A minimal set-valued strong derivative for vector-valued Lipschitz functions

A set-valued derivative for a function at a point is a set of linear transformations whichapproximates the function near the point. This is stated precisely, and it is shown that, in general, there is not a unique minimal set-valued derivative for functions in the family of closed convex sets of linear transformations. For Lipschitz functions, a construction is given for a specific set-valued derivative, which reduces to the usual derivative when the function is strongly differentiable, and which is shown to be the unique minimal set-valued derivative within a certain subfamily of the family of closed convex sets of linear transformations. It is shown that this constructed set may be larger than Clarke's and Pourciau's set-valued derivatives, but that no irregularity is introduced.The author would like to thank Professor H. Halkin for numerous discussions of the material contained here.     

Lower estimates of circuit complexity in the basis of antichain functions

The antichain function is a characteristic function of an antichain in the Boolean cube. The set of antichain functions is an infinite complete basis. We study the computational complexity of Boolean functions over an antichain functional basis. In this paper we prove an asymptotic lower bound of order $\sqrt n $ for the computational complexity of a linear function, a majority function, and almost all Boolean functions of n variables.     

A parametric successive underestimation method for convex multiplicative programming problems

This paper addresses itself to the algorithm for minimizing the product of two nonnegative convex functions over a convex set. It is shown that the global minimum of this nonconvex problem can be obtained by solving a sequence of convex programming problems. The basic idea of this algorithm is to embed the original problem into a problem in a higher dimensional space and to apply a branch-and-bound algorithm using an underestimating function. Computational results indicate that our algorithm is efficient when the objective function is the product of a linear and a quadratic functions and the constraints are linear. An extension of our algorithm for minimizing the sum of a convex function and a product of two convex functions is also discussed.     

Geometry of cotangent bundle of Heisenberg group

In this paper, the classification of left invariant Riemannian metrics on the cotangent bundle of the (2n+1)-dimensional Heisenberg group up to the action of the automorphism group is presented. Moreover, it is proved that the complex structure on this group is unique, and the corresponding pseudo-Kähler metrics are described and shown to be Ricci flat. It is known that this algebra admits an ad-invariant metric of neutral signature. Here, the uniqueness of such metric is proved.     

Evidential reasoning in large partially ordered sets

The Dempster-Shafer theory of belief functions has proved to be a powerful formalism for uncertain reasoning. However, belief functions on a finite frame of discernment Ω are usually defined in the power set 2^Ω, resulting in exponential complexity of the operations involved in this framework, such as combination rules. When Ω is linearly ordered, a usual trick is to work only with intervals, which drastically reduces the complexity of calculations. In this paper, we show that this trick can be extrapolated to frames endowed with an arbitrary lattice structure, not necessarily a linear order. This principle makes it possible to apply the Dempster-Shafer framework to very large frames such as the power set, the set of partitions, or the set of preorders of a finite set. Applications to multi-label classification, ensemble clustering and preference aggregation are demonstrated.     

Uniqueness and nonuniqueness of the solution to the problem of determining the source in the heat equation

An initial–boundary value problem for the two-dimensional heat equation with a source is considered. The source is the sum of two unknown functions of spatial variables multiplied by exponentially decaying functions of time. The inverse problem is stated of determining two unknown functions of spatial variables from additional information on the solution of the initial–boundary value problem, which is a function of time and one of the spatial variables. It is shown that, in the general case, this inverse problem has an infinite set of solutions. It is proved that the solution of the inverse problem is unique in the class of sufficiently smooth compactly supported functions such that the supports of the unknown functions do not intersect. This result is extended to the case of a source involving an arbitrary finite number of unknown functions of spatial variables multiplied by exponentially decaying functions of time.     

Homogenization and Geometric Linearization for Multi-Well Energies

The concept of linear elasticity is to assume that the stored energy has one-well structure and to consider small displacements. The energy function is then expanded around the equilibrium state and higher-order terms are neglected. It was proved in [3] that this concept really gives results that provide a good approximation for the actual problem. On the other hand, for materials with fine periodic structure we accomplish a simplification of the problem in an analogous manner by homogenizing the energy. It was recently shown by Müller and Neukamm in [8] that both processes, homogenization and linearization, are interchangeable for such elastic energies. If we consider an energy with multiple wells then, under some reasonable conditions, it is also possible to (geometrically) linearize the problem. The second author proved that quasiconvexification of the limit function yields the desired result. We present that in this case both processes still commute. This is not a priori clear since the proof by Müller and Neukamm significantly rests upon the one-well structure and properties of quadratic forms. They also provide an example when the statement does not hold. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Linear maps of holomorphic functions on commutative algebras

Generalized function theory on finite dimensional linear associative algebras over the fieldo freals has pro- duced many forms o f generalized regular1ty Of particularin - terest are the forms o fregularity known as S-regularity and F- regularity. It is well known that functions regular in one sense need not be regular in the other. Here we prove a transformation theorem which allows the formation o f F-regular functions from S- regular functions. Furthermore, a classi flcation theorem is proved that determines the algebras for which this transformation property holds.     

Approximate Separability of the Green's Function of the Helmholtz Equation in the High Frequency Limit

The minimum number of terms that are needed in a separable approximation for a Green's function reveals the intrinsic complexity of the solution space of the underlying differential equation. It also has implications for whether low‐rank structures exist in the linear system after numerical discretization. The Green's function for a coercive elliptic differential operator in divergence form was shown to be highly separable [2], and efficient numerical algorithms exploiting low‐rank structures of the discretized systems were developed. In this work, a new approach to study the approximate separability of the Green's function of the Helmholtz equation in the high‐frequency limit is developed. We show (1) lower bounds based on an explicit characterization of the correlation between two Green's functions and a tight dimension estimate for the best linear subspace to approximate a set of decorrelated Green's functions, (2) upper bounds based on constructing specific separable approximations, and (3) sharpness of these bounds for a few case studies of practical interest. © 2018 Wiley Periodicals, Inc.     

Effort estimates through project complexity

This paper reports the results obtained from use of project complexity parameters in modeling effort estimates. It highlights the attention that complexity has recently received in the project management area. After considering that traditional knowledge has consistently proved to be prone to failure when put into practice on actual projects, the paper endorses the belief that there is a need for more open-minded and novel approaches to project management. With a view to providing some insight into the opportunities that integrate complexity concepts into model building offers, we extend the work previously undertaken on the complexity dimension in project management. We do so analyzing the results obtained with classical linear models and artificial neural networks when complexity is considered as another managerial parameter. For that purpose, we have used the International Software Benchmarking Standards Group data set. The results obtained proved the benefits of integrating the complexity of the projects at hand into the models. They also addressed the need of a complex system, such as artificial neural networks, to capture the fine nuances of the complex systems to be modeled, the projects.     

Complexity Estimation for an Algorithm of Searching for Zero of a Piecewise Linear Convex Function

It is known that the problem of the orthogonal projection of a point to the standard simplex can be reduced to solution of a scalar equation. In this article, the complexity is analyzed of an algorithm of searching for zero of a piecewise linear convex function which is proposed in [30]. The analysis is carried out of the best and worst cases of the input data for the algorithm. To this end, the largest and smallest numbers of iterations of the algorithm are studied as functions of the size of the input data. It is shown that, in the case of equality of elements of the input set, the algorithm performs the smallest number of iterations. In the case of different elements of the input set, the number of iterations is maximal and depends rather weakly on the particular values of the elements of the set. The results of numerical experiments with random input data of large dimension are presented.     

On the full and block-decoupling of nonlinear functions

We review a method that decouples multivariate functions into linear combinations of a set of univariate (or simpler multivariate) functions of transformed variables. In this way the given nonlinear multiple-input-multiple-output function is decoupled into a structure having simpler parallel internal branches that are linked by linear transformations to the original inputs and outputs. The procedure collects first-order information by evaluating the Jacobian matrix of the given function in a set of points. These matrices are stacked into a three-way tensor, whose decomposition reveals the decoupled representation. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Euler Obstruction and Defects of Functions on Singular Varieties

Several authors have proved Lefschetz type formulas for thelocal Euler obstruction. In particular, a result of this typehas been proved that turns out to be equivalent to saying thatthe local Euler obstruction, as a constructible function, satisfiesthe local Euler condition (in bivariant theory) with respectto general linear forms. The purpose of the paper is to determinewhat prevents the local Euler obstruction from satisfying thelocal Euler condition with respect to functions which are singularat the considered point. This is measured by an invariant (or‘defect’) of such functions. An interpretation ofthis defect is given in terms of vanishing cycles, which allowsit to be calculated algebraically. When the function has anisolated singularity, the invariant can be defined geometrically,via obstruction theory. This invariant unifies the usual conceptsof the Milnor number of a function and the local Euler obstructionof an analytic set.     

Any 2-asummable bipartite function is weighted threshold

Possible characterizations of which positive boolean functions are weighted threshold were studied in the 60s and 70s. It is known that a boolean function is weighted threshold if and only if it is k-asummable for every value of k. Furthermore, for some particular subfamilies of functions (those with up to eight variables, and graph functions), it is known that a function is weighted threshold if and only if it is 2-asummable.In this work we prove that bipartite functions also satisfy this property: a bipartite function is weighted threshold if and only if it is 2-asummable. In a bipartite function the set of variables can be partitioned in two classes, such that all the variables in the same class play exactly the same role in the function.     

Arithmetic functions and the Cauchy product

It is known that under the Dirichlet product, the set of arithmetic functions in several variables becomes a unique factorization domain. A. Zaharescu and M. Zaki proved an analog of the ABC conjecture in this ring and showed that there exists a non-trivial solution to the Fermat equation $$z^n=x^n+y^n$$ ($$n\ge 3$$). It is also known that under the Cauchy product, the set of arithmetic functions becomes a unique factorization domain. In this paper, we consider the ring of arithmetic functions in several variables under the Cauchy product and prove an analog of the ABC conjecture in this ring to show that there exists a non-trivial solution to the Fermat equation $$z^n=x^n+y^n$$ ($$n\ge 3$$).     

The Embedding Theory of Native Spaces

In the theory of radial basis functions, mathematicians use linear combinations of the translates of the radial basis functions as interpolants. The set of these linear combinations is a normed vector space. This space can be completed and become a Hilbert space, called native space, which is of great importance in the last decade. The native space then contains some abstract elements which are not linear combinations of radial basis functions. The meaning of these abstract elements is not fully known. This paper presents some interpretations for the these elements. The native spaces are embedded into some well-known spaces. For example, the Sobolev-space is shown to be a native space. Since many differential equations have solutions in the Sobolev-space, we can therefore approximate the solutions by linear combinations of radial basis functions. Moreover, the famous question of the embedding of the native space into L²(Ω) is also solved by the author.     

On orlicz sequence spaces III

It is proved that the set ofp's such thatl _p is isomorphic to a subspace of a given Orlicz spacel _Fforms an interval. Some examples and properties of minimal Orlicz sequence spaces are presented. It is proved that an Orlicz function space (different froml ₂) is not isomorphic to a subspace of an Orlicz sequence space. Finally it is shown (under a certain restriction) that if two Orlicz function spaces are isomorphic, then they are identical (i.e. consist of the same functions).     

Delaunay configurations and multivariate splines: A generalization of a result of B. N. Delaunay

In the 1920s, B. N. Delaunay proved that the dual graph of the Voronoi diagram of a discrete set of points in a Euclidean space gives rise to a collection of simplices, whose circumspheres contain no points from this set in their interior. Such Delaunay simplices tessellate the convex hull of these points. An equivalent formulation of this property is that the characteristic functions of the Delaunay simplices form a partition of unity. In the paper this result is generalized to the so-called Delaunay configurations. These are defined by considering all simplices for which the interiors of their circumspheres contain a fixed number of points from the given set, in contrast to the Delaunay simplices, whose circumspheres are empty. It is proved that every family of Delaunay configurations generates a partition of unity, formed by the so-called simplex splines. These are compactly supported piecewise polynomial functions which are multivariate analogs of the well-known univariate B-splines. It is also shown that the linear span of the simplex splines contains all algebraic polynomials of degree not exceeding the degree of the splines.

     

Characterization of kernel functions associated with operator algebraic Riccati equations for linear delay systems

In infinite time quadratic control and stochastic filtering problems for linear delay systems, operator algebraic Riccati equations play a very important role. However, since these are abstract operator equations, it is very useful, in analyzing their structure, to be able to characterize the kernel functions associated with the solutions of the operator Riccati equations. The kernel functions are given by the unique solution of a set of coupled differential equations. By comparing these kernel equations with similar ones available in the literature, it is shown that this characterization result is somewhat stronger than previously known results. Possible extentions to systems with control, observation, as well as state delays are also pointed out.