Which finitely generated Abelian groups admit isomorphic Cayley graphs?

We show that Cayley graphs of finitely generated Abelian groups are rather rigid. As a consequence we obtain that two finitely generated Abelian groups admit isomorphic Cayley graphs if and only if they have the same rank and their torsion parts have the same cardinality. The proof uses only elementary arguments and is formulated in a geometric language.     

Which weights on ℝ admit Jackson Theorems?admit Jackson Theorems?

Let W: ?→(0,∞) be continuous. DoesW admit a classical Jackson Theorem? That is, does there exist a sequence $\{ \eta _n \} _{n = 1}^\infty $ of positive numbers with limit 0 such that for 1≤p≤∞, $\mathop {\inf }\limits_{\deg (P) \le n} ||(f - P)W||_{L_p (R)} \le \eta n||f'W||_{L_p (R)} $ for all absolutely continuousf with $||f'W||_{L_p (R)} $ finite? We show that such a theorem is true iff both $\mathop {\lim }\limits_{\chi \to \infty } W(\chi )\int_0^\chi {W^{ - 1} } = 0$ and $\mathop {\lim }\limits_{\chi \to \infty } W^{ - 1} (\chi )\int_\chi ^\infty W = 0,$ with analogous limits asx→?∞. In particular,W(x)=exp(?|x|) does not admit a Jackson theorem of this type. We also construct weights that admit anL ₁ but not anL _∞ Jackson theorem (or conversely).     

When is the Isbell topology a group topology?

Conditions on a topological space X under which the space C(X,R) of continuous real-valued maps with the Isbell topology κ is a topological group (topological vector space) are investigated. It is proved that the addition is jointly continuous at the zero function in Cκ(X,R) if and only if X is infraconsonant. This property is (formally) weaker than consonance, which implies that the Isbell and the compact-open topologies coincide. It is shown the translations are continuous in Cκ(X,R) if and only if the Isbell topology coincides with the fine Isbell topology. It is proved that these topologies coincide if X is prime (that is, with at most one non-isolated point), but do not even for some sums of two consonant prime spaces.     

Metaplectic operators for finite abelian groups and ?

The Segal-Shale-Weil representation associates to a symplectic transformation of the Heisenberg group an intertwining operator, called metaplectic operator. We develop an explicit construction of metaplectic operators for the Heisenberg group H(G) of a finite abelian group G, an important setting in finite time-frequency analysis. Our approach also yields a simple construction for the multivariate Euclidean case G = ?^d.     

Which finite simple groups are unit groups?

We prove that if G$G$ is a finite simple group which is the unit group of a ring, then G$G$ is isomorphic to: (a) a cyclic group of order 2; or (b) a cyclic group of prime order 2^k−1$2^k-1$ for some k$k$; or (c) a projective special linear group _PSLn(_F2)${PSL}_n\left({\mathbb{F}}_2\right)$ for some n≥3$n\ge 3$. Moreover, these groups do all occur as unit groups. We deduce this classification from a more general result, which holds for groups G$G$ with no non-trivial normal 2-subgroup.     

The number of all transformation groups on an infinite set

Let X be an infinite set, T(X)={f∈X~X:f is a bijection}, C={G:G is a transformation group on X}={G:G is a subgroup of T(X)}.Then |C|=2~(2~(|x|)). Inproof of this result, the AC is used.     

What is an infinite sequence?

<正>Series will be introduced briefly in the following paper.It is important in calculus for Newton's idea of representing functions as sums of infinite series.In finding areas,he often integrated a function by expressing it as series.Now,we'll learn infinite sequences.     

What is an infinite series?

<正>Finite sequences and series have defined first and last terms,otherwise,infinite sequences and series continue indefinitely.Infinite series need tools from mathematical analysis,which are also widely used in other quantitative disciplines such as physics,computer science,and finance.Now we have an infinite sequence a_1,a_2,     

Which algebraic groups are Picard varieties?

We show that every connected commutative algebraic group over an algebraically closed field of characteristic 0 is the Picard variety of some projective variety having only finitely many non-normal points.In contrast,no Witt group of dimension at least 3 over a perfect field of prime characteristic is isogenous to a Picard variety obtained by this construction.     

When must pure extensions of countable abelian groups necessarily split?

Necessary and sufficient conditions are given for the group of pure extensions of a countable abelian group by a countable abelian group to equal zero.

     

On the Erd?s-Ginzburg-Ziv constant of finite abelian groups of high rank

Let G be a finite abelian group. The Erd?s-Ginzburg-Ziv constant s(G) of G is defined as the smallest integer l∈N such that every sequence S over G of length |S|?l has a zero-sum subsequence T of length |T|=exp(G). If G has rank at most two, then the precise value of s(G) is known (for cyclic groups this is the theorem of Erd?s-Ginzburg-Ziv). Only very little is known for groups of higher rank. In the present paper, we focus on groups of the form , with n,r∈N and n?2, and we tackle the study of s(G) with a new approach, combining the direct problem with the associated inverse problem.     

Finite p-groups all of whose cyclic subgroups A, Bwith A ∩ B ≠ {1} generate an abelian group

We characterize the title p-groups for all primes p.     

What is the probability an automorphism fixes a group element?

Extending the notion of probability to the automorphisms of a group, we find the probability of an arbitrarily chosen automorphism of a group fixing an arbitrary element of the group.     

When is a -adic power series an endomorphism of a formal group?

If is a noninvertible endomorphism of a formal group, then we have that commutes with an invertible series and is Galois over for all . We shall prove that the converse of this statement is also true.

     

The Inequality Process Versus the Saved Wealth Model: Which Is the More Likely to Imply an Analogue of Thermodynamics in Social Science?

The Inequality Process (IP) and the Saved Wealth Model (SW) are theories of income distribution. The IP's social science metatheory requires its stationary distribution to fit the distribution of labor income conditioned on education. The SW is a modification of the particle system model of the kinetic theory of gases (KTG), the basis of gas thermodynamics. The IP is a particle system similar to the SW and KTG. This article shows that the IP passes the empirical test required of it by social science theory better than the SW. The IP's advantage increases as the U.S. labor force becomes more educated. The IP may the better bet to imply an analogue of thermodynamics in social science.