scholarly journals Zero-Sum Coefficient Derivations in Three Variables of Triangular Algebras

2016 ◽  
Vol 8 (5) ◽  
pp. 37
Author(s):  
Youngsoo Kim ◽  
Byunghoon Lee
Keyword(s):  
Standard Form ◽  
Sufficient Conditions ◽  
Multilinear Polynomial ◽  
Triangular Algebra ◽  
Triangular Algebras ◽  
Zero Sum ◽  
Lie Derivations

Under mild assumptions Benkovi\v{c} showed that an $f$-derivation of a triangular algebra is a derivation when the sum of the coefficients of the multilinear polynomial $f$ is nonzero. We investigate the structure of $f$-derivations of triangular algebras when $f$ is of degree 3 and the coefficient sum is zero. The zero-sum coeffient derivations include Lie derivations (degree 2) and Lie triple derivations (degree 3), which have been previously shown to be not necessarily derivations but in standard form, i.e., the sum of a derivation and a central map. In this paper, we present sufficient conditions on the coefficients of $f$ to ensure that any $f$-derivations are derivations or are in standard form.<br /><br />

scholarly journals Nonlinear Lie derivations of triangular algebras

2010 ◽  
Vol 432 (11) ◽  
pp. 2953-2960 ◽  
Author(s):  
Weiyan Yu ◽  
Jianhua Zhang
Keyword(s):  
Triangular Algebras ◽  
Lie Derivations

scholarly journals Generalized Higher Derivations on Lie Ideals of Triangular Algebras

2017 ◽  
Vol 25 (1) ◽  
pp. 35-53
Author(s):  
Mohammad Ashraf ◽  
Nazia Parveen ◽  
Bilal Ahmad Wani
Keyword(s):  
Commutative Ring ◽  
Lie Ideal ◽  
Additive Subgroup ◽  
Triangular Algebra ◽  
Higher Derivation ◽  
Square Closed Lie Ideal ◽  
Triangular Algebras

Abstract Let be the triangular algebra consisting of unital algebras A and B over a commutative ring R with identity 1 and M be a unital (A; B)-bimodule. An additive subgroup L of A is said to be a Lie ideal of A if [L;A] ⊆ L. A non-central square closed Lie ideal L of A is known as an admissible Lie ideal. The main result of the present paper states that under certain restrictions on A, every generalized Jordan triple higher derivation of L into A is a generalized higher derivation of L into A.

Functional Reward Markov Decision Processes: Theory and Applications

2017 ◽  
Vol 26 (03) ◽  
pp. 1760014
Author(s):  
Paul Weng ◽  
Olivier Spanjaard
Keyword(s):  
Markov Decision Processes ◽  
Infinite Horizon ◽  
Standard Form ◽  
Sufficient Conditions ◽  
Decision Processes ◽  
Markov Decision ◽  
Standard Models ◽  
Reward Functions ◽  
Planning Problems ◽  
Horizon Case

Markov decision processes (MDP) have become one of the standard models for decisiontheoretic planning problems under uncertainty. In its standard form, rewards are assumed to be numerical additive scalars. In this paper, we propose a generalization of this model allowing rewards to be functional. The value of a history is recursively computed by composing the reward functions. We show that several variants of MDPs presented in the literature can be instantiated in this setting. We then identify sufficient conditions on these reward functions for dynamic programming to be valid. We also discuss the infinite horizon case and the case where a maximum operator does not exist. In order to show the potential of our framework, we conclude the paper by presenting several illustrative examples.

scholarly journals Characterizations of Lie derivations of triangular algebras

2011 ◽  
Vol 435 (5) ◽  
pp. 1137-1146 ◽  
Author(s):  
Peisheng Ji ◽  
Weiqing Qi
Keyword(s):  
Triangular Algebras ◽  
Lie Derivations

Nonlinear Lie triple derivations by local actions on triangular algebras

2021 ◽  
Author(s):  
Xingpeng Zhao
Keyword(s):  
Commutative Ring ◽  
Triangular Algebra ◽  
Mild Conditions ◽  
Derivation Formula ◽  
Triangular Algebras

Let [Formula: see text] be a triangular algebra over a commutative ring [Formula: see text]. In this paper, under some mild conditions on [Formula: see text], we prove that if [Formula: see text] is a nonlinear map satisfying [Formula: see text] for any [Formula: see text] with [Formula: see text]. Then [Formula: see text] is almost additive on [Formula: see text], that is, [Formula: see text] Moreover, there exist an additive derivation [Formula: see text] of [Formula: see text] and a nonlinear map [Formula: see text] such that [Formula: see text] for [Formula: see text], where [Formula: see text] for any [Formula: see text] with [Formula: see text].

scholarly journals The Elementary Divisors, Associated with 0, of a Singular M-matrix

1956 ◽  
Vol 10 (3) ◽  
pp. 108-122 ◽  
Author(s):  
Hans Schneider
Keyword(s):  
Standard Form ◽  
Characteristic Root ◽  
Sufficient Conditions ◽  
Elementary Divisor ◽  
Absolute Value ◽  
Elementary Divisors ◽  
Characteristic Roots ◽  
Index Pairs ◽  
Negative Characteristic ◽  
Necessary And Sufficient

1. Many investigations have been concerned with a squaro matrix P with non-negative coefficients (elements). It is remarkable that many interesting properties of P are determined by the set Σ of index pairs of positive (i.e. non-zero) coefficients of P, the actual values of these coefficients being irrelevant. Thus, for example, the number of characteristic roots equal in absolute value to the largest non-negative characteristic root p depends on Σ alone, if P is irreducible. If P is reducible, then Σ determines the standard forms of P (cf. § 3). The multiplicity of p depends on Σ, and on the set S of indices of those submatrices in the diagonal in a standard form of P which have p as a characteristic root. It has apparently not been considered before whether Σ and S also determine the elementary divisors associated with p. We shall show that, in general, the elementary divisors do not depend on these sets alone, but that necessary and sufficient conditions may be found in terms of Σ and S (a) for the elementary divisors associated with p to be simple, and (b) that there is only one elementary divisor associated with p.

SOLVABILITY OF LINEAR-QUADRATIC DIFFERENTIAL GAMES ASSOCIATED WITH PURSUIT-EVASION PROBLEMS

2008 ◽  
Vol 10 (04) ◽  
pp. 481-515 ◽  
Author(s):  
JOSEF SHINAR ◽  
VLADIMIR TURETSKY ◽  
VALERY Y. GLIZER ◽  
EDUARD IANOVSKY
Keyword(s):  
Quadratic Differential ◽  
Sufficient Conditions ◽  
Lebesgue Integral ◽  
Linear Quadratic ◽  
Pursuit Evasion ◽  
Linear Quadratic Differential Games ◽  
Matrix Differential ◽  
Zero Sum ◽  
Generalized Cost ◽  
Riccati Matrix

A finite horizon zero-sum linear-quadratic differential game with a generalized cost functional, involving a Lebesgue integral with a measure that has both discrete and distributed parts, is considered. Sufficient conditions for the solvability of such a game are established in terms of the eigenvalues of an integral operator in Hilbert space. The game solution is based on solving an impulsive Riccati matrix differential equation. These results are applied for two games associated with pursuit-evasion problems. Illustrative examples are presented.

scholarly journals Subgame Maxmin Strategies in Zero-Sum Stochastic Games with Tolerance Levels

2021 ◽  
Author(s):  
János Flesch ◽  
P. Jean-Jacques Herings ◽  
Jasmine Maes ◽  
Arkadi Predtetchinski
Keyword(s):  
Stochastic Games ◽  
Sufficient Conditions ◽  
Payoff Function ◽  
Surprising Result ◽  
Countable State ◽  
Finite Action ◽  
Zero Sum ◽  
Necessary And Sufficient ◽  
Action Spaces ◽  
Special Case

AbstractWe study subgame $$\phi $$ ϕ -maxmin strategies in two-player zero-sum stochastic games with a countable state space, finite action spaces, and a bounded and universally measurable payoff function. Here, $$\phi $$ ϕ denotes the tolerance function that assigns a nonnegative tolerated error level to every subgame. Subgame $$\phi $$ ϕ -maxmin strategies are strategies of the maximizing player that guarantee the lower value in every subgame within the subgame-dependent tolerance level as given by $$\phi $$ ϕ . First, we provide necessary and sufficient conditions for a strategy to be a subgame $$\phi $$ ϕ -maxmin strategy. As a special case, we obtain a characterization for subgame maxmin strategies, i.e., strategies that exactly guarantee the lower value at every subgame. Secondly, we present sufficient conditions for the existence of a subgame $$\phi $$ ϕ -maxmin strategy. Finally, we show the possibly surprising result that each game admits a strictly positive tolerance function $$\phi ^*$$ ϕ ∗ with the following property: if a player has a subgame $$\phi ^*$$ ϕ ∗ -maxmin strategy, then he has a subgame maxmin strategy too. As a consequence, the existence of a subgame $$\phi $$ ϕ -maxmin strategy for every positive tolerance function $$\phi $$ ϕ is equivalent to the existence of a subgame maxmin strategy.

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