Invariance principle on the slice

This thesis focuses on the interplay of extremal combinatorics and finite geometry. Combinatorics is concerned with discrete (and usually finite) objects. Extremal combinatorics studies how large or how small a collection of finite objects can be under certain restrictions. Those objects can be sets, graphs, vectors, etc. These questions are often motivated by problems in information theory and computer science. Another branch of combinatorics is finite geometry over finite fields of order q. Although there is no field of order 1, certain substructures in finite geometry can be interpreted as versions of projective spaces for q = 1. A triangle in a projective plane is a good example of this phenomenon. Following an introductory chapter, Chapter 2 gives an overview of some classical problems and their q-analogues in extremal combinatorics. The most recent results for the q-analogues of Sperner's theorem, the Erdös-Ko-Rado theorem and several versions of Bollobás's theorem are described. For the q-analogue of the Hilton-Milner theorem we give some new results. We also give a new bound for the minimum size of the q-analogue of small maximal cliques. We conclude that sometimes results for a q-analogue can be obtained by using the same technique as in the original problem. In some cases the answer to the problem is even identical to that of the original problem. In other cases techniques used for the q-analogue could be used for improving bounds for the original problem. The third chapter describes the known results for the chromatic number of the Kneser graphs and gives new bounds for the chromatic number of the q-analogue. The vertices of a Kneser graph are subsets (of a fixed size) of a set, whereas two vertices are adjacent if they are disjoint in their subset representation. In 1955 M. Kneser conjectured the chromatic number of those graphs. In 1978 this was proven correct by L. Lóvasz. Two small cases in the projective space q-analogue were solved in 2001 by J. Eisfeld, L. Storme and P. Sziklai and in 2006 by A. Chowdhury, C. Godsil and G. Royle. We describe an asymptotic result for all cases (except for one parameter family, where we give a partial proof) using the bounds from the q-analogue of the Hilton-Milner theorem. Chapters 4, 5 and 6 describe other q-analogues of the Kneser graphs. In Chapter 4 we define a family of Kneser type graphs over pairs of incident points and hyperplanes of a projective space. We describe large maximal cocliques in these graphs and prove that for small dimensions these are the largest possible cocliques. We conjecture that this is the case for all dimensions. In Chapter 5 we extend the Kneser graphs to the case of finite polar spaces instead of finite projective spaces and in some cases give descriptions and chromatic numbers of these graphs. Chapter 6 describes the generalization of Kneser graphs over coset spaces of Chevalley groups (parameterized by q) with respect to parabolic subgroups. This encompasses all previous cases and extends to some interesting new cases. First we describe these graphs when q = 1 and give chromatic numbers for some families. Then we consider the graphs defined over coset spaces of Chevalley groups for general q with respect to parabolic subgroups and study what results of the q = 1 case can be translated to the general case. In this thesis we found different possibilities for the connection between the bounds for the q = 1 case and the case for general q. In some cases the bounds for the general case are identical to the bounds for q = 1. In other cases we obtain the bounds for q = 1 by taking the limit for q¿1 in the general bound. Other cases show a completely different bound in both cases.

Many of the currently best-known approximation algorithms for NP-hard optimization problems are based on Linear Programming (LP) and Semi-definite Programming (SDP) relaxations. Given its power, this class of algorithms seems to contain the most favourable candidates for outperforming the current state-of-the-art approximation guarantees for NP-hard problems, for which there still exists a gap between the inapproximability results and the approximation guarantees that we know how to achieve in polynomial time. In this thesis, we address both the power and the limitations of these relaxations, as well as the connection between the shortcomings of these relaxations and the inapproximability of the underlying problem. In the first part, we study the limitations of LP relaxations of well-known graph problems such as the Vertex Cover problem and the Independent Set problem. We prove that any small LP relaxation for the aforementioned problems, cannot have an integrality gap strictly better than $2$ and $\omega(1)$, respectively. Furthermore, our lower bound for the Independent Set problem also holds for any SDP relaxation. Prior to our work, it was only known that such LP relaxations cannot have an integrality gap better than $1.5$ for the Vertex Cover Problem, and better than $2$ for the Independent Set problem. In the second part, we study the so-called knapsack cover inequalities that are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield LP relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. We address this issue and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities. In the last part, we show a close connection between structural hardness for k-partite graphs and tight inapproximability results for scheduling problems with precedence constraints. This connection is inspired by a family of integrality gap instances of a certain LP relaxation. Assuming the hardness of an optimization problem on k-partite graphs, we obtain a hardness of $2-\varepsilon$ for the problem of minimizing the makespan for scheduling with preemption on identical parallel machines, and a super constant inapproximability for the problem of scheduling on related parallel machines. Prior to this result, it was only known that the first problem does not admit a PTAS, and the second problem is NP-hard to approximate within a factor strictly better than 2, assuming the Unique Games Conjecture.

We prove the endpoint case of a conjecture of Khot and Moshkovitz related to the unique games conjecture, less a small error. Let n ≥ 2. Suppose a subset Ω of n‐dimensional Euclidean space satisfies −Ω = Ωc and Ω + v = Ωc (up to measure zero sets) for every standard basis vector . For any and for any q ≥ 1, let and let . For any x ∈ ∂Ω, let N(x) denote the exterior normal vector at x such that ‖N(x)‖2 = 1. Let . Our main result shows that B has the smallest Gaussian surface area among all such subsets Ω, less a small error: In particular, Standard arguments extend these results to a corresponding weak inequality for noise stability. Removing the factor 6 × 10−9 would prove the endpoint case of the Khot‐Moshkovitz conjecture. Lastly, we prove a Euclidean analogue of the Khot and Moshkovitz conjecture. The full conjecture of Khot and Moshkovitz provides strong evidence for the truth of the unique games conjecture, a central conjecture in theoretical computer science that is closely related to the P versus NP problem. So, our results also provide evidence for the truth of the unique games conjecture. Nevertheless, this paper does not prove any case of the unique games conjecture.

Many engineering applications face the problem of bounding the expected value of a quantity of interest (performance, risk, cost, etc.) that depends on stochastic uncertainties whose probability distribution is not known exactly. Optimal uncertainty quantification (OUQ) is a framework that aims at obtaining the best bound in these situations by explicitly incorporating available information about the distribution. Unfortunately, this often leads to non-convex optimization problems that are numerically expensive to solve. This thesis emphasizes on efficient numerical algorithms for OUQ problems. It begins by investigating several classes of OUQ problems that can be reformulated as convex optimization problems. Conditions on the objective function and information constraints under which a convex formulation exists are presented. Since the size of the optimization problem can become quite large, solutions for scaling up are also discussed. Finally, the capability of analyzing a practical system through such convex formulations is demonstrated by a numerical example of energy storage placement in power grids. When an equivalent convex formulation is unavailable, it is possible to find a convex problem that provides a meaningful bound for the original problem, also known as a convex relaxation. As an example, the thesis investigates the setting used in Hoeffding's inequality. The naive formulation requires solving a collection of non-convex polynomial optimization problems whose number grows doubly exponentially. After structures such as symmetry are exploited, it is shown that both the number and the size of the polynomial optimization problems can be reduced significantly. Each polynomial optimization problem is then bounded by its convex relaxation using sums-of-squares. These bounds are found to be tight in all the numerical examples tested in the thesis and are significantly better than Hoeffding's bounds.

Co-channel interference is a major factor in limiting the capacity and link quality in cellular communications. As the co-channel interference is modeled by lognormal distribution, sum of the co-channel interferences of neighboring cells is represented by the sum of lognormal Random Variables (RVs) which has no closed-form expression. Assuming independent, identically distributed (iid) RVs, the sum of lognormal RVs has been approximated by another log-normally distributed RV in the literature. In practice, the co-channel interference RVs have identical standard deviations (SDs) and different means. In this paper, first a new method based on curve fitting is proposed to approximate the sum of two log-normally distributed RV's with identical SDs and different means. Then a recursive method using the surface fitting is developed for approximating the sum of more than two lognormal RVs. Results show that the proposed method can approximate the first and the second moments of the resulting RV very well.

As a primary branch of Operations Research, Queueing Theory models and analyzes engineering systems with random fluctuations. With the development of internet and computation techniques, the engineering systems today are much bigger in scale and more complicated in structure than 20 years ago, which raises numerous new problems to researchers in the field of queueing theory. The aim of this thesis is to explore new methods and tools, from both algorithmic and analytical perspectives, that are useful to solve such problems. In Chapter 1 and 2, we introduce some techniques of asymptotic analysis that are relatively new to queueing applications in order to give more accurate probabilistic characterization of queueing models with large scale and complicated structure. In particular, Chapter 1 gives the first functional large deviation result for infinite-server system with general inter-arrival and service times. The functional approach we use enables a nice description of the whole system over the entire time horizon of interest, which is important in real problems. In Chapter 2, we construct a queueing model for the so-called limit order book that is used in main financial markets worldwide. We use an asymptotic approach called multi-scale modeling to disentangle the complicated dependence among the elements in the trading system and to reduce the model dimensionality. The asymptotic regime we use is inspired by empirical observations and the resulting limit process explains and reproduces stylized features of real market data. Chapter 2 also provides a nice example of novel applications of queueing models in systems, such as the electronic trading system, that are traditionally outside the scope of queueing theory. Chapter 3 and 4 focus on stochastic simulation methods for performance evaluation of queueing models where analytic approaches fail. In Chapter 3, we develop a perfect sampling algorithm to generate exact samples from the stationary distribution of stochastic fluid networks in polynomial time. Our approach can be used for time-varying networks with general inter-arrival and service times, whose stationary distributions have no analytic expression. In Chapter 4, we focus on the stochastic systems with continuous random fluctuations, for instance, the workload arrives to the system in continuous flow like a Levy process. We develop a general framework of simulation algorithms featuring a deterministic error bound and an almost square root convergence rate. As an application, we apply this framework to estimate the stationary distributions of reflected Brownian motions and the performance of our algorithm is better than existing prevalent numeric methods.

This thesis presents two original ontributions on orthogonal frequency division multiplexing (OFDM). The first is a thorough investigation of the error performance and channel capacity of asymmetrically clipped optical OFDM (ACO-OFDM). ACO-OFDM is a modulation scheme designed specifically for optical communications. The second concerns the use of multiple-input multiple-output (MIMO) OFDM in radio frequency (RF) systems. A new scheme which uses polynomial cancellation coding (PCC) OFDM with time offsets is proposed. This new scheme is robust to both carrier frequency offset (CFO) and channel estimation errors. OFDM has been used extensively to reduce the effect of dispersion in wired and wireless RF systems. Although dispersion is also a problem in optical wireless systems, OFDM has not until recently been considered for use in these. This is because optical wireless systems use intensity modulation (IM) and so the transmitted signals must be nonnegative and as a result conventional OFDM signals which are bipolar cannot be used directly. ACO-OFDM and DC biased optical OFDM (DCO-OFDM) are two non-negative forms of OFDM developed for optical systems. Both are robust to ISI but ACO-OFDM is in most cases much more power efficient. In this thesis results are presented for the performance of ACO-OFDM, DCO-OFDM and conventional optical modulation schemes such as on-off keying (OOK) and pulse position modulation (PPM) for both dispersive and non-dispersive channels. These show that ACO-OFDM is more tolerant to dispersion and requires less optical power than the other modulation schemes. To investigate the theoretical limits of an ACO-OFDM system, closed form analytical expressions for the channel capacity of ACO-OFDM are derived both for a flat channel and for representative dispersive optical wireless channels. The capacity of ACO-OFDM cannot be compared with the general case of an intensity modulated signal, because no general solution for this has yet been found. Instead the mutual information of an ACO-OFDM signal is calculated and compared with two other signal distributions: an exponential distribution and a clipped-Gaussian distribution. The combination of OFDM and MIMO has been used in several radio standards. However, MIMO OFDM systems like single-input single-output (SISO) OFDM systems are very sensitive to CFO. An improved MIMO PCC OFDM system is proposed in this thesis. In this system, the signals transmitted from different antennae have different time offsets. At the receiver, OFDM symbols are demodulated by using each of these time offsets in a matched detection window. This system shows better performance than a MIMO PCC OFDM system without time offsets in two cases: when there are channel estimation errors and in MIMO systems with more transmit antennae than receive antennae.

Sparse polynomial chaos expansions have recently emerged in uncertainty quantification analysis as a tool to solve high dimensional problems, e.g. stochastic problems involving a few dozens to a few hundred random variables. Based on penalized regression analysis and the so-called least angle regression algorithm the method has proven efficiency in a number of applications. The approach was so far rather limited to scalar output quantities, e.g. quantities of interest that are post-processed from the solution of a stochastic partial differential equation (SPDE). In this paper we extend this approach to vector output quantities in order to obtain the complete solution field. This is carried out by using principal component analysis before computing the PC expansions of the various components. As a whole a complete non intrusive framework is obtained that is only based on a set of deterministic solutions of the underlying deterministic problem. The approach is illustrated by the computation of the displacement field of a tension rod with lognormal, spatially variable Young’s modulus. The problem exhibits 62 stochastic dimensions.

The following results are obtained for the metric of a sign sensitive weight: necessary and sufficient conditions imposed on the weight in order to ensure the fulfillment of the complete analogue of Jackson's theorem on the estimate of the best approximation of an arbitrary continuous function by means of its modulus of continuity; the analogue of Bernstein's inequality on the estimate of the derivative of a trigonometric polynomial; the analogue of Stechkin's theorem on the connection between the modulus of continuity of a function and the rate of its approximation by polynomials; the analogue of Dolzhenko's inequality on the estimate of the variation of a rational function; and the analogue of Dolzhenko's theorem on the estimate of the variation of a function by means of its best rational approximation.

In this study, the prediction errors of various spatial interpolation methods used to model values at unmeasured locations was compared and the accuracy of these predictions was evaluated. The root mean square (RMS) was calculated by processing different parameters associated with spatial interpolation by using techniques such as inverse distance weighting, kriging, local polynomial interpolation and radial basis function to known elevation data of the east coastal area under the same condition. As a result, a circular model of simple kriging reached the smallest RMS value. Prediction map using the multiquadric method of a radial basis function was coincident with the spatial distribution obtained by constructing a triangulated irregular network of the study area through the raster mathematics. In addition, better interpolation results can be obtained by setting the optimal power value provided under the selected condition.

In typical multiagent teamwork settings, the teammates are either programmed together, or are otherwise provided with standard communication languages and coordination protocols. In contrast, this paper presents an ad hoc team setting in which the teammates are not pre-coordinated, yet still must work together in order to achieve their common goal(s). We represent a specific instance of this scenario, in which a teammate has limited action capabilities and a fixed and known behavior, as a finite-horizon, cooperative k-armed bandit. In addition to motivating and studying this novel ad hoc teamwork scenario, the paper contributes to the k-armed bandits literature by characterizing the conditions under which certain actions are potentially optimal, and by presenting a polynomial dynamic programming algorithm that solves for the optimal action when the arm payoffs come from a discrete distribution.

This paper discusses the methods of algebraic functions on the settlement limit of indeterminate forms. Limit is a mathematical method used to describe the effect of variable functions move closer to a point on a particular function. The goal is to determine the specific characteristics of the methods of algebraic functions on the settlement limit of indeterminate forms. The methods used is a special method that consists of factoring methods, methods of multiplication herd, and the division with the highest rank method, and the method of L' Hopital. By using the methods mentioned above, then we obtain the specific characteristics of the methods of algebraic functions settlement limit on the indeterminate form are 1. Factoring methods: (a) The numerator is the result of squaring the binomial equation of the denominator. Where, (i) If the binomial equation using the difference operation, then its points is positive. (ii) If the binomial equations using addition operation, then its points is negative. (b) the constant factor of each of the numerator and denominator equally. One of which is a three tribal equation in the form ax2 + bx + c , the value of b it is the sum of the value of the constant multiplication. (c) If the numerator and denominator are both binomial equations with difference operation, where the variables and constants in the numerator is the square of the variable and constant denominator. Or conversely, variables and constants in the denominator is the square of the variable and constant numerator. (d) If the equation is a difference of two fractions where the denominator is a binomial variables and constants in the denominator is the result of one of the squaring of the variables and constants in the denominator of the other. (e) If the equation in the form of fragments which all contain the same variables. (f) If the equation is in the form of two parts, where the coefficients and constants in the numerator or the same multiplication factor, so that when the coefficients and constants removed it will form one of the factors that equation equal to that of the denominator. 2. A herd Multiplication method: (a) If the equation is the difference of two roots. (b) If the equation in the form of the difference or fractions containing amount root form. Both the numerator and the denominator. 3. Distribution by Top Rank methods: (a) If the equation in the form ∞/∞. (b) If a polynomial equation of degree 3, 4, 5,... etc. Which is hard to be factored. 4. L' Hopital method: if the limit point is substituted then produces the indeterminate form 0/0, ∞/∞, ∞-∞ , except if the derivative of the function at the root of the denominator in the form similar to the form factor function in the numerator. Vice versa, the derivative of the function at the root of the numerator shaped form factor similar to the function in the denominator.

Graphons are analytic objects associated with convergent sequences of dense graphs. Finitely forcible graphons, that is, those determined by finitely many subgraph densities, are of particular interest because of their relation to various problems in extremal combinatorics and theoretical computer science. Lovász and Szegedy conjectured that the topological space of typical vertices of a finitely forcible graphon always has finite dimension, which would have implications on the minimum number of parts in its weak ε-regular partition. We disprove the conjecture by constructing a finitely forcible graphon with the space of typical vertices that has infinite dimension.

The algebraic theory of automata was created by Schützenberger and Chomsky over 50 years ago and there has since been a great deal of development. Classical work on the theory to noncommutative power series has been augmented more recently to areas such as representation theory, combinatorial mathematics and theoretical computer science. This book presents to an audience of graduate students and researchers a modern account of the subject and its applications. The algebraic approach allows the theory to be developed in a general form of wide applicability. For example, number-theoretic results can now be more fully explored, in addition to applications in automata theory, codes and non-commutative algebra. Much material, for example, Schützenberger's theorem on polynomially bounded rational series, appears here for the first time in book form. This is an excellent resource and reference for all those working in algebra, theoretical computer science and their areas of overlap.

Computability is one of the fundamental notions of mathematics and computer science, trying to capture the effective content of mathematics and its applications. Computability Theory explores the frontiers and limits of effectiveness and algorithmic methods. It has its origins in Godel's Incompleteness Theorems and the formalization of computability by Turing and others, which later led to the emergence of computer science as we know it today. Computability Theory is strongly connected to other areas of mathematics and theoretical computer science. The core of this theory is the analysis of relative computability and the induced degrees of unsolvability; its applications are mainly to Kolmogorov complexity and randomness as well as mathematical logic, analysis and algebra. Current research in computability theory stresses these applications and focuses on algorithmic randomness, computable analysis, computable model theory, and reverse mathematics (proof theory). Recent advances in these research directions have revealed some deep interactions not only among these areas but also with the core parts of computability theory. The goal of this Dagstuhl Seminar is to bring together researchers from all parts of computability theory and related areas in order to discuss advances in the individual areas and the interactions among those.