scholarly journals Paraconsistent Differential Calculus (Part I): First-Order Paraconsistent Derivative

2014 ◽  
Vol 05 (06) ◽  
pp. 904-916 ◽  
Author(s):  
João Inácio Da Silva Filho
Keyword(s):  
Differential Calculus ◽  
First Order

scholarly journals Restrictive metric regularity and generalized differential calculus in Banach spaces

2004 ◽  
Vol 2004 (50) ◽  
pp. 2653-2680 ◽  
Author(s):  
Boris S. Mordukhovich ◽  
Bingwu Wang
Keyword(s):  
Banach Spaces ◽  
Differential Calculus ◽  
Metric Regularity ◽  
Sufficient Conditions ◽  
Normal Cones ◽  
First Order ◽  
Nonconvex Sets ◽  
Generalized Differential ◽  
Necessary And Sufficient ◽  
Generalized Differential Calculus

We consider nonlinear mappingsf:X→Ybetween Banach spaces and study the notion ofrestrictive metric regularityoffaround some pointx¯, that is, metric regularity offfromXinto the metric spaceE=f(X). Some sufficient as well as necessary and sufficient conditions for restrictive metric regularity are obtained, which particularly include an extension of the classical Lyusternik-Graves theorem in the case whenfis strictly differentiable atx¯but its strict derivative∇f(x¯)is not surjective. We develop applications of the results obtained and some other techniques in variational analysis to generalized differential calculus involving normal cones to nonsmooth and nonconvex sets, coderivatives of set-valued mappings, as well as first-order and second-order subdifferentials of extended real-valued functions.

scholarly journals DIVERGENCES ON PROJECTIVE MODULES AND NON-COMMUTATIVE INTEGRALS

2011 ◽  
Vol 08 (04) ◽  
pp. 885-896 ◽  
Author(s):  
TOMASZ BRZEZIŃSKI
Keyword(s):  
Commutative Algebra ◽  
Hopf Algebras ◽  
Differential Calculus ◽  
Module Structure ◽  
Projective Modules ◽  
Finitely Generated ◽  
Integration By Parts ◽  
Left Module ◽  
First Order

A method of constructing (finitely generated and projective) right module structure on a finitely generated projective left module over an algebra is presented. This leads to a construction of a first-order differential calculus on such a module which admits a hom-connection or a divergence. Properties of integrals associated to this divergence are studied, in particular the formula of integration by parts is derived. Specific examples include inner calculi on a non-commutative algebra, the Berezin integral on the supercircle and integrals on Hopf algebras.

scholarly journals Korevaar–Schoen’s energy on strongly rectifiable spaces

2021 ◽  
Vol 60 (6) ◽  
Author(s):  
Nicola Gigli ◽  
Alexander Tyulenev
Keyword(s):  
Energy Density ◽  
Lower Semicontinuity ◽  
Differential Calculus ◽  
Representation Formula ◽  
Target Space ◽  
List Type ◽  
First Order ◽  
Schmidt Norm ◽  
Sobolev Maps ◽  
Limit Energy

AbstractWe extend Korevaar–Schoen’s theory of metric valued Sobolev maps to cover the case of the source space being an $$\mathsf{RCD}$$ RCD space. In this situation it appears that no version of the ‘subpartition lemma’ holds: to obtain both existence of the limit of the approximated energies and the lower semicontinuity of the limit energy we shall rely on: the fact that such spaces are ‘strongly rectifiable’ a notion which is first-order in nature (as opposed to measure-contraction-like properties, which are of second order). This fact is particularly useful in combination with Kirchheim’s metric differentiability theorem, as it allows to obtain an approximate metric differentiability result which in turn quickly provides a representation for the energy density, the differential calculus developed by the first author which allows, thanks to a representation formula for the energy that we prove here, to obtain the desired lower semicontinuity from the closure of the abstract differential. When the target space is $$\mathsf{CAT}(0)$$ CAT ( 0 ) we can also identify the energy density as the Hilbert-Schmidt norm of the differential, in line with the smooth situation.

scholarly journals THE 3D SPIN GEOMETRY OF THE QUANTUM TWO-SPHERE

2010 ◽  
Vol 22 (08) ◽  
pp. 963-993 ◽  
Author(s):  
SIMON BRAIN ◽  
GIOVANNI LANDI
Keyword(s):  
Quantum Group ◽  
Differential Calculus ◽  
Arbitrary Order ◽  
Three Dimensional ◽  
Spectral Triple ◽  
Frame Bundle ◽  
Order Condition ◽  
Spin Geometry ◽  
First Order ◽  
Calculus Formula

We study a three-dimensional differential calculus [Formula: see text] on the standard Podleś quantum two-sphere [Formula: see text], coming from the Woronowicz 4D+ differential calculus on the quantum group SU q(2). We use a frame bundle approach to give an explicit description of [Formula: see text] and its associated spin geometry in terms of a natural spectral triple over [Formula: see text]. We equip this spectral triple with a real structure for which the commutant property and the first order condition are satisfied up to infinitesimals of arbitrary order.

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

Stochasting modeling of planetary ring systems

1984 ◽  
Vol 75 ◽  
pp. 461-469 ◽  
Author(s):  
Robert W. Hart
Keyword(s):  
Maximum Entropy ◽  
Ring System ◽  
The Sun ◽  
Ultimate State ◽  
Interparticle Interactions ◽  
Ring Systems ◽  
First Order ◽  
Planetary Ring ◽  
Insight Into

ABSTRACTThis paper models maximum entropy configurations of idealized gravitational ring systems. Such configurations are of interest because systems generally evolve toward an ultimate state of maximum randomness. For simplicity, attention is confined to ultimate states for which interparticle interactions are no longer of first order importance. The planets, in their orbits about the sun, are one example of such a ring system. The extent to which the present approximation yields insight into ring systems such as Saturn's is explored briefly.

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