Sequence X1 Research Articles

Some inequalities concerning cross-intersecting families of integer sequences

Let [n]={1,2,…,n} and let [n]k be the family of integer sequences (x1,x2,…,xk) with 1≤xi≤n, i=1,2,…,k. Two families F,G⊂[n]k are called cross-intersecting if F and G agree in at least one position for every F∈F and G∈G. A family F⊂[n]k is called non-trivial if for every i∈[k] there exist (x1,x2,…,xk), (y1,y2,…,yk)∈F such that xi≠yi. In the present paper, we show that if F,G⊂[n]k are non-empty cross-intersecting and n≥2, then|F|+|G|≤1+nk−(n−1)k. If F,G⊂[n]k are both non-trivial, cross-intersecting and n≥2, then|F|+|G|≤nk−2(n−1)k+(n−2)k+2. We also establish a similar inequality for non-empty cross-intersecting families of generalized integer sequences.

First-Order Convex Fitting and Its Application to Economics and Optimization

This paper studies a function fitting problem which we coin first-order convex fitting (FCF): given any two vector sequences x1, ..., xT and p1, ..., pT, when is it possible to efficiently construct a convex function f(x) that ``fits'' the two sequences in the first-order sense, i.e, the (sub)gradient of f(xi) equals precisely pi, for all i = 1, ..., T? Despite a basic question of convex analysis, FCF has surprisingly been overlooked in the past literature. With an efficient constructive proof, we provide a clean answer to this question: FCF is possible if and only if the two sequences are permutation stable: p1 * x1 + ... + pT * xT is greater than or equal to p1 * x’1 + ... + pT * x’T where x’1, ..., x’T is any permutation of x1, ..., xT. We demonstrate the usefulness of FCF in two applications. First, we study how it can be used as an empirical risk minimization procedure to learn the original convex function. We provide efficient PAC-learnability bounds for special classes of convex functions learned via FCF, and demonstrate its application to multiple economic problems where only function gradients (as opposed to function values) can be observed. Second, we empirically show how it can be used as a surrogate to significantly accelerate the minimization of the original convex function.

On the asymptotic distribution of randomly weighted averages of random vectors

Let X1,X2,…,Xn be a sequence of mutually independent continuous random vectors with respective positive definite covariance matrices Σ1,Σ2,…,Σn. The main purpose of this article is to derive the asymptotic distribution of Sn:n, a randomly weighted average of the sequence X1,X2,…,Xn, as n→∞. The random weights are the cuts of (0, 1) by an increasing ordered array of the ordered statistics of n independent and identically uniformly distributed random variables. We prove that under certain assumptions on the covariance matrices, n(Sn:n−μ) converges in distribution to the multivariate normal distribution with zero mean and covariance matrix 2Σ, where Σ is the limit of the expression in terms of Σis. Finally, we give an application for the multivariate randomly weighted averages in modeling (height, general intelligence) parents’ gene compositions given to their offspring. The simulation results which are based on three distributions (Bivariate t, Dirichlet and Normal) which confirm the main theorem of the paper.

Freely adjoining monoidal duals

AbstractGiven a monoidal category $\mathcal C$ with an object J, we construct a monoidal category $\mathcal C[{J^ \vee }]$ by freely adjoining a right dual ${J^ \vee }$ to J. We show that the canonical strong monoidal functor $\Omega :\mathcal C \to \mathcal C[{J^ \vee }]$ provides the unit for a biadjunction with the forgetful 2-functor from the 2-category of monoidal categories with a distinguished dual pair to the 2-category of monoidal categories with a distinguished object. We show that $\Omega :\mathcal C \to \mathcal C[{J^ \vee }]$ is fully faithful and provide coend formulas for homs of the form $\mathcal C[{J^ \vee }](U,\,\Omega A)$ and $\mathcal C[{J^ \vee }](\Omega A,U)$ for $A \in \mathcal C$ and $U \in \mathcal C[{J^ \vee }]$. If ${\rm{N}}$ denotes the free strict monoidal category on a single generating object 1, then ${\rm{N[}}{{\rm{1}}^ \vee }{\rm{]}}$ is the free monoidal category Dpr containing a dual pair – ˧ + of objects. As we have the monoidal pseudopushout $\mathcal C[{J^ \vee }] \simeq {\rm{Dpr}}{{\rm{ + }}_{\rm{N}}}\mathcal C$, it is of interest to have an explicit model of Dpr: we provide both geometric and combinatorial models. We show that the (algebraist’s) simplicial category Δ is a monoidal full subcategory of Dpr and explain the relationship with the free 2-category Adj containing an adjunction. We describe a generalization of Dpr which includes, for example, a combinatorial model Dseq for the free monoidal category containing a duality sequence X0 ˧ X1 ˧ X2 ˧ … of objects. Actually, Dpr is a monoidal full subcategory of Dseq.

Deciding the point-to-fixed-point problem for skew tent maps on an interval

We consider a family of skew tent maps fa on the unit interval, determined by the parameter a, with 0<a<1. We give a decision procedure, that on input a and a point x0 in the unit interval, determines whether or not the sequence x0,fa(x0),fa2(x0),... of iterates of fa on x0 reaches one of the two fixed points of fa after a finite number of iterations.

Global Behavior in Functional Iteration Problems

Functional iteration, the process of forming a sequence x0, x1 = f(x0), x2 = f(x1) = f(f(x0)) = f2(x0), . . ., by repeated application of a function f, is fundamental to the approximation of solutions to nonlinear equations and plays an important role in the study of nonlinear dynamics. In this note we examine another aspect of functional iteration: the global behavior of the sequence f, f2, f3, . . ., fn, . . . of composite functions as n → ∞. This topic can be presented to students at many levels using computer graphics to stimulate conjectures that can then be investigated theoretically.

Multipopulation Spin Models: A View from Large Deviations Theoretic Window

This paper studies large deviations properties of vectors of empirical means and measures generated as follows. Consider a sequence X1,X2,…,Xn of independent and identically distributed random variables partitioned into d-subgroups with sizes n1,…,nd. Further, consider a d-dimensional vector mn whose coordinates are made up of the empirical means of the subgroups. We prove the following. (1) The sequence of vector of empirical means mn satisfies large deviations principle with rate n and rate function I, when the sequence X1,X2,…,Xn is Rl valued, with l≥1. (2) Similar large deviations results hold for the corresponding sequence of vector of empirical measures Ln if Xi’s, i=1,2,…,n, take on finitely many values. (3) The rate functions for the above large deviations principles are convex combinations of the corresponding rate functions arising from the large deviations principles of the coordinates of mn and Ln. The probability distributions used in the convex combinations are given by α=(α1,…,αd)=limn→∞1/n(n1,…,nd). These results are consequently used to derive variational formula for the thermodynamic limit for the pressure of multipopulation Curie-Weiss (I. Gallo and P. Contucci (2008), and I. Gallo (2009)) and mean-field Pott’s models, via a version of Varadhan’s integral lemma for an equicontinuous family of functions. These multipopulation models serve as a paradigm for decision-making context where social interaction and other socioeconomic attributes of individuals play a crucial role.

Cramér-type moderate deviations for intermediate trimmed means

ABSTRACTIn this article, we establish Cramér-type moderate deviation results for (intermediate) trimmed means Tn = n− 1∑n − mni = kn + 1Xi: n, where Xi: n's are the order statistics corresponding to the first n observations in a sequence X1, X2, … of independent identically distributed random variables with F. We consider two cases of intermediate and heavy trimming. In the former case, when max (αn, βn) → 0 (αn = kn/n, βn = mn/n) and min (kn, mn) → ∞ as n → ∞, we obtain our results under a natural moment assumption and a mild condition on the rate at which αn and βn tend to zero. In the latter case, we do not impose any moment conditions on F, instead, we require some smoothness of F− 1 in an open set containing the limit points of the trimming sequences αn, 1 − βn.

Asymptotic behaviour in the robot rendezvous problem

This paper presents a natural extension of the results obtained by Feintuch and Francis in (2012a,b) concerning the so-called robot rendezvous problem. In particular, we revisit a known necessary and sufficient condition for convergence of the solution in terms of Cesàro convergence of the translates Skx0, k≥0, of the sequence x0 of initial positions under the right-shift operator S, thus shedding new light on questions left open in Feintuch and Francis (2012a,b). We then present a new proof showing that a certain stronger ergodic condition on x0 ensures that the corresponding solution converges to its limit at the optimal rate O(t−1/2) as t→∞. After considering a natural two-sided variant of the robot rendezvous problem already studied in Feintuch and Francis (2012a) and in particular proving a new quantified result in this case, we conclude by relating the robot rendezvous problem to a more realistic model of vehicle platoons.

An Improved Bound for the Star Discrepancy of Sequences in the Unit Interval

Abstract It is known that there is a constant c &gt; 0 such that for every sequence x 1, x 2, . . . in [0, 1) we have for the star discrepancy D N * $D_N^* $ of the first N elements of the sequence that ND N * ≥ c ⋅ log ⁡ N $ND_N^* \ge c \cdot \log N$ holds for infinitely many N. Let c ∗ be the supremum of all such c with this property. We show c ∗ &gt; 0.065664679 . . . , thereby slightly improving the estimates known until now.

Estimates of accuracy of the Poisson approximation for the distribution of number of runs of long string repetitions in a Markov chain

Abstract Let X 0, X 1, … be a simple ergodic Markov chain with a finite set of states and ξ̃ n, k (s) be a number of runs of k-fold repetitions of strings having length s. Estimates of accuracy of the Poisson approximation for the distribution of ξ n, k (s) in the sequence X 0, X 1, …, X n+s−1 are obtained, these estimates are uniform over k.

Parking Functions, Shi Arrangements, and Mixed Graphs

The Shi arrangement is the set of all hyperplanes in ℝn of the form xj − xk = 0 or 1 for 1 ≤ j < k ≤ n. Shi observed in 1986 that the number of regions (i.e., connected components of the complement) of this arrangement is (n + 1)n−1. An unrelated combinatorial concept is that of a parking function, i.e., a sequence (x1, x2, …, xn) of positive integers that, when rearranged from smallest to largest, satisfies xk ≤ k. (There is an illustrative reason for the term parking function.) It turns out that the number of parking functions of length n also equals (n + 1)n−1, a result given by Konheim and Weiss in 1966. A natural problem consists of finding a bijection between the n-dimensional Shi arrangement and the parking functions of length n. Pak and Stanley (1996) and Athanasiadis and Linusson (1999) gave such (quite different) bijections. We will shed new light on the former bijection by taking a scenic route through certain mixed graphs.

Fibonacci goes magic

1 The phenomenon Let p be a prime number. We write Zp for the set {0, 1, . . . , p−1} of residues modulo p, and we consider the usual addition and multiplication modulo p onZp . It will be important in the sequel that Zp , provided with these operations, is a field, i.e., one calculates as in the usual number system. (E.g., for x = 0, the number 1/x is defined as that element y ∈ Zp for which x · y = 1. In the case p = 7, for example, one has 1/5 = 3. And −x means that y such that x + y = 0. E.g., −1 = p − 1 for x ∈ Zp .) Now let numbers a, b ∈ Zp be given. They generate a sequence x0, x1, . . . in Zp by x0 := a, x1 := b, xn := xn−1 + xn−2 mod p for p ≥ 2. Note that this is the usual Fibonacci sequence modulo p in the case (a, b) = (0, 1).

On the star discrepancy of sequences in the unit interval

It is known that there is a constant c>0 such that for every sequence x1,x2,… in [0,1) we have for the star discrepancy DN∗ of the first N elements of the sequence that NDN∗≥c⋅logN holds for infinitely many N. Let c∗ be the supremum of all such c with this property. We show c∗>0.0646363, thereby improving the until now known estimates.

CHARACTERIZATIONS OF OPTIMAL POLICIES IN A GENERAL STOPPING PROBLEM AND STABILITY ESTIMATING

We consider an optimal stopping problem for a general discrete-time process X1, X2, …, Xn, … on a common measurable space. Stopping at time n (n = 1, 2, …) yields a reward Rn(X1, …, Xn) ≥ 0, while if we do not stop, we pay cn(X1, …, Xn) ≥ 0 and keep observing the process. The problem is to characterize all the optimal stopping times τ, i.e., such that maximize the mean net gain: $$E(R_\tau(X_1,\dots,X_\tau)-\sum_{n=1}^{\tau-1}c_n(X_1,\dots,X_n)).$$ We propose a new simple approach to stopping problems which allows to obtain not only sufficient, but also necessary conditions of optimality in some natural classes of (randomized) stopping rules.In the particular case of Markov sequence X1, X2, … we estimate the stability of the optimal stopping problem under perturbations of transition probabilities.

Diophantine approximation generalized

In this paper we study the set of x ∈ [0, 1] for which the inequality |x − x n | < z n holds for infinitely many n = 1, 2, .... Here x n ∈ [0, 1) and z n s> 0, z n → 0, are sequences. In the first part of the paper we summarize known results. In the second part, using the theory of distribution functions of sequences, we find the asymptotic density of n for which |x − x n | < z n , where x is a discontinuity point of some distribution function of x n . Generally, we also prove, for an arbitrary sequence x n , that there exists z n such that the density of n = 1, 2, ..., x n → x, is the same as the density of n = 1, 2, ..., |x − x n | < z n , for x ∈ [0, 1]. Finally we prove, using the longest gap d n in the finite sequence x 1, x 2, ..., x n , that if d n ≤ z n for all n, z n → 0, and z n is non-increasing, then |x − x n | < z n holds for infinitely many n and for almost all x ∈ [0, 1].

Generalizations of some zero sum theorems

Given an abelian group G of order n, and a finite non-empty subset A of integers, the Davenport constant of G with weightA, denoted by DA(G), is defined to be the least positive integer t such that, for every sequence (x1,..., xt) with xi ∈ G, there exists a non-empty subsequence \((x_{j_1},\ldots, x_{j_l})\) and ai ∈ A such that \(\sum_{i=1}^{l}a_ix_{j_i} = 0\). Similarly, for an abelian group G of order n, EA(G) is defined to be the least positive integer t such that every sequence over G of length t contains a subsequence \((x_{j_1} ,\ldots, x_{j_n})\) such that \(\sum_{i=1}^{n}a_ix_{j_i} = 0\), for some ai ∈ A. When G is of order n, one considers A to be a non-empty subset of {1,..., n − 1 }. If G is the cyclic group \({\Bbb Z}/n{\Bbb Z}\), we denote EA(G) and DA(G) by EA(n) and DA(n) respectively.

Topological complexity of the telescope

We use an alternative definition of topological complexity to show that the topological complexity of the mapping telescope of a sequence X1→f1X2→f2X3→f3⋯ is bounded above by 2max{TC(Xi);i=1,2,…}.

On the Relation between Realizable and Nonrealizable Cases of the Sequence Prediction Problem

A sequence x1,...,xn,... of discrete-valued observations is generated according to some unknown probabilistic law (measure) μ. After observing each outcome, one is required to give conditional prob...

Hardy-type spaces on certain noncompact manifolds and applications

In this paper we consider a complete connected noncompact Riemannian manifold M with Ricci curvature bounded from below, positive injectivity radius and spectral gap b. We introduce a sequence X1(M), X2(M), … of new Hardy spaces on M, the sequence Y1(M), Y2(M), … of their dual spaces, and show that these spaces may be used to obtain endpoint estimates for purely imaginary powers of the Laplace–Beltrami operator and for more general spectral multipliers associated to the Laplace–Beltrami operator ℒ on M. Under the additional condition that the volume of the geodesic balls of radius r is controlled by C r α e 2 b r for some nonnegative real number α and for all large r, we prove also an endpoint result for the first-order Riesz transform ∇ ℒ−1/2. In this case, the kernels of the operators ℒiu and ∇ ℒ−1/2 are singular both on the diagonal and at infinity. In particular, these results apply to Riemannian symmetric spaces of the noncompact type.