Bending invariant representations for surfaces

We derive new representations for the generalised Jacobian of a locally Lipschitz map between finite dimensional real Euclidean spaces as the lower limit (i.e., limit inferior) of the classical derivative of the map where it exists. The new representations lead to significantly shorter proofs for the basic properties of the subgradient and the generalised Jacobian including the chain rule. We establish that a sequence of locally Lipschitz maps between finite dimensional Euclidean spaces converges to a given locally Lipschitz map in the L-topology—that is, the weakest refinement of the sup norm topology on the space of locally Lipschitz maps that makes the generalised Jacobian a continuous functional—if and only if the limit superior of the sequence of directional derivatives of the maps in a given vector direction coincides with the generalised directional derivative of the given map in that direction, with the convergence to the limit superior being uniform for all unit vectors. We then prove our main result that the subspace of Lipschitz C ∞ maps between finite dimensional Euclidean spaces is dense in the space of Lipschitz maps equipped with the L-topology, and, for a given Lipschitz map, we explicitly construct a sequence of Lipschitz C ∞ maps converging to it in the L-topology, allowing global smooth approximation of a Lipschitz map and its differential properties. As an application, we obtain a short proof of the extension of Green’s theorem to interval-valued vector fields. For infinite dimensions, we show that the subgradient of a Lipschitz map on a Banach space is upper continuous, and, for a given real-valued Lipschitz map on a separable Banach space, we construct a sequence of Gateaux differentiable functions that converges to the map in the sup norm topology such that the limit superior of the directional derivatives in any direction coincides with the generalised directional derivative of the Lipschitz map in that direction.

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.

Normal functioning of the brain depends on the homeostasis (∼ steady state) of its various physiological sub-systems, one of which is the intracranial pressure (ICP) dynamic system. The ICP dynamic system of an injured brain is susceptible to various acute changes that should ideally be detected by ICP monitoring even for comatose patients. However, the status quo of ICP monitoring solely targets mean ICP. We aimed to demonstrate a novel approach to detect acute deviation from steady state of an ICP dynamic system in an absence of significant mean ICP changes. We hypothesized that steady state of ICP dynamic systems is reflected as ICP pulses of similar mean ICP levels resembling each other for a given subject. A general framework was used to derive such a steady-state indicator that can accommodate different metrics of inter-pulse distance and different statistics of the distance histograms. In addition to conventional Euclidean distance and Pearson correlation, geodesic distance between pulses was introduced as a novel metric. These different ways of calculating steady-state indicators under the proposed framework were evaluated on three types of continuous ICP recordings: (1) those between two consecutive brain imaging studies that demonstrated acute ventricular enlargement for slit ventricle syndrome (SVS) patients undergoing a trial of shunt externalization and clamping (SVS+); (2) those between consecutive brain imaging studies from the SVS patients under the same trial but without ventricular enlargement (SVS−); (3) overnight recordings from normal pressure hydrocephalus (NPH) patients. It was observed that only the standard deviation of geodesic distance correctly differentiated between SVS+ and SVS− and between SVS+ and NPH while avoiding discriminating between SVS− and NPH. It was also found that 45% SVS+ cases had a multimodal geodesic distance histogram while none of SVS− and 3.8% of NPH cases had such a multimodal histogram. Pulses with a large number of distant pulses for the five multimodal-histogram SVS+ cases fell in short time windows indicating that acute ventricular changes may have occurred in these confined time windows during which no significant changes of mean ICP were observed. In contrast, the pulses with a large number of distant pulses for the two multimodal-histogram NPH cases did not cluster temporally. In conclusion, the geodesic inter-pulse distance is a promising metric to quantify distance intrinsic to the underneath geometric structure of ICP signals and hence is a more suitable way to derive a steady-state indicator of an ICP dynamic system.

We study classes of mappings between finite and infinite dimensional Banach spaces that are monotone and mappings which are differences of monotone mappings (DM). We prove a Radó–Reichelderfer estimate for monotone mappings in finite dimensional spaces that remains valid for DM mappings. This provides an alternative proof of the Fréchet differentiability a.e. of DM mappings. We establish a Morrey-type estimate for the distributional derivative of monotone mappings. We prove that a locally DM mapping between finite dimensional spaces is also globally DM. We introduce and study a new class of the so-called UDM mappings between Banach spaces, which generalizes the concept of curves of finite variation.

Non-Euclidean dissimilarity measures can be well suited for building representation spaces that are more beneficial for pattern classification systems than the related Euclidean ones [1,2]. A non-Euclidean representation space is however cumbersome for training classifiers, as many statistical techniques rely on the Euclidean inner product that is missing there. In this paper we report our findings on the applicability of corrections that transform a non-Euclidean representation space into a Euclidean one in which similar or better classifiers can be trained. In a case-study based on four principally different classifiers we find out that standard correction procedures fail to construct an appropriate Euclidean space, equivalent to the original non-Euclidean one.

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.

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.

By a system (X, T) we mean a continuous selfmap T: X → X of a separable metrizable space X and by its linearization a topological embedding i: X → E into a linear system (E, L) consisting of a Hilbert or Euclidean space E and a continuous linear operator L: E → E and satisfying the equivariancy condition L ○ i = i ○ T. Our main results concern linearization by systems (E, L) in which the norm of L is < 1. By a weakening of the equivariancy condition L ○ i = i ○ T, we show that a system which does not admit a finite dimensional linearization may still be linearized in this modified sense in a finite dimensional space.

It is shown that the stochastic counterpart of the classical fixed point theorem for continuous maps in a finite dimensional Euclidean space (“Brouwer’s theorem”) is not, in general, true. This result implies, in particular, that a careful choice of invariant sets in the stochastic version of Brouwer’s theorem is necessary in the theory of stochastic nonlinear operators.

The systematic study of various mean-values of nonnegative valued functions is due in large part to the work of J. L. W. V. Jensen [12], A. Kolmogoroff [11], and B. Jessen [13]. In this paper we present a similar theory of certain means of functions whose values are taken from a collection of convex bodies in a finite dimensional Euclidean space. The range of nonnegative valued functions is totally ordered while the collection of convex bodies to be considered is partially ordered by set inclusion; corresponding to inequalities, we shall have inclusions, here to be called inequalities, between mean-values. Thus the discussion will furnish an example of a partially ordered system whose algebraic and topological structure is sufficient to admit a fairly elaborate theory of inequalities analogous to that of the real numbers. The first section is a resume of those parts of the work of Jensen, Kolmogoroff, and Jessen for which analogues will be developed. ?2 treats of pertinent material about star and convex bodies. Certain families of star bodies and of convex bodies are defined in the third section; these latter play the role of nonnegative valued functions. Two systems of power means of such families are described. Each system is, in its way, anialogous to the power means of nonnegative valued functions. In ?4 we discuss some special cases of power means including elementary means and the Riemaiin-Minkowski integrals of A. Dinghas [2], [3]. Also some rotation invariants of a convex body are described as power means of special families determined by the convex body. The fifth section begins with crucial properties of the means and contains an extremal characterization for the two systems of means defined in the third section. Analogues of Jensen's and Jessen's inequalities make up ?6. We also mentioni the limiting cases of power means as the power index becomes infinite, positively and negatively. As an applicationl of Jessen's inequality, a Brunn-Minkowski type theorem is proved. In the final section we discuss some further systems of meanls of convex bodies. 1. The definition of the elementary power means extends in a natural way to include power means of certain functions f which have finite, nonnegative values and whose domain T is a compact topological group equipped with its Haar measure. Iff is bounded and measurable over its domain, its pth- power mean is defined by

Finite groups admitting grassmannian 4-designs

In 1965, Shepard introduced a general interpolation formula for arbitrarily spaced data over any finite dimensional Euclidean space. The deficiencies in his initial results have been corrected and later derivations have been widely applied to surface fitting problems in aviation design and geology.This paper describes a new research direction in Shepard's method and a new application to computer graphics. In the first, the convex weighting functions are generalized by the creation of a user-defined mathematical domain which is independent of the surface. In the second, the original formula is applied to blend mapping projections for aviation visual simulation and an algorithm for rapid evaluation is presented.

Construction of unimodal random probability measures on finite dimensional Euclidean space is considered. The approach based on Bayesian nonparametric models and Convexity Theory. Specifically, the proposed model makes use of the special properties of convex sets and Choquet’s theorem. As a result, we get random probability measures that admit derivatives almost everywhere in R d .

It is well known that any stochastically continuous real valued stochastic process with independent increments defined on a compact time interval can be decomposed into a sum of independent processes, one of which is Gaussian with continuous sample paths, and the remainder of which have sample paths which are continuous except at a finite number of points with the discontinuities occurring at Poisson time points. The purpose of this note is to announce a proof of the above theorem in the case where the process takes values in an abelian group G. The detailed proof will appear elsewhere. The basic ideas of the proof in the case when G is finite dimensional Euclidean space are contained in Chapter VI of Gikhman and Skorohod (1965).

The concept of monotonicity and its various extensions existing in the literature in the classical setting are generalized in finite dimensional Euclidean spaces. With examples, as support to the proposed structure, it is shown when the traditional development fails but the proposed one works. As an application, the computation of probability under pairwise disjointness and mutual independence for a finite number of sets is shown to be tied to the proposed framework. Characterizations are given in terms of generalized convexity and its extension. Some basic results are shown which may point to applications to some of the popular areas of research in theoretical mathematical programming