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After proving that numerical sequences such as Fibonacci numbers and prime numbers, can be generated as sequences of equilibrium points of an ideal half-infinite one-dimensional distribution of electric charges, a model of the distribution of primes on the x-axis is proposed, where primes ρ(n) are considered as quantum particles oscillating around the sequence of stationary points r(n) of the Lennard-Jones-like potential of the single-particle Hamiltonian. A particle-counting function πQ(x) is defined over the many-particle system, in the same way as the prime-counting function π(x). Through the application of the Hellmann-Feynman theorem, the existence of a solution n(x) ≈ li(x), coinciding asymptotically with the logarithmic integral function, is proved for the counting function of stationary points, together with the quantum equivalent of the Prime Number Theorem for the particle-counting function πQ(x). The conditions on the sequence of energy eigenvalues E(n) of the system, so that the Riemann hypothesis holds for the function πQ(x), are derived, thus proving the quantum theory results in the same bound of the position of the prime-particle in the n-th quantum state as that derived through analytical methods for the n-th prime. The model suggests new conjectures and statistical procedures which are applied to study the sign of the difference π(x)-li(x) and to explore the region beyond the Skewes number. The data predicted by the theory find confirmation when compared with those known in the literature. A new statistical local test of the Riemann hypothesis, based on the difference π(x)-li(x), is proposed. [ABSTRACT FROM AUTHOR]

Let f be a Hecke-Maass cusp form for SL(3,ℤ) with Fourier coeffcients Af (m, n), and let ϕ(x) be a C∞-function supported on [1, 2] with derivatives bounded by ϕ(j)(x)≪j 1: We prove an asymptotic formula for the nonlinear Σn≡lmodqAf (m, n)ϕ(n/X)e(3(kn)1/3/q), where e(z) = e2πiz and k ∈ℤ+. [ABSTRACT FROM AUTHOR]

We show that minimizers of the Heitmann-Radin energy (Heitmann and Radin in J Stat Phys 22(3):281-287, 1980) are unique if and only if the particle number N belongs to an infinite sequence whose first thirty-five elements are 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 30, 33, 37, 40, 44, 48, 52, 56, 61, 65, 70, 75, 80, 85, 91, 96, 102, 108, 114, 120 (see the paper for a closed-form description of this sequence). The proof relies on the discrete differential geometry techniques introduced in De Luca and Friesecke (Crystallization in two dimensions and a discrete Gauss-Bonnet Theorem, 2016). [ABSTRACT FROM AUTHOR]

Let λ = (λn)n≥1be a nondecreasing sequence of positive numbers tending to infinity such that λ1 = 1 and λn+1 ≥ λn + 1 for all n, and let In = [n -- λn + 1; n] for n = 1; 2;,.... Then for any given nonzero sequence µ, we define by Δ+(µ) the operator that generalizes the operator of the First difference and is defined by Δ+(µ)xk = µk(xk -- xk+1). In this article, for any given integer r ≤ 1, we deal with the λ+r(µ)-statistical convergence that generalizes in a certain sense the well-known λE-rstatistical convergence, main results consist in determining sets of sequences χ and χ' of the form sξ0 satisfying χ ⊂ [V; λ]0(Δ+r(µ)) ⊂ χ' and sets κ and κ' of the form sξ satisfying χ ⊂ [V, λ]∞(Δ+r(µ)) ≥ κ'. This study is justified since the infinite matrix associated with the operator Δ+r(µ) cannot be explicitly calculated for all r. [ABSTRACT FROM AUTHOR]

In this paper, we study the log-behavior of a new sequence $\{S_{n}\} _{n=0}^{\infty}$ , which was defined by Z-W Sun. We find that the sequence is log-convex by using the interlacing method. Additionally, we consider ratio log-behavior of $\{S_{n}\}_{n=0}^{\infty}$ and find the sequences $\{S_{n+1}/S_{n}\}_{n=0}^{\infty}$ and $\{\sqrt[n]{S_{n}}\} _{n=1}^{\infty}$ are log-concave. Our results give an affirmative answer to a conjecture of Z-W Sun on the ratio monotonicity of this new sequence. [ABSTRACT FROM AUTHOR]

We consider synthesis problems for a discrete event model of a real time distributed computational system. The model reflects information exchange events between software units of the system and is intended to construct special test sequences that detect faults in these exchanges. [ABSTRACT FROM AUTHOR]

Testing of reactive systems is challenging because long input sequences are often needed to drive them into a state to test a desired feature. This is particularly problematic in on-target testing, where a system is tested in its real-life application environment and the amount of time required for resetting is high. This article presents an approach to discovering a test case chain-a single software execution that covers a group of test goals and minimizes overall test execution time. Our technique targets the scenario in which test goals for the requirements are given as safety properties. We give conditions for the existence and minimality of a single test case chain and minimize the number of test case chains if a single test case chain is infeasible. We report experimental results with our ChainCover tool for C code generated from Simulink models and compare it to state-of-the-art test suite generators. [ABSTRACT FROM AUTHOR]

We establish sufficient conditions for the Marcinkiewicz-Zygmund type weak law of large numbers for a linear process $${\{X_k:k\in\mathbb Z\}}$$ defined by $${X_k=\sum_{j=0}^\infty\psi_j\varepsilon_{k-j}}$$ for $${k\in\mathbb Z}$$ , where $${\{\psi_j:j\in\mathbb Z\}\subset\mathbb R}$$ and $${\{\varepsilon_k:k\in\mathbb Z\}}$$ are independent and identically distributed random variables such that $${{x^p\Pr\{|\varepsilon_0| > x\}\to 0}}$$ as $${{x\to \infty}}$$ with $${1 < p < 2}$$ and $${E \varepsilon_0=0}$$ . We use an abstract norming sequence that does not grow faster than $${n^{1/p}}$$ if $${\sum|\psi_j| < \infty}$$ . If $${\sum|\psi_j|=\infty}$$ , the abstract norming sequence might grow faster than $${n^{1/p}}$$ as we illustrate with an example. Also, we investigate the rate of convergence in the Marcinkiewicz-Zygmund type weak law of large numbers for the linear process. [ABSTRACT FROM AUTHOR]

For a simple finite graph G denote by [InlineEquation not available: see fulltext.] the number of ways of partitioning the vertex set of G into k non-empty independent sets (that is, into classes that span no edges of G). If $$E_n$$ is the graph on n vertices with no edges then [InlineEquation not available: see fulltext.] coincides with [InlineEquation not available: see fulltext.], the ordinary Stirling number of the second kind, and so we refer to [InlineEquation not available: see fulltext.] as a graph Stirling number. Harper showed that the sequence of Stirling numbers of the second kind, and thus the graph Stirling sequence of $$E_n$$ , is asymptotically normal-essentially, as n grows, the histogram of [InlineEquation not available: see fulltext.], suitably normalized, approaches the density function of the standard normal distribution. In light of Harper's result, it is natural to ask for which sequences $$(G_n)_{n \ge 0}$$ of graphs is there asymptotic normality of [InlineEquation not available: see fulltext.]. Thanh and Galvin conjectured that if for each n, $$G_n$$ is acyclic and has n vertices, then asymptotic normality occurs, and they gave a proof under the added condition that $$G_n$$ has no more than $$o(\sqrt{n/\log n})$$ components. Here we settle Thanh and Galvin's conjecture in the affirmative, and significantly extend it, replacing 'acyclic' in their conjecture with 'co-chromatic with a quasi-threshold graph, and with negligible chromatic number'. Our proof combines old work of Navon and recent work of Engbers, Galvin and Hilyard on the normal order problem in the Weyl algebra, and work of Kahn on the matching polynomial of a graph. [ABSTRACT FROM AUTHOR]

We study the rate of convergence in the strong law of large numbers expressed in terms of complete convergence of Baum-Katz type for sequences of random variables satisfying Petrov's condition. [ABSTRACT FROM AUTHOR]

A sequence A of nonnegative integers is called complete if all sufficiently large integers can be represented as the sum of distinct terms taken form A. For a sequence $${S=\{s_{1}, s_{2}, \dots\}}$$ of positive integers and a positive real number α, let S denote the sequence $${\{\lfloor\alpha s_{1}\rfloor, \lfloor\alpha s_{2}\rfloor, \dots\}}$$ , where $${\lfloor x \rfloor}$$ denotes the greatest integer not greater than x. Let $${{U_S = \{\alpha \mid S_\alpha} \, is complete\}}$$ . Hegyvári [] proved that if $${\lim_{n\to\infty} (s_{n+1}-s_{n})=+ \infty}$$ , $${s_{n+1} < \gamma s_{n}}$$ for all integers $${n \geqq n_{0}}$$ , where $${1 < \gamma < 2}$$ , and $${U_{S}\ne\emptyset}$$ , then $${\mu(U_{S}) > 0}$$ , where $${\mu(U_{S})}$$ is the Lebesgue measure of U. Yong-Gao Chen and the first author [] proved that, if $${s_{n+1} < \gamma s_{n}}$$ for all integers $${n \geqq n_{0}}$$ , where $${1 < \gamma \leqq 7/4=1.75}$$ , then $${\mu(U_{S}) > 0}$$ . In this paper, we prove that the conclusion holds for $${1 < \gamma \leqq \sqrt[4]{13}=1.898\dots\;}$$ . [ABSTRACT FROM AUTHOR]

Using annular domains introduced in Part II (Leung, Calc Var PDE 46:1-29, ), we present constructive results on blow-up sequences of infinite number of solutions for the (prescribed and fixed) conformal scalar curvature equation on $$S^n$$ ( $$n\ge 6$$ ), including aggregated and towering blow-ups. The constructions make use of the Lyapunov-Schmidt reduction method, and count on the hyperbolic structure on the collection of standard bubbles. [ABSTRACT FROM AUTHOR]

We investigate a multi-index analogue of unconditional convergence theorem of Tandori. Namely, we formulate a condition regarding a multiindex sequence of numbers ( a) which ensures that for all orthonormal fields ( $${\Phi_n}$$ ) the series $${\sum_{n} a_n\Phi_n}$$ is unconditionally almost everywhere convergent. We consider convergence in Pringsheim's and in regular sense. Our condition is in a way optimal. [ABSTRACT FROM AUTHOR]

We study the complementary design problem, which is to express the uniformity pattern of a q-level design in terms of that of its complementary design. Here, a pair of complementary designs form a design in which all the Hamming distances of any two distinct runs are the same, and the uniformity pattern proposed by H. Qin, Z. Wang, and K. Chatterjee [J. Statist. Plann. Inference, 2012, 142: 1170-1177] comes from discrete discrepancy for q-level designs. Based on relationships of the uniformity pattern between a pair of complementary designs, we propose a minimum projection uniformity rule to assess and compare q-level factorials. [ABSTRACT FROM AUTHOR]

In this note, we consider the trees (caterpillars) that minimize the number of subtrees among trees with a given degree sequence. This is a question naturally related to the extremal structures of some distance based graph invariants. We first confirm the expected fact that the number of subtrees is minimized by some caterpillar. As with other graph invariants, the specific optimal caterpillar is nearly impossible to characterize and depends on the degree sequence. We provide some simple properties of such caterpillars as well as observations that will help finding the optimal caterpillar. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the structures of extremal trees which have the minimal number of subtrees in the set of all trees with a given degree sequence. In particular, the extremal trees must be caterpillar and but in general not unique. Moreover, all extremal trees with a given degree sequence $${\pi = (d_1, \ldots, d_{5}, 1, \ldots, 1)}$$ have been characterized. [ABSTRACT FROM AUTHOR]

The balancing numbers originally introduced by Behera and Panda [] as solutions of a Diophantine equation on triangular numbers possess many interesting properties. Many of these properties are comparable to certain properties of Fibonacci numbers, while some others are more interesting. Wall [] studied the periodicity of Fibonacci numbers modulo arbitrary natural numbers. The periodicity of balancing numbers modulo primes and modulo terms of certain sequences exhibits beautiful results, again, some of them are identical with corresponding results of Fibonacci numbers, while some others are more fascinating. An important observation concerning the periodicity of balancing numbers is that, the period of this sequence coincides with the modulus of congruence if the modulus is any power of 2. There are three known primes for which the period of the sequence of balancing numbers modulo each prime is equal to the period modulo its square, while for the Fibonacci sequence, till date no such prime is available. [ABSTRACT FROM AUTHOR]

Let { X, n ≥ 1} be a sequence of negatively associated random variables. The aim of this paper is to establish some limit theorems of negatively associated sequence, which include the L-convergence theorem and Marcinkiewicz-Zygmund strong law of large numbers. Furthermore, we consider the strong law of sums of order statistics, which are sampled from negatively associated random variables. [ABSTRACT FROM AUTHOR]

A composition of a positive integer in which a part of size n may be assigned one of n colors is called an n-color composition. Let a denote the number of n-color compositions of the integer m. It is known that a= F for all m≥1, where F denotes the Fibonacci number defined by F= F+ F if m≥2, with F=0 and F=1. A statistic is studied on the set of n-color compositions of m, thus providing a polynomial generalization of the sequence F. The statistic may be described, equivalently, in terms of statistics on linear tilings and lattice paths. The restriction to the set of n-color compositions having a prescribed number of parts is considered and an explicit formula for the distribution is derived. We also provide q-generalizations of relations between a and the number of self-inverse n-compositions of 2 m+1 or 2 m. Finally, we consider a more general recurrence than that satisfied by the numbers a and note some particular cases. [ABSTRACT FROM AUTHOR]

For an integer k ≥ 2, let $(L_{n}^{(k)})_{n}$ be the k-Lucas sequence which starts with 0,...,0,2,1 ( k terms) and each term afterwards is the sum of the k preceding terms. In 2000, Luca ( Port. Math. 57(2) (2000) 243-254) proved that 11 is the largest number with only one distinct digit (the so-called repdigit) in the sequence $(L_{n}^{(2)})_{n}$. In this paper, we address a similar problem in the family of k-Lucas sequences. We also show that the k-Lucas sequences have similar properties to those of k-Fibonacci sequences and occur in formulae simultaneously with the latter. [ABSTRACT FROM AUTHOR]

In the paper, we investigate the complete convergence and complete moment convergence for the maximal partial sum of martingale difference sequence. Especially, we get the Baum-Katz-type Theorem and Hsu-Robbins-type Theorem for martingale difference sequence. As an application, a strong law of large numbers for martingale difference sequence is obtained. [ABSTRACT FROM AUTHOR]

A pebbling move on a graph removes two pebbles at a vertex and adds one pebble at an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed. In this new move, one pebble each is removed at vertices v and w adjacent to a vertex u, and an extra pebble is added at vertex u. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using rubbling moves. The rubbling number is the smallest number m needed to guarantee that any vertex is reachable from any pebble distribution of m pebbles. The optimal rubbling number is the smallest number m needed to guarantee a pebble distribution of m pebbles from which any vertex is reachable. We give bounds for rubbling and optimal rubbling numbers. In particular, we find an upper bound for the rubbling number of n-vertex, diameter d graphs, and estimates for the maximum rubbling number of diameter 2 graphs. We also give a sharp upper bound for the optimal rubbling number, and sharp upper and lower bounds in terms of the diameter. [ABSTRACT FROM AUTHOR]

A Keith number is a positive integer N with the decimal representation a a⋯ a such that k ≥ 2 and N appears in the sequence that starts with a, a,..., a and for which each term afterwards is the sum of the k preceding terms. In 2007, Klazar and Luca [M. Klazar and F. Luca, Counting Keith numbers, J. Integer Seq., 10(2):Article 07.2.2, 2007] proved that there are only finitely many Keith numbers with only one distinct digit (so-called repdigits). In this paper, we prove that there are no Keith numbers which are repdigits. [ABSTRACT FROM AUTHOR]

Let $${\mathcal{F}}$$ be a separable uniformly bounded family of measurable functions on a standard measurable space $${(X, \mathcal{X})}$$, and let $${N_{[]}(\mathcal{F}, \varepsilon, \mu)}$$ be the smallest number of $${\varepsilon}$$ -brackets in L( μ) needed to cover $${\mathcal{F}}$$. The following are equivalent: It follows that universal Glivenko-Cantelli classes are uniformity classes for general sequences of almost surely convergent random measures. [ABSTRACT FROM AUTHOR]

Let $${\mathcal{A}}$$ be a finite subset of $${\mathbb{N}}$$ containing 0, and let f ( n) denote the number of ways to write n in the form $${\sum \varepsilon _{j}2^{j}}$$ , where $${\varepsilon _{j} \epsilon \mathcal{A}}$$ . We show that there exists a computable $${T = T (\mathcal{A})}$$ so that the sequence ( f ( n) mod 2) is periodic with period T. Variations and generalizations of this problem are also discussed. [ABSTRACT FROM AUTHOR]

To count the number of chemical compositions of a particular mass, we consider an alphabet $${\mathop{\bf a}}$$ with a mass function which assigns a mass to each letter in $${\mathop{\bf a}}$$ . We then compute the mass of a word (an ordered sequence of letters) by adding the masses of the constituent letters. Our main interest is to count the number of words that have a particular mass, where we ignore the order of the letters within the word. We show first that counting the number of words of a given mass has a geometric interpretation, whose solutions are called Ehrhart quasi-polynomials, a class of functions defined on integers. These special functions are 'periodic' in the sense that they use the same polynomial every λ steps. In addition to discovering the connection between counting compositions and Ehrhart quasi-polynomials, we also find number theoretic results that greatly reduce the number of candidates for the period, λ. Finally, we illustrate the usefulness of these results and the use of a software library named barvinok (by Verdoolaege et al.) by applying them to eight different classes of chemical compositions, including organic molecules, peptides, DNA, and RNA. [ABSTRACT FROM AUTHOR]

We study the problem of optimally controlling the solution of the obstacle problem in a domain perforated by small periodically distributed holes. The solution is controlled by the choice of a perforated obstacle which is to be chosen in such a fashion that the solution is close to a given profile and the obstacle is not too irregular. We prove existence, uniqueness and stability of an optimal obstacle and derive necessary and sufficient conditions for optimality. When the number of holes increase indefinitely we determine the limit of the sequence of optimal obstacles and solutions. This limit depends strongly on the rate at which the size of the holes shrink. [ABSTRACT FROM AUTHOR]

We investigate the partition property of $${\mathcal{P}_{\kappa}\lambda}$$. Main results of this paper are as follows: (1) If λ is the least cardinal greater than κ such that $${\mathcal{P}_{\kappa}\lambda}$$ carries a ( λ, 2)-distributive normal ideal without the partition property, then λ is $${\Pi^1_n}$$-indescribable for all n < ω but not $${\Pi^2_1}$$ -indescribable. (2) If cf( λ) ≥ κ, then every ineffable subset of $${\mathcal{P}_{\kappa}\lambda}$$ has the partition property. (3) If cf( λ) ≥ κ, then the completely ineffable ideal over $${\mathcal{P}_{\kappa}\lambda}$$ has the partition property. [ABSTRACT FROM AUTHOR]

An inaccessible cardinal κ is supercompact when ( κ, λ)-ITP holds for all λ ≥ κ. We prove that if there is a model of ZFC with two supercompact cardinals, then there is a model of ZFC where simultaneously $${(\aleph_2, \mu)}$$ -ITP and $${(\aleph_3, \mu')}$$ -ITP hold, for all $${\mu\geq \aleph_2}$$ and $${\mu'\geq \aleph_3}$$ . [ABSTRACT FROM AUTHOR]

The notion of ideal convergence is a generalization of statistical convergence which has been intensively investigated in last few years. For an admissible ideal ∮ ⊂ ℔ × ℔, the aim of the present paper is to introduce the concepts of ∮-convergence and ∮-convergence for double sequences on probabilistic normed spaces (PN spaces for short). We give some relations related to these notions and find condition on the ideal ∮ for which both the notions coincide. We also define ∮ -Cauchy and ∮-Cauchy double sequences on PN spaces and show that ∮ -convergent double sequences are ∮-Cauchy on these spaces. We establish example which shows that our method of convergence for double sequences on PN spaces is more general. [ABSTRACT FROM AUTHOR]

Let f( x) be the product of several linear polynomials with integer coefficients. In this paper, we obtain the estimate log lcm ( f(1),..., f( n))∼ An as n→∞, where A is a constant depending on f. [ABSTRACT FROM AUTHOR]

For a set A, let P( A) be the set of all finite subset sums of A. We prove that if a sequence B={ b< b<⋯} of integers satisfies b≧11 and b≧3 b+5 ( n=1,2,...), then there exists a sequence of positive integers A={ a< a<⋯} for which P( A)=ℕ∖ B. On the other hand, if a sequence B={ b< b<⋯} of positive integers satisfies either b=10 or b=3 b+4, then there is no sequence A of positive integers for which P( A)=ℕ∖ B. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of badly approximable numbers in problems involving lacunary or sublacunary sequences. The principal results depend upon Peres-Schlag's method. [ABSTRACT FROM AUTHOR]

We first introduce a new notion called statistical convergence of order α and primarily show that it gives rise to a decreasing chain of closed linear subspaces of the space of all bounded real sequences with sup norm which never coincides with the class of convergent sequences and in fact their intersection properly contains the class of convergent sequences. We then show that the same method can be applied for double sequences also and introduce the notion of statistical convergence of order ( α, β). [ABSTRACT FROM AUTHOR]

In the classic Josephus problem, elements 1,2,…, n are placed in order around a circle and a skip value k is chosen. The problem proceeds in n rounds, where each round consists of traveling around the circle from the current position, and selecting the kth remaining element to be eliminated from the circle. After n rounds, every element is eliminated. Special attention is given to the last surviving element, denote it by j. We generalize this popular problem by introducing a uniform number of lives ℓ, so that elements are not eliminated until they have been selected for the ℓth time. We prove two main results: 1) When n and k are fixed, then j is constant for all values of ℓ larger than the nth Fibonacci number. In other words, the last surviving element stabilizes with respect to increasing the number of lives. 2) When n and j are fixed, then there exists a value of k that allows j to be the last survivor simultaneously for all values of ℓ. In other words, certain skip values ensure that a given position is the last survivor, regardless of the number of lives. For the first result we give an algorithm for determining j (and the entire sequence of selections) that uses O( n2) arithmetic operations. “un gatto ha sette vite” [ABSTRACT FROM AUTHOR]

Let { X} be i.i.d. positive random variables and let {λ} be a sequence of nonnegative nonincreasing numbers. We continue to examine conditions under which the log Laplace transform of $ \sum\limits_{j \geqslant 1} {{\lambda_j}} {X_j} $ has a closed form of asymptotics at infinity. The behavior of $ \mathop {{\sup }}\limits_{j \geqslant 1} {\lambda_j}{X_j} $ is also studied. Bibliography: 14 titles. [ABSTRACT FROM AUTHOR]

We investigate linear relations between pattern sequences in a 〈 q, r〉-numeration system, and give a basis of the module generated by pattern sequences for words of length not exceeding l. The expressions of pattern sequences using the basis are also studied. Similar results are obtained for the module generated by all pattern sequences. [ABSTRACT FROM AUTHOR]

We consider the classical Kolmogorov condition for strong law of large numbers for sequences of dependent random variables; the so-called φ-mixing and Rademacher-Menchoff condition for ρ-mixing sequences. [ABSTRACT FROM AUTHOR]

Two methods to test whether a given polynomial has two distinct roots whose quotient is a root of unity are discussed. [ABSTRACT FROM AUTHOR]

n addition sequence problem is given a set of numbers X = { n, n, . . . , n}, what is the minimal number of additions needed to compute all m numbers starting from 1? This problem is NP-complete. In this paper, we present a branch and bound algorithm to generate an addition sequence with a minimal number of elements for a set X by using a new strategy. Then we improve the generation by generalizing some results on addition chains ( m = 1) to addition sequences and finding what we will call a presumed upper bound for each n, 1 ≤ j ≤ m, in the search tree. [ABSTRACT FROM AUTHOR]

Let g ≥ 2 be an integer and let ( u) be a sequence of integers which satisfies a relation u = h( n) u for a rational function h( X). For example, various combinatorial numbers as well as their products satisfy relations of this type. Here, we show that under some mild technical assumptions the number of nonzero digits of u in base g is large on a set of n of asymptotic density 1. We also extend this result to a class of sequences satisfying relations of second order u = h( n) u + h( n) u with two nonconstant rational functions $${h_1(X), h_2(X) \in {\mathbb Q} [X]}$$. This class includes the Apéry, Delannoy, Motzkin, and Schröder numbers. [ABSTRACT FROM AUTHOR]

Let S be a set of r red points and b= r+2 δ blue points in general position in the plane, with δ≥0. A line ℓ determined by them is balanced if in each open half-plane bounded by ℓ the difference between the number of blue points and red points is δ. We show that every set S as above has at least r balanced lines. The proof is a refinement of the ideas and techniques of Pach and Pinchasi (Discrete Comput. Geom. 25:611-628, ), where the result for δ=0 was proven, and introduces a new technique: sliding rotations. [ABSTRACT FROM AUTHOR]

Prime chains are sequences $$p_{1}, \ldots , p_{k}$$ of primes for which $${p_{j+1} \equiv 1}$$ (mod p) for each j. We introduce three new methods for counting long prime chains. The first is used to show that $${N(x; p) = O_{\varepsilon}(x^{1+\varepsilon})}$$, where N( x; p) is the number of chains with p = p and $${p_k \leq p_x}$$. The second method is used to show that the number of prime chains ending at p is ≍ log p for most p. The third method produces the first nontrivial upper bounds on H( p), the length of the longest chain with p = p, valid for almost all p. As a consequence, we also settle a conjecture of Erdős, Granville, Pomerance and Spiro from 1990. A probabilistic model of H( p), based on the theory of branching random walks, is introduced and analyzed. The model suggests that for most $${p \leq x}$$, H( p) stays very close to e log log x. [ABSTRACT FROM AUTHOR]

For an odd integer n ≥ 3, an odd prime p ≡ 3(mod4) and $${d=\frac{p^n+1}{p+1}-\frac{p^n-1}{2}}$$, the value distribution of the exponential sum $${\sum\limits_{x\in \mathbb{F}_{p^n}^{\,*}}\omega^{Tr^n_1(x-\gamma x^{d})}\,(\gamma\in \mathbb{F}_{p^n}^{*})}$$ is completely determined in this paper, where ω is a primitive complex p-th root of unity. This improves the results of Müller (1999) and Hu et al. (2001) about the crosscorrelation of sequences with the decimation factor d. [ABSTRACT FROM AUTHOR]

The Bratteli diagram is an infinite graph which reflects the structure of projections in an AF-algebra. We prove that every strictly ergodic unimodular Bratteli diagram of rank 2 g+ m−1 gives rise to a minimal geodesic lamination with the m principal regions on a hyperbolic surface of genus g≥1. The proof is based on a Morse theory of the recurrent geodesics on the hyperbolic surfaces. [ABSTRACT FROM AUTHOR]

In general category, the group of self-equivalences of sum object can decompose as the product of its two subgroups under certain conditions and also several split short exact sequences are obtained. Subsequently, we apply the above results to the category of space under a space and get some results which are dual to those in fibrewise category. [ABSTRACT FROM AUTHOR]

The q-Catalan numbers studied by Carlitz and Riordan are polynomials in q with nonnegative coefficients. They evaluate, at q = 1, to the Catalan numbers: 1, 1, 2, 5, 14,..., a log-convex sequence. We use a combinatorial interpretation of these polynomials to prove a q-log-convexity result. The sequence of q-Catalan numbers is not q-log-convex in the narrow sense used by other authors, so our work suggests a more flexible definition of q-log convex be adopted. [ABSTRACT FROM AUTHOR]