Smaller superconcentrators of density 28

An N-superconcentrator is a directed, acyclic graph with N input nodes and N output nodes such that every subset of the inputs and every subset of the outputs of same cardinality can be connected by node-disjoint paths. It is known that linear-size and bounded-degree superconcentrators exist. Here it is proved that such superconcentrators exist (by a random construction of certain expander graphs as building blocks) having density 28 (where the density is the number of edges divided by N). The best known density before this paper was 34.2 [U. Schöning, Construction of expanders and superconcentrators using Kolmogorov complexity, J. Random Structures Algorithms 17 (2000) 64-77] or 33 [L.A. Bassalygo, Personal communication, 2004].     

Lower bounds for convex programmes

An important aspect associated with the solution of any mathematical programme; is a sensitivity analysis applied to the various parameters of the programme. For the case of linear programmes, the notion of sensitivity analysis is fully developed. However, for non-linear programmes there is a scarcity of useful results. It is the purpose of this paper to establish lower bounds on the solution of a convex programme when the parameters of the programme are allowed to vary. Such bounds may be useful, for example, in establishing the minimal effects of inflation on optimal cost estimations.     

Distribution-free bounds for relational classification

Statistical relational learning (SRL) is a subarea in machine learning which addresses the problem of performing statistical inference on data that is correlated and not independently and identically distributed (i.i.d.)—as is generally assumed. For the traditional i.i.d. setting, distribution-free bounds exist, such as the Hoeffding bound, which are used to provide confidence bounds on the generalization error of a classification algorithm given its hold-out error on a sample size of N. Bounds of this form are currently not present for the type of interactions that are considered in the data by relational classification algorithms. In this paper, we extend the Hoeffding bounds to the relational setting. In particular, we derive distribution-free bounds for certain classes of data generation models that do not produce i.i.d. data and are based on the type of interactions that are considered by relational classification algorithms that have been developed in SRL. We conduct empirical studies on synthetic and real data which show that these data generation models are indeed realistic and the derived bounds are tight enough for practical use.     

Complexity bounds for zero-test algorithms

In this paper, we analyze the complexity of a zero-test for expressions built from formal power series solutions of first order differential equations with non-degenerate initial conditions. We will prove a doubly exponential complexity bound. This bound establishes a power series analogue for “witness conjectures”.     

Performance bounds for nonhomogeneous queues

This paper considers nonhomogeneous M(t)/M(t)1 queues which can model systems such as communications networks. For such systems, bounds on moment-generating functions and on the tail distribution of the queue process are obtained. These bounds are useful for characterizing the quality of service a system can provide to its users. An approach utilizing the theory of differential equations is adapted. The bounds given in this paper are tighter than those previously available. In fact, the bounds can be made arbitrarily tight given sufficient computational effort     

Effective bounds for P-recursive sequences

We describe an algorithm that takes as input a complex sequence (_un)$\left(u_n\right)$ given by a linear recurrence relation with polynomial coefficients along with initial values, and outputs a simple explicit upper bound (_vn)$\left(v_n\right)$ such that |_un|≤_vn$\left|u_n\right|\le v_n$ for all n$n$. Generically, the bound is tight, in the sense that its asymptotic behaviour matches that of _un$u_n$. We discuss applications to the evaluation of power series with guaranteed precision.     

Lower bounds for context-free grammars

Ellul, Krawetz, Shallit and Wang prove an exponential lower bound on the size of any context-free grammar generating the language of all permutations over some alphabet. We generalize their method and obtain exponential lower bounds for many other languages, among them the set of all squares of given length, and the set of all words containing each symbol at most twice.     

Lower bounds for asynchronous consensus

Impossibility results and best-case lower bounds are proved for the number of message delays and the number of processes required to reach agreement in an asynchronous consensus algorithm that tolerates non-Byzantine failures. General algorithms exist that achieve these lower bounds in the normal case, when the response time of non-faulty processes and the transmission delay of messages they send to one another are bounded. Our theorems allow algorithms to do better in certain exceptional cases, and such algorithms are presented. Two of these exceptional algorithms may be of practical interest.     

Recoding-derived bounds for input encoding

This paper uses theory and experiment to help explain why state assignment algorithms which use two-level-based cost measures often give good multi-level logic implementations. First, we develop theorems that give conditions under which an input encoding that results in multi-cube functions in the minimized Boolean network can be re-encoded to change the multi-cube functions into smaller functions to produce a smaller network. Second, we measure the properties of some typical finite-state machines to determine how well they fit the requirements of the theorems. The good fit between the requirements of the theorems and the properties of typical state machines helps explain why state assignment algorithms designed for two-level-logic implementations are relatively successful in designing state assignments for multi-level logic.     

Robustness bounds for LQ regulators

A robustification procedure for LQ state feedback design is presented. Such a procedure consists of choosing the state and input weighting matrices according to the kind of uncertainties on the system. Both structured and norm-bounded additive uncertainties are addressed, and upper bounds for the uncertainties that do not destabilize the closed-loop system are presented. Connections with the quadratic stabilizability problem are established     

Reliability bounds for the stress-strength model

Stress-strength model has been applied in a wide variety of engineering areas for the purpose of reliability design. However, there is no suitable approach for the model's reliability bound is found yet since the difficulty is arising from many parameters incorporating in the systems. In this paper, a conservative reliability bound is constructed by modeling the problem as a mathematical programming. A numerical example is presented for illustrative the methodology.     

Performance bounds for estimation under uncertainty

For the class of linear Gauss-Markov systems with binary parameters uncertainty, the minimum variance estimate of the state and associated covariance of error expressions were derived in a closed form in [1, 2]. In this paper expressions for the conditional and unconditional covariances of error matrices are presented for the M-ary case. Useful upper and lower bounds for the unconditional covariance of error are also presented which are valid, on the average, for any measurement sequence. A geophysical seismic data filtering example applying these results is also given.     

Combinatorial shell bounds for generalization ability

We consider a problem of supervised statistical learning for a finite population and obtain PAC generalization bounds within the combinatorial PAC framework. Combinatorial counterparts of Langford shell bounds are obtained and are shown to be either the particular case of the Occam razor bound or a variant of Vapnik-Chervonenkis bound and similarly loose in both cases. The reasons for looseness of shell bounds are analyzed, the bounds are compared experimentally.     

Error bounds for Loop subdivision surfaces

In this paper, we obtain the error bounds on the distance between a Loop subdivision surface and its control mesh. Both local and global bounds are derived by means of computing and analysing the control meshes with two rounds of refinement directly. The bounds can be expressed with the maximum edge length of all triangles in the initial control mesh. Our results can be used as posterior estimates and also can be used to predict the subdivision depth for any given tolerance.     

Frequency-restricted norm bounds for interval systems

We investigate bounds of frequency-restricted norms (FRN), a generalization of H_∞-norm, for SISO LTI interval systems, where the numerator and the denominator are interval polynomials. Though the H_∞-norm of interval systems can be determined by 16 Kharitonov-type systems, we can show that this fact is not true in general by a counterexample. A special class of systems for which the FRN is bounded by fixed extremal systems is derived. On the basis of this result, an example of PI-type controller design considering both parametric and unstructured perturbations is illustrated by a parameter space design method.