Classical Bound Research Articles

Optimal memory-aware Sensor Network Gossiping (or how to break the Broadcast lower bound)

Gossiping is a well-studied problem in Radio Networks. However, due to the strong resource limitations of sensor nodes, previous solutions are frequently not feasible in Sensor Networks. In this paper, we study the Gossiping problem in the restrictive context of Sensor Networks. We present a distributed algorithm that completes Gossiping with high probability in a Sensor Network of unknown topology and adversarial start-up. This algorithm exploits the geometry of sensor node distributions to achieve an optimal running time of Θ(D+Δ), where D is the diameter and Δ the maximum degree of the network. Given that any algorithm for Gossiping also solves the Broadcast problem, this result shows that the classical Broadcast lower bound of Kushilevitz and Mansour does not hold if nodes are allowed to do preprocessing. The proposed algorithm requires that a linear number of messages be stored and transmitted per unit time. We also show an optimal distributed algorithm that solves the problem in linear time for the case where only a constant number of messages can be stored.

Instantaneous Relaying: Optimal Strategies and Interference Neutralization

In a multi-user wireless network equipped with multiple relay nodes, some relays are more intelligent than other relay nodes. The intelligent relays are able to gather channel state information, perform linear processing and forward signals whereas the dumb relays is only able to serve as amplifiers. As the dumb relays are oblivious to the source and destination nodes, the wireless network can be modeled as a relay network with *smart instantaneous relay* only: the signals of source-destination arrive at the same time as source-relay-destination. Recently, instantaneous relaying is shown to improve the degrees-of-freedom of the network as compared to classical cut-set bound. In this paper, we study an achievable rate region and its boundary of the instantaneous interference relay channel in the scenario of (a) uninformed non-cooperative source-destination nodes (source and destination nodes are not aware of the existence of the relay and are non-cooperative) and (b) informed and cooperative source-destination nodes. Further, we examine the performance of interference neutralization: a relay strategy which is able to cancel interference signals at each destination node in the air. We observe that interference neutralization, although promise to achieve desired degrees-of-freedom, may not be feasible if relay has limited power. Simulation results show that the optimal relay strategies improve the achievable rate region and provide better user-fairness in both uninformed non-cooperative and informed cooperative scenarios.

Apparent and effective mechanical properties of linear matrix-inclusion random composites: Improved bounds for the effective behavior

This paper is devoted to the derivation of improved bounds for the effective behavior of linear elastic matrix-inclusion composites based on a strategy which is inspired by both the works of Huet (1990) and Danielsson et al. (2007). As shown by the former author, the effective properties of random linear composites can be bounded by ensemble averages of their apparent elastic moduli defined on square (or cubic) volume elements (VEs) and computed with either affine displacement Boundary Conditions (BC) or uniform traction BC. However, in the case of a large contrast of the constituents, the discrepancy between the upper and lower bounds remains significant, even for large values of the VE size. This occurs because the contribution to the total potential (or complementary) energy of the particles (or pores) which intersect the edges of the VE becomes unphysically very large when uniform BC are directly applied to the particles. To avoid such limitations, we considerer non-square (or non-cubic) VEs consisting in simply connex assemblages of cells, each cell being composed of an inclusion surrounded by the matrix, thus forbidding any direct application of BC to the particles. Such VEs are generated by extending the scheme proposed by Danielsson et al. (2007) in the context of periodic random microstructures to fully random microstructures. By applying the classical energy bounding theorems to the non-square VEs, new bounds for the effective behavior are derived. Their application to a two-phase composite composed of an isotropic matrix and aligned identical fibers randomly distributed in the transverse plane leads to sharper bounds which converge quickly with the VE size, even for infinite contrasts.

Eigenvalues and expansion of bipartite graphs

We prove lower bounds on the largest and second largest eigenvalue of the adjacency matrix of connected bipartite graphs and give necessary and sufficient conditions for equality. We give several examples of classes of graphs that are optimal with respect to the bounds. We prove that BIBD-graphs are characterized by their eigenvalues. Finally we present a new bound on the expansion coefficient of (c, d)-regular bipartite graphs and compare that with with a classical bound.

Quantum mechanics can reduce the complexity of classical models

Mathematical models are an essential component of quantitative science. They generate predictions about the future, based on information available in the present. In the spirit of simpler is better; should two models make identical predictions, the one that requires less input is preferred. Yet, for almost all stochastic processes, even the provably optimal classical models waste information. The amount of input information they demand exceeds the amount of predictive information they output. Here we show how to systematically construct quantum models that break this classical bound, and that the system of minimal entropy that simulates such processes must necessarily feature quantum dynamics. This indicates that many observed phenomena could be significantly simpler than classically possible should quantum effects be involved.

Lower bound of effective yield strength domain for random heterogeneous materials

A random heterogeneous material is represented by a finite family of microstructures, the environment of each microstructure being subjected to a perfect mix condition whose validity is justified both theoretically and numerically. Necessary conditions, satisfied by any permissible strain rate field on a representative and unlimited domain of the material, are highlighted. They enable one to obtain, by solving a discrete infmax problem, a lower bound of the effective yield strength domain of the material, that is rigorous and more predictive than the classical bound of Reuss. An analytical application on a porous medium illustrates the methodology.

Numerical and statistical analysis of permeability of concrete as a random heterogeneous composite

This paper investigates the concrete permeability through a numerical and statistical approach. Concrete is considered as a random heterogeneous composite of three phases: aggregates, interfacial transition zones (ITZ) and matrix. The paper begins with some classical bound and estimate theories applied to concrete permeability and the influence of ITZ on these bound and estimate values is discussed. Numerical samples for permeability analysis are established through random aggregate structure (RAS) scheme, each numerical sample containing randomly distributed aggregates coated with ITZ and dispersed in a homogeneous matrix. The volumetric fraction of aggregates is fixed and the size distribution of aggregates observes Fuller's curve. Then finite element method is used to solve the steady permeation problem on 2D numerical samples and the overall permeability is deduced from flux-pressure relation. The impact of ITZ on overall permeability is analyzed in terms of ITZ width and contrast ratio between ITZ and matrix permeabilities. Hereafter, 3680 samples are generated for 23 sample sizes and 4 contrast ratios, and statistical analysis is performed on the permeability dispersion in terms of sample size and ITZ characteristics. By sample theory, the size of representative volume element (RVE) for permeability is then quantified considering sample realization number and expected error. Concluding remarks are provided for the impact of ITZ on concrete permeability and its statistical characteristics.

Maximal exponents of polyhedral cones (I)

Let K be a proper (i.e., closed, pointed, full convex) cone in R n . An n × n matrix A is said to be K-primitive if there exists a positive integer k such that A k ( K ∖ { 0 } ) ⊆ int K ; the least such k is referred to as the exponent of A and is denoted by γ ( A ) . For a polyhedral cone K, the maximum value of γ ( A ) , taken over all K-primitive matrices A, is called the exponent of K and is denoted by γ ( K ) . It is proved that if K is an n-dimensional polyhedral cone with m extreme rays then for any K-primitive matrix A, γ ( A ) ⩽ ( m A − 1 ) ( m − 1 ) + 1 , where m A denotes the degree of the minimal polynomial of A, and the equality holds only if the digraph ( E , P ( A , K ) ) associated with A (as a cone-preserving map) is equal to the unique (up to isomorphism) usual digraph associated with an m × m primitive matrix whose exponent attains Wielandt's classical sharp bound. As a consequence, for any n-dimensional polyhedral cone K with m extreme rays, γ ( K ) ⩽ ( n − 1 ) ( m − 1 ) + 1 . Our work answers in the affirmative a conjecture posed by Steve Kirkland about an upper bound of γ ( K ) for a polyhedral cone K with a given number of extreme rays.

Lower Bounds for $q$-ary Codes with Large Covering Radius

Let $K_q(n,R)$ denote the minimal cardinality of a $q$-ary code of length $n$ and covering radius $R$. Recently the authors gave a new proof of a classical lower bound of Rodemich on $K_q(n,n-2)$ by the use of partition matrices and their transversals. In this paper we show that, in contrast to Rodemich's original proof, the method generalizes to lower-bound $K_q(n,n-k)$ for any $k>2$. The approach is best-understood in terms of a game where a winning strategy for one of the players implies the non-existence of a code. This proves to be by far the most efficient method presently known to lower-bound $K_q(n,R)$ for large $R$ (i.e. small $k$). One instance: the trivial sphere-covering bound $K_{12}(7,3)\geq 729$, the previously best bound $K_{12}(7,3)\geq 732$ and the new bound $K_{12}(7,3)\geq 878$.

A 3D fourth order fabric tensor approach of anisotropy in granular media

We propose a micromechanical approach for granular media, with a particular account of the texture-induced anisotropy and of the strain localization rule. The approach is mainly based on the consideration of a fourth order fabric tensor able to capture general anisotropy which can be induced by complex distribution of contacts. Incorporation of this fourth order fabric tensor in a suitable homogenization scheme allows to determine the corresponding macroscopic elastic properties of the granular material. For this purpose, in addition to the classical Voigt upper bound, a new kinematics-based localization rule is proposed. It generalizes the one formulated by Cambou et al. [B. Cambou, Ph. Dubujet, F. Emeriault, F. Sidoroff, Eur. J. Mech. A/Solids 14 (1995) 225–276] in the case of an isotropic contact distribution. The results of the complete model compare well to numerical simulations results when available [C.S. Chang, C.L. Liao, Appl. Mech. Rev. 47 (1 Part 2) (1994) 197–207] (case of isotropic distribution of contacts). Finally, the interest of the fourth order fabric tensor based approach combined with the proposed localization rule is shown for different distributions of contacts by comparing its predictions to those given by a second order fabric tensor approach.

Extremal problems on triangle areas in two and three dimensions

The study of extremal problems on triangle areas was initiated in a series of papers by Erdős and Purdy in the early 1970s. In this paper we present new results on such problems, concerning the number of triangles of the same area that are spanned by finite point sets in the plane and in 3-space, and the number of distinct areas determined by the triangles. In the plane, our main result is an O ( n 44 / 19 ) = O ( n 2.3158 ) upper bound on the number of unit-area triangles spanned by n points, which is the first breakthrough improving the classical bound of O ( n 7 / 3 ) from 1992. We also make progress in a number of important special cases. We show that: (i) For points in convex position, there exist n-element point sets that span Ω ( n log n ) triangles of unit area. (ii) The number of triangles of minimum (nonzero) area determined by n points is at most 2 3 ( n 2 − n ) ; there exist n-element point sets (for arbitrarily large n) that span ( 6 / π 2 − o ( 1 ) ) n 2 minimum-area triangles. (iii) The number of acute triangles of minimum area determined by n points is O ( n ) ; this is asymptotically tight. (iv) For n points in convex position, the number of triangles of minimum area is O ( n ) ; this is asymptotically tight. (v) If no three points are allowed to be collinear, there are n-element point sets that span Ω ( n log n ) minimum-area triangles (in contrast to (ii), where collinearities are allowed and a quadratic lower bound holds). In 3-space we prove an O ( n 17 / 7 β ( n ) ) = O ( n 2.4286 ) upper bound on the number of unit-area triangles spanned by n points, where β ( n ) is an extremely slowly growing function related to the inverse Ackermann function. The best previous bound, O ( n 8 / 3 ) , is an old result of Erdős and Purdy from 1971. We further show, for point sets in 3-space: (i) The number of minimum nonzero area triangles is at most n 2 + O ( n ) , and this is worst-case optimal, up to a constant factor. (ii) There are n-element point sets that span Ω ( n 4 / 3 ) triangles of maximum area, all incident to a common point. In any n-element point set, the maximum number of maximum-area triangles incident to a common point is O ( n 4 / 3 + ε ) , for any ε > 0 . (iii) Every set of n points, not all on a line, determines at least Ω ( n 2 / 3 / β ( n ) ) triangles of distinct areas, which share a common side.

Bounds and Majorization Relations for the Zeros of Polynomials

We use matrix inequalities to prove several bounds and majorization relations for the zeros of polynomials. Our results generalize the classic bound of Montel and improve some other known bounds.

Perturbative Analysis of the Method of Particular Solutions for Improved Inclusion of High-Lying Dirichlet Eigenvalues

The Dirichlet eigenvalue or “drum” problem in a domain $\Omega\subset\mathbb{R}^2$ becomes numerically challenging at high eigenvalue (frequency) E. In this regime the method of particular solutions (MPS) gives spectral accuracy for many domain shapes. It requires a number of degrees of freedom scaling as $\sqrt{E}$, the number of wavelengths on the boundary, in contrast to direct discretization for which this scaling is E. Our main result is an inclusion bound on eigenvalues that is a factor $O(\sqrt{E})$ tighter than the classical bound of Moler–Payne and that is optimal in that it reflects the true slopes of curves appearing in the MPS. We also present an MPS variant that cures a normalization problem in the original method, while evaluating basis functions only on the boundary. This method is efficient at high frequencies, where we show that, in practice, our inclusion bound can give three extra digits of eigenvalue accuracy with no extra effort.

Lower bounds on covering codes via partition matrices

Let K q ( n , R ) denote the minimal cardinality of a q-ary code of length n and covering radius R. Let σ q ( n , s ; r ) denote the minimal cardinality of a q-ary code of length n, which is s-surjective with radius r. In order to lower-bound K q ( n , n − 2 ) and σ q ( n , s ; s − 2 ) we introduce partition matrices and their transversals. Our approach leads to a short new proof of a classical bound of Rodemich on K q ( n , n − 2 ) and to the new bound K q ( n , n − 2 ) ⩾ 3 q − 2 n + 2 , improving the first iff 5 ⩽ n < q ⩽ 2 n − 4 . We determine K q ( q , q − 2 ) = q − 2 + σ 2 ( q , 2 ; 0 ) if q ⩽ 10 . Moreover, we obtain the new powerful recursive bound K q + 1 ( n + 1 , R + 1 ) ⩾ min { 2 ( q + 1 ) , K q ( n , R ) + 1 } .

Entropy of Operators or why Matrix Multiplication is Hard for Depth-Two Circuits

We consider unbounded fanin depth-2 circuits with arbitrary boolean functions as gates. We define the entropy of an operator f:{0,1}n →{0,1}m as the logarithm of the maximum number of vectors distinguishable by at least one special subfunction of f. Our main result is that every depth-2 circuit for f requires at least entropy(f) wires. This is reminiscent of a classical lower bound of Nechiporuk on the formula size, and gives an information-theoretic explanation of why some operators require many wires. We use this to prove a tight estimate Θ(n 3) of the smallest number of wires in any depth-2 circuit computing the product of two n by n matrices over any finite field. Previously known lower bound for this operator was Ω(n 2log n).

Fast winning strategies in Maker–Breaker games

We consider unbiased Maker–Breaker games played on the edge set of the complete graph K n on n vertices. Quite a few such games were researched in the literature and are known to be Maker's win. Here we are interested in estimating the minimum number of moves needed for Maker in order to win these games. We prove the following results, for sufficiently large n: (1) Maker can construct a Hamilton cycle within at most n + 2 moves. This improves the classical bound of 2 n due to Chvátal and Erdős [V. Chvátal, P. Erdős, Biased positional games, Ann. Discrete Math. 2 (1978) 221–228] and is almost tight; (2) Maker can construct a perfect matching (for even n) within n / 2 + 1 moves, and this is tight; (3) For a fixed k ⩾ 3 , Maker can construct a spanning k-connected graph within ( 1 + o ( 1 ) ) k n / 2 moves, and this is obviously asymptotically tight. Several other related results are derived as well.

Analytic and Asymptotic Analysis of Bayesian CramÉr–Rao Bound for Dynamical Phase Offset Estimation

In this paper, we present a closed-form expression of a Bayesian Cramer-Rao lower bound for the estimation of a dynamical phase offset in a non-data-aided BPSK transmitting context. This kind of bound is derived considering two different scenarios: a first expression is obtained in an offline context, and then a second expression in an online context logically follows. The SNR-asymptotic expressions of this bound drive us to introduce a new asymptotic bound, namely the asymptotic Bayesian Cramer-Rao Bound. This bound is close to the classical Bayesian bound but is easier to evaluate.

Cross-entropy-based method to analyse iterative decoding

Cross-entropy (CE) in information theory is introduced as a method to analyse iterative decoding. The maximum a posteriori probability decoding algorithm is shown to minimise the CE between the a priori and the extrinsic information under given code constraints. The error-correcting ability of the constituent decoders is evaluated in terms of CE. Analysis on turbo decoding is carried out based on theoretical findings from several aspects, including analysis of convergence rate, derivation of the Eb/N0 threshold for convergence, evaluation of error performance in the ‘error-floor’ region and a design example of asymmetric turbo codes. Compared with conventional methods, the new technique provides stricter prediction on the Eb/N0 threshold for convergence and quicker error performance evaluation. An asymmetric turbo code designed with the guidance of our new method exhibits more than 0.1 dB of gain over that guided by classical bounding techniques in both high and low bit error rate regions. Particularly, since no information on the source is required and the density of the a priori/extrinsic information can be arbitrary, the new technique is valuable for both offline design and online evaluation in practical systems.

Higher-order corrections to mass-charge relation of extremal black holes

We investigate the hypothesis that the higher-derivative corrections always make extremal non-supersymmetric black holes lighter than the classical bound and self-repulsive. This hypothesis was recently formulated in the context of the so-called swampland program. One of our examples involves an extremal heterotic black hole in four dimensions. We also calculate the effect of general four-derivative terms in Maxwell-Einstein theories in D dimensions. The results are consistent with the conjecture.

Relay Networks With Delays

The paper investigates the effect of link delays on the capacity of relay networks. The relay-with-delay is defined as a relay channel with relay encoding delay d isin Z of units, or equivalently, a delay of units on the link from the sender to the relay, zero delay on the links from the transmitter to the receiver and from the relay to the receiver, and zero relay encoding delay. Two special cases are studied. The first is the relay-with-unlimited look-ahead, where each relay transmission can depend on its entire received sequence, and the second is the relay-without-delay, where the relay transmission can depend only on current and past received symbols, i.e., d=0. Upper and lower bounds on capacity for these two channels that are tight in some cases are presented. It is shown that the cut-set bound for the classical relay channel, corresponding to the case where d=1, does not hold for the relay-without-delay. Further, it is shown that instantaneous relaying can be optimal and can achieve higher rates than the classical cut-set bound. Capacity for the classes of degraded and semi-deterministic relay-with-unlimited-look-ahead and relay-without-delay are established. These results are then extended to the additive white Gaussian noise (AWGN) relay-with-delay case, where it is shown that for any dles0, capacity is achieved using amplify-and-forward when the channel from the sender to the relay is sufficiently weaker than the other two channels. In addition, it is shown that a superposition of amplify-and-forward and decode-and-forward can achieve higher rates than the classical cut-set bound. The relay-with-delay model is then extended to feedforward relay networks. It is shown that capacity is determined only by the relative delays of paths from the sender to the receiver and not by their absolute delays. A new cut-set upper bound that generalizes both the classical cut-set bound for the classical relay and the upper bound for the relay-without-delay on capacity is established.