Multicriteria optimization with a multiobjective golden section line search

This work presents an algorithm for multiobjective optimization that is structured as: (i) a descent direction is calculated, within the cone of descent and feasible directions, and (ii) a multiobjective line search is conducted over such direction, with a new multiobjective golden section segment partitioning scheme that directly finds line-constrained efficient points that dominate the current one. This multiobjective line search procedure exploits the structure of the line-constrained efficient set, presenting a faster compression rate of the search segment than single-objective golden section line search. The proposed multiobjective optimization algorithm converges to points that satisfy the Kuhn-Tucker first-order necessary conditions for efficiency (the Pareto-critical points). Numerical results on two antenna design problems support the conclusion that the proposed method can solve robustly difficult nonlinear multiobjective problems defined in terms of computationally expensive black-box objective functions.     

The p-adic Approach to Wolstenholme's Theorem

The p-adic Approach to Wolstenholme's Theorem@孙琦@洪绍方&&     

Critical p-adic L-functions

We attach p-adic L-functions to critical modular forms and study them. We prove that those L-functions fit in a two-variables p-adic L-function defined locally everywhere on the eigencurve.     

On p-adic classification

A p-adic modification of the split-LBG classification method is presented in which first clusterings and then cluster centers are computed which locally minimize an energy function. The outcome for a fixed dataset is independent of the prime number p with finitely many exceptions. The methods are applied to the construction of p-adic classifiers in the context of learning.     

Universal p-adic series

We establish the analogue of the original Fekete Theorem in the context of p-adic analysis.     

p-adic dynamic systems

Dynamic systems in non-Archimedean number fields (i.e., fields with non-Archimedean valuations) are studied. Results are obtained for the fields of p-adic numbers and complex p-adic numbers. Simple p-adic dynamic systems have a very rich structure—attractors, Siegel disks, cycles, and a new structure called a “fuzzy cycle”. The prime number p plays the role of a parameter of the p-adic dynamic system. Changing p radically changes the behavior of the system: attractors may become the centers of Siegel disks, and vice versa, and cycles of different lengths may appear or disappear. Alexander von Humboldt Fellowship and SFB 237 Essen-Bochum-Düsseldorf, on leave from Moscow State Institute of Electronic Engineering. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 114, No. 3, pp. 349–365, March, 1998.     

p-adic Fourier transforms

In [3]-[7] Woodcock developed a Fourier theory for continuously differentiable functions defined on the set of p-adic integers. In this paper his theory is continued by giving a characterization of the image of the Fourier transformation. Also a special form of continuity of the inverse Fourier transformation is proved and, as an application, the Fourier transform of an antiderivative of a function is calculated.     

Radius of Convergence of p-adic Connections and the p-adic Rolle Theorem

We apply the theory of the radius of convergence of a p-adic connection [2] to the special case of the direct image of the constant connection via a finite morphism of compact p-adic curves, smooth in the sense of rigid geometry. We detail in sections 1 and 2, how to obtain convergence estimates for the radii of convergence of analytic sections of such a finite morphism. In the case of an étale covering of curves with good reduction, we get a lower bound for that radius, corollary 3.3, and obtain, via corollary 3.7, a new geometric proof of a variant of the p-adic Rolle theorem of Robert and Berkovich, theorem 0.2. We take this opportunity to clarify the relation between the notion of radius of convergence used in [2] and the more intrinsic one used by Kedlaya [16, Def. 9.4.7.].     

The Picard–Lefschetz formula for p-adic cohomology

In this paper, we discuss a p-adic analogue of the Picard–Lefschetz formula. For a family with ordinary double points over a complete discrete valuation ring of mixed characteristic (0,p), we construct vanishing cycle modules which measure the difference between the rigid cohomology groups of the special fiber and the de Rham cohomology groups of the generic fiber. Furthermore, the monodromy operators on the de Rham cohomology groups of the generic fiber are described by the canonical generators of the vanishing cycle modules in the same way as in the case of the ℓ-adic (or classical) Picard–Lefschetz formula. For the construction and the proof, we use the logarithmic de Rham–Witt complexes and those weight filtrations investigated by Mokrane (Duke Math. J. 72(2):301–337, 1993).      

Conditional p-adic probability logic

In this paper we present the proof-theoretical approach to p-adic valued conditional probabilistic logics. We introduce two such logics denoted by ${\mathit{CPL}}_{{\mathbf{Z}}_{\mathbf{p}}}$ and ${\mathit{CPL}}_{{\mathbf{Q}}_{\mathbf{p}}}^{\mathit{fin}}$. Each of these logics extends classical propositional logic with a list of binary (conditional probability) operators. Formulas are interpreted in Kripke-like models that are based on p-adic probability spaces. Axiomatic systems with infinitary rules of inference are given and proved to be sound and strongly complete. The decidability of the satisfiability problem for each logic is proved.     

On p-adic mathematical physics

A brief review of some selected topics in p-adic mathematical physics is presented. The text was submitted by the authors in English.     

On periodic p-adic distributions

In this paper we introduce a notion of periodic p-adic distribution defined on ? _p — the set of p-adic integers. This periodicity depends on a partition of ? _p . For several concrete partitions we describe corresponding periodic p-adic distributions. Moreover, we construct a periodic p-adic measure.     

Finite and p-adic Polylogarithms

The finite nth polylogarithm li _n (z) /p(z) is defined as _k=1 ^p–1 z ^k /k ⁿ . We state and prove the following theorem. Let Li _k : _{p p} be the p-adic polylogarithms defined by Coleman. Then a certain linear combination F _n of products of polylogarithms and logarithms, with coefficients which are independent of p, has the property that p ^1–n DF _n (z) reduces modulo p>n+1 to li _n–1((z)), where D is the Cathelineau operator z(1–z)d/dz and is the inverse of the p-power map. A slightly modified version of this theorem was conjectured by Kontsevich. This theorem is used by Elbaz-Vincent and Gangl to deduce functional equations of finite polylogarithms from those of complex polylogarithms.