Quasi-Invariant and Pseudo-Differentiable Measures with Values in Non-Archimedean Fields on a Non-Archimedean Banach Space

Quasi-invariant and pseudo-differentiable measures on a Banach space X over a non-Archimedean locally compact infinite field with a non-trivial valuation are defined and constructed. Measures are considered with values in non-Archimedean fields, for example, the field Q _p of p-adic numbers. Theorems and criteria are formulated and proved about quasi-invariance and pseudo-differentiability of measures relative to linear and non-linear operators on X. Characteristic functionals of measures are studied. Moreover, the non-Archimedean analogs of the Bochner-Kolmogorov and Minlos-Sazonov theorems are investigated. Infinite products of measures are considered and the analog of the Kakutani theorem is proved. Convergence of quasi-invariant and pseudo-differentiable measures in the corresponding spaces of measures is investigated.__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 149–199, 2003.     

Büchi’s problem for ultrametric meromorphic functions inside an open disk

Let K be a complete ultrametric algebraically closed field of characteristic zero such as ℂ _p . Büchi’s problem was solved for p-adic meromorphic functions in the whole field K. Here we show similar conclusions for meromorphic functions in an open disk that are not quotients of bounded analytic functions. The main method is the secondMain Theorem for p-adic meromorphic functions inside a disk, a specific p-adic theorem.     

Some aspects of <Emphasis Type="Italic">m</Emphasis>-adic analysis and its applications to <Emphasis Type="Italic">m</Emphasis>-adic stochastic processes

In this paper we consider a generalization of analysis on p-adic numbers field to the m case of m-adic numbers ring. The basic statements, theorems and formulas of p-adic analysis can be used for the case of m-adic analysis without changing. We discuss basic properties of m-adic numbers and consider some properties of m-adic integration and m-adic Fourier analysis. The class of infinitely divisible m-adic distributions and the class of m-adic stochastic Levi processes were introduced. The special class of m-adic CTRW process and fractional-time m-adic random walk as the diffusive limit of it is considered. We found the asymptotic behavior of the probability measure of initial distribution support for fractional-time m-adic random walk.     

The BC-system and L-functions

In these lectures we survey some relations between L-functions and the BC-system, including new results obtained in collaboration with C. Consani. For each prime p and embedding σ of the multiplicative group of an algebraic closure of \mathbb F_p{\mathbb {F}_p} as complex roots of unity, we construct a p-adic indecomposable representation π_σ of the integral BC-system. This construction is done using the identification of the big Witt ring of [`(\mathbb F)]_p{\bar{\mathbb F}_p} and by implementing the Artin–Hasse exponentials. The obtained representations are the p-adic analogues of the complex, extremal KMS_∞ states of the BC-system. We use the theory of p-adic L-functions to determine the partition function. Together with the analogue of the Witt construction in characteristic one, these results provide further evidence towards the construction of an analogue, for the global field of rational numbers, of the curve which provides the geometric support for the arithmetic of function fields.     

Approximations simultanées de nombres algébriques de Q p par des rationnels

 In this paper, we prove that if β₁,…, β _n are p-adic numbers belonging to an algebraic number field K of degree n + 1 over Q such that 1, β₁,…,β _n are linearly independent over Z, there exist infinitely many sets of integers (q ₀,…, q _n ), with q ₀ ≠ 0 and

Prüfer’s ideal numbers as Gelfand’s maximal ideals

Polyadic arithmetics is a branch of mathematics related to p-adic theory. The authors suggest two non-classical models for the Prüfer profinite completion Z of the ring Z. Firstly, letA be the algebra of all periodic functions on Z, then Z can be defined as the ring of all non-zero morphisms A → C with convolution ring operations equipped with some natural topology. Secondly, let (E, μ) be the maximal ideal space for the Banach algebra of almost periodic functions on Z with the Gelfand topology μ. One can define rings operations in (E, μ) which turn it into a topological ring isomorphic to Z.     

A combinatorial principle and endomorphism rings I: Onp-groups

Two lines of research are involved here. One is a combinatorial principle, proved in ZFC for many cardinals (e.g., any λ = λ^ℵ ₀) enabling us to prove things which have been proven using the diamond or for strong limit cardinals of uncountable cofinality. The other direction is building abelian groups with few endomorphisms and/or prescribed rings of endomorphisms. We prove that for a ringR, whose additive group is thep-adic completion of a freep-adic module,R is isomorphic to the endomorphism ring of some separable abelianp-groupG divided by the ideal of small endomorphisms, withG of power λ for any λ = λ^ℵ ₀≧|R|. Dedicated to the memory of Abraham Robinson on the tenth anniversary of his death The author would like to thank the United States-Israel Binational Science Foundation for partially supporting this research.     

On zeta functions and families of Siegel modular forms

Let p be a prime, and let G = \textS\textp_g( \mathbbZ ) \Gamma = {\text{S}}{{\text{p}}_g}\left( \mathbb{Z} \right) be the Siegel modular group of genus g. This paper is concerned with p-adic families of zeta functions and L-functions of Siegel modular forms; the latter are described in terms of motivic L-functions attached to Sp _g ; their analytic properties are given. Critical values for the spinor L-functions are discussed in relation to p-adic constructions. Rankin’s lemma of higher genus is established. A general conjecture on a lifting of modular forms from GSp_2m   × GSp_2m to GSp_4m (of genus g = 4 m) is formulated. Constructions of p-adic families of Siegel modular forms are given using Ikeda–Miyawaki constructions.     

Symmetry in data mining and analysis: A unifying view based on hierarchy

Data analysis and data mining are concerned with unsupervised pattern finding and structure determination in data sets. The data sets themselves are explicitly linked as a form of representation to an observational, or otherwise empirical, domain of interest. “Structure” has long been understood as symmetry which can take many forms with respect to any transformation, including point, translational, rotational, and many others. Symmetries directly point to invariants that pinpoint intrinsic properties of the data and of the background empirical domain of interest. As our data models change, so too do our perspectives on analyzing data. The structures in data surveyed here are based on hierarchy, represented as p-adic numbers or an ultrametric topology.     

Laurent rings

This is a study of ring-theoretic properties of a Laurent ring over a ring A, which is defined to be any ring formed from the additive group of Laurent series in a variable x over A, such that left multiplication by elements of A and right multiplication by powers of x obey the usual rules, and such that the lowest degree of the product of two nonzero series is not less than the sum of the lowest degrees of the factors. The main examples are skew-Laurent series rings A((x; ϕ)) and formal pseudo-differential operator rings A((t ⁻¹; δ)), with multiplication twisted by either an automorphism ϕ or a derivation δ of the coefficient ring A (in the latter case, take x = t ⁻¹). Generalized Laurent rings are also studied. The ring of fractional n-adic numbers (the localization of the ring of n-adic integers with respect to the multiplicative set generated by n) is an example of a generalized Laurent ring. Necessary and/or sufficient conditions are derived for Laurent rings to be rings of various standard types. The paper also includes some results on Laurent series rings in several variables. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 3, pp. 151–224, 2006.     

From image processing to topological modelling with <Emphasis Type="Italic">p</Emphasis>-adic numbers

Encoding the hierarchical structure of images by p-adic numbers allows for image processing and computer vision methods motivated from arithmetic physics. The p-adic Polyakov action leads to the p-adic diffusion equation in low level vision. Hierarchical segmentation provides another way of p-adic encoding. Then a topology on that finite set of p-adic numbers yields a hierarchy of topological models underlying the image. In the case of chain complexes, the chain maps yield conditions for the existence of a hierarchy, and these can be expressed in terms of p-adic integrals. Such a chain complex hierarchy is a special case of a persistence complex from computational topology, where it is used for computing persistence barcodes for shapes. The approach is motivated by the observation that using p-adic numbers often leads to more efficient algorithms than their real or complex counterparts.     

Structure,classification, and conformal symmetry of elementary particles over non-archimedean space-time

It is well known that at distances shorter than Planck length, no length measurements are possible. The Volovich hypothesis asserts that at sub-Planckian distances and times, spacetime itself has a non-Archimedean geometry. We discuss the structure of elementary particles, their classification, and their conformal symmetry under this hypothesis. Specifically, we investigate the projective representations of the p-adic Poincaré and Galilean groups, using a new variant of the Mackey machine for projective unitary representations of semidirect products of locally compact and second countable (lcsc) groups. We construct the conformal spacetime over p-adic fields and discuss the imbedding of the p-adic Poincaré group into the p-adic conformal group. Finally, we show that the massive and the so called eventually masssive particles of the Poincaré group do not have conformal symmetry. The whole picture bears a close resemblance to what happens over the field of real numbers, but with some significant variations.     

On infinite products of non-Archimedean measure spaces

We study generalized ‘probabilistic measures’ taking values in non-Archimedean fields (in particular, fields of p-adic numbers). We prove the theorem on the existence of probability on a product of non-Archimedean probabilistic spaces.     

p-adic physics below and above planck scales

We present a review and also new possible applications of p-adic numbers to pre-space-time physics. It is shown that instead of the extension IRⁿ → Q_pⁿ, which is usually implied in p-adic quantum field theory, it is possible to build a model based on the IRⁿ → Q_p, where p=n+2 extension and get rid of loop divergences. It is also shown that the concept of mass naturally arises in p-adic models as inverse transition probability with a dimensional constant of proportionality.     

On the theory of groups with generalized minimality condition for closed subgroups

We prove that a topological Abelian locally compact group with generalized minimality condition for closed subgroups is a group of one of the following types: 1) a group with minimality condition for closed subgroups, 2) an additive group of theJ _p -ring of integerp-adic numbers, 3) an additive groupR _p of the field ofp-adic numbers (p is a prime number). Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 3, pp. 398–409, March, 1999.     

Criteri jacobiani di regolarità ed eccellenza per anelli di serie ristrette

Riassunto Nel presente lavoro studiamo il problema seguente: seA è un anellom-adico eccellente, è pure eccellente l’anello (A, m) {X ₁,…,X _n } delle serie ristrette? I risultati che presentiamo sono relativi ad anelli baseA di char. 0, ma con quozienti di char.p>0; essi sono basati su tecniche che sfruttano fortemente i criteri jacobiani di regolarità. Perciò ci riferiamo sempre ad anelli base ?di tipo analitico? (nel senso di [5], def. 1.1), cioè ad anelli muniti di un numero sufficiente di derivazioni, tale da garantire la validità di criteri jacobiani. Proviamo che, seA è locale regolare di tipo analitico eA/pA è finito come modulo su (A/pA) ^p , allora (A, m) {X ₁,…,X _n } è eccellente; evitando così, rispetto a [10] (teorema 10), ogni condizione restrittiva sui corpi residui. Estendiamo inoltre in modo naturale la definizione di anello di tipo analitico al caso non locale e troviamo condizioni necessarie e sufficienti per la permanenza dell’analiticità nel passaggio alle serie ristrette. Proviamo infine che gli anelli analitici sono eccellenti se lo sono modulop.

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Summary In the present paper we investigate the following problem: ifA is anm-adic excellent ring, is the restricted power series ring(A, m){X ₁,…,X _n } also excellent? Here we are able to produce some results when the basic ringsA are of char. 0, but with quotients of char.p>0. We need to use techniques which exploit strongly the jacobian criteria of regularity; hence we limit ourselves to the class of the basic rings ?of analytic type? (in the sense of [5], def. 1.1), i.e. rings with enough derivations to make jacobian criteria true. We prove that, ifA is a regular local ring of analytic type andA/pA is a finite module over (A/pA)^p, then (A, m){X ₁,…,X _n } is excellent. With regard to [10], theorem 10, we can avoid every restrictive condition over the residue fields. Moreover we extend in a natural way the definition of a ring of analytic type in the case of a regular domain of char. 0 not necessarily local and we find necessary and sufficient conditions for the permanence of the analytic property in the passage to the restricted power series. At last we prove that the analytic rings are excellent if they are so modulop.

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Lavoro eseguito nell’ambito dell’attività dei Gruppi di Ricerca Matematica del CNR (GNSAGA).     

Factorisation Theorems for T-decomposable Measures on Groups

 For a real or p-adic unipotent algebraic group G, given a T∈ Hom(G, G) and T-decomposable measure on G which is either ‘full’ or symmetric, we get a decomposition , where μ₀ is T-invariant and , and this decomposition is unique upto a shift. We also show that ν₀ is T-decomposable under some additional sufficient condition and give a counter example to justify this. We generalise the above to power bounded operators on p-adic Banach spaces. We also prove some convergence-of-types theorems on p-adic groups as well as Banach spaces. (Received 21 October 2000; in revised form 21 February 2001)     

<Emphasis Type="Italic">p</Emphasis>-Adic Brownian motion over ℚ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

In this paper, we generalize the result of Bikulov and Volovich (1997) and construct a p-adic Brownian motion over ℚ _p . First, we construct directly a p-adic white noise over ℚ _p by using a specific complete orthonormal system of (ℚ _p ). A p-adic Brownian motion over ℚ _p is then constructed by the Paley-Wiener method. Finally, we introduce a p-adic random walk and prove a theorem on the approximation of a p-adic Brownian motion by a p-adic random walk.     

Mumford dendrograms and discrete p-adic symmetries

In this article, we present an effective encoding of dendrograms by embedding them into the Bruhat-Tits trees associated to p-adic number fields. As an application, we show how strings over a finite alphabet can be encoded in cyclotomic extensions of ℚ _p and discuss p-adic DNA encoding. The application leads to fast p-adic agglomerative hierarchic algorithms similar to the ones recently used e.g. by A. Khrennikov and others. From the viewpoint of p-adic geometry, to encode a dendrogram X in a p-adic field K means to fix a set S of K-rational punctures on the p-adic projective line ℙ₁. To ℙ₁ \ S is associated in a natural way a subtree inside the Bruhat-Tits tree which recovers X, a method first used by F. Kato in 1999 in the classification of discrete subgroups of PGL₂(K). Next, we show how the p-adic moduli space of ℙ¹ with n punctures can be applied to the study of time series of dendrograms and those symmetries arising from hyperbolic actions on ℙ¹. In this way, we can associate to certain classes of dynamical systems a Mumford curve, i.e. a p-adic algebraic curve with totally degenerate reduction modulo p. Finally, we indicate some of our results in the study of general discrete actions on ℙ¹, and their relation to p-adic Hurwitz spaces. The text was submitted by the author in English.