Introduction. Let A be a commutative semi-simple Banach algebra and let A(.4) be the set of nonzero multiplicative functionals on A. Denote by A' the strongly closed span of A(^4) and by A" the Banach space adjoint of A'. Modifying a construction of R. Arens [l] we introduce a multiplication in A" under which A" becomes a commutative Banach algebra. A is algebraically isomorphic to a subalgebra...

Let k be an algebraically closed field of characteristic p > 0. The purpose of this paper is to describe the Hopf algebra of a finite commutative infinitesimal unipotent k-group scheme which is uniserial, i.e., which has a unique composition series. As there is only one simple finite commutative infinitesimal unipotent group scheme (namely αp := ker {F : Ga → Ga} , with Ga being the additive gr...

We investigate the isomorphism problem in the setting of definable sets (equivalent to sets with atoms): given two definable relational structures, are they related by a definable isomorphism? Under mild assumptions on the underlying structure of atoms, we prove decidability of the problem. The core result is parameter-elimination: existence of an isomorphism definable with parameters implies e...

We extend the concept of graph isomorphisms to multilayer networks, and we identify multiple types of isomorphisms. For example, in multilayer networks with a single “aspect” (i.e., type of layering), permuting vertex labels, layer labels, and both of types of layers each yield a different type of isomorphism. We discuss how multilayer network isomorphisms naturally lead to defining isomorphism...

We investigate the relative complexity of the graph isomorphism problem (GI) and problems related to the reconstruction of a graph from its vertex-deleted or edge-deleted subgraphs (in particular, deck checking (DC) and legitimate deck (LD) problems). We show that these problems are closely related for all amounts c ≥ 1 of deletion: 1. GI ≡iso VDCc, GI ≡iso EDCc, GI ≤m LVDc, and GI ≡iso LEDc. 2...

While in general it is not known whether there is a polynomial time algorithm to decide whether two given graphs are isomorphic, there are polynomial-time algorithms for certain subsets of graphs, including but not limited to planar graphs and graphs with bounded valence. In this thesis, we will give a brief introduction on the Graph Isomorphism Problem and its relations to complexity theory. W...

Two basic tools of free probability are the R-transform and the S-transform. These transforms were introduced by Voiculescu in the 1980s, and are used to understand the addition and the multiplication of two free random variables respectively. The R-transform has a natural and very useful multi-variable extension describing the addition of two free k-tuples of random variables, but the problem ...

This paper discusses the instability of eleven nonlinear state space models that underly exponential smoothing. Hyndman et al. (2002) proposed a framework of 24 state space models for exponential smoothing, including the well-known simple exponential smoothing, Holt’s linear and Holt-Winters’ additive and multiplicative methods. This was extended to 30 models with Taylor’s (2003) damped multipl...

The group isomorphism problem consists in deciding whether two groups G and H given by their multiplication tables are isomorphic. An algorithm for group isomorphism attributed to Tarjan runs in time n, c.f. [Mil78]. Miller and Monk showed in [Mil79] that group isomorphism can be many-one reduced to isomorphism testing for directed graphs. For groups with n elements, the graphs have valence at ...

In this paper we examine the hard problems underlying asymmetric pairings, their precise relationships and how they affect a number of existing protocols. Furthermore, we present a new model for the elliptic curve groups used in asymmetric pairings, which allows both an efficient pairing and an efficiently computable isomorphism.