Logarithmic derivatives of solutions to linear differential equations

Given an ordinary differential field of characteristic zero, it is known that if and satisfy linear differential equations with coefficients in , then is algebraic over . We present a new short proof of this fact using Gröbner basis techniques and give a direct method for finding a polynomial over that satisfies. Moreover, we provide explicit degree bounds and extend the result to fields with positive characteristic. Finally, we give an application of our method to a class of nonlinear differential equations.

     

Optical phenomena in photoelastic models by the rotation of principal axes

The basic equations of three-dimensional photoelasticity are derived in a form which is simpler than that of equations known previously. Using the matrix representation of the solution of these equations, it is also shown that when rotation of principal axes is present there always exist two perpendicular directions of polarizer by which the light emerging from the model is linearly polarized. These polarization directions of the incident and emergent light are named primary and secondary characteristic directions, respectively. The experimental determination of characteristic directions, as well as of the phase retardation, gives three equations on every light path to determine the stress components in a three-dimensional model. A general algorithm of the method of characteristic directions is presented, and its application by determination of stress in shells by normal and tangential incidence is described. A further extension of the method to the general axisymmetric problem has been suggested.     

On the Positivity of the Coefficients of a Certain Polynomial Defined by Two Positive Definite Matrices

It is shown that the polynomialp(t) = Tr[(A+tB)^m]has positive coefficients when m = 6 and A and B are any two 3-by-3 complex Hermitian positive definite matrices. This case is the first that is not covered by prior, general results. This problem arises from a conjecture raised by Bessis, Moussa, and Villani in connection with a long-standing problem in theoretical physics. The full conjecture, as shown recently by Lieb and Seiringer, is equivalent to p(t) having positive coefficients for any m and any two n-by-n positive definite matrices. We show that, generally, the question in the real case reduces to that of singular A and B, and this is a key part of our proof.     

Optical theory of the multilayer-reflection technique for three-dimensional photoelastic studies

Using the theory of characteristic directions developed previously by the author^{2, 3} optical phenomena by the multilayer-reflection technique¹ in the general case are studied. Recursive formulas which enable successive determination of the parameters of all the photo-elastic layers without any complication of the experimental technique are derived.     

An elementary and constructive solution to Hilbert's 17th Problem for matrices

We give a short and elementary proof of a theorem of Procesi, Schacher and (independently) Gondard, Ribenboim that generalizes a famous result of Artin. Let be an symmetric matrix with entries in the polynomial ring . The result is that if is positive semidefinite for all substitutions , then can be expressed as a sum of squares of symmetric matrices with entries in . Moreover, our proof is constructive and gives explicit representations modulo the scalar case.

     

Solvability of symmetric word equations in positive definite letters

Let S(X, B) be a symmetric (‘palindromic’) wordin two letters X and B. A theorem due to Hillar and Johnsonstates that for each pair of positive definite matrices B andP, there is a positive definite solution X to the word equationS(X, B)=P. They also conjectured that these solutions are finiteand unique. In this paper, we resolve a modified version ofthis conjecture by showing that the Brouwer degree of such anequation is equal to 1 (in the case of real matrices). It followsthat, generically, the number of solutions is odd (and thusfinite) in the real case. Our approach allows us to addressthe more subtle question of uniqueness by exhibiting equationswith multiple real solutions, as well as providing a secondproof of the result of Hillar and Johnson in the real case.     

Polynomial recurrences and cyclic resultants

Let be an algebraically closed field of characteristic zero and let . The -th cyclic resultant of is

   Res

A generic monic polynomial is determined by its full sequence of cyclic resultants; however, the known techniques proving this result give no effective computational bounds. We prove that a generic monic polynomial of degree is determined by its first cyclic resultants and that a generic monic reciprocal polynomial of even degree is determined by its first of them. In addition, we show that cyclic resultants satisfy a polynomial recurrence of length . This result gives evidence supporting the conjecture of Sturmfels and Zworski that resultants determine . In the process, we establish two general results of independent interest: we show that certain Toeplitz determinants are sufficient to determine whether a sequence is linearly recurrent, and we give conditions under which a linearly recurrent sequence satisfies a polynomial recurrence of shorter length.

     

Hidden Hypergraphs,Error-Correcting Codes,and Critical Learning in Hopfield Networks

In 1943, McCulloch and Pitts introduced a discrete recurrent neural network as a model for computation in brains. The work inspired breakthroughs such as the first computer design and the theory of finite automata. We focus on learning in Hopfield networks, a special case with symmetric weights and fixed-point attractor dynamics. Specifically, we explore minimum energy flow (MEF) as a scalable convex objective for determining network parameters. We catalog various properties of MEF, such as biological plausibility, and then compare to classical approaches in the theory of learning. Trained Hopfield networks can perform unsupervised clustering and define novel error-correcting coding schemes. They also efficiently find hidden structures (cliques) in graph theory. We extend this known connection from graphs to hypergraphs and discover n-node networks with robust storage of 2Ω(n1−ϵ) memories for any ϵ>0. In the case of graphs, we also determine a critical ratio of training samples at which networks generalize completely.