2 Basic group theory 2 2.1 The definition of a group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Group homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.3 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.4 Cosets and quotient spaces . . . . . . . . . . . . . . . . . . . . . . . . . . ....

We study the existence of cubic quotients of finite-dimensional quasi-normed spaces, that is, quotients well isomorphic to `∞ for some k. We give two results of this nature. The first guarantees a proportional dimensional cubic quotient when the envelope is cubic; the second gives an estimate for the size of a cubic quotient in terms of a measure of nonconvexity of the quasi-norm.

We prove that the quotient of integer homology cobordism group by subgroup generated Seifert fibered spaces is infinitely generated.

Affine Cartesian codes are defined by evaluating multivariate polynomials at a cartesian product of finite subsets field. In this work we examine properties these as batch codes. We consider the recovery sets to be points aligned on specific direction and buckets derived from cosets subspace ambient space evaluation points. able prove that under conditions, an affine code is satisfy query size ...

Hypertoric varieties are quaternionic analogues of toric varieties, important for their interaction with the combinatorics of matroids as well as for their prominent place in the rapidly expanding field of algebraic symplectic and hyperkähler geometry. The aim of this survey is to give clear definitions and statements of known results, serving both as a reference and as a point of entry to this...

in this paper, we introduce the concepts of right, left and product stabilizers on hoops and study some properties and the relation between them.  and we try to find that how they can be equal and investigate that under what condition they can be filter, implicative filter, fantastic and positive implicative filter. also, we prove that  right and product stabilizers are filters and if they are ...