Abstract. In archimedean **analysis** Tauberian operators and operators having property N were deﬁned by Kalton and Wilansky (1976). We give several characterizations of **p**-**adic** Tauberian operators and operators having property N in terms of basic sequences. And, as its applications, we give some equivalent relations between these operators and **p**-**adic** semi-Fredholm operators.

14 Read more

The **p**-**adic** numbers are a counterintuitive arithmetic system, which were …rstly introduced by the Kummer in 1850. Then, the German mathematician, Kurt Hensel (1861-1941) developed the **p**-**adic** numbers in a paper concerned with the development of algebraic numbers in power series in circa 1897, cf. [25]. There are numbers of all kinds such as natural, rational, real, complex, **p**-**adic**, quantum numbers. The **p**-**adic** numbers are less well known than the others, however these numbers play a main role in number theory and the related topics in mathematics. Whereas, mentioned **p**-**adic** numbers have penetrated some mathematical areas, among algebraic number theory, algebraic geometry, algebraic topolgy and **analysis**, the foregoing numbers are now well-established in mathematical …eld and are used also by physicists. In conjunction with the introduction of these numbers, some mathematicians and physicists started to investigate new scienti…c tools utilizing their useful and positive properties. Some e¤ects of these new researches have emerged in mathematics and physics such as **p**-**adic** **analysis**, string theory, **p**-**adic** quantum mechanics, quantum …eld theory, representation theory, algebraic geometry, complex systems, dynamical systems, genetic codes and so on (cf. [1-3; 6-16; 18-24; 25]). The one of the most important tool of these investigations is **p**-**adic** gamma function which is …rstly described by Yasou Morita in about 1975 (cf. [18]). Intense research activities in such an area as **p**-**adic** gamma function is principally motivated by their importance in **p**-**adic** **analysis**. Therefore, in recent fourty years, **p**-**adic** gamma function and its generalizations have been investigated and studied extensively by many mathematicians (cf. [6; 8-11; 13-16; 18; 20; 21; 25]).

Show more
17 Read more

In recent years, studying degenerate versions of various special polynomials and numbers has attracted many mathematicians and has been carried out by several diﬀerent methods like generating functions, combinatorial approaches, umbral calculus, **p**-**adic** **analysis**, and diﬀerential equations. In this paper, we introduced degenerate type 2 Bernoulli polynomi- als, fully degenerate type 2 Bernoulli polynomials, and degenerate type 2 Euler polynomi- als, and their corresponding numbers, as degenerate and type 2 versions of Bernoulli and Euler numbers. We investigated those polynomials and numbers by means of bosonic and fermionic **p**-**adic** integrals and derived some identities, distribution relations, Witt type formulas, and analogues for the Bernoulli interpretation of powers of the ﬁrst m positive integers in terms of Bernoulli polynomials. In more detail, our main results are as follows. As to the analogues for the Bernoulli interpretation of power sums, in Theorem 2.6 we expressed powers of the ﬁrst m odd integers in terms of type 2 Bernoulli polynomials b n (x), in Theorem 2.11 alternating sum of powers of the ﬁrst m odd integers in terms of

Show more
15 Read more

In recent years, **p**-**adic** ﬁelds have been introduced into some aspects of mathematical physics. There are a lot of articles where diﬀerent applications of the **p**-**adic** **analysis** in the string theory, quantum mechanics, stochastics, the theory of dynamical systems, cognitive sciences, and psychology are studied [–] (see also the references therein). As a conse- quence, new mathematical problems have emerged, among them, the study of harmonic **analysis** on a **p**-**adic** ﬁeld has been drawing more and more concern (cf. [–] and the references therein).

Show more
12 Read more

In this article we shall be concerned with metaplectic groups of type 1. As we point out in Proposition 2.2 below, there are many groups G for which all metaplectic covers of G are of type 1, so this restriction is not too onerous. Our methods will not apply to type 2 covers, since such covers do not possess genuine metaplectic forms of cohomological type. However we note that various authors have dealt with the **p**-**adic** interpolation of metaplectic forms of type 2 in certain cases, using very different methods (see for example [14,21,22]).

48 Read more

We write B rig;Q þ **p** for the ring of power series f ðpÞ A Q **p** J **p** K such that f ðX Þ converges everywhere on the open unit **p**-**adic** disc. Equip B rig; þ Q **p** with actions of G and a Frobenius operator j by g:p ¼ ðp þ 1Þ wðgÞ 1 and jðpÞ ¼ ðp þ 1Þ **p** 1. We can then deﬁne a left in- verse c of j satisfying

27 Read more

splits at one archimedean place). For most of the levels this homology group does not contain any rational Hecke eigenline, and thus one does not expect an elliptic curve of that conductor. For the levels in which there are rational lines, we have computed the L-invariant of each line, and tried to recognize an algebraic curve over K whose L-invariant matches up to high **p**-**adic** precision and whose conductor is N.

33 Read more

The **p**-**adic** numbers have been applied in the string theory, turbulence theory, statistical mechanics, quantum mechanics, and so forth (see [1, 9, 10] for detail). In the past few years, there is an increasing interest in the study of harmonic **analysis** on **p**-**adic** ﬁeld (see [5–8] for detail).

13 Read more

Let D be a complemented closed subspace of E, linearly homeomorphic to is a homeoa dual Banach space F’.. morphism and there exists a projection of D" onto with.[r]

Here, F denotes both the algebra of formal power series in t and the vector space of all linear functionals on **P** , and so an element f (t) of F will be thought of as both a formal power series and a linear functional (see [, ]). We will call F the umbral algebra. The umbral calculus is the study of umbral algebra (see [, ]).

11 Read more

for semi-linear and semi-algebraic sets. In particular, we give a characterization of definable functions in Section 3.1, and in Section 3.2, we give some examples to show that classification by definable bijection is not quite as simple as it is for semi-algebraic sets. (It was shown by Cluckers [1] that any two infinite **p**-**adic** semi-algebraic sets are isomorphic if and only if they have the same dimension.)

25 Read more

In recent years, there has been a growing interest on **p**-**adic** numbers. The lat- ter may answer some questions in Physics. Besides the number of papers in this area shows the interest of **p**-**adic** numbers to answer some questions in physics, as in string theory (connected with **p**-**adic** quantum field) and in the other natural sciences in which there are complicated fractal behaviors and hierarchical struc- tures (turbulence theory, dynamical systems, statistical physics, biology, see (Kozyrev (2008), Hua-chieh (2001), Dragovich (2009)). Particularly 2-**adic** num- bers will be useful for computer construction see (Klapper (1994)). Cianciri (1994) presented the main ideas to interpret a quantum mechanical state by means of **p**-**adic** statistics. He was interested in limits of probabilities when the number of trials approaches infinity. However, these limits are considered with respect to the **p**-**adic** metric. Khrennikov (1998) found a new asymptotic of the classical Bernoulli probabilities. In Khrennikov (1993), he developed the theory of **p**-**adic** probability to describe the statistical information processes. Kamizono (2007) defined the symmetric stochastic integrals with respect to **p**-**adic** Brownian motion and provided a sufficient condition for its existence. The properties of the trajectories of a **p**-**adic** Wiener process were studied using Vladimirov’s **p**-**adic** differentiation operator, see (Bikulov and Volovichb (1997)).

Show more
18 Read more

Based on results on proper smooth schemes over k , we expect certain relations between the **p** -**adic** completion of the étale motivic cohomology with compact supports (resp. **p** -**adic** completion of the étale Borel-Moore homology) with rigid cohomology with compact support (resp. dual of rigid cohomology with compact support), as stated below in Conjecture 1.3.1 (resp. Conjecture 1.3.2). These relations hold in the case of proper-smooth schemes as shown in [FM18, Proposition 7.21].

67 Read more

REPKA, J., Germs associated to regular unipotent classes in p-adic SLn, Canad.. REGAWSKI, J., An application of the building to orbital integrals, Compositio Math.[r]

11 Read more

If f is non-ordinary (the Hecke eigenvalue of f at **p** has valuation > 0) then both roots of the Hecke polynomial satisfy this condition, but if f is ordinary, then there is one root with valuation k − 1 (“critical slope”), to which the classical modular symbol constructions do not apply. Two approaches exist to rectify this injustice to the ordinary forms by constructing a critical-slope **p**-**adic** L-function. Firstly, there is an approach using **p**-**adic** modular symbols [15, 14, 2]. Secondly, there is an approach using Kato’s Euler system [9] and Perrin-Riou’s **p**-**adic** regulator map [13] (cf. [4, Remarque 9.4]). Although it is natural to conjecture that the objects arising from these two constructions coincide (cf. [14, Remark 9.7]), and the results of [10] are strong evidence for this conjecture, prior to the present work this was not known in a single example.

Show more
23 Read more

Thus we have a sharp lower bound on the 2-**adic** valuations of Lee weights of words in Abelian codes over Z /4 Z . We first compare this result with the specialization to Z /4 Z of Wilson’s result (Theorem 1.5). The new result is stronger for Z /4 Z -codes because ` ss (C) ≥ `(C). Indeed, there exists a sequence of codes such that ` ss (C) − `(C) is unbounded (see Proposition 4.22). Since Wilson’s result is stronger than that of Calderbank, Li, and Poonen, this new theorem is stronger than theirs as well. We should also compare this with the result of Helleseth, Kumar, Moreno, and Shanbhag, which applies only to the special case when our Abelian group A is cyclic of order 2 n − 1 for some n > 1. The new theorem states that the 2-**adic** valuations of weights are bounded below by ` ss (C) + 1 = bω ss (C)/2c, while

Show more
231 Read more

[NSW00] J¨ urgen Neukirch, Alexander Schmidt, and Kay Wingberg, Cohomology of number fields, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, Springer-Verlag, Berlin, 2000. MR 1737196. [Pas05] Vicentiu Pasol, **P**-**adic** modular symbols attached to C. M. forms, Ph.D. thesis, Boston

46 Read more

Some identities of Korobov type polynomials associated with p adic integrals on Zp$\mathbb{Z} {p}$ Kim and Kim Advances in Difference Equations (2015) 2015 282 DOI 10 1186/s13662 015 0602 8 R E S E A[.]

13 Read more

A b s t r ac t . The paper concerns the Sylow **p**-subgroups of automorphism groups of level homogeneous rooted trees. We recall and summarize the results obtained by L. Kaluzhnin on the structure of Sylow **p**-subgroups of isometry groups of ultrametric Cantor **p**-spaces in terms of automorphism groups of rooted trees. Most of the paper should be viewed as a systematic topical survey, however we include some new ideas in last sections.

20 Read more

The algebraic (in)dependence between elements of the form x, exp(x) in the **p**−**adic** domain plays a fundamental role in the **p**−**adic** Transcendental Number Theory. Many results have been made towards this direction. For example, in 1932 K.Mahler, [N], proved that exp(α) is transcendental over Q for any non-zero algebraic element α ∈ E (the domain of convergence of the exponential function). In 2008, Yu.V. Nesterenko proved that if α 1 , ..., α n ∈ E are algebraic over Q and form a basis of a finite extension of degree n of Q . Then, there exist at least b n 2 c among the elements exp(α 1 ), ...., exp(α n ) which are

Show more
12 Read more