A Trace Bound for the Hereditary Discrepancy

The $\gamma_2$ norm of a real $m\times n$ matrix $A$ is the minimum number $t$ such that the column vectors of $A$ are contained in a $0$-centered ellipsoid $E\subseteq\mathbb{R}^m$ which in turn is contained in the hypercube $[-t, t]^m$. We prove that this classical quantity approximates the \emph{hereditary discrepancy} $\mathrm{herdisc} A$ as follows: $\gamma_2(A) = {O(\log m)}\cdot \mathrm{herdisc} A$ and $\mathrm{herdisc} A = O(\sqrt{\log m}\,)\cdot\gamma_2(A) $. Since $\gamma_2$ is polynomial-time computable, this gives a polynomial-time approximation algorithm for hereditary discrepancy. Both inequalities are shown to be asymptotically tight. We then demonstrate on several examples the power of the $\gamma_2$ norm as a tool for proving lower and upper bounds in discrepancy theory. Most notably, we prove a new lower bound of $\Omega(\log^{d-1} n)$ for the \emph{$d$-dimensional Tusnady problem}, asking for the combinatorial discrepancy of an $n$-point set in $\mathbb{R}^d$ with respect to axis-parallel boxes. For $d>2$, this improves the previous best lower bound, which was of order approximately $\log^{(d-1)/2}n$, and it comes close to the best known upper bound of $O(\log^{d+1/2}n)$, for which we also obtain a new, very simple proof.

Tolerance Stack up Analysis for Angularity of Components and their Assembly

A polynomial function of order p is used to approximate a trend surface of a variable y in an area with coordinates u, v. The determination of the coefficients of the approximating polynomial requires the solution of p simultaneous ‘normal’ equations. A method is shown by which a ‘well-balanced’ matrix of this system of equations can be established to avoid complications in the computation. Gauss's method of solution facilitates the computation of all coefficients and the sum of squares ss for each term added to the polynomial. The step-by-step variation of the variance provides a means of judging which terms should be included in the final polynomial trend function. If the data are distributed in clusters over the map area, it is necessary to apply a ‘block system’ and attach different weights to the data. It is shown how such weighted data are handled.

Stabilization of polynomial systems is revisited in this paper. Here, we proposed a new sufficient condition for the controller design, with guaranteed bounds on the input magnitudes. In the proposed synthesis condition, the system matrices and the Lyapunov matrices are separated, and hence parameterization of the resulting controller is independent of the Lyapunov matrices. Moreover, the proposed condition is convex in the decision variables and solvable via the sum-of-squares (SOS) technique. Main contributions of the proposed approach are twofold. Firstly, parameter-dependent Lyapunov functions can be considered in robust controller design for uncertain polynomial systems. Secondly, static output-feedback design can be addressed in a computationally efficient manner.

This paper studies the idea of answering range searching queries using simple data structures. The only data structure we need is the Delaunay Triangulation of the input points. The idea is to first locate a vertex of the (arbitrary) query polygon {\cal q} and walk along the boundary of the polygon in the Delaunay Triangulation and report all the points enclosed by the query polygon. For a set of uniformly distributed random points in 2-D and a query polygon the expected query time of this algorithm is O(n^{1/3} + Q + {\bf E}K + L_{r}n^{1/2}), where Q is the size of the query polygon {\cal q}, {\bf E}K = O(n\bcdot area({\cal q})) is the expected number of output points, L_{r} is a parameter related to the shape of the query polygon {\cal q} and n, and L_{r} is always bounded by the sum of the edge lengths of {\cal q}. Theoretically, when L_{r} = O(1/n^{1/6}) the expected query time is O(n^{1/3} + Q + {\bf E}K), which improves the best known average query time for general range searching. Besides the theoretical meaning, the good property of this algorithm is that once the Delaunay Triangulation is given, no additional preprocessing is needed. In order to obtain empirical results, we design a new algorithm for generating random simple polygons within a given domain. Our empirical results show that the constant coefficient of the algorithm is small, at least for the special (practical) cases when the query polygon is either a triangle (simplex range searching) or an axis-parallel box (orthogonal range searching) and for the general case when the query polygons are generated by our new polygon-generating algorithms and their sizes are relatively small.

For the tensor principal component analysis (tensor PCA) problem, we propose a new hierarchy of increasingly powerful algorithms with increasing runtime. Our hierarchy is analogous to the sum-of-squares (SOS) hierarchy but is instead inspired by statistical physics and related algorithms such as belief propagation and AMP (approximate message passing). Our level-ℓ algorithm can be thought of as a linearized message-passing algorithm that keeps track of ℓ-wise dependencies among the hidden variables. Specifically, our algorithms are spectral methods based on the Kikuchi Hessian , which generalizes the well-studied Bethe Hessian to the higher-order Kikuchi free energies. It is known that AMP, the flagship algorithm of statistical physics, has substantially worse performance than SOS for tensor PCA. In this work, we ‘redeem’ the statistical physics approach by showing that our hierarchy gives a polynomial-time algorithm matching the performance of SOS. Our hierarchy also yields a continuum of subexponential-time algorithms, and we prove that these achieve the same (conjecturally optimal) tradeoff between runtime and statistical power as SOS. Our proofs are much simpler than prior work, and also apply to the related problem of refuting random k -XOR formulas. The results we present here apply to tensor PCA for tensors of all orders, and to k -XOR when k is even. Our methods suggest a new avenue for systematically obtaining optimal algorithms for Bayesian inference problems, and our results constitute a step toward unifying the statistical physics and sum-of-squares approaches to algorithm design.

This paper addresses the problem of establishing robust exponential stability of 2D mixed continuous-discrete-time systems affected by uncertainty. Specifically, it is supposed that the matrices of the system are polynomial functions of an uncertain vector constrained over a semialgebraic set. First, it is shown that robust exponential stability is equivalent to the existence of a complex Lyapunov functions depending polynomially on the uncertain vector and an additional parameter of degree not greater than a known quantity. Second, a condition for establishing robust exponential stability is proposed via convex optimization by exploiting sums-of-squares (SOS) matrix polynomials. This condition is sufficient for any chosen degree of the complex Lyapunov function candidate, and is also necessary for degrees sufficiently large.

This paper presents a first attempt at using the spectral moments method (SMM) to solve Maxwell's equations in twisted anisotropic media in the presence of defects. This numerical method, previously developed in condensed matter physics, allows computation of Green functions for very large systems. The dynamic matrix of the discretized system is built from the medium parameters. Green functions, calculated for a given source, representing a point source at infinity and given receiver, are developed as a continued fraction whose coefficients are related to the moments and directly computed from the dynamic matrix. In this study we compute the light transmitted through thin surface-stabilized ferroelectric liquid crystal cells with a chevron structure and a twisted director distribution. The efficiency and accuracy of the method are analysed by comparing the results obtained by SMM with the analytical solution obtained using the Jones matrix formalism. Finally, we apply SMM to compute the transmitted light with different director configurations. We show, by comparisons with experimental data, that the simplest director configuration is certainly the most probable.

Spectral methods have indisputable advantages over numerical methods in solving various problems of mathematical physics. The advantages are the high convergence and accuracy of approximate solutions, which is most relevant for calculating the strength of aerospace machinery. The problem of choosing basic functions inevitably arises when applying spectral methods. The problem is that, in addition to providing exponential convergence, the system of basic functions has to satisfy some other requirements: stability of approximate solutions and procedures for their obtaining, reduction of calculations, convenience, and some more. This paper compares eleven basic systems: the systems constructed in the form of linear combinations of Legendre polynomials that satisfy either only the main boundary conditions or the main and natural ones, similar systems constructed using Chebyshev polynomials, and the functions proposed by Khalilov S. A., systems of Lagrange – Lobatto interpolation polynomials using the Legendre and Chebyshev interpolation points, system of trigonometric functions, exponentiation, and system of finite functions of the finite element method. The convergence speed of the approximate solution to the exact one, the error in the equations of boundary value problems, and the condition numbers of matrices of linear algebraic equation systems, which arise when using variational, projection and collocation methods, were compared. The study performed on three test problems modeling beam bending: classic beam bending under unevenly distributed load, bending of additionally stretched beam on the elastic basis and geometrically nonlinear bending. The impact of the Gibbs effect on the approximate solution convergence is investigated. Among the considered basic systems, the system of basis functions in the form of linear combinations of Legendre polynomials has proved to be the best, as they satisfy all boundary conditions. This basis leads to the highest speed at which approximate solution approaches the exact one, the error in the equations approaches to zero, and also it has the smallest increase in the condition number with the increase in the order of SLE matrices, which appear due to variational and projection methods. The finite functions of the finite element method have proved to be the worst in terms of accuracy and convergence.

TIME DOMAIN CONSIDERATIONS IN OPTIMAL ESTIMATION FOR BANDLIMITED SIGNALS

Orthogonal space-time block codes (OSTBCs) have attracted much attention owing to their simple code construction, maximal diversity gain, and low maximum-likelihood (ML) detection complexity when channel state information (CSI) is available at the receiver. This paper addresses the problem of ML OSTBC detection with unknown CSI. Focusing on the binary and quaternary PSK constellations, we show that blind ML OSTBC detection can be simplified to a Boolean quadratic program (BQP). From an optimization viewpoint the BQP is still a computationally hard problem, and we propose two alternatives for dealing with this inherent complexity. First, we consider the semidefinite relaxation (SDR) approach, which leads to a suboptimal, but accurate, blind ML detection algorithm with an affordable worst-case computational cost. We also consider the sphere decoding approach, which leads to an exact blind ML detection algorithm that remains computationally expensive in the worst case, but generally incurs a reasonable average computational cost. For the two algorithms, we study implementation methods that can significantly reduce the computational complexity. Simulation results indicate that the two blind ML detection algorithms are competitive, in that the bit error performance of the two algorithms is almost the same and is noticeably better than that of some other existing blind detectors. Moreover, numerical studies show that the SDR algorithm provides better complexity performance than the sphere decoder in the worst-case sense, and vice versa in the average sense.

It is a problem of some interest in Electromagnetism to determine the natural modes of decay, and the corresponding persistencies, of free currents in a given conductor. When this has been solved it is an easy matter to find the currents induced by given varying electromotive forces. The general theory for a system of linear circuits is of course well known. If the variables y . 1 , y . 2 , . . . y . n , which specify the currents, be so chosen that the electrokinetic energy T and the dissipation-function F are both expressed by sums of squares, say 2T = L 1 y . 1 2 + L 2 y . 2 2 + . . + L n y . n 2 , 2F = R 1 y . 1 2 + R 2 y . 2 2 + . . + R n y . n 2 , then y 1 , y 2 , . . y n are for the present purpose the “normal coordinates” of the system; and the equations of motion of electricity are of the form Lÿ + Ry . = E

The major difficulty of accurate worst case tolerance analysis is to identify the worst case parameter sets. Traditional vertex analysis presumes the monotonicity between circuit response and circuit parameters, and uses nominal sensitivity information to select the vertices of parameter space as the worst case parameter sets. However, under parameter variations, the sensitivity itself is uncertain within an interval range, furthermore, the worst case parameter sets may not be the vertices of parameter space. In this paper, the relationship among monotonicity, sensitivity, and worst case parameter sets is investigated in the context of parameter variations. An efficient yet accurate approach for linear circuit worst case tolerance analysis is proposed. The underlying idea is that if the circuit response is monotonic with respect to the changes in a specific circuit parameter over the parameter space, then the worst case parameter sets are located at the corners of that parameter. The monotonicity is identified by the sensitivity band computation, and is used to generate uncertainty-reduced simulation problems. Experimental results strongly meet the expectation for accuracy and efficiency.

A simplified integrating model is developed for soil-pile dynamic interaction system under seismic loading. Different from previous work, this simplified model constructs the whole system stiffness matrix by combining the stiffness matrix of pile and near region soil system according to the volumetric ratio of pile group and soil system. This simplified model can avoid deviation brought by previous soil-pile surface model and can felicitously simulate the co-working mechanism between soil and piles. The feasibility of the proposed approach is assessed by comparing with in-site observation result of real project.

We develop a new framework that extends the quantum walk framework of Magniez, Nayak, Roland, and Santha, by utilizing the idea of quantum data structures to construct an efficient method of nesting quantum walks. Surprisingly, only classical data structures were considered before for searching via quantum walks.The recently proposed learning graph framework of Belovs has yielded improved upper bounds for several problems, including triangle finding and more general subgraph detection. We exhibit the power of our framework by giving a simple explicit constructions that reproduce both the O(n35/27) and O(n9/7) learning graph upper bounds (up to logarithmic factors) for triangle finding, and discuss how other known upper bounds in the original learning graph framework can be converted to algorithms in our framework. We hope that the ease of use of this framework will lead to the discovery of new upper bounds.