An oscillation theory for second-order integral differential equations

The primary purpose of this paper is to give an oscillation theory for second-order integral differential equations. It is shown that this theory follows in a natural way as “a corollary” from the more abstract approximation theory of quadratic forms given previously by the author. Thus, our ideas are primarily constructive and quantitative as opposed to the usual qualitative methods. We also note that the usual oscillation theory for second-order differential equations follows directly by our methods. Furthermore, our methods provide a unified theory for eigenvalue problems, optimization problems, and numerical approximation problems within this setting.In Section 1 we give the preliminaries for the remainder of the paper. In Section 2 we define the basic quadratic form and integral differential equation and give the relationships between them. These relationships are used (in Section 3) to give a theory of oscillation in our setting and some basic oscillation results. Finally, in Section 4 we give some deeper oscillation results.To emphasize the unifying methods of our ideas, this paper is presented as a companion paper to “A Numerical Approximation Theory for Second Order Integral Differential Equations.”     

Elliptic quadratic forms,focal points,and a generalized theory of oscillation

The purpose of this paper is to generalize the theory, methods, and results for oscillation of second-order normal ordinary differential equations. This purpose is obtained by use of a theory of quadratic forms on Hilbert spaces given by Hestenes and the author.In particular, the ideas of this paper may be applied to second-order abnormal problems of differential equations, higher-order control problems, integral and partial differential equations, abstract approximation problems, and to finite dimensional approximations which lead to meaningful computer algorithms.For expository purposes some examples are included. Finally we show that specific existence and comparison theorems for the second-order case may be generalized to the 2nth-order case.     

Solving Standard Quadratic Optimization Problems via Linear, Semidefinite and Copositive Programming

The problem of minimizing a (non-convex) quadratic function over the simplex (the standard quadratic optimization problem) has an exact convex reformulation as a copositive programming problem. In this paper we show how to approximate the optimal solution by approximating the cone of copositive matrices via systems of linear inequalities, and, more refined, linear matrix inequalities (LMI's). In particular, we show that our approach leads to a polynomial-time approximation scheme for the standard quadratic optimzation problem. This is an improvement on the previous complexity result by Nesterov who showed that a 2/3-approximation is always possible. Numerical examples from various applications are provided to illustrate our approach.     

Weight-dependent congruence properties of modular forms

In this paper, we study congruence properties of modular forms in various ways. By proving a weight-dependent congruence property of modular forms, we give some sufficient conditions, in terms of the weights of modular forms, for a modular form to be non-p-ordinary. As applications of our main theorem we derive a linear relation among coefficients of new forms. Furthermore, congruence relations among special values of Dedekind zeta functions of real quadratic fields are derived.     

Effective Estimates on Integral Quadratic Forms: Masser’s Conjecture,Generators of Orthogonal Groups,and Bounds in Reduction Theory

In this paper we prove a conjecture of David Masser on small height integral equivalence between integral quadratic forms. Using our resolution of Masser’s conjecture we show that integral orthogonal groups are generated by small elements which is essentially an effective version of Siegel’s theorem on the finite generation of these groups. We also obtain new estimates on reduction theory and representation theory of integral quadratic forms. Our line of attack is to make and exploit the connections between certain problems about quadratic forms and group actions, whence we may study the problem in the well-developed theory of homogeneous dynamics, arithmetic groups, and the spectral theory of automorphic forms.     

BCS Model Hamiltonian of the Theory of Superconductivity as a Quadratic Form

Bogolyubov proved that the average energies (per unit volume) of the ground states for the BCS Hamiltonian and the approximating Hamiltonian asymptotically coincide in the thermodynamic limit. In the present paper, we show that this result is also true for all excited states. We also establish that, in the thermodynamic limit, the BCS Hamiltonian and the approximating Hamiltonian asymptotically coincide as quadratic forms.     

Reduction theory over quadratic imaginary fields

Hurwitz developed a reduction theory for real binary quadratic forms of positive discriminant based on least-remainder continued fractions. For each quadratic imaginary field k, we develop a similar theory for complex binary quadratic forms of nonzero discriminant. This uses a Markov partition for the geodesic flow over the quotient of hyperbolic 3-space by the Bianchi group B_k. When k has a Euclidean algorithm, our theory is based on least-remainder continued fractions.     

Degrees of functionals

In this paper we will discuss some problems of degree-theoretic nature in connection with recursion in normal objects of higher types.Harrington [2] and Loewenthal [6] have proved some results concerning Post's problem and the Minimal Pair Problem, using recursion modulo subindividuals. Our degrees will be those obtained from Kleene-recursion modulo individuals. To solve our problems we then have to put some extra strength to ZFC. We will first assume V = L, and then we restrict ourselves to the situation of a recursive well-ordering and Martin's axiom.We assume familiarity with recursion theory in higher types as presented in Kleene [3]. Further backround is found in Harrington [2], Moldestad [9] and Normann [11]. We will survey the parts of these papers that we need.In Section 1 we give the general background for the arguments used later. In Section 2 we prove some lemmas assuming V = L. In section 3, assuming V = L we solve Post's problem and another problem using the finite injury method. We will thereby describe some of the methods needed for the more complex priority argument of Section 4 where we give a solution of the minimal pair problem for extended r.e. degress of functionals.In Section 5 we will see that if Martin's Axiom holds and we have a minimal well-ordering of tp (1) recursive in ³E, we may use the same sort of arguments as in parts 3 and 4.     

Convexity Conditions and the Legendre-Fenchel Transform for the Product of Finitely Many Positive Definite Quadratic Forms

While the product of finitely many convex functions has been investigated in the field of global optimization, some fundamental issues such as the convexity condition and the Legendre-Fenchel transform for the product function remain unresolved. Focusing on quadratic forms, this paper is aimed at addressing the question: When is the product of finitely many positive definite quadratic forms convex, and what is the Legendre-Fenchel transform for it? First, we show that the convexity of the product is determined intrinsically by the condition number of so-called ‘scaled matrices’ associated with quadratic forms involved. The main result claims that if the condition number of these scaled matrices are bounded above by an explicit constant (which depends only on the number of quadratic forms involved), then the product function is convex. Second, we prove that the Legendre-Fenchel transform for the product of positive definite quadratic forms can be expressed, and the computation of the transform amounts to finding the solution to a system of equations (or equally, finding a Brouwer’s fixed point of a mapping) with a special structure. Thus, a broader question than the open “Question 11” in Hiriart-Urruty (SIAM Rev. 49, 225–273, 2007) is addressed in this paper.     

Hidden conic quadratic representation of some nonconvex quadratic optimization problems

The problem of minimizing a quadratic objective function subject to one or two quadratic constraints is known to have a hidden convexity property, even when the quadratic forms are indefinite. The equivalent convex problem is a semidefinite one, and the equivalence is based on the celebrated S-lemma. In this paper, we show that when the quadratic forms are simultaneously diagonalizable (SD), it is possible to derive an equivalent convex problem, which is a conic quadratic (CQ) one, and as such is significantly more tractable than a semidefinite problem. The SD condition holds for free for many problems arising in applications, in particular, when deriving robust counterparts of quadratic, or conic quadratic, constraints affected by implementation error. The proof of the hidden CQ property is constructive and does not rely on the S-lemma. This fact may be significant in discovering hidden convexity in some nonquadratic problems.     

Banach ideals of operators with applications

We present some results concerning the general theory of Banach ideals of operators and give several applications to Banach space theory. We give, in Section 3, new proofs of several recent results, as well as new operator characterizations of the $\text{L}$_p-spaces of Lindenstrauss and Pelczynski. In Section 4 we prove that the space of absolutely summing operators from E to F is reflexive if both E and F are reflexive and E has the approximation property. Section 5 concerns Hilbert spaces. In particular, we compute the relative projection constant of Hilbert spaces in L_p(μ)-spaces.     

On the uniform asymptotic stability for a linear integro-differential equation of Volterra type

In this paper, we give a necessary and sufficient condition on the uniform asymptotic stability of the zero solution of a linear integro-differential equation of Volterra type where the ordinary part is ax(t). We put emphasis on the case a>0. The proofs of our results are carried out by using the root analysis of the characteristic equation. In Section 5 we give some conjectures.     

Time-dependent operator matrices and inhomogeneous Cauchy problems

The theory of operator matrices has been applied recently in various fields (cf. [4], [9], [10]). In particular, it is possible to solve inhomogeneous abstract Cauchy problems using the theory of operator matrices on appropriate product spaces (see [9]). For nonautonomous Cauchy problems, however, it seems that there is still lacking a systematic theory of operator matrices. As a first step towards such a theory, in this paper we will show that nonautonomous inhomogeneous Cauchy problems can be solved by using operator matrices. First we will give simplified proofs of known wellposedness results for these problems (Section 2). In Section 3 we use the results of Section 2 for a discussion of nonautonomous Cauchy problems in the context of abstract operator matrices. This paper is part of a research project supported by the Deutsche Forschungsgemeinschaft DFG. The support of DAAD is gratefully acknowledged. The authors wish to thank Rainer Nagel and Klaus Jochen Engel for helpful comments.     

Statistical inference for the optimal approximating model

In the setting of high-dimensional linear models with Gaussian noise, we investigate the possibility of confidence statements connected to model selection. Although there exist numerous procedures for adaptive (point) estimation, the construction of adaptive confidence regions is severely limited (cf. Li in Ann Stat 17:1001–1008, 1989). The present paper sheds new light on this gap. We develop exact and adaptive confidence regions for the best approximating model in terms of risk. One of our constructions is based on a multiscale procedure and a particular coupling argument. Utilizing exponential inequalities for noncentral χ ²-distributions, we show that the risk and quadratic loss of all models within our confidence region are uniformly bounded by the minimal risk times a factor close to one.     

New efficient numerical procedures for solving stochastic variational problems with a priori maximum pointwise error estimates

In a previous paper we gave a new formulation and derived the Euler equations and other necessary conditions to solve strong, pathwise, stochastic variational problems with trajectories driven by Brownian motion. Thus, unlike current methods which minimize the control over deterministic functionals (the expected value), we find the control which gives the critical point solution of random functionals of a Brownian path and then, if we choose, find the expected value.This increase in information is balanced by the fact that our methods are anticipative while current methods are not. However, our methods are more directly connected to the theory and meaningful examples of deterministic variational theory and provide better means of solution for free and constrained problems. In addition, examples indicate that there are methods to obtain nonanticipative solutions from our equations although the anticipative optimal cost function has smaller expected value.In this paper we give new, efficient numerical methods to find the solution of these problems in the quadratic case. Of interest is that our numerical solution has a maximal, a priori, pointwise error of O(h3/2) where h is the node size. We believe our results are unique for any theory of stochastic control and that our methods of proof involve new and sophisticated ideas for strong solutions which extend previous deterministic results by the first author where the error was O(h²).We note that, although our solutions are given in terms of stochastic differential equations, we are not using the now standard numerical methods for stochastic differential equations. Instead we find an approximation to the critical point solution of the variational problem using relations derived from setting to zero the directional derivative of the cost functional in the direction of simple test functions.Our results are even more significant than they first appear because we can reformulate stochastic control problems or constrained calculus of variations problems in the unconstrained, stochastic calculus of variations formulation of this paper. This will allow us to find efficient and accurate numerical solutions for general constrained, stochastic optimization problems. This is not yet being done, even in the deterministic case, except by the first author.     

Dynamic lot-sizing with price changes and price-dependent holding costs

This paper proposes a new formulation of the dynamic lot-sizing problem with price changes which considers the unit inventory holding costs in a period as a function of the procurement decisions made in previous periods. In Section 1, the problem is defined and some of its fundamental properties are identified. A dynamic programming approach is developed to solve it when solutions are restricted to sequential extreme flows, and results from location theory are used to derive an O(T²) algorithm which provides a provably optimal solution of an integer linear programming formulation of the general problem. In Section 2, a heuristic is developed for the case where the inventory carrying rates and the order costs are constant, and where the item price can change once during the planning horizon. Permanent price increases, permanent price decreases and temporary price reductions are considered. In Section 3, extensive testing of the various optimal and heuristic algorithms is reported. Our results show that, in this context, the two following intuitive actions usually lead to near optimal solutions: accumulate stock at the lower price just prior to price increase and cut short on orders when a price decrease is imminent.     

Primes represented by binary quadratic forms

In 1882 Weber showed that any primitive binary quadratic form with integral coefficients represents infinitely many primes in any arithmetic progression consistent with the generic characters of the form. In this paper it is shown that for any two primitive integral binary quadratic forms with unequal but fundamental discriminants, there is an infinite set of prime numbers p in any arithmetic progression consistent with the generic characters of the forms such that both forms represent p.     

On the norm theorem for semisingular quadratic forms

The aim of this paper is to prove some results concerning the norm theorem for semisingular quadratic forms, i.e., those which are neither nonsingular nor totally singular. More precisely, we will give necessary conditions in order that an irreducible polynomial, possibly in more than one variable, is a norm ofa semisingular quadratic form, and we prove that our conditions are sufficient if the polynomial is given by a quadratic form which represents 1. As a consequence, we extend the Cassels-Pflster subform theorem to the case of semisingular quadratic forms.     

Geometry in Grassmannians and a generalization of the dilogarithm

In this paper we introduce some polyhedra in Grassman manifolds which we call Grassmannian simplices. We study two aspects of these polyhedra: their combinatorial structure (Section 2) and their relation to harmonic differential forms on the Grassmannian (Section 3). Using this we obtain results about some new differential forms, one of which is the classical dilogarithm (Section 1). The results here unite two threads of mathematics that were much studied in the 19th century. The analytic one, concerning the dilogarithm, goes back to Leibnitz (1696) and Euler (1779) and the geometric one, concerning Grassmannian simplices, can be traced to Binet (1811). In Section 4, we give some of this history along with some recent related results and open problems. In Section 0, we give as an introduction an account in geometric terms of the simplest cases.