We previously proposed evolutionary fuzzy systems of playing Ms.PacMan for the competitions. As a consequence of the evolution, reflective action rules such that PacMan tries to eat pills effectively until ghosts come close to PacMan are acquired. Such rules works well. However, sometimes it is too reflective so that PacMan go toward ghosts by herself in longer corridors. In this paper, a criti...

a module $m$ is called $emph{h}$-cofinitely supplemented if for every cofinite submodule $e$ (i.e. $m/e$ is finitely generated) of $m$ there exists a direct summand $d$ of $m$ such that $m = e + x$ holds if and only if $m = d + x$, for every submodule $x$ of $m$. in this paper we study factors, direct summands and direct sums of $emph{h}$-cofinitely supplemented modules. let $m$ be an $emph{h}$...

In this paper, we give characterizations of retro Banach frames in Banach spaces. The notion of almost exact retro Banach frame is defined and a characterization of retro Banach frame has been given. Also results exhibiting relationship between frames, almost exact retro Banach frames and Riesz bases has been proved. Finally, we give some perturbation results of retro Banach frames and an almos...

Let $\mathfrak{g}_{0}$ be a simple Lie algebra of type ADE and let $U'_{q}(\mathfrak{g})$ the corresponding untwisted quantum affine algebra. We show that there exists an action braid group $B(\mathfrak{g}_{0})$ on Grothendieck ring $\mathcal{K}_{t}(\mathfrak{g})$ Hernandez-Leclerc’s category $\mathcal{C}_{\mathfrak{g}}^{0}$. Focused case $A_{N-1}$, we construct family monoidal autofunctors $\{...

A W-algebra action is constructed via Hecke transformations on the equivariant Borel–Moore homology of Hilbert scheme points a nonreduced plane in three-dimensional affine space. The resulting W-module then identified to vacuum module. construction based generalization ADHM as well W-action moduli space instantons by Schiffmann and Vasserot.

Let A be a Banach algebra and X a Banach A-bimodule, the derivation D : A → X is semi-inner if there are ξ, μ ∈ X such that D(a) = a.ξ − μ.a, (a ∈ A). A is called semi-amenable if every derivation D : A → X∗ is semi-inner. The dual Banach algebra A is Connes semi-amenable (resp. approximately semi-amenable) if, every D ∈ Z1w _ (A,X), for each normal, dual Banach A-bimodule X, is semi -inner (re...

A general Fatou Lemma is established for a sequence of Gelfand integrable functions from a vector Loeb space to the dual of a separable Banach space or, with a weaker assumption on the sequence, a Banach lattice. A corollary sharpens previous results in the …nite dimensional setting even for the case of scalar measures. Counterexamples are presented to show that the results obtained here are sh...

In this paper, we introduce the new concept of biprojectivity of a Banach algebra modulo an ideal, as a generalization of this notion in the classical case. By using it , we obtain some necessary and sufficient conditions for contractibility of Banach algebras modulo an ideal. In particular we characterize the contractibility of quotient Banach algebras. Also we study the relationship between t...