LOWER BOUNDS FOR BOXICITY

An axis-parallel b-dimensional box is a Cartesian product R-1 x R-2 x ... x R-b where R-i is a closed interval of the form a(i),b(i)] on the real line. For a graph G, its boxicity box(G) is the minimum dimension b, such that G is representable as the intersection graph of boxes in b-dimensional space. Although boxicity was introduced in 1969 and studied extensively, there are no significant results on lower bounds for boxicity. In this paper, we develop two general methods for deriving lower bounds. Applying these methods we give several results, some of which are listed below: 1. The boxicity of a graph on n vertices with no universal vertices and minimum degree delta is at least n/2(n-delta-1). 2. Consider the g(n,p) model of random graphs. Let p <= 1 - 40logn/n(2.) Then with high `` probability, box(G) = Omega(np(1 - p)). On setting p = 1/2 we immediately infer that almost all graphs have boxicity Omega(n). Another consequence of this result is as follows: For any positive constant c < 1, almost all graphs on n vertices and m <= c((n)(2)) edges have boxicity Omega(m/n). 3. Let G be a connected k-regular graph on n vertices. Let lambda be the second largest eigenvalue in absolute value of the adjacency matrix of G. Then, the boxicity of G is a least (kappa(2)/lambda(2)/log(1+kappa(2)/lambda(2))) (n-kappa-1/2n). 4. For any positive constant c 1, almost all balanced bipartite graphs on 2n vertices and m <= cn(2) edges have boxicity Omega(m/n).

Lower bounds for testing triangle-freeness in Boolean functions

Given a Boolean function , we say a triple (x, y, x + y) is a triangle in f if . A triangle-free function contains no triangle. If f differs from every triangle-free function on at least points, then f is said to be -far from triangle-free. In this work, we analyze the query complexity of testers that, with constant probability, distinguish triangle-free functions from those -far from triangle-free. Let the canonical tester for triangle-freeness denotes the algorithm that repeatedly picks x and y uniformly and independently at random from , queries f(x), f(y) and f(x + y), and checks whether f(x) = f(y) = f(x + y) = 1. Green showed that the canonical tester rejects functions -far from triangle-free with constant probability if its query complexity is a tower of 2's whose height is polynomial in . Fox later improved the height of the tower in Green's upper bound to . A trivial lower bound of on the query complexity is immediate. In this paper, we give the first non-trivial lower bound for the number of queries needed. We show that, for every small enough , there exists an integer such that for all there exists a function depending on all n variables which is -far from being triangle-free and requires queries for the canonical tester. We also show that the query complexity of any general (possibly adaptive) one-sided tester for triangle-freeness is at least square root of the query complexity of the corresponding canonical tester. Consequently, this means that any one-sided tester for triangle-freeness must make at least queries.

Lower bounds on Ricci flow invariant curvatures and geometric applications

We consider Ricci flow invariant cones C in the space of curvature operators lying between the cones ``nonnegative Ricci curvature'' and ``nonnegative curvature operator''. Assuming some mild control on the scalar curvature of the Ricci flow, we show that if a solution to the Ricci flow has its curvature operator which satisfies R + epsilon I is an element of C at the initial time, then it satisfies R + epsilon I is an element of C on some time interval depending only on the scalar curvature control. This allows us to link Gromov-Hausdorff convergence and Ricci flow convergence when the limit is smooth and R + I is an element of C along the sequence of initial conditions. Another application is a stability result for manifolds whose curvature operator is almost in C. Finally, we study the case where C is contained in the cone of operators whose sectional curvature is nonnegative. This allows us to weaken the assumptions of the previously mentioned applications. In particular, we construct a Ricci flow for a class of (not too) singular Alexandrov spaces.

Lower Bounds for Sums of Powers of Low Degree Univariates

We consider the problem of representing a univariate polynomial f(x) as a sum of powers of low degree polynomials. We prove a lower bound of Omega(root d/t) for writing an explicit univariate degree-d polynomial f(x) as a sum of powers of degree-t polynomials.

Bounds on Probability of Error in Decision Feedback Equalizers

Evaluation of the probability of error in decision feedback equalizers is difficult due to the presence of a hard limiter in the feedback path. This paper derives the upper and lower bounds on the probability of a single error and multiple error patterns. The bounds are fairly tight. The bounds can also be used to select proper tap gains of the equalizer.

Model independent bounds on the modulus of the pion form factor on the unitarity cut below the omega pi threshold

We calculate upper and lower bounds on the modulus of the pion electromagnetic form factor on the unitarity cut below the omega pi inelastic threshold, using as input the phase in the elastic region known via the Fermi-Watson theorem from the pi pi P-wave phase shift, and a suitably weighted integral of the modulus squared above the inelastic threshold. The normalization at t = 0, the pion charge radius and experimental values at spacelike momenta are used as additional input information. The bounds are model independent, in the sense that they do not rely on specific parametrizations and do not require assumptions on the phase of the form factor above the inelastic threshold. The results provide nontrivial consistency checks on the recent experimental data on the modulus available below the omega pi threshold from e(+)e(-) annihilation and tau-decay experiments. In particular, at low energies the calculated bounds offer a more precise description of the modulus than the experimental data.

Cramer-Rao-Type Bounds for Sparse Bayesian Learning

In this paper, we derive Hybrid, Bayesian and Marginalized Cramer-Rao lower bounds (HCRB, BCRB and MCRB) for the single and multiple measurement vector Sparse Bayesian Learning (SBL) problem of estimating compressible vectors and their prior distribution parameters. We assume the unknown vector to be drawn from a compressible Student-prior distribution. We derive CRBs that encompass the deterministic or random nature of the unknown parameters of the prior distribution and the regression noise variance. We extend the MCRB to the case where the compressible vector is distributed according to a general compressible prior distribution, of which the generalized Pareto distribution is a special case. We use the derived bounds to uncover the relationship between the compressibility and Mean Square Error (MSE) in the estimates. Further, we illustrate the tightness and utility of the bounds through simulations, by comparing them with the MSE performance of two popular SBL-based estimators. We find that the MCRB is generally the tightest among the bounds derived and that the MSE performance of the Expectation-Maximization (EM) algorithm coincides with the MCRB for the compressible vector. We also illustrate the dependence of the MSE performance of SBL based estimators on the compressibility of the vector for several values of the number of observations and at different signal powers.

Asymptotic Bounds on the Power-Delay Tradeoff for Fading Point-to-Point Links From Geometric Bounds on the Stationary Distribution of the Queue Length

The optimal power-delay tradeoff is studied for a time-slotted independently and identically distributed fading point-to-point link, with perfect channel state information at both transmitter and receiver, and with random packet arrivals to the transmitter queue. It is assumed that the transmitter can control the number of packets served by controlling the transmit power in the slot. The optimal tradeoff between average power and average delay is analyzed for stationary and monotone transmitter policies. For such policies, an asymptotic lower bound on the minimum average delay of the packets is obtained, when average transmitter power approaches the minimum average power required for transmitter queue stability. The asymptotic lower bound on the minimum average delay is obtained from geometric upper bounds on the stationary distribution of the queue length. This approach, which uses geometric upper bounds, also leads to an intuitive explanation of the asymptotic behavior of average delay. The asymptotic lower bounds, along with previously known asymptotic upper bounds, are used to identify three new cases where the order of the asymptotic behavior differs from that obtained from a previously considered approximate model, in which the transmit power is a strictly convex function of real valued service batch size for every fade state.

An Algorithm to Decide Feasibility of Linear Integer Constraints Occurrng in Decision Tables

Abstract-To detect errors in decision tables one needs to decide whether a given set of constraints is feasible or not. This paper describes an algorithm to do so when the constraints are linear in variables that take only integer values. Decision tables with such constraints occur frequently in business data processing and in nonnumeric applications. The aim of the algorithm is to exploit. the abundance of very simple constraints that occur in typical decision table contexts. Essentially, the algorithm is a backtrack procedure where the the solution space is pruned by using the set of simple constrains. After some simplications, the simple constraints are captured in an acyclic directed graph with weighted edges. Further, only those partial vectors are considered from extension which can be extended to assignments that will at least satisfy the simple constraints. This is how pruning of the solution space is achieved. For every partial assignment considered, the graph representation of the simple constraints provides a lower bound for each variable which is not yet assigned a value. These lower bounds play a vital role in the algorithm and they are obtained in an efficient manner by updating older lower bounds. Our present algorithm also incorporates an idea by which it can be checked whether or not an (m - 2)-ary vector can be extended to a solution vector of m components, thereby backtracking is reduced by one component.

Phased-Mission Analysis for Evaluating the Effectiveness of Aerospace Computing-Systems

Effectiveness evaluation of aerospace fault-tolerant computing systems used in a phased-mission environment is rather tricky and difficult because of the interaction of its several degraded performance levels with the multiple objectives of the mission and the use environment. Part I uses an approach based on multiobjective phased-mission analysis to evaluate the effectiveness of a distributed avionics architecture used in a transport aircraft. Part II views the computing system as a multistate s-coherent structure. Lower bounds on the probabilities of accomplishing various levels of performance are evaluated.

On the growth of the Bergman kernel near an infinite-type point

We study diagonal estimates for the Bergman kernels of certain model domains in C-2 near boundary points that are of infinite type. To do so, we need a mild structural condition on the defining functions of interest that facilitates optimal upper and lower bounds. This is a mild condition; unlike earlier studies of this sort, we are able to make estimates for non-convex pseudoconvex domains as well. Thisn condition quantifies, in some sense, how flat a domain is at an infinite-type boundary point. In this scheme of quantification, the model domains considered below range-roughly speaking-from being mildly infinite-type'' to very flat at the infinite-type points.

DMT of Multi-hop Cooperative Networks-Part II: Layered and Multi-Antenna Networks

We consider single-source, single-sink (ss-ss) multi-hop relay networks, with slow-fading Rayleigh links. This two part paper aims at giving explicit protocols and codes to achieve the optimal diversity-multiplexing tradeoff (DMT) of two classes of multi-hop networks: K-parallel-path (KPP) networks and Layered networks. While single-antenna KPP networks were the focus of the first part, we consider layered and multi-antenna networks in this second part. We prove that a linear DMT between the maximum diversity d(max). and the maximum multiplexing gain of 1 is achievable for single-antenna fully-connected layered networks under the half-duplex constraint. This is shown to be equal to the optimal DMT if the number of relaying layers is less than 4. For the multiple-antenna case, we provide an achievable DMT, which is significantly better than known lower bounds for half duplex networks. Along the way, we compute the DMT of parallel MIMO channels in terms of the DMT of the component channel. For arbitrary ss-ss single-antenna directed acyclic networks with full-duplex relays, we prove that a linear tradeoff between maximum diversity and maximum multiplexing gain is achievable using an amplify-and-forward (AF) protocol. Explicit short-block-length codes are provided for all the proposed protocols. Two key implications of the results in the two-part paper are that the half-duplex constraint does not necessarily entail rate loss by a factor of two as previously believed and that simple AN protocols are often sufficient to attain the best possible DMT.

DMT of Multi-hop Cooperative Networks-Part I: K-Parallel-Path Networks

We consider single-source, single-sink multi-hop relay networks, with slow-fading Rayleigh fading links and single-antenna relay nodes operating under the half-duplex constraint. While two hop relay networks have been studied in great detail in terms of the diversity-multiplexing tradeoff (DMT), few results are available for more general networks. In this two-part paper, we identify two families of networks that are multi-hop generalizations of the two hop network: K-Parallel-Path (KPP) networks and Layered networks. In the first part, we initially consider KPP networks, which can be viewed as the union of K node-disjoint parallel paths, each of length > 1. The results are then generalized to KPP(I) networks, which permit interference between paths and to KPP(D) networks, which possess a direct link from source to sink. We characterize the optimal DMT of KPP(D) networks with K >= 4, and KPP(I) networks with K >= 3. Along the way, we derive lower bounds for the DMT of triangular channel matrices, which are useful in DMT computation of various protocols. As a special case, the DMT of two-hop relay network without direct link is obtained. Two key implications of the results in the two-part paper are that the half-duplex constraint does not necessarily entail rate loss by a factor of two, as previously believed and that, simple AF protocols are often sufficient to attain the best possible DMT.

Four generations: SUSY and SUSY breaking

We revisit four generations within the context of supersymmetry. Wecompute the perturbativity limits for the fourth generation Yukawa couplings and show that if the masses of the fourth generation lie within reasonable limits of their present experimental lower bounds, it is possible to have perturbativity only up to scales around 1000 TeV. Such low scales are ideally suited to incorporate gauge mediated supersymmetry breaking, where the mediation scale can be as low as 10-20 TeV. The minimal messenger model, however, is highly constrained. While lack of electroweak symmetry breaking rules out a large part of the parameter space, a small region exists, where the fourth generation stau is tachyonic. General gauge mediation with its broader set of boundary conditions is better suited to accommodate the fourth generation.