Related papers: First Order Optimum Calculi
MV-algebras are an algebraic semantics for Lukasiewicz logic and MV-algebras generated by a finite chain are Heyting algebras where the Godel implication can be written in terms of De Morgan and Moisil's modal operators. In our work, a…
We consider systems of ordinary differential equations with quadratic homogeneous right hand side. We give a new simple proof of a result already obtained in [8,10] which gives the necessary conditions for the existence of polynomial first…
A number of categories is presented that are algebraically complete and cocomplete, i.e., every endofunctor has an initial algebra and a terminal coalgebra. For all finitary (and, more generally, all precontinuous) set functors the initial…
A number of categories is presented that are algebraically complete and cocomplete, i.e., every endofunctor has an initial algebra and a terminal coalgebra. For all finitary (and, more generally, all precontinuous) set functors the initial…
First-order methods for minimization and saddle point (min-max) problems are widely used for solving large-scale problems, in particular arising in machine learning. The majority of works obtain favorable complexity guarantees of such…
Most of theoretical physics is based on the mathematics of functions of a real or a complex variable; yet we frequently are drawn to try extending our reach to include quaternions. The non-commutativity of the quaternion algebra poses…
This paper presents new first-order methods for achieving optimal oracle complexities in convex optimization with convex functional constraints. Oracle complexities are measured by the number of function and gradient evaluations. To achieve…
A free differential for an arbitrary associative algebra is defined as a differential with a uniqueness property. The existence problem for such a differential is posed. The notion of optimal calculi for given commutation rules is…
The notion of singular reduction operators, i.e., of singular operators of nonclassical (conditional) symmetry, of partial differential equations in two independent variables is introduced. All possible reductions of these equations to…
We introduce the concept of a homogeneity supermanifold, which is, roughly speaking, a supermanifold equipped with a privileged atlas whose coordinates carry prescribed (real) homogeneity degrees. This structure defines a sheaf of graded…
In this paper we give definitions of basic concepts such as symmetries, first integrals, Hamiltonian and recursion operators suitable for ordinary differential equations on associative algebras, and in particular for matrix differential…
Differential calculi are obtained for quantum homogeneous spaces by extending Woronowicz' approach to the present context. Representation theoretical properties of the differential calculi are investigated. Connections on quantum…
The usual approach to developing and analyzing first-order methods for smooth convex optimization assumes that the gradient of the objective function is uniformly smooth with some Lipschitz constant $L$. However, in many settings the…
We define a new congruence relation on the set of integers, leading to a group similar to the multiplicative group of integers modulo $n$. It makes use of a symmetry almost omnipresent in modular multiplications and halves the number of…
This note tries to show that a re-examination of a first course in analysis, using the more sophisticated tools and approaches obtained in later stages, can be a real fun for experts, advanced students, etc. We start by going to the…
We introduce the notion of bounded quasi-inversion closed semiprime f-algebras and we prove that, if A is such an algebra, then any intermediate algebra in A is an order ideal of A. This extends a recent result by Dominguez who has dealt…
We introduce basic aspects of new operator method, which is very suitable for practical solving differential equations of various types. The main advantage of the method is revealed in opportunity to find compact exact operator solutions of…
Modular data are commonly studied in mathematics and physics. A modular datum defines a finite-dimensional representation of the modular group $SL_2(\mathbb{Z})$. In this paper, we show that there is a one-to-one correspondence between…
Starting from involutive BE algebras, we redefine the orthomodular algebras, by introducing the notion of implicative-orthomodular algebras. We investigate properties of implicative-orthomodular algebras, and give characterizations of these…
We study covariant differential calculus on the quantum spheres S_q^2N-1. Two classification results for covariant first order differential calculi are proved. As an important step towards a description of the noncommutative geometry of the…