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 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.     

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.     

The supercuspidal representations of <Emphasis Type=Italic>p</Emphasis>-adic classical groups

Let G be a unitary, symplectic or special orthogonal group over a locally compact non-archimedean local field of odd residual characteristic. We construct many new supercuspidal representations of G, and Bushnell–Kutzko types for these representations. Moreover, we prove that every irreducible supercuspidal representation of G arises from our constructions.     

Van der Put basis and <Emphasis Type=Italic>p</Emphasis>-adic dynamics

We consider a function g: Z _p → Z _p and its the van der Put series. Then we get a criteria of Haar’s measure preserving compatible p-adic functions which, actually, need not be uniformly differentiable modulo p. This is used to study ergodicity of p-adic dynamical systems [2, 16].     

<Emphasis Type=Italic>p</Emphasis>-adic interpolation of Taylor coefficients of modular forms

In this paper we show that the Taylor coefficients of a Hecke eigenform at a CM-point, suitably modified, form a sequence of algebraic numbers that satisfy the Kubota–Leopoldt generalization of the Kummer congruences for primes that split in the imaginary quadratic field associated with a CM-point. More generally, we show that these numbers are moments of a certain p-adic measure. In addition, we write down explicitly the “Euler factor” at p in terms of the p th Hecke eigenvalue of the modular form in question and certain data of the CM-point. P. Guerzhoy is supported by NSF grant DMS-0700933.     

A derivative on the field of <Emphasis Type=Italic>p</Emphasis>-adic numbers

Using the fact that the multiplicative group of units in a p-adic field is a compact Vilenkin group, we introduce a new pointwise derivative on ℚ _p ^* and investigate basic properties of this derivative.     

Asymptotic properties of combinatorial optimization problems in <Emphasis Type=Italic>p</Emphasis>-adic space

In this article we consider two well known combinatorial optimization problems (travel-ling salesman and minimum spanning tree), when n points are randomly distributed in a unit p-adic ball of dimension d. We investigate an asymptotic behavior of their solutions at large number of n. It was earlier found that the average lengths of the optimal solutions in both problems are of order n ^1−1/d . Here we show that standard deviations of the optimal lengths are of order n ^1/2−1/d if d > 1, and prove that large number laws are valid only for special subsequences of n.     

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 cyclic <Emphasis Type=Italic>p</Emphasis>-gonal Riemann surfaces with several <Emphasis Type=Italic>p</Emphasis>-gonal morphisms

Let p be a prime number, p > 2. A closed Riemann surface which can be realized as a p-sheeted covering of the Riemann sphere is called p-gonal, and such a covering is called a p-gonal morphism. If the p-gonal morphism is a cyclic regular covering, the Riemann surface is called a cyclic p-gonal Riemann surface. Accola showed that if the genus is greater than (p − 1)² the p-gonal morphism is unique. Using the characterization of p-gonality by means of Fuchsian groups we show that there exists a uniparametric family of cyclic p-gonal Riemann surfaces of genus (p − 1)² which admit two p-gonal morphisms. In this work we show that these uniparametric families are connected spaces and that each of them is the Riemann sphere without three points. We study the Hurwitz space of pairs (X, f), where X is a Riemann surface in one of the above families and f is a p-gonal morphism, and we obtain that each of these Hurwitz spaces is a Riemann sphere without four points.     

<Emphasis Type=Italic>L</Emphasis><Subscript><Emphasis Type=Italic>p</Emphasis></Subscript>-mixed intersection bodies

In this paper the author first introduce a new concept of L _p -dual mixed volumes of star bodies which extends the classical dual mixed volumes. Moreover, we extend the notions of L _p intersection body to L _p -mixed intersection body. Inequalities for L _p -dual mixed volumes of L _p -mixed intersection bodies are established and the results established here provide new estimates for these type of inequalities. This work was supported by the Natural Science Foundation of Zhejiang Province of China (Grant No. Y605065) and the Foundation of the Education Department of Zhejiang Province of China (Grant No. 20050392)     

Existence of nontrivial solutions for <Emphasis Type=Italic>p</Emphasis>-Laplacian-Like equations

In this paper, we study the p-Laplacian-Like equations involving Hardy potential or involving critical exponent and prove the existence of one or infinitely many nontrivial solutions. The results of the equations discussed can be applied to a variety of different fields in applied mechanics.     

Approximations for the <Emphasis Type=Italic>M</Emphasis>/<Emphasis Type=Italic>GI</Emphasis>/<Emphasis Type=Italic>N</Emphasis>+<Emphasis Type=Italic>GI</Emphasis> type call center

In this paper, we propose approximations to compute the steady-state performance measures of the M/GI/N+GI queue receiving Poisson arrivals with N identical servers, and general service and abandonment-time distributions. The approximations are based on scaling a single server M/GI/1+GI queue. For problems involving deterministic and exponential abandon times distributions, we suggest a practical way to compute the waiting time distributions and their moments using the Laplace transform of the workload density function. Our first contribution is numerically computing the workload density function in the M/GI/1+GI queue when the abandon times follow general distributions different from the deterministic and exponential distributions. Then we compute the waiting time distributions and their moments. Next, we scale-up the M/GI/1+GI queue giving rise to our approximations to capture the behavior of the multi-server system. We conduct extensive numerical experiments to test the speed and performance of the approximations, which prove the accuracy of their predictions.      

On <Emphasis Type=Italic>l</Emphasis><Subscript>2,<Emphasis Type=Italic>p</Emphasis></Subscript>-circle numbers

Circle numbers are defined to reflect the Euclidean area-content and, for p ≠ 2, suitably defined non-Euclidean circumference properties of the l _2,p -circles, p ∈ [1, ∞]. The resulting function is continuous, increasing, and takes all values from [2, 4]. The actually chosen dual l _2,p -geometry for measuring the arc-length is closely connected with a generalization of the method of indivisibles of Cavalieri and Torricelli in the sense that integrating such arc-lengths means measuring area content. Moreover, this approach enables one to look in a new way into the co-area formula of measure theory which says that integrating Euclidean arc-lengths does not yield area content except for p = 2. The new circle numbers play a natural role, e.g., as norming constants in geometric measure representation formulae for p-generalized uniform probability distributions on l _2,p -circles.     

Group invariance and <Emphasis Type=Italic>L</Emphasis><Superscript><Emphasis Type=Italic>p</Emphasis></Superscript>-bounded operators

The Hilbert and Riesz transforms can be characterized up to scalar as the translation invariant operators that satisfy additionally certain relative invariance of conformal transformation groups. In this article, we initiate a systematic study of translation invariant operators from group theoretic viewpoints, and formalize a geometric condition that characterizes specific multiplier operators uniquely up to scalar by means of relative invariance of affine subgroups. After providing some interesting examples of multiplier operators having “large symmetry”, we classify which of these examples can be extended to continuous operators on L ^p (R ⁿ ) (1 < p < ∞). T. Kobayashi was partially supported by Grant-in-Aid for Scientific Research 18340037, Japan Society for the Promotion of Science. A. Nilsson was partially supported by Japan Society for the Promotion of Science.