Narrow Results

Suppose $K$ is a finite field extension of ${\mathbb{Q}}_p$ containing a primitive $p$th root of unity. Let $K_{<p}$ be a maximal $p$-extension of $K$ with the Galois group of period $p$ and nilpotent class $<p$. In this paper, we develop formalism which allows us to study the structure of ${\mathrm{\Gamma }}_{<p}=\mathrm{\text{Gal}}(K_{<p}/K)$ via methods of Lie theory. In particular, we introduce an explicit construction of a Lie ${𝔽}_p$-algebra $L$ and an identification ${\mathrm{\Gamma }}_{<p}=G(L)$, where $G(L)$ is a $p$-group obtained from the elements of $L$ via the Campbell–Hausdorff composition law. In the next paper, we apply this formalism to describe the ramification filtration ${\{{\mathrm{\Gamma }}_{<p}^{(v)}\}}_{v\ge 0}$ and an explicit form of the Demushkin relation for ${\mathrm{\Gamma }}_{<p}$.

Suppose $K$ is a finite field extension of ${\mathbb{Q}}_p$ containing a primitive $p$th root of unity. Let ${\mathrm{\Gamma }}_{<p}$ be the maximal quotient of period $p$ and nilpotent class $<p$ of the Galois group of a maximal $p$-extension of $K$. We describe the ramification filtration ${\{{\mathrm{\Gamma }}_{<p}^{(v)}\}}_{v\ge 0}$ and relate it to an explicit form of the Demushkin relation for ${\mathrm{\Gamma }}_{<p}$. The results are given in terms of Lie algebras attached to the appropriate $p$-groups by the classical equivalence of the categories of $p$-groups and Lie algebras of nilpotent class $<p$.

This paper investigates the Weierstrass type function in local fields whose graph is a chaotic repelling set of a discrete dynamical system, and proves that their exists a linear connection between the orders of its p-adic calculus and the dimensions of the corresponding graphs.

The Weierstrass nowhere differentiable function has been studied often as example of functions whose graphs are fractals in ${ℝ}^2$. This paper investigates the Weierstrass-type function in the $p$-adic local field $K_p$ whose graph is a repelling set of a discrete dynamical system, and proves that there exists a linear connection between the orders of the $p$-adic calculus and the dimensions of the corresponding graphs.

Femtosecond nonlinear optical response of BaTiO₃ and ZrO₂ matrices doped with gold nanoparticle were investigated by off-resonance optical Kerr effect. The experimental results show that the third-order nonlinear optical susceptibility of the composite films strongly depends on metal particle size and matrix refractive index. A calculation based on Lorenz–Mie scattering theory was performed to explain the experimental results. Additionally, optimization of nonlinearity was obtained by considering the particle size and matrices refractive index.

Domain switch, phase transition and electrochemical mechanisms introduced traditional smart materials to various technologies such as piezoelectric ceramics, shape memory alloys and elastor/polymer used in actuators, sensors, etc. From the studies on carbon nanotues, graphite and other nanoscaled materials, we show that due to the quantum effect induced response at atomic or molecular level, all matters in this scale would exhibit intelligent behaviors, and buildings based on this are expected to construct nanointelligent systems which can be believed to improve the current technologies.

This paper is an investigation of shift invariant subspaces of L²(G), where G is a locally compact abelian group, or in general a local field, with a compact open subgroup. In this paper we state necessary and sufficient conditions for shifts of an element of L²(G) to be an orthonormal system or a Parseval frame. Also we show that each shift invariant subspace of L²(G) is a direct sum of principle shift invariant subspaces of L²(G) generated by Parseval frame generators.

A multiresolution analysis (MRA) on local fields of positive characteristic was defined by Shah and Abdullah for which the translation set is a discrete set which is not a group. In this paper, we continue the study based on this nonstandard setting and introduce vector-valued nonuniform multiresolution analysis (VNUMRA) where the associated subspace V₀ of L²(K, ℂ^M) has an orthonormal basis of the form {Φ (x - λ)}_λ∈Λ where Λ = {0, r/N} + 𝒵, N ≥ 1 is an integer and r is an odd integer such that r and N are relatively prime and 𝒵 = {u(n) : n ∈ ℕ₀}. We establish a necessary and sufficient condition for the existence of associated wavelets and derive an algorithm for the construction of VNUMRA on local fields starting from a vector refinement mask G(ξ) with appropriate conditions. Further, these results also hold for Cantor and Vilenkin groups.

A constructive algorithm based on the theory of spectral pairs for constructing nonuniform wavelet basis in $L^2(ℝ)$ was considered by Gabardo and Nashed [Nonuniform multiresolution analysis and spectral pairs, J. Funct. Anal.158 (1998) 209–241]. In this setting, the associated translation set is a spectrum $\mathrm{\Lambda }$ which is not necessarily a group nor a uniform discrete set, given $\mathrm{\Lambda }=\{0,r/N\}+2\mathbb{Z},$ where $N\ge 1$ (an integer) and $r$ is an odd integer with $1\le r\le 2N-1$ such that $r$ and $N$ are relatively prime and $\mathbb{Z}$ is the set of all integers. The objective of this paper is to construct nonuniform wavelet frame on local fields. A necessary condition and four sufficient conditions for nonuniform wavelet frame on local fields are given.

Regarding convolutional codes as polynomial analogues of arithmetic lattices, we derive a Poisson–Jacobi formula for their trivariate weight enumerator. The proof is based on harmonic analysis on locally compact abelian groups as developed in Tate's thesis to derive the functional equation of the zeta function.

Let n be an integer and p be a prime number. An important problem in number theory is to classify the degree n extensions of the p-adic numbers through their arithmetic invariants. The most difficult cases arise when p divides n and n is composite. In this paper, we consider the case n = 12 and p = 3; the degrees n < 12 having previously been determined.

We consider the group ${\mathrm{\text{SL}}}_2(K)$, where $K$ is a local non-archimedean field of characteristic two. We prove that the depth of any irreducible representation of ${\mathrm{\text{SL}}}_2(K)$ is larger than the depth of the corresponding Langlands parameter, with equality if and only if the $L$-parameter is essentially tame. We also work out a classification of all $L$-packets for ${\mathrm{\text{SL}}}_2(K)$ and for its non-split inner form, and we provide explicit formulae for the depths of their $L$-parameters.