The rank of a 2 × 2 × 2 tensor

As computing power increases, many more problems in engineering and data analysis involve computation with tensors, or multi-way data arrays. Most applications involve computing a decomposition of a tensor into a linear combination of rank-1 tensors. Ideally, the decomposition involves a minimal number of terms, i.e. computation of the rank of the tensor. Tensor rank is not a straight-forward extension of matrix rank. A constructive proof based on an eigenvalue criterion is provided that shows when a 2?×?2?×?2 tensor over ? is rank-3 and when it is rank-2. The results are extended to show that n?×?n?×?2 tensors over ? have maximum possible rank n?+?k where k is the number of complex conjugate eigenvalue pairs of the matrices forming the two faces of the tensor cube.     

On the classification of nonsingular 2×2×2×2 hypercubes

As a first step in the classification of nonsingular 2×2×2×2 hypercubes up to equivalence, we resolve the case where the base field is finite and the hypercubes can be written as a product of two 2×2×2 hypercubes. (Nonsingular hypercubes were introduced by D. Knuth in the context of semifields. Where semifields are related to hypercubes of dimension 3, this paper considers the next case, i.e., hypercubes of dimension 4.) We define the notion of ij-rank (with 1 ≤ i < j ≤ 4) and prove that a hypercube is the product of two 2×2×2 hypercubes if and only if its 12-rank is at most 2. We derive a ‘standard form’ for nonsingular 2×2×2×2 hypercubes of 12-rank less than 4 as a first step in the classification of such hypercubes up to equivalence. Our main result states that the equivalence class of a nonsingular 2×2×2×2 hypercube M of 12-rank 2 depends only on the value of an invariant δ ₀(M) which derives in a natural way from the Cayley hyperdeterminant det₀ M and another polynomial invariant det M of degree 4. As a corollary we prove that the number of equivalence classes is (q + 1)/2 or q/2 depending on whether the order q of the field is odd or even.     

A note on the ranks of 2 × 2 × 2 and 2 × 2 × 2 × 2 tensors

Using rank-1 reduction formula and the vector space spanned by the real rank-1 matrices, we present a different way to show that the maximum possible rank of the 2?×?2?×?2 tensors over the real field is 3. Following, we obtain that the maximum rank of the 2?×?2?×?2?×?2 tensors over the real field is less than or equal to 5 and propose another way to show that the maximum rank of the 2?×?2?×?2?×?2 tensors over the complex field is 4, except one special case.     

Development of a 2D Isoparametric Finite-Element Model Based on Reddy’s Third-Order Theory for the Bending Behavior Analysis of Composite Laminated Plates

Mechanics of Composite Materials - The objective of this work is to propose a simple efficient finite element, based on Reddy’s third-order shear deformation theory, for analyzing the bending...     

Behavior of the Fourier–Walsh coefficients of a corrected function

We prove that, given a sequence {ak}_k=1^∞ with a_k ↓ 0 and {ak}_k=1^∞ ? l₂, reals 0 < ε < 1 and p ∈ [1, 2], and f ∈ L^p(0, 1), we can find f ∈ L^p(0, 1) with mes{f ≠ f < ε whose nonzero Fourier–Walsh coefficients c_k(f) are such that |c_k(f)| = a_k for k ∈ spec(f).     

Even 2 × 2 Submatrices of a Random Zero-One Matrix

Consider an m×n zero-one matrix A. An s×t submatrix of A is said to be even if the sum of its entries is even. In this paper, we focus on the case m=n and s=t=2. The maximum number M(n) of even 2×2 submatrices of A is clearly and corresponds to the matrix A having, e.g., all ones (or zeros). A more interesting question, motivated by Turán numbers and Hadamard matrices, is that of the minimum number m(n) of such matrices. It has recently been shown that for some constant B. In this paper we show that if the matrix A=A_n is considered to be induced by an infinite zero one matrix obtained at random, then where E_n denotes the number of even 2×2 submatrices of A_n. Results such as these provide us with specific information about the tightness of the concentration of E_n around its expected value of Acknowledgments. The research of both authors was supported by NSF Grant DMS-0139291 and conducted at East Tennessee State University during the Summer of 2002, when Johnson was a student in Godboles Research Experiences for Undergraduates Program. The valuable suggestions of two anonymous referees are gratefully acknowledged.     

Behavior of the Solution of the Klein—Gordon Equation with a Localized Initial Condition

Theoretical and Mathematical Physics - We consider the Klein—Gordon equation with a localized initial condition and describe the transition of the solution from localized to rapidly...     

Perturbation of spectra for a class of 2×2 operator matrices

In this paper, we study the perturbation of spectra for 2 × 2 operator matrices such as M _X = ( _{0 B} ^{A X} ) and M _Z = ( _{Z B} ^{A C} ) on the Hilbert space H ?? K and the sets $\bigcap\limits_{X \in \mathcal{B}(K,H)} {P_\sigma (M_X )} ,\bigcap\limits_{X \in \mathcal{B}(K,H)} {R_\sigma (M_X )} $ and $\bigcap\limits_{Z \in \mathcal{B}(H,K)} {\sigma (M_Z )} ,\bigcap\limits_{Z \in \mathcal{B}(H,K)} {P_\sigma (M_Z )} ,\bigcap\limits_{Z \in \mathcal{B}(H,K)} {R_\sigma (M_Z )} ,\bigcap\limits_{Z \in \mathcal{B}(H,K)} {C_\sigma (M_Z )} $ , where R(C) is a closed subspace, are characterized     

Convergence of approximate solutions of the Cauchy problem for a 2 × 2 nonstrictly hyperbolic system of conservation laws

A convergence theorem for the vanishing viscosity method and for the Lax-Friedrichs schemes, applied to a nonstrictly hyperbolic and nongenuinely nonlinear system is established. Using the theory of compensated compactness we prove convergence of a subsequence in the strong topology.     

Investigation of the problem of the flow of a compressible viscous fluid past a bounded body in the hölder classes of functions

It is shown that the stationary exterior problem for the equations of motion of a viscous compressible fluid is uniquely solvable in weighted Hölder spaces if the exterior forces and the value of the velocity at infinity are sufficiently small. As a weight function, a power function (1+|x|)^?m, m>0, is taken. The proof, which is carried out by the method of decomposition, relies on the estimates of singular integrals and solutions of the transport equations in weighted spaces. Bibliography: 5 titles.     

Defining Relations for the Algebra of Invariants of 2×2 Matrices

We obtain defining relations of the algebra of invariants of the classical subgroups of GL ₂(C) acting by simultaneous conjugation on m-tuples of 2×2 complex matrices. The sets of defining relations look uniformly for all m2 and are derived by translation of classical results on invariant theory of orthogonal groups in the language of 2×2 matrix invariants, combined with arguments of representation theory of the general linear group GL _m (C) and ideas coming from the theory of algebras with polynomial identities.     

Computation of the commutator of 2 × 2 matrices via five multiplications

Algorithms for computing the commutator AB ? BA of 2 × 2 matrices A and B are proposed that involve five multiplications.     

Continuous Singularity of the Weak Limit of Convolution Powers of a Discrete Probability Measure on 2 × 2 Stochastic Matrices

Let be a probability measure on n 2 × 2 stochastic matrices, n an arbitrary positive integer, and = (w) lim _n ⁿ , such that the support of consists of 2 × 2 stochastic matrices of rank one, and as such, can be regarded as a probability measure on [0, 1]. We present simple sufficient conditions for to be continuous singular w.r.t. the Lebesgue measure on [0, 1]. We also determine , given .     

On the Distribution of Weakly Composite Numbers

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On the Growth of Composite Entire Functions

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