Exponential Bound Research Articles

Novel Range Wise Optimization of the Exponential Bounds on the Gaussian Q Function and its Applications in Communications Theory

Novel Range Wise Optimization of the Exponential Bounds on the Gaussian Q Function and its Applications in Communications Theory

Extremely Accurate, Generic and Tractable Exponential Bounds for the Gaussian Q Function With Applications in Communication Systems

In this letter, we apply the Fejer’s Rule of interpolatory integration in a modified manner to obtain a very generic, tractable and extremely accurate approximation for the Gaussian <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> function. We further show the utility of the proposed approximation by simplifying the error performance metrics like average symbol error probability (ASEP) of popular digital modulation techniques over a very useful κ-μ shadowed fading channel. This eliminates the need of mathematical softwares which require very high processing power (a constraint at the receiver side) yielding simplified and cost effective receiver’s design. In addition, an insight on the asymptotic ASEP (and diversity order thereof) is also highlighted in this letter.

Collapsing the Tower - On the Complexity of Multistage Stochastic IPs

In this paper we study the computational complexity of solving a class of block structured integer programs (IPs) - so called multistage stochastic IPs. A multistage stochastic IP is an IP of the form min { c ⊺ x ∣ Ax = b , x ≥ 0 , x integral} where the constraint matrix A consists of small block matrices ordered on the diagonal line and for each stage there are larger blocks with few columns connecting the blocks in a tree like fashion. Over the last years there was enormous progress in the area of block structured IPs. For many of the known block IP classes - such as n -fold, tree-fold, and two-stage stochastic IPs, nearly matching upper and lower bounds are known concerning their computational complexity. One of the major gaps that remained however was the parameter dependency in the running time for an algorithm solving multistage stochastic IPs. Previous algorithms require a tower of t exponentials, where t is the number of stages. In contrast, only a double exponential lower bound was known based on the exponential time hypothesis. In this paper we show that the tower of t exponentials is actually not necessary. We show an improved running time of \(2^{(d\left\Vert A \right\Vert _\infty)^{\mathcal {O}(d^{3t+1})}} \cdot rn\log ^{\mathcal {O}(2^d)}(rn) \) for the algorithm solving multistage stochastic IPs, where d is the sum of columns in the connecting blocks and rn is the number of rows. Hence, we obtain the first bound by an elementary function for the running time of an algorithm solving multistage stochastic IPs. In contrast to previous works, our algorithm has only a triple exponential dependency on the parameters and only doubly exponential for every constant t . By this we come very close to the known double exponential bound that holds already for two-stage stochastic IPs, i.e. multistage stochastic IPs with two stages. The improved running time of the algorithm is based on new bounds for the proximity of multistage stochastic IPs. The idea behind the bound is based on generalization of a structural lemma originally used for two-stage stochastic IPs. While the structural lemma requires iteration to be applied to multistage stochastic IPs, our generalization directly applies to inherent combinatorial properties of multiple stages. Already a special case of our lemma yields an improved bound for the Graver complexity of multistage stochastic IPs.

Rational solutions to the variants of Erdős–Selfridge superelliptic curves

For the superelliptic curves of the form [Formula: see text] with [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] a prime, Das, Laishram, Saradha and Edis showed that the superelliptic curve has no rational points for [Formula: see text]. In fact, the double exponential bound, obtained in these papers is far from the reality. In this paper, we study the superelliptic curves for small values of [Formula: see text]. In particular, we explicitly solve the above equation for [Formula: see text]

A nonlinear observer for bilinear systems in block form

This paper presents the design of a high-gain nonlinear observer for bilinear systems in the block form. We prove the uniform exponential stability of the observer error by finding a concrete exponential bound. In particular, we extend the result of O. I. Goncharov on the exponential stability of high-gain bilinear observers into observers of block forms. Our work results in a generalization of that work. We also study and simulate two examples of observers for bioreactor systems in two and three dimensions. Finally, the results show the effectiveness of the proposed approach.

Exponential bound of the integral of Hermite functions product with Gaussian weight

In this paper we derive a bound on the integral of a product of two Hermite-Gaussian functions with a Gaussian weight. We prove that such integrals decay exponentially in the difference of the indices of the Hermite-Gaussian functions. Such integrals arise naturally in mathematical physics and applied mathematics. The estimate is applied to a variational problem related to a Strichartz functional.

On public-coin zero-error randomized communication complexity

Improving the exponential bound of [5], we show that the largest possible gap between the deterministic communication complexity and the public-coin zero-error randomized communication complexity is at most polynomial. Previously, such a bound was known only in the private-coin model. The proof combines the approach of Gavinsky and Lovett [4] with new ideas.

Optimization of the Exponential Bounds on the Gaussian Q Function Using Interior Point Algorithm and Its Application in Communication Theory

Optimization of the Exponential Bounds on the Gaussian <i>Q</i> Function Using Interior Point Algorithm and Its Application in Communication Theory

New Integral Representations for the Fox–Wright Functions and Its Applications II

In this paper our aim is to establish new integral representations for the Fox–Wright function $${}_{p}\Psi_{q}[^{(\alpha_{p},A_{p})}_{(\beta_{q},B_{q})}|z]$$ when $$\mu=\sum_{j=1}^{q}\beta_{j}-\sum_{k=1}^{p}\alpha_{k}+\frac{p-q}{2}=-m,\;\;m\in\mathbb{N}_{0}.$$ In particular, closed-form integral expressions are derived for the four parameter Wright function under a special restriction on parameters. Exponential bounding inequalities are derived for a class of the Fox–Wright function. Moreover, complete monotonicity property is presented for these functions.

Determinizing monitors for HML with recursion

We examine the determinization of monitors for HML with recursion. We demonstrate that every monitor is equivalent to a deterministic one, which is at most doubly exponential in size with respect to the original monitor. When monitors are described as CCS-like processes, this doubly exponential bound is optimal. When (deterministic) monitors are described as finite automata (or as their labeled transition systems), then they can be exponentially more succinct than their CCS process form.

On a Problem of Danzer

LetCbe a bounded convex object in ℝd, and letPbe a set ofnpoints lying outsideC. Further, letcp,cqbe two integers with 1 ⩽cq⩽cp⩽n- ⌊d/2⌋, such that everycp+ ⌊d/2⌋ points ofPcontain a subset of sizecq+ ⌊d/2⌋ whose convex hull is disjoint fromC. Then our main theorem states the existence of a partition ofPinto a small number of subsets, each of whose convex hulls are disjoint fromC. Our proof is constructive and implies that such a partition can be computed in polynomial time.In particular, our general theorem implies polynomial bounds for Hadwiger--Debrunner (p,q) numbers for balls in ℝd. For example, it follows from our theorem that whenp&gt;q= (1+β)⋅d/2 for β &gt; 0, then any set of balls satisfying the (p,q)-property can be hit byO((1+β)2d2p1+1/βlogp) points. This is the first improvement over a nearly 60 year-old exponential bound of roughlyO(2d).Our results also complement the results obtained in a recent work of Keller, Smorodinsky and Tardos where, apart from improvements to the bound on HD(p,q) for convex sets in ℝdfor various ranges ofpandq, a polynomial bound is obtained for regions with low union complexity in the plane.

Minimizing Lundberg inequality for ruin probability under correlated risk model by investment and reinsurance

This paper investigates optimal investment and reinsurance policies for an insurance company under a correlated risk model with common Poisson shocks. The goal of the insurance company is to minimize the ultimate ruin probability. By the dynamic programming principle, the Hamilton–Jacobi–Bellman (HJB for short) equation associated with this control problem is obtained. Since there is no explicit solution to the HJB equation, this paper alternates to find the minimal exponential upper bound of the ruin probability. The exponential upper bound of ruin probability is also called Lundberg inequality. Minimizing Lundberg inequality is equal to finding the maximal Lundberg coefficient. It turns out that the optimal investment and reinsurance polices are constant policies. Some numerical examples are given to illustrate the impact of the dependent structure and the investment chance on the upper bound.

Parameterized Games and Parameterized Automata

We introduce a way to parameterize automata and games on finite graphs with natural numbers. The parameters are accessed essentially by allowing counting down from the parameter value to 0 and branching depending on whether 0 has been reached. The main technical result is that in games, a player can win for some values of the parameters at all, if she can win for some values below an exponential bound. For many winning conditions, this implies decidability of any statements about a player being able to win with arbitrary quantification over the parameter values. While the result seems broadly applicable, a specific motivation comes from the study of chains of strategies in games. Chains of games were recently suggested as a means to define a rationality notion based on dominance that works well with quantitative games by Bassett, Jecker, P., Raskin and Van den Boogard. From the main result of this paper, we obtain generalizations of their decidability results with much simpler proofs. As both a core technical notion in the proof of the main result, and as a notion of potential independent interest, we look at boolean functions defined via graph game forms. Graph game forms have properties akin to monotone circuits, albeit are more concise. We raise some open questions regarding how concise they are exactly, which have a flavour similar to circuit complexity. Answers to these questions could improve the bounds in the main theorem.

Convergence to Stationary Measures in Nonlinear Fokker–Planck–Kolmogorov Equations

The convergence of solutions of nonlinear Fokker–Planck–Kolmogorov equations to stationary solutions is studied. Broad sufficient conditions for convergence in variation with an exponential bound are obtained.

Qualification Conditions in Semialgebraic Programming

For an arbitrary finite family of semialgebraic/definable functions, we consider the corresponding inequality constraint set and we study qualification conditions for perturbations of this set. In particular we prove that all positive diagonal perturbations, save perhaps a finite number of them, ensure that any point within the feasible set satisfies the Mangasarian--Fromovitz constraint qualification. Using the Milnor--Thom theorem, we provide a bound for the number of singular perturbations when the constraints are polynomial functions. Examples show that the order of magnitude of our exponential bound is relevant. Our perturbation approach provides a simple protocol to build sequences of “regular” problems approximating an arbitrary semialgebraic/definable problem. Applications to sequential quadratic programming methods and sum of squares relaxation are provided.

Sharp logarithmic estimates for positive dyadic shifts

The paper contains the study of sharp logarithmic estimates for positive dyadic shifts A given on probability spaces (X,μ) equipped with a tree-like structure. For any K>0 we determine the smallest constant L=L(K) such that∫E|Af|dμ≤K∫RΨ(|f|)dμ+L(K)⋅μ(E), where Ψ(t)=(t+1)log⁡(t+1)−t, E is an arbitrary measurable subset of X and f is an integrable function on X. The proof exploits Bellman function method: we extract the above estimate from the existence of an appropriate special function, enjoying certain size and concavity-type conditions. As a corollary, a dual exponential bound is obtained.

Derivatives of Parsing Expression Grammars

This paper introduces a new derivative parsing algorithm for recognition of parsing expression grammars. Derivative parsing is shown to have a polynomial worst-case time bound, an improvement on the exponential bound of the recursive descent algorithm. This work also introduces asymptotic analysis based on inputs with a constant bound on both grammar nesting depth and number of backtracking choices; derivative and recursive descent parsing are shown to run in linear time and constant space on this useful class of inputs, with both the theoretical bounds and the reasonability of the input class validated empirically. This common-case constant memory usage of derivative parsing is an improvement on the linear space required by the packrat algorithm.

Cohomology of SL2 and related structures

ABSTRACTLet SL2 be an algebraic group defined over an algebraically closed field k of characteristic p > 0. In this paper, we provide a closed formula for for Weyl SL2-modules V(m) when n ≤ 2p − 3. For n > 2p − 3, an exponential bound, only depending on n, is obtained for . Analogous results are also established for the extension spaces between Weyl modules V(m1) and V(m2). As a by-product, our results and techniques give explicit upper bounds for the dimensions of cohomology of the Specht modules of symmetric groups, and the cohomology of simple modules of SL2 and the finite group of Lie type .

Exponential stability of perturbed superstable systems in Hilbert spaces

We study exponential stability of superstable systems in Hilbert spaces under perturbations. Formulas to calculate or to estimate the exponential growth bound of the perturbed systems are derived via which sufficient conditions on exponential stability are established. The obtained results are applied to a partial differential equation governing the vibration of a smart beam made of self-straining material. Several numerical simulations are given.

On Regularity Lemmas and their Algorithmic Applications

Szemerédi's regularity lemma and its variants are some of the most powerful tools in combinatorics. In this paper, we establish several results around the regularity lemma. First, we prove that whether or not we include the condition that the desired vertex partition in the regularity lemma is equitable has a minimal effect on the number of parts of the partition. Second, we use an algorithmic version of the (weak) Frieze–Kannan regularity lemma to give a substantially faster deterministic approximation algorithm for counting subgraphs in a graph. Previously, only an exponential dependence for the running time on the error parameter was known, and we improve it to a polynomial dependence. Third, we revisit the problem of finding an algorithmic regularity lemma, giving approximation algorithms for several co-NP-complete problems. We show how to use the weak Frieze–Kannan regularity lemma to approximate the regularity of a pair of vertex subsets. We also show how to quickly find, for each ε′&gt;ε, an ε′-regular partition withkparts if there exists an ε-regular partition withkparts. Finally, we give a simple proof of the permutation regularity lemma which improves the tower-type bound on the number of parts in the previous proofs to a single exponential bound.