Notion Of Discrepancy Research Articles

Beyond the Erdős discrepancy problem in function fields

We characterize the limiting behavior of partial sums of multiplicative functions f:Fq[t]→S1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:\\mathbb {F}_q[t]\\rightarrow S^1$$\\end{document}. In contrast to the number field setting, the characterization depends crucially on whether the notion of discrepancy is defined using long intervals, short intervals, or lexicographic intervals. Concerning the notion of short interval discrepancy, we show that a completely multiplicative f:Fq[t]→{-1,+1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:\\mathbb {F}_q[t]\\rightarrow \\{-1,+1\\}$$\\end{document} with q odd has bounded short interval sums if and only if f coincides with a “modified" Dirichlet character to a prime power modulus. This confirms the function field version of a conjecture over Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}$$\\end{document} that such modified characters are extremal with respect to partial sums. Regarding the lexicographic discrepancy, we prove that the discrepancy of a completely multiplicative sequence is always infinite if we define it using a natural lexicographic ordering of Fq[t]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_{q}[t]$$\\end{document}. This answers a question of Liu and Wooley. Concerning the long sum discrepancy, it was observed by the Polymath 5 collaboration that the Erdős discrepancy problem admits infinitely many completely multiplicative counterexamples on Fq[t]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_q[t]$$\\end{document}. Nevertheless, we are able to classify the counterexamples if we restrict to the class of modified Dirichlet characters. In this setting, we determine the precise growth rate of the discrepancy, which is still unknown for the analogous problem over the integers.

Powers of Hamilton Cycles of High Discrepancy are Unavoidable

The Pósa-Seymour conjecture asserts that every graph on n vertices with minimum degree at least (1−1/(r +1))n contains the r-th power of a Hamilton cycle. Komlós, Sárközy and Szemerédi famously proved the conjecture for large n. The notion of discrepancy appears in many areas of mathematics, including graph theory. In this setting, a graph G is given along with a 2-coloring of its edges. One is then asked to find in G a copy of a given subgraph with a large discrepancy, i.e., with significantly more than half of its edges in one color. For r > 2, we determine the minimum degree threshold needed to find the r-th power of a Hamilton cycle of large discrepancy, answering a question posed by Balogh, Csaba, Pluhár and Treglown. Notably, for r > 3, this threshold approximately matches the minimum degree requirement of the Pósa-Seymour conjecture.

Parameters of Codes for the Binary Asymmetric Channel

We introduce two notions of discrepancy between binary vectors, which are not metric functions in general but nonetheless capture the mathematical structure of the binary asymmetric channel. These lead to two new fundamental parameters of binary error-correcting codes, both of which measure the probability that the maximum likelihood decoder fails. We then derive various bounds for the cardinality and weight distribution of a binary code in terms of these new parameters, giving examples of codes meeting the bounds with equality.

A discrepancy version of the Hajnal–Szemerédi theorem

AbstractA perfect Kr-tiling in a graph G is a collection of vertex-disjoint copies of the clique Kr in G covering every vertex of G. The famous Hajnal–Szemerédi theorem determines the minimum degree threshold for forcing a perfect Kr-tiling in a graph G. The notion of discrepancy appears in many branches of mathematics. In the graph setting, one assigns the edges of a graph G labels from {‒1, 1}, and one seeks substructures F of G that have ‘high’ discrepancy (i.e. the sum of the labels of the edges in F is far from 0). In this paper we determine the minimum degree threshold for a graph to contain a perfect Kr-tiling of high discrepancy.

On the Discrepancies of Graphs

In the literature, the notion of discrepancy is used in several contexts, even in the theory of graphs. Here, for a graph $G$ with each edge labelled $-1$ or $1$, we consider a family $\mathcal{S}_G$ of subgraphs of a certain type, such as spanning trees or Hamiltonian cycles. As usual, we seek for bounds on the sum of the labels that hold for all elements of $\mathcal{S}_G$, for every labeling.

Extremal distributions of discrepancy functions

The irregularities of a distribution of N points in the unit interval are often measured with various notions of discrepancy. The discrepancy function can be defined with respect to intervals of the form [0,t)⊂[0,1) or arbitrary subintervals of the unit interval. In the former case, it is a well known fact in discrepancy theory that the N-element point set in the unit interval with the lowest L2 or L∞ norm of the discrepancy function is the centered regular grid ΓN≔2n+12N:n=0,1,…,N−1. We show a stronger result on the distribution of discrepancy functions of point sets in [0,1], which basically says that the distribution of the discrepancy function of ΓN is in some sense minimal among all N-element point sets. As a consequence, we can extend the above result to rearrangement-invariant norms, including Lp, Orlicz and Lorentz norms.We study the same problem for the discrepancy notions with respect to arbitrary subintervals. In this case, we will observe that we have to deal with integrals of convolutions of functions. To this end, we prove a general upper bound on such expressions, which might be of independent interest as well.

The Stolarsky Principle and Energy Optimization on the Sphere

The classical Stolarsky invariance principle connects the spherical cap $$L^2$$ discrepancy of a finite point set on the sphere to the pairwise sum of Euclidean distances between the points. In this paper, we further explore and extend this phenomenon. In addition to a new elementary proof of this fact, we establish several new analogs, which relate various notions of discrepancy to different discrete energies. In particular, we find that the hemisphere discrepancy is related to the sum of geodesic distances. We also extend these results to arbitrary measures on the sphere and arbitrary notions of discrepancy and apply them to problems of energy optimization and combinatorial geometry and find that, surprisingly, the geodesic distance energy behaves differently than its Euclidean counterpart.

Log terminal singularities

ABSTRACTThe aim of this work is to simplify the definitions related to the study of singularities of normal varieties initiated in [3] and [8]. We introduce a notion of discrepancy for normal varieties, and we define log terminal+ singularities. We use finite generation to relate these new singularities with log terminal singularities (in the sense of [3]).

Information-Theoretic Lower Bounds on the Oracle Complexity of Stochastic Convex Optimization

Relative to the large literature on upper bounds on complexity of convex optimization, lesser attention has been paid to the fundamental hardn4516420ess of these problems. Given the extensive use of convex optimization in machine learning and statistics, gaining an understanding of these complexity-theoretic issues is important. In this paper, we study the complexity of stochastic convex optimization in an oracle model of computation. We introduce a new notion of discrepancy between functions, and use it to reduce problems of stochastic convex optimization to statistical parameter estimation, which can be lower bounded using information-theoretic methods. Using this approach, we improve upon known results and obtain tight minimax complexity estimates for various function classes.

Quasi‐random hypergraphs revisited

AbstractThe quasi‐random theory for graphs mainly focuses on a large equivalent class of graph properties each of which can be used as a certificate for randomness. For k ‐graphs (i.e., k ‐uniform hypergraphs), an analogous quasi‐random class contains various equivalent graph properties including the k ‐discrepancy property (bounding the number of edges in the generalized induced subgraph determined by any given (k ‐ 1) ‐graph on the same vertex set) as well as the k ‐deviation property (bounding the occurrences of “octahedron”, a generalization of 4 ‐cycle). In a 1990 paper (Chung, Random Struct Algorithms 1 (1990) 363‐382), a weaker notion of l ‐discrepancy properties for k ‐graphs was introduced for forming a nested chain of quasi‐random classes, but the proof for showing the equivalence of l ‐discrepancy and l ‐deviation, for 2 ≤ l < k, contains an error. An additional parameter is needed in the definition of discrepancy, because of the rich and complex structure in hypergraphs. In this note, we introduce the notion of (l,s) ‐discrepancy for k ‐graphs and prove that the equivalence of the (k,s) ‐discrepancy and the s ‐deviation for 1 ≤ s ≤ k. We remark that this refined notion of discrepancy seems to point to a lattice structure in relating various quasi‐random classes for hypergraphs. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011

Coverage-guided test generation for continuous and hybrid systems

In this paper, we describe a formal framework for conformance testing of continuous and hybrid systems, using the international standard `Formal Methods in Conformance Testing' FMCT. We propose a novel test coverage measure for these systems, which is defined using the star discrepancy notion. This coverage measure is used to quantify the validation `completeness'. It is also used to guide input stimulus generation by identifying the portions of the system behaviors that are not adequately examined. We then propose a test generation method, which is based on a robotic motion planning algorithm and is guided by the coverage measure. This method was implemented in a prototype tool that can handle high dimensional systems (up to 100 dimensions).

On the density and discrepancy of a 2D point set with applications to thermal analysis of VLSI chips

In this era of giga-scale integration, thermal analysis has become one of the hot topics in VLSI chip design. Active thermal sources may be abstracted as a set of weighted points on a 2D chip-floor. The conventional notion of discrepancy that deals with the congestion properties of a set of scattered points may not be able to capture properly all real-life instances in this context. In this paper, we have introduced a new concept, called the density of a region to study some of the properties of the distribution of these weighted points. We prove several counter-intuitive results concerning the properties of the regions that have maximum or minimum density. We then outline algorithms for recognizing these regions. We also compare the attributes of density with the existing concept of discrepancy.

Tractability properties of the weighted star discrepancy

Tractability properties of various notions of discrepancy have been intensively studied in the last decade. In this paper we consider the so-called weighted star discrepancy which was introduced by Sloan and Woźniakowski. We show that under a very mild condition on the weights one can obtain tractability with s -exponent zero ( s is the dimension of the point set). In the case of product weights we give a condition such that the weighted star discrepancy is even strongly tractable. Furthermore, we give a lower bound for the weighted star discrepancy for a large class of weights. This bound shows that for such weights one cannot obtain strong tractability.

Scheduling streaming flows on the downlink shared channel of UMTS

AbstractIn Universal Mobile Telecommunication Systems (UMTS), the Downlink Shared Channel (DSCH) may be used to provide streaming services. The traffic model for streaming services is different from the continuously backlogged model used in much of the literature. Each connection specifies a required service rate over an interval of time. In this paper, we are interested in determining how k DSCH frames should be allocated among a set of I connections. We need a scheduler that is channel‐aware, so that channels presently enjoying low fading losses can be exploited to achieve higher aggregate throughput. On the other hand, the scheduler is also required to be fair, so that each connection obtains a throughput as close as possible to what it requires. We introduce the notion of discrepancy to capture the inherent trade‐off between aggregate throughput and fairness. We show that the discrepancy criterion provides a flexible means for balancing efficiency, as measured by aggregate throughput, and fairness. Our problem, then, is to schedule mobiles so as to minimize the discrepancy over the control horizon. We provide a simple low‐complexity heuristic called ITEM that is provably optimal in certain cases. In particular, we show that ITEM is optimal when applied in the UMTS context. Next, we compare the performance of ITEM with that of other algorithms, and show that it performs better in terms of both fairness and aggregate throughput. Thus, ITEM provides benefits in both dimensions—fairness and efficiency—and is therefore a promising algorithm for scheduling streaming connections. Copyright © 2007 John Wiley & Sons, Ltd.

Discrepancy estimates for point sets on the s-dimensional Sierpińnski carpet

In a recent paper Cristea and Tichy introduced several types of discrepancies of point sets on the s-dimensional Sierpí nski carpet and proved various relations between these discrepancies. In the present paper we prove a general lower bound for those discrepancies in terms of N, the cardinality of the point set, and we give a probabilistic proof for the existence of point sets with "small" discrepancy. Furthermore we consider a van der Corput type construction of point sets on C s and determine the exact order of convergence of various notions of discrepancy. Finally, Carpet-Walsh functions are defined to prove an Erdős-Turán-Koksma inequality which we apply to digital point sets on the carpet.

Bounds for the weighted Lp discrepancy and tractability of integration

Quite recently Sloan and Woźniakowski (J. Complexity 14 (1998) 1) introduced a new notion of discrepancy, the so-called weighted L p discrepancy of points in the d-dimensional unit cube for a sequence γ=( γ 1, γ 2,…) of weights. In this paper we prove a nice formula for the weighted L p discrepancy for even p. We use this formula to derive an upper bound for the average weighted L p discrepancy. This bound enables us to give conditions on the sequence of weights γ such that there exists N points in [0,1) d for which the weighted L p discrepancy is uniformly bounded in d and goes to zero polynomially in N −1. Finally we use these facts to generalize some results from Sloan and Woźniakowski (1998) on (strong) QMC-tractability of integration in weighted Sobolev spaces.

45.2: Moiré‐Free Collimating Light Guide with Low‐Discrepancy Dot Patterns

AbstractUsing the notion of discrepancy, we developed a new technique for generating uniformly distributed dot patterns. We designed and prototyped a light guide having prismatic grooves and micro scatterers on the surface. We experimentally confirmed that the new pattern of the micro scatterers effectively prevents moiré patterns, and improves the luminance homogeneity.

Le jargon, figure du multiple

RéSUMé This article considers slang and occupational jargon starting from the notion of enunciative and rhetorical discrepancy, in an attempt to differentiate these two discursive practices, while showing their resemblance as well. This notion of discrepancy stresses the heterogeneity of language, the variety of its registers and creative methods. It also shows the different ways in which the subject puts his mark on the language, because language is the result of speech.

Organizational Vision: The Other Side of the Coin

Executive Summary Vision is the essence of leadership that brings about change in the life of organizations. present paper argues that three critical components, the qualities of an organization's leader, the attributes of the followers and the nature of the environment must be taken into consideration by the leader in articulating an organizational vision. paper offers classification of organizational visions and discusses some likely consequences of disregarding them. Some examples are drawn from educational settings. Implications and conclusions about the gains and losses that leaders with different operational orientations may face when articulating are also discussed. ********** There is no more powerful engine driving an organization toward excellence and long-range success than an attractive worthwhile, and achievable of the future, widely shared. (Nanus 1992) Vision is one of the key elements of effective leadership in organizational settings. As indicated by Bennis and Nanus (1985), lack of clear is major cause of the declining effectiveness of many organizations in recent years. This assertion is shared by many scholars (e.g., Peters & Waterman, 1982; Terry, 1993) who have argued for the importance of organizational and, as such, for the need to update visions periodically because of on-going changes that are taking place in an ever-evolving environment. Organizations should follow the direction set forth in the and, because the serves as compass to the organization's members, it must not only be defined in general terms and be inspirational, but must also accurately reflect the internal qualities of the organization and its interactions with the environment. Because of its great importance, has been studied by scholars from both educational settings (e.g., Bell & Harrison, 1995; Holmes, 1993; Sergiovanni, 1990, 1996) and non-educational ones (e.g., Nanus, 1992; Terry, 1993; Yukl, 1994). Sergiovanni (1990) argues, that vision gets the most attention in the leadership literature (p. 57) because it has been credited for the success of organizational leadership on the one hand, and blamed for organizational leadership failure, on the other. Since is essential for organizational effectiveness and is used as major leadership tool, its articulation requires sufficient degree of clarity. Conceptualizing vision: as A has been conceived variously as just dream, and at other times ... as concrete as written mission statement (Yukl, 1994, p. 363), a thing of imagination (Bell & Harrisonn 1995, p. 2), or as mental image (Holmes, 1993, p. 16). These notions portray somewhat unrealistic and intangible view of the concept. Practically speaking, vision, in organizational settings, is concrete idea that describes what needs to be achieved by the organization's members and how this should be done (Creemers, 1997). Therefore, that mainly reflects vague dreams or thing(s) of imagination may be detached from reality. Conger, Kanungo and Associates (1989) discuss the notion of discrepancy between the goal (the vision) that the leader aspires for the organization to achieve and the status (the circumstances): The more idealized or utopian the future goal advocated by the leader, the more discrepant it becomes in relation to the status quo (p. 85). From their point of view, the qualities of are determined by the magnitude of the discrepancy between the future goal and the present circumstances: the greater the discrepancy, the greater the probability that the is exceptional and not ordinary. Trethowan (1991) elaborates on the analogy of as dream. He argues that only that is an informed dream can be beneficial to an organization. Such takes into consideration what both the members of the organization and its customers think and understand of it and the situation in which it is to be realized. …

Roth’s estimate of the discrepancy of integer sequences is nearly sharp

Letg be a coloring of the set {1, ...,N} = [1,N] in red and blue. For each arithmetic progressionA in [1,N], consider the absolute value of the difference of the numbers of red and of blue members ofA. LetR(g) be the maximum of this number over all arithmetic progression (thediscrepancy ofg). Set $$R(N) = \mathop {\min }\limits_g R(g)$$ over all two-coloringsg. A remarkable result of K. F. Roth gives*R(N)≫N 1/4. On the other hand, Roth observed thatR(N)≪N 1/3+ɛ and suggested that this bound was nearly sharp. A. Sarkozy disproved this by provingR(N)≪N 1/3+ɛ. We prove thatR(N)=N 1/4+o(1) thus showing that Roth’s original lower bound was essentially best possible. Our result is more general. We introduce the notion ofdiscrepancy of hypergraphs and derive an upper bound from which the above result follows.