The finite <Emphasis Type=Italic>p</Emphasis>-groups with <Emphasis Type=Italic>p</Emphasis> conjugacy classes of non-normal subgroups

Let ν(G) be the number of conjugacy classes of non-normal subgroups of a finite group G. The finite groups for which ν(G) ≤ 2 were determined by Dedekind and by Schmidt in the early times of group theory. On the other hand, if G is a finite p-group, La Haye and Rhemtulla have proved that either ν(G) ≤ 1 or ν(G) ≥ p. In this note, we determine all finite p-groups satisfying ν(G) = p for p > 2.     

On finite groups with special Sylow <Emphasis Type=Italic>p</Emphasis>-subgroups

We find the cases in which a finite p-soluble group with a special Sylow p-subgroup has p-length 1.     

On the uniqueness of (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">h</Emphasis>)-gonal automorphisms of Riemann surfaces

Let X be a compact Riemann surface of genus g ≥  2. A cyclic subgroup of prime order p of Aut(X) is called properly (p, h)-gonal if it has a fixed point and the quotient surface has genus h. We show that if p > 6h + 6, then a properly (p, h)-gonal subgroup of Aut(X) is unique. We also discuss some related results.     

The <Emphasis Type=Italic>p</Emphasis>-adic quantum differential

We propose the Fredholm module structure on the space L₂(ℙ¹(ℚ_p)). We study the properties of the corresponding quantum Connes differential.     

A necessary condition for non-abelian finite <Emphasis Type=Italic>p</Emphasis>-groups with second centre of order <Emphasis Type=Italic>p</Emphasis><Superscript>2</Superscript>

Let G be a non-abelian finite p-group such that |Z ₂(G)| = p ². In this paper we prove that each maximal subgroup M ≠ C _G (Z ₂(G)) is non-abelian and has cyclic centre.     

A polynomial <Emphasis Type=Italic>p</Emphasis>-adic dynamical system

We completely describe the Siegel discs and attractors for the p-adic dynamical system f(x) = x ²ⁿ⁺¹ + axⁿ⁺¹ on the space of complex p-adic numbers.     

Solutions of <Emphasis Type=Italic>p</Emphasis>-adic string equations

We review several purely mathematical results concerning boundary value problems for nonlinear pseudodifferential equations for p-adic closed and open strings in the tree approximation in the case d = 1. For the solutions of these problems, we present formulas establishing the relations between the numbers of their zeros, the multiplicities of the zeros, and the numbers indicating how many times the solutions change sign.     

Nonlocal General Boundary Value Problems of Elliptic Type in <Emphasis Type=Italic>L</Emphasis><Superscript><Emphasis Type=Italic>p</Emphasis></Superscript> Cases

This paper is devoted to the study of operational second-order differential equations of elliptic type with nonregular coefficient-operator boundary conditions. In the framework of UMD spaces, we give some new results by using semi-groups and interpolation theory. We define two types of solutions: semi-strict and strict solutions. We then characterizes the existence and uniqueness of such solutions.     

The <Emphasis Type=Italic>p</Emphasis>-Daugavet property for function spaces

A natural extension of the Daugavet property for p-convex Banach function spaces and related classes is analysed. As an application, we extend the arguments given in the setting of the Daugavet property to show that no reflexive space falls into this class.     

Adaptive learning and <Emphasis Type=Italic>p</Emphasis>-best response sets

A product set of strategies is a p-best response set if for each agent it contains all best responses to any distribution placing at least probability p on his opponents’ profiles belonging to the product set. A p-best response set is minimal if it does not properly contain another p-best response set. We study a perturbed joint fictitious play process with bounded memory and sample and a perturbed independent fictitious play process as in Young (Econometrica 61:57–84, 1993). We show that in n-person games only strategies contained in the unique minimal p-best response set can be selected in the long run by both types of processes provided that the rate of perturbations and p are sufficiently low. For each process, an explicit bound of p is given and we analyze how this critical value evolves when n increases. Our results are robust to the degree of incompleteness of sampling relative to memory.     

About the computation of the signature of surface singularities <Emphasis Type=Italic>z</Emphasis><Superscript><Emphasis Type=Italic>N</Emphasis></Superscript> + <Emphasis Type=Italic>g</Emphasis>(<Emphasis Type=Italic>x,y</Emphasis>) = 0

In this article we describe our experiences with a parallel Singular implementation of the signature of a surface singularity defined by z ^N + g(x; y) = 0.     

Periodic wavelets on the <Emphasis Type=Italic>p</Emphasis>-adic Vilenkin group

Using the Walsh-Dirichlet type kernel, we construct periodic wavelets on the p-adic Vilenkin group. These wavelets are similar to the trigonometric wavelets which were introduced by C. K. Chui and H. N. Mhaskar [1]. Results on the corresponding fast algorithms for decomposition and reconstruction are also discussed.