Group algebras are Hopf algebras, and their Hopf structure plays crucial roles in representation theory and cohomology of groups. A Hopf algebra is an algebra A (say over a field k) that has a comultiplication (∆ : A → A ⊗k A) generalizing the diagonal map on group elements, an augmentation (ε : A → k) generalizing the augmentation on a group algebra, and an antipode (S : A → A) generalizing th...

A general A∞-infinity bialgebra is a DG module (H, d) equipped with a family of structurally compatible operations ωj,i : H ⊗i → H,where i, j ≥ 1 and i+ j ≥ 3 (see [6]). In special A∞-bialgebras, ωj,i = 0 whenever i, j ≥ 2, and the remaining operationsmi = ω1,i and ∆j = ωj,1 define the underlying A∞-(co)algebra substructure. Thus special A∞-bialgebras have the form (H, d,mi,∆j)i,j≥2 subject to ...

Let $mathbf{c}_0$ be the real vector space of all real sequences which converge to zero. For every $x,yin mathbf{c}_0$, it is said that $y$ is block diagonal majorized by $x$ (written $yprec_b x$) if there exists a block diagonal row stochastic matrix $R$ such that $y=Rx$. In this paper we find the possible structure of linear functions $T:mathbf{c}_0rightarrow mathbf{c}_0$ preserving $prec_b$.

for an n-by-n complex matrix a in a block form with the (possibly) nonzero blocks only on the diagonal above the main one, we consider two other matrices whose nonzero entries are along the diagonal above the main one and consist of the norms or minimum moduli of the diagonal blocks of a. in this paper, we obtain two inequalities relating the numeical radii of these matrices and also determine ...

Abstract It is proved that a map ${\varphi }\colon R\to S$ of commutative Noetherian rings essentially finite type and flat locally complete intersection if only $S$ proxy small as bimodule. This means the thick subcategory generated by module over enveloping algebra $S\otimes _RS$ contains perfect complex supported fully on diagonal ideal. in spirit classical result }$ smooth bimodule; to say,...

in this paper, we nd the relationships between module contractibility of abanach algebra and its ideals. we also prove that module contractibility ofa banach algebra is equivalent to module contractibility of its module uniti-zation. finally, we show that when a maximal group homomorphic image ofan inverse semigroup s with the set of idempotents e is nite, the moduleprojective tensor product ...

would like to thank Mark Watson for correcting a mistake in his summary of Cantor’s argument. 1A note on citations: Mises’s article appeared in German in 1920. An English translation, “Economic Calculation in the Socialist Commonwealth,” appeared in Hayek’s (1990) collection Collectivist Economic Planning in 1935. 2The current paper assumes a basic familiarity with the socialist calculation deb...

We have developed a three level implicit method for solution of the Helmholtz equation. Using the cubic spline in space and finite difference in time directions. The approach has been modied to drive Numerov type nite difference method. The method yield the tri-diagonal linear system of algebraic equations which can be solved by using a tri-diagonal solver. Stability and error estimation of the...