Related papers: A few calculus rules for chain differentials
A systematic theory is introduced for calculating the derivatives of quaternion matrix function with respect to quaternion matrix variables. The proposed methodology is equipped with the matrix product rule and chain rule and it is able to…
We consider the differentiation of the value function for parametric optimization problems. Such problems are ubiquitous in Machine Learning applications such as structured support vector machines, matrix factorization and min-min or…
In this paper we provide a rigorous mathematical foundation for continuous approximations of a class of systems with piece-wise continuous functions. By using techniques from the theory of differential inclusions, the underlying piece-wise…
We study the fundamental problem of the calculus of variations with variable order fractional operators. Fractional integrals are considered in the sense of Riemann-Liouville while derivatives are of Caputo type.
We define an abstract framework called {\it discrete finite differences embedding} which can be used to obtain discrete analogue of formal functional relations in the spirit of category theory. For ordinary differential equations we exhibit…
We establish an analogue of the first fundamental theorem of calculus for functions defined on the Wasserstein space of probability measures. Precisely, we show that if a function on the Wasserstein space is sufficiently regular in the…
The paper concerns the second-order generalized differentiation theory of variational analysis and new applications of this theory to some problems of constrained optimization in finitedimensional spaces. The main attention is paid to the…
We study the structure possessed by the Goodwillie derivatives of a pointed homotopy functor of based topological spaces. These derivatives naturally form a bimodule over the operad consisting of the derivatives of the identity functor. We…
A variational principle is introduced to provide a new formulation and resolution for several boundary value problems with a variational structure. This principle allows one to deal with problems well beyond the weakly compact structure. As…
We give an overview of differential cohomology from the point of view of algebraic topology. This includes a survey of several different definitions of differential cohomology groups, a discussion of differential characteristic classes, an…
In this paper we provide a definition of fractional gradient operators, related to directional derivatives. We develop a fractional vector calculus, providing a probabilistic interpretation and mathematical tools to treat multidimensional…
We explain why the macroscopic fluctuations of deterministic local collision dynamics should be characterized by a non strictly convex functional.
This expository paper presents elementary proofs of four basic results concerning derivatives of quasi-convex functions. They are combined into a fifth theorem which is simple to apply and adequate in many cases. Along the way we establish…
This article characterizes conjugates and subdifferentials of convex integral functionals over linear spaces of cadlag stochastic processes. The approach is based on new measurability results on the Skorokhod space and new interchange rules…
The General Curve Lemma is a tool of Infinite-Dimensional Analysis, which enables refined studies of differentiability properties of mappings between real locally convex spaces. In this article, we generalize the General Curve Lemma in two…
We prove a version of the Stokes formula for differential forms on locally convex spaces. The main tool used for proving this formula is the surface layer theorem proved in another paper by the author. Moreover, for differential forms of a…
In this article, we further explore convex functions by revealing new bounds, resulting from stronger convexity behavior. In particular, we define the so called radical convex functions and study their properties. We will see that such…
A first order inference system, called R-calculus, is defined to develop the specifications. It is used to eliminate the laws which is not consistent with the user's requirements. The R-calculus consists of the structural rules, an axiom, a…
We identity the optimal non-infinitesimal direction of descent for a convex function. An algorithm is developed that can theoretically minimize a subset of (non-convex) functions.
The paper deals with the problem of reconstructing the topological structure of a network of dynamical systems. A distance function is defined in order to evaluate the "closeness" of two processes and a few useful mathematical properties…