On Σ-subsets of naturals over abelian groups

We obtain conditions for the Σ-definability of a subset of the set of naturals in the hereditarily finite admissible set over a model and for the computability of a family of such subsets. We prove that: for each e-ideal I there exists a torsion-free abelian group A such that the family of e-degrees of Σ-subsets of ω in $\mathbb{H}\mathbb{F}(A)$ coincides with I; there exists a completely reducible torsion-free abelian group in the hereditarily finite admissible set over which there exists no universal Σ-function; for each principal e-ideal I there exists a periodic abelian group A such that the family of e-degrees of Σ-subsets of ω in $\mathbb{H}\mathbb{F}(A)$ coincides with I.     

On ac-number quantum τ-function

We first review the properties of the conventional -functions of the KP and Toda-lattice hierarchies. A straightforward generalization is then discussed. It corresponds to passing from differential to finite-difference equations; it does not involve, however, the concept of an operator-valued -function nor the one associated with non-Cartanian (levelk 1) algebras. The present study can be useful for better understandingq-free fields and their relation to ordinary free fields.E-mail: mironov@td.fian.free.net, mironov@grotte.teorfys.uu.se; morozov@vxitep.itep.msk.su, morozov@vxdesy.desy.de; vinet@ere.umontreal.ca, vinet@lps.umontreal.ca. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 1, pp. 119–131, July, 1994.     

On Σ-V Rings

We discuss various properties of a ring over which each simple module is Σ-injective.     

On 3-dimensional Riemannian Σ-spaces

The following result is proved: a 3-dimensional connected and simply connected Riemannian manifold admitting a reduced -structure (in the sense of O. Loos) is either a Riemannian symmetric space or it is isometric to a unimodular Lie group with a left-invariant Riemannian metric. At the same time, we give first nontrivial examples of Riemannian -spaces, which are not symmetric of finite order.     

On basic properties of the Ramanujan τ-function

It is proved that the values of the Ramanujan function constitute an additive basis for the set of integers of order 8012.     

On Σ-rigid presentations of the real order

For arbitrary tuples of real parameters \(\bar p\) , we prove the existence and effective infiniteness of the class of the linear orders on ? of type 〈?,<〉 which are Σ-definable over \(\mathbb{H}\mathbb{F}(\mathbb{R})\) with parameters \(\bar p\) and have no nontrivial Σ-definable self-embeddings with parameters \(\bar p\) .     

On a Matrix Equation AX＋XB=C over a Skew Field

In this paper we study a matrix equation AX BX=C （Ⅰ）over an arbitrary skew field,and give a consistency criterion of （Ⅰ） and an explicit expression of general solutions of （Ⅰ）.A convenient,simple and practical method of solving （Ⅰ） is also given.As a particular case,we also give a simple method of finding a system of fundamental solutions of a homogeneous system of right linear equations over a skew field.     

On the Relation of Σ-Reducibility Between Admissible Sets

Reducibility on admissible sets is studied which is a stronger version of the usual -presentability of models. One of its informal prototypes is the interpretability of one computational device in the other. We obtain criteria of reducibility for recursively listed and pure sets, introduce the notion of jump, and prove exact boundaries for the ordinals of jumps. We also show that this reducibility is lifted to -superstructures. Several results are proven on the relations of this reducibility to some known reducibilities.     

On a System of Martix Equations over an Arbitrary Skew Field

ＯｎａＳｙｓｔｅｍｏｆＭａｒｔｉｘＥｑｕａｔｉｏｎｓｏｖｅｒａｎＡｒｂｉｔｒａｒｙＳｋｅｗＦｉｅｌｄ￥ＷａｎｇＱｉｎｇｗｅｎ（Ｄｅｐｔ．ｏｆＭａｔｈ．，ＣｈａｎｇｗｅｉＴｅａｃｈｅｒｓＣｏｌｌｅｙｅ，Ｗｅｉｆａｎｇ，Ｗｅｉｆａｎｇ，２６１０４３）Ａｂ...     

On a relation between the universal Teichmüller space and the Grunsky operator

Being a Strebel point gives a sufficient condition for that the extremal Beltrami coefficient is uniquely determined in a Teichmiiller equivalence class. We consider how Strebel points are characterized. In this paper, we will give a new characterization of Strebel points in a certain subset of the universal Teichmfiller space by a property of the Grunsky operator.     

Presentability of Groups of Σ-Definable Permutations over Admissible Sets

It is proved that a group of all -permutations of an arbitrary locally countable, recursively listed, admissible set is not -presentable over that set. An example is constructed showing that this group could be presentable provided that the condition of being locally countable is dropped.     

On t-Dimension over Strong Mori Domains

In this note we prove that if R is a strong Mori domain with t-dim R = n and with countably many prime v-ideals, then there is a chain of rings between R and R^w R1=R belong to R2……belong to Rn lohtuin in R^w such that each R, is also a strong Mori domain and t-dim Rk=n - k ＋ 1 for k = 1,2,..., n.     

On measure-preserving functions over ?3

This paper is devoted to (discrete) p-adic dynamical systems, an important domain of algebraic and arithmetic dynamics [31]?C[41], [5]?C[8]. In this note we study properties of measurepreserving dynamical systems in the case p = 3. This case differs crucially from the case p = 2. The latter was studied in the very detail in [43]. We state results on all compatible functions which preserve measure on the space of 3-adic integers, using previous work of A. Khrennikov and author of present paper, see [24]. To illustrate one of the obtained theorems we describe conditions for the 3-adic generalized polynomial to be measure-preserving on ?₃. The generalized polynomials with integral coefficients were studied in [17, 33] and represent an important class of T-functions. In turn, it is well known that T-functions are well-used to create secure and efficient stream ciphers, pseudorandom number generators.     

On the index sets of Σ-subsets of the real numbers

We compute the levels of complexity in analytical and arithmetical hierarchies for the sets of the Σ-formulas defining in the hereditarily finite superstructure over the ordered field of the reals the classes of open, closed, clopen, nowhere dense, dense subsets of ? ⁿ , first category subsets in ? ⁿ as well as the sets of pairs of Σ-formulas corresponding to the relations of set equality and inclusion which are defined by them. It is also shown that the complexity of the set of the Σ-formulas defining connected sets is at least Π ₁ ¹ .     

On Polynomial Functions over Finite Commutative Rings

Let R be an arbitrary finite commutative local ring. In this paper, we obtain a necessary and sufficient condition for a function over R to be a polynomial function. Before this paper, necessary and sufficient conditions for a function to be a polynomial function over some special finite commutative local rings were obtained.     

What is a universal computing machine?

A computer is classically formalised as a universal Turing machine or a similar device. However over the years a lot of research has focused on the computational properties of dynamical systems other than Turing machines, such cellular automata, artificial neural networks, mirrors systems, etc.In this paper we propose a unifying formalism derived from a generalisation of Turing’s arguments. Then we review some of universal systems proposed in the literature and show that are particular case of this formalism. Finally, we review some of the attempts to understand the relation between dynamical and computational properties of a system.