(i) For any two disjoint cocircuits and in it holds that .

(ii) For any two disjoint cocircuits and in it holds that .

Part (ii) implies Ore's Theorem, a well-known theorem giving sufficient conditions for the existence of a hamilton cycle in a graph. As an application of part (i), it is shown that if M is a k-connected regular matroid and has cocircumference c^*2k, then there is a circuit which intersects each cocircuit of size c^*−k+2 or greater.We also extend a theorem of Dirac for graphs by showing that for any k-connected binary matroid M having no -minor, it holds that for any k cocircuits of M there is a circuit which intersects them.     

Sparse inertially arbitrary patterns

An n-by-n sign pattern is a matrix with entries in {+,-,0}. An n-by-n nonzero pattern is a matrix with entries in {*,0} where * represents a nonzero entry. A pattern is inertially arbitrary if for every set of nonnegative integers n₁,n₂,n₃ with n₁+n₂+n₃=n there is a real matrix with pattern having inertia (n₁,n₂,n₃). We explore how the inertia of a matrix relates to the signs of the coefficients of its characteristic polynomial and describe the inertias allowed by certain sets of polynomials. This information is useful for describing the inertia of a pattern and can help show a pattern is inertially arbitrary. Britz et al. [T. Britz, J.J. McDonald, D.D. Olesky, P. van den Driessche, Minimal spectrally arbitrary sign patterns, SIAM J. Matrix Anal. Appl. 26 (2004) 257–271] conjectured that irreducible spectrally arbitrary patterns must have at least 2n nonzero entries; we demonstrate that irreducible inertially arbitrary patterns can have less than 2n nonzero entries.     

Endpoint estimates for some maximal operators associated to the circular conical surface

For and m>0 we consider the maximal operator given by

     

Periodic Schur functions and slit discs

A simply connected domain is called a slit disc if minus a finite number of closed radial slits not reaching the origin. A slit disc is called rational (rationally placed) if the lengths of all its circular arcs between neighboring slits (the arguments of the slits) are rational multiples of 2π. The conformal mapping of onto , (0)=0, ^′(0)>0, extends to a continuous function on mapping it onto . A finite union E of closed non-intersecting arcs e_k on is called rational if for every k, ν_E(e_k) being the harmonic measures of e_k at ∞ for the domain . A compact E is rational if and only if there is a rational slit disc such that . A compact E essentially supports a measure with periodic Verblunsky parameters if and only if for a rationally placed . For any tuple (α₁,…,α_g+1) of positive numbers with ∑_kα_k=1 there is a finite family of closed non-intersecting arcs e_k on such that ν_E(e_k)=α_k. For any set and any >0 there is a rationally placed compact such that the Lebesgue measure |EE^*| of the symmetric difference EE^* is smaller than .     

Parabolic mean values and maximal estimates for gradients of temperatures

We aim to prove inequalities of the form for solutions of on a domain Ω=D×R⁺, where δ(x,t) is the parabolic distance of (x,t) to parabolic boundary of Ω, is the one-sided Hardy-Littlewood maximal operator in the time variable on R⁺, is a Calderón-Scott type d-dimensional elliptic maximal operator in the space variable on the domain D in Rd, and 0<λ<k<λ+d. As a consequence, when D is a bounded Lipschitz domain, we obtain estimates for the Lp(Ω) norm of δ2n−λn(∇2,1)u in terms of some mixed norm for the space with denotes the Besov norm in the space variable x and where .     

New examples of maximal space like surfaces in the anti-de Sitter space

Let be the three-dimensional anti-de Sitter space. In this paper we will construct new examples of complete maximal space like surfaces in . Moreover, we will prove that any complete maximal space like surface in with principal curvatures ±κ bounded away from zero must be isometric to the hyperbolic cylinder. Since the new examples that we have constructed have exactly two principal curvatures everywhere, we conclude that the condition on the principal curvatures on the previous result, i.e. the condition |κ(m)|>c>0, cannot be replaced by the condition |κ(m)|>0.     

Algebraic independence of the values of Mahler functions associated with a certain continued fraction expansion

It is proved that the function , which can be expressed as a certain continued fraction, takes algebraically independent values at any distinct nonzero algebraic numbers inside the unit circle if the sequence {R_k}_k?0 is the generalized Fibonacci numbers.     

Forbidden configurations and repeated induction

For a given k×? matrix F, we say a matrix A has no configurationF if no k×? submatrix of A is a row and column permutation of F. We say a matrix is simple if it is a (0,1)-matrix with no repeated columns. We define as the maximum number of columns in an m-rowed simple matrix which has no configuration F. A fundamental result of Sauer, Perles and Shelah, and Vapnik and Chervonenkis determines exactly, where K_k denotes the k×2^k simple matrix. We extend this in several ways. For two matrices G,H on the same number of rows, let [G∣H] denote the concatenation of G and H. Our first two sets of results are exact bounds that find some matrices B,C where and . Our final result provides asymptotic boundary cases; namely matrices F for which is O(m^p) yet for any choice of column α not in F, we have is Ω(m^p+1). This is evidence for a conjecture of Anstee and Sali. The proof techniques in this paper are dominated by repeated use of the standard induction employed in forbidden configurations. Analysis of base cases tends to dominate the arguments. For a k-rowed (0,1)-matrix F, we also consider a function which is the minimum number of columns in an m-rowed simple matrix for which each k-set of rows contains F as a configuration.     

Distance irredundance and connected domination numbers of a graph

Let k be a positive integer and G be a connected graph. This paper considers the relations among four graph theoretical parameters: the k-domination number γ_k(G), the connected k-domination number ; the k-independent domination number and the k-irredundance number ir_k(G). The authors prove that if an ir_k-set X is a k-independent set of G, then , and that for k?2, if ir_k(G)=1, if ir_k(G) is odd, and if ir_k(G) is even, which generalize some known results.     

Log-concavity and LC-positivity

A triangle {a(n,k)}_0?k?n of nonnegative numbers is LC-positive if for each r, the sequence of polynomials is q-log-concave. It is double LC-positive if both triangles {a(n,k)} and {a(n,n−k)} are LC-positive. We show that if {a(n,k)} is LC-positive then the log-concavity of the sequence {xk} implies that of the sequence {zn} defined by , and if {a(n,k)} is double LC-positive then the log-concavity of sequences {xk} and {yk} implies that of the sequence {zn} defined by . Examples of double LC-positive triangles include the constant triangle and the Pascal triangle. We also give a generalization of a result of Liggett that is used to prove a conjecture of Pemantle on characteristics of negative dependence.