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.

Spectral solutions to the Korteweg-de-Vries and nonlinear Schrodinger equations

In this paper, we present the solutions of 1-D and 2-D non-linear partial differential equations with initial conditions. We approach the solutions in time domain using two methods. We first solve the equations using Fourier spectral approximation in the spatial domain and secondly we compare the results with the approximation in the spatial domain using orthogonal functions such as Legendre or Chebyshev polynomials as their basis functions. The advantages and the applicability of the two different methods for different types of problems are brought out by considering 1-D and 2-D nonlinear partial differential equations namely the Korteweg-de-Vries and nonlinear Schrodinger equation with different potential function. (C) 2015 Elsevier Ltd. All rights reserved.

An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Formulas

We show here a 2(Omega(root d.log N)) size lower bound for homogeneous depth four arithmetic formulas. That is, we give an explicit family of polynomials of degree d on N variables (with N = d(3) in our case) with 0, 1-coefficients such that for any representation of a polynomial f in this family of the form f = Sigma(i) Pi(j) Q(ij), where the Q(ij)'s are homogeneous polynomials (recall that a polynomial is said to be homogeneous if all its monomials have the same degree), it must hold that Sigma(i,j) (Number of monomials of Q(ij)) >= 2(Omega(root d.log N)). The above mentioned family, which we refer to as the Nisan-Wigderson design-based family of polynomials, is in the complexity class VNP. Our work builds on the recent lower bound results 1], 2], 3], 4], 5] and yields an improved quantitative bound as compared to the quasi-polynomial lower bound of 6] and the N-Omega(log log (N)) lower bound in the independent work of 7].

Counting zero kernel pairs over a finite field

Helmke et al. have recently given a formula for the number of reachable pairs of matrices over a finite field. We give a new and elementary proof of the same formula by solving the equivalent problem of determining the number of so called zero kernel pairs over a finite field. We show that the problem is, equivalent to certain other enumeration problems and outline a connection with some recent results of Guo and Yang on the natural density of rectangular unimodular matrices over F-qx]. We also propose a new conjecture on the density of unimodular matrix polynomials. (C) 2016 Elsevier Inc. All rights reserved.

FAST AND ACCURATE BILATERAL FILTERING USING GAUSS-POLYNOMIAL DECOMPOSITION

The bilateral filter is a versatile non-linear filter that has found diverse applications in image processing, computer vision, computer graphics, and computational photography. A common form of the filter is the Gaussian bilateral filter in which both the spatial and range kernels are Gaussian. A direct implementation of this filter requires O(sigma(2)) operations per pixel, where sigma is the standard deviation of the spatial Gaussian. In this paper, we propose an accurate approximation algorithm that can cut down the computational complexity to O(1) per pixel for any arbitrary sigma (constant-time implementation). This is based on the observation that the range kernel operates via the translations of a fixed Gaussian over the range space, and that these translated Gaussians can be accurately approximated using the so-called Gauss-polynomials. The overall algorithm emerging from this approximation involves a series of spatial Gaussian filtering, which can be efficiently implemented (in parallel) using separability and recursion. We present some preliminary results to demonstrate that the proposed algorithm compares favorably with some of the existing fast algorithms in terms of speed and accuracy.

Interactions of Penny-Shaped Cracks in Three-Dimensional Solids

The interaction of arbitrarily distributed penny-shaped cracks in three-dimensional solids is analyzed in this paper. Using oblate spheroidal coordinates and displacement functions, an analytic method is developed in which the opening and the sliding displacements on each crack surface are taken as the basic unknown functions. The basic unknown functions can be expanded in series of Legendre polynomials with unknown coefficients. Based on superposition technique, a set of governing equations for the unknown coefficients are formulated from the traction free conditions on each crack surface. The boundary collocation procedure and the average method for crack-surface tractions are used for solving the governing equations. The solution can be obtained for quite closely located cracks. Numerical examples are given for several crack problems. By comparing the present results with other existing results, one can conclude that the present method provides a direct and efficient approach to deal with three-dimensional solids containing multiple cracks.

Static Study of Cantilever Beam stiction under Electrostatic Force Influence

The model and analysis of the cantilever beam adhesion problem under the action of electrostatic force are given. Owing to the nonlinearity of electrostatic force, the analytical solution for this kind of problem is not available. In this paper, a systematic method of generating polynomials which are the exact beamsolutions of the loads with different distributions is provided. The polynomials are used to approximate the beam displacement due to electrostatic force. The equilibrium equation offers an answer to how the beam deforms but no information about the unstuck length. The derivative of the functional with respect to the unstuck length offers such information. But to compute the functional it is necessary to know the beam deformation. So the problem is iteratively solved until the results are converged. Galerkin and Newton-Raphson methods are used to solve this nonlinear problem. The effects of dielectric layer thickness and electrostatic voltage on the cantilever beamstiction are studied.The method provided in this paper exhibits good convergence. For the adhesion problem of cantilever beam without electrostatic voltage, the analytical solution is available and is also exactly matched by the computational results given by the method presented in this paper.

迟滞系统随机响应问题基于函数级数的解法

本文提出了一个三次多项式的迟滞模型,并用Wiener-Hermite函数级数展开的方法求解,得到了不同阻尼及不同的非线性强度时系统响应的均方值σ(t)。不仅从理论上而且通过实验证实了用这种方法求解迟滞系统的响应是行之有效的、简便的。

Bending solution of high-order refined shear deformation-theory for rectangular composite plates

A new high-order refined shear deformation theory based on Reissner's mixed variational principle in conjunction with the state- space concept is used to determine the deflections and stresses for rectangular cross-ply composite plates. A zig-zag shaped function and Legendre polynomials are introduced to approximate the in-plane displacement distributions across the plate thickness. Numerical results are presented with different edge conditions, aspect ratios, lamination schemes and loadings. A comparison with the exact solutions obtained by Pagano and the results by Khdeir indicates that the present theory accurately estimates the in-plane responses.

Randomness-efficient curve sampling

Curve samplers are sampling algorithms that proceed by viewing the domain as a vector space over a finite field, and randomly picking a low-degree curve in it as the sample. Curve samplers exhibit a nice property besides the sampling property: the restriction of low-degree polynomials over the domain to the sampled curve is still low-degree. This property is often used in combination with the sampling property and has found many applications, including PCP constructions, local decoding of codes, and algebraic PRG constructions.

The randomness complexity of curve samplers is a crucial parameter for its applications. It is known that (non-explicit) curve samplers using O(log N + log(1/δ)) random bits exist, where N is the domain size and δ is the confidence error. The question of explicitly constructing randomness-efficient curve samplers was first raised in [TU06] where they obtained curve samplers with near-optimal randomness complexity.

In this thesis, we present an explicit construction of low-degree curve samplers with optimal randomness complexity (up to a constant factor) that sample curves of degree (m log_q(1/δ))^O(1) in F_q^m. Our construction is a delicate combination of several components, including extractor machinery, limited independence, iterated sampling, and list-recoverable codes.

Efficient methods for stochastic optimal control

The Hamilton Jacobi Bellman (HJB) equation is central to stochastic optimal control (SOC) theory, yielding the optimal solution to general problems specified by known dynamics and a specified cost functional. Given the assumption of quadratic cost on the control input, it is well known that the HJB reduces to a particular partial differential equation (PDE). While powerful, this reduction is not commonly used as the PDE is of second order, is nonlinear, and examples exist where the problem may not have a solution in a classical sense. Furthermore, each state of the system appears as another dimension of the PDE, giving rise to the curse of dimensionality. Since the number of degrees of freedom required to solve the optimal control problem grows exponentially with dimension, the problem becomes intractable for systems with all but modest dimension.

In the last decade researchers have found that under certain, fairly non-restrictive structural assumptions, the HJB may be transformed into a linear PDE, with an interesting analogue in the discretized domain of Markov Decision Processes (MDP). The work presented in this thesis uses the linearity of this particular form of the HJB PDE to push the computational boundaries of stochastic optimal control.

This is done by crafting together previously disjoint lines of research in computation. The first of these is the use of Sum of Squares (SOS) techniques for synthesis of control policies. A candidate polynomial with variable coefficients is proposed as the solution to the stochastic optimal control problem. An SOS relaxation is then taken to the partial differential constraints, leading to a hierarchy of semidefinite relaxations with improving sub-optimality gap. The resulting approximate solutions are shown to be guaranteed over- and under-approximations for the optimal value function. It is shown that these results extend to arbitrary parabolic and elliptic PDEs, yielding a novel method for Uncertainty Quantification (UQ) of systems governed by partial differential constraints. Domain decomposition techniques are also made available, allowing for such problems to be solved via parallelization and low-order polynomials.

The optimization-based SOS technique is then contrasted with the Separated Representation (SR) approach from the applied mathematics community. The technique allows for systems of equations to be solved through a low-rank decomposition that results in algorithms that scale linearly with dimensionality. Its application in stochastic optimal control allows for previously uncomputable problems to be solved quickly, scaling to such complex systems as the Quadcopter and VTOL aircraft. This technique may be combined with the SOS approach, yielding not only a numerical technique, but also an analytical one that allows for entirely new classes of systems to be studied and for stability properties to be guaranteed.

The analysis of the linear HJB is completed by the study of its implications in application. It is shown that the HJB and a popular technique in robotics, the use of navigation functions, sit on opposite ends of a spectrum of optimization problems, upon which tradeoffs may be made in problem complexity. Analytical solutions to the HJB in these settings are available in simplified domains, yielding guidance towards optimality for approximation schemes. Finally, the use of HJB equations in temporal multi-task planning problems is investigated. It is demonstrated that such problems are reducible to a sequence of SOC problems linked via boundary conditions. The linearity of the PDE allows us to pre-compute control policy primitives and then compose them, at essentially zero cost, to satisfy a complex temporal logic specification.

Topological strings, double affine Hecke algebras, and exceptional knot homology

In this thesis, we consider two main subjects: refined, composite invariants and exceptional knot homologies of torus knots. The main technical tools are double affine Hecke algebras ("DAHA") and various insights from topological string theory.

In particular, we define and study the composite DAHA-superpolynomials of torus knots, which depend on pairs of Young diagrams and generalize the composite HOMFLY-PT polynomials from the full HOMFLY-PT skein of the annulus. We also describe a rich structure of differentials that act on homological knot invariants for exceptional groups. These follow from the physics of BPS states and the adjacencies/spectra of singularities associated with Landau-Ginzburg potentials. At the end, we construct two DAHA-hyperpolynomials which are closely related to the Deligne-Gross exceptional series of root systems.

In addition to these main themes, we also provide new results connecting DAHA-Jones polynomials to quantum torus knot invariants for Cartan types A and D, as well as the first appearance of quantum E6 knot invariants in the literature.

像差对星间激光通信相互对准误差的影响

We analyze mutual alignment errors due to wave-front aberrations. To solve the central obscured problem, we introduce modified Zernike polynomials, which are a set of complete orthogonal polynomials. It is found that different aberrations have different effects on mutual alignment errors. Some aberrations influence only the line of sight, while some aberrations influence both the line of sight and the intensity distributions. (c) 2005 Optical Society of America

Symmetric representation of an integral domain over a subdomain

Let F(θ) be a separable extension of degree n of a field F. Let Δ and D be integral domains with quotient fields F(θ) and F respectively. Assume that Δ ᴝ D. A mapping φ of Δ into the n x n D matrices is called a Δ/D rep if (i) it is a ring isomorphism and (ii) it maps d onto dI_n whenever d ϵ D. If the matrices are also symmetric, φ is a Δ/D symrep.

Every Δ/D rep can be extended uniquely to an F(θ)/F rep. This extension is completely determined by the image of θ. Two Δ/D reps are called equivalent if the images of θ differ by a D unimodular similarity. There is a one-to-one correspondence between classes of Δ/D reps and classes of Δ ideals having an n element basis over D.

The condition that a given Δ/D rep class contain a Δ/D symrep can be phrased in various ways. Using these formulations it is possible to (i) bound the number of symreps in a given class, (ii) count the number of symreps if F is finite, (iii) establish the existence of an F(θ)/F symrep when n is odd, F is an algebraic number field, and F(θ) is totally real if F is formally real (for n = 3 see Sapiro, “Characteristic polynomials of symmetric matrices” Sibirsk. Mat. Ž. 3 (1962) pp. 280-291), and (iv) study the case D = Z, the integers (see Taussky, “On matrix classes corresponding to an ideal and its inverse” Illinois J. Math. 1 (1957) pp. 108-113 and Faddeev, “On the characteristic equations of rational symmetric matrices” Dokl. Akad. Nauk SSSR 58 (1947) pp. 753-754).

The case D = Z and n = 2 is studied in detail. Let Δ’ be an integral domain also having quotient field F(θ) and such that Δ’ ᴝ Δ. Let φ be a Δ/Z symrep. A method is given for finding a Δ’/Z symrep ʘ such that the Δ’ ideal class corresponding to the class of ʘ is an extension to Δ’ of the Δ ideal class corresponding to the class of φ. The problem of finding all Δ/Z symreps equivalent to a given one is studied.

Some generalizations of commutativity for linear transformations on a finite dimensional vector space

Let L be the algebra of all linear transformations on an n-dimensional vector space V over a field F and let A, B, ƐL. Let A_i+1 = A_iB - BA_i, i = 0, 1, 2,…, with A = A_o. Let f_k (A, B; σ) = A_2K+1 - ^σ1^A2K-1 ^{+ σ}2^A2K-3 -… +(-1)^Kσ_KA₁ where σ = (σ₁, σ₂,…, σ_K), σ_i belong to F and K = k(k-1)/2. Taussky and Wielandt [Proc. Amer. Math. Soc., 13(1962), 732-735] showed that f_n(A, B; σ) = 0 if σ_i is the i^th elementary symmetric function of (β₄- β_s)², 1 ≤ r ˂ s ≤ n, i = 1, 2, …, N, with N = n(n-1)/2, where β₄ are the characteristic roots of B. In this thesis we discuss relations involving f_k(X, Y; σ) where X, Y Ɛ L and 1 ≤ k ˂ n. We show: 1. If F is infinite and if for each X Ɛ L there exists σ so that f_k(A, X; σ) = 0 where 1 ≤ k ˂ n, then A is a scalar transformation. 2. If F is algebraically closed, a necessary and sufficient condition that there exists a basis of V with respect to which the matrices of A and B are both in block upper triangular form, where the blocks on the diagonals are either one- or two-dimensional, is that certain products X₁, X₂…X_r belong to the radical of the algebra generated by A and B over F, where X_i has the form f₂(A, P(A,B); σ), for all polynomials P(x, y). We partially generalize this to the case where the blocks have dimensions ≤ k. 3. If A and B generate L, if the characteristic of F does not divide n and if there exists σ so that f_k(A, B; σ) = 0, for some k with 1 ≤ k ˂ n, then the characteristic roots of B belong to the splitting field of g_k(w; σ) = w^2K+1 - σ₁w^2K-1 + σ₂w^2K-3 - …. +(-1)^K σ_Kw over F. We use this result to prove a theorem involving a generalized form of property L [cf. Motzkin and Taussky, Trans. Amer. Math. Soc., 73(1952), 108-114]. 4. Also we give mild generalizations of results of McCoy [Amer. Math. Soc. Bull., 42(1936), 592-600] and Drazin [Proc. London Math. Soc., 1(1951), 222-231].