Monotonically Controlled Mappings

Recurrence formula for some higher order evolution equations

Linear algebra is considered an essential mathematical theory that has many engineering applications. While many theorem provers support linear spaces, they only consider finite dimensional spaces. In addition, available libraries only deal with real vectors, whereas complex vectors are extremely useful in many fields of engineering. In this paper, we propose a new linear space formalization which covers both finite and infinite dimensional complex vector spaces, implemented in HOL-Light. We give the definition of a linear space and prove many properties about its operations, e.g., addition and scalar multiplication. We also formalize a number of related fundamental concepts such as linearity, hermitian operation, self-adjoint, and inner product space. Using the developed linear algebra library, we were able to implement basic definitions about quantum mechanics and use them to verify a quantum beam splitter, an optical device that has many applications in quantum computing.

Background. The area formulae are well known for surfaces embedded into a finite-dimensional Euclidean space. However, in the case of an infinite-dimensional Banach manifold, such formulae cannot be used. Thus, a problem of finding an alternative approach to the surface measures construction appears, that, on the one hand, leads to classical results in finite-dimensional case, and on the other hand, can be used for infinite-dimensional Banach manifolds. Objective. The aim of the paper is to get a construction of surface measure induced by the Lebesgue measure and the associated form for a parametrically defined surface embedded into finite-dimensional Euclidean space. Show the consistency of surface area calculation by this construction with an area calculated by using well-known classical formulae. Methods. Basic results of mathematical analyses, measure theory and differential geometry are used. Results. An alternative construction of surface measures induced by the Lebesgue measure on surfaces in finite-dimensional space is obtained. It is shown that such approach is consistent with the classical definition of the surface area. Conclusions. The construction of surface measures suggested for infinite-dimensional spaces is a generalization of the classical approach in finite-dimensional spaces. Therefore further investigation of the described approach seems to be reasonable.

Let X be a finite or infinite dimensional complex Banach space. We characterize the bounded weighted composition operators between the Bloch type space and the Hardy space on the unit ball of X, extending several known results for finite dimensional domains.

Finite dimensional approximation and Newton-based algorithm for stochastic approximation in Hilbert space

This work studies the structural properties and convergence approach of chance-constrained optimization of boundary-value elliptic partial differential equation systems (CCPDEs). The boundary conditions are random input functions deliberated from the boundary of the partial differential equation (PDE) system and in the infinite-dimensional reflexive and separable Banach space. The structural properties of the chance constraints studied in this paper are continuity, closedness, compactness, convexity, and smoothness of probabilistic uniform or pointwise state constrained functions and their parametric approximations. These are open issues even in the finite-dimensional Banach space. Thus, it needs finite-dimensional and smooth parametric approximation representations. We propose a convex approximation approach to nonconvex CCPDE problems. When the approximation parameter goes to zero from the right, the solutions of the relaxation and compression approximations converge asymptotically to the optimal solution of the original CCPDE. Due to the convexity of the problem, a global solution exists for the proposed approximations. Numerical results are provided to demonstrate the plausibility and applicability of the proposed approach.

There are many characterizations which distinguish finite-dimensional normed linear spaces from infinite-dimensional normed linear spaces. Perhaps the best known of these is the compactness of the unit ball. Recently, V. Klee [1] showed that in any infinite-dimensional normed linear space there exists a decreasing sequence of unbounded but linearly bounded closed convex sets whose intersection is empty. We will give here a somewhat similar condition which holds in all infinite-dimensional normed linear spaces but does not hold in any finite-dimensional space. We begin with the following terminology. U will denote the unit ball {x: ||x|| S 1 } and S the unit sphere {x: ||x|| = 1 }. A shell will be any set of the form {x: ri? IIx-xalI ? r2 for 0 1 such that (x+p U)Cr (-x+p U) CO, and S will be called finitely nonfiat if there exists a finite number of points {xi} '1 in S such that n(xj+pU) Co. A collection of sets has the finite intersection property if the intersection of the sets in any finite subcollection is not empty.

Natural generalizations of planar results on volume ratios with respect to convex bodies in finite dimensional Banach spaces are given. The underlying paper contains inequalities for the volume of simplices with edges of equal length in a Banach space.

SOME RESULTS RELATED TO THE SEPARABLE QUOTIENT PROBLEM

Characterization of simultaneous similarity for commuting m m - tuples of operators is an open problem even in finite-dimensional spaces; known as “A wild problem in linear algebra”. In this paper we offer a criterion for simultaneous similarity of m m -tuples of k k -cyclic commuting operators on an arbitrary Banach space. Moreover, we obtain an additional equivalence condition in the case of finite dimensional Banach spaces, which extends the result found by Shekhtman [Math. Stat. 1 (2013), pp. 157–161] for pairs of cyclic commuting matrices. We also present two applications of our results, one in the case of general multiplication operators on Banach spaces of analytic function, and one for m m -tuples of commuting square matrices.

Over the past years, a significant interest has developed in the study of holomorphic functions defined on a domain in an infinite-dimensional Banach space and of their constituents (via Taylor expansions), the homogeneous polynomials. Many of the questions that have been studied have arisen from considerations of infinitedimensional topology and from standard function algebra questions [CCG]. Recently, there has been an interest in connecting the well-developed theory of the geometry of Banach spaces with the function theory questions that have been studied classically, and some progress has been made in this direction [ACG; D; F; CCG; CGJ]. In addition, connections between properties of polynomials and geometry of the unit ball has been of interest (see [GJL] for a survey of this topic). The present work is an attempt to study some of the properties of bounded analytic functions on the unit ball of an infinite-dimensional Banach space. In particular, we are interested in understanding something of boundary behavior; we combine techniques from the several fields to investigate it, especially with regard to the interplay with convexity and smoothness. Many of the results here apply to the classical “nice” reflexive spaces, such as lp and Lp (1 < p <∞). It is almost certain that there is much more to be learned even about the Hilbert space case. We consider the boundary behavior of H∞ functions on B, the open unit ball of an infinite-dimensional complex Banach space that has the geometric properties of uniform convexity, uniform smoothness, or both. By uniform smoothness, we mean uniform (real) Frechet differentiability of the norm, with the space considered as a real Banach space. Uniform convexity will mean that the dual is uniformly smooth; since spaces with either property are reflexive, this definition is complete. To be specific however, we state the following (after [LT]).

The asymptotic behavior of the evolution families is a widely interesting topic in mathematics over time. In 1930, O. Perron was the first one who established the connection between the asymptotic behavior of the solution of the homogenous differential equation and the associated non-homogeneous equation, in finite dimensional spaces. Further, the result was extended for infinite dimensional spaces. The case of dynamical systems described by evolution processes was studied by C. Chicone and Y. Latushkin. One of the most remarkable results in the theory of stability of dynamical systems has been obtained by R. Datko in 1970 for the particular case of C0-semigroups. Practically, R. Datko defines a characterization for uniform exponential stability of the C0-semigroups. Later, it was proved that a similar characterization is also valid for two-parameter evolution families.In this paper we obtain different versions of a well-known theorem of R. Datko for uniform and nonuniform exponential bounded evolution families. More precisely, we obtain theorems that characterize the nonuniform and uniform exponential stability of evolution families with uniform and nonuniform exponential growth. We show that, if we choose K dependent of t0 in the form of Datko's theorem used by C. Stoica and M. Megan, we obtain a result of nonuniform exponential stability, which is no longer possible in the original form of Datko's theorem.In conclusion, we generalize the results initially obtained by Datko (1972) and Preda and Megan (1985), by presenting some sufficient conditions for the nonuniform exponential stability of evolution families with nonuniform exponential growth.

In a recent series of short papers [2, 3, 41 I have been discussing the relationships of uniform convexity with certain other properties of normed vector spaces. In this paper I propose to discuss relationships between uniform convexity, factor spaces, and conjugate spaces. Because of the large number of special results which are needed for two dimensional spaces-and which are false in general-the paper will be divided into two parts: In Part I (??2-4) B is two dimensional; in Part II (??5-7) this restriction is removed and B may be any normed vector space. To recall the definition, B is said to be uniformly convex if there exists a function 6 such that 0 e, is a modulus of convexity for B. Note that if 51 is nowhere greater than 5, a modulus of convexity for B, then 5l is a modulus of convexity for B. Because of the pointwise nature of the definition of uniform convexity, it is clear that B is uniformly convex with modulus of convexity a if and only if all the two dimensional subspaces of B have the common modulus of convexity 6. One of the most useful results of this investigation (Theorem 5.5) is the complementary fact that B is uniformly convex if and only if all the two dimensional factor spaces of B have a common modulus of convexity. In the study of the effect of uniform convexity of B on the nature of the conjugate space B*, this result makes it possible to reduce the problem to the study of two dimensional spaces. In such a space B uniform convexity is equivalent to strict convexity: that is, a two (or finite) dimensional space is uniformly convex if and only if there does not exist a line segment of positive length all of whose points are of norm one. It has been observed [1, Footnote 13] that such a line segment on the unit sphere is equivalent to the existence of a sharp edge on the unit sphere in B*. The attempt to describe a sharp edge of the unit sphere in a finite dimensional space B in terms of the norm in that space leads to the condition that there exists a k > 0 such that for any e > 0 a pair of points bi and b2 exists such that II b -b2 11 k 11 bl-be 11 . Contradicting this suggests the following condition, a sort of dual concept to uniform convexity. A space B is said to be uniformly flattened if there exists a function v7 positive for 0 < e ? 2 such that lim.b.o 7(e) = 0 while (211 b + b2 11)/i b2 11 < 7(e) if J]b11 = JJ b1 = 1 and 11 bb2 ? || ; 375

This paper shows if $X$ is an infinite dimensional Banach space, $X$ contains a linearly independent arc. Also based on the continuum hypothesis, that if $X$ is an infinite dimensional Banach space and card $X = c$, then $X$ contains a dense arcwise connected Hamel basis.

The notion of the shell of aHilbert spa ce opera tor, which is auseful genera liza tion (proposed by Wielandt) of the numerical range, is extended to operators in spaces with an indefinite inner product. For the most part, finite dimensional spaces are considered. Geometric properties of shells (convexity, boundedness, being a subset of a line, etc.) are described, as well as shells of operators in two dimensional indefinite inner product spaces. For normal operators, it is conjectured that the shell is convex and its closure is polyhedral; the conjecture is proved for indefinite inner product spaces of dimension at most three, and for finite dimensional inner product spaces with one positive eigenvalue.