Smooth representations of involutive algebra groups over non-archimedean local fields

An algebra group over a field $F$ is a group of the form $G = 1+J$ where $J$ is a finite-dimensional nilpotent associative $F$-algebra. A theorem of M. Boyarchenko asserts that, in the case where $F$ is a non-archimedean local field, every irreducible smooth representation of $G$ is admissible and smoothly induced by a one-dimensional smooth representation of some algebra subgroup of $G$. If $J$ is a nilpotent algebra endowed with an involution $\sigma:J\to J$, then $\sigma$ naturally defines a group automorphism of $G$, and we may consider the fixed point subgroup $C_{G}(\sigma)$. Assuming that $F$ has characteristic different from $2$, we extend Boyarchenko's result and show that every irreducible smooth representation of $C_{G}(\sigma)$ is admissible and smoothly induced by a one-dimensional smooth representation of a subgroup of the form $C_{H}(\sigma)$ where $H$ is an $\sigma$-invariant algebra subgroup of $G$. As a particular case, the result holds for maximal unipotent subgroups of the classical Chevalley groups defined over $F$.Comment: arXiv admin note: text overlap with arXiv:1910.1463

A symplectic extension map and a new Shubin class of pseudo-differential operators

For an arbitrary pseudo-differential operator $A:\mathcal{S}(\mathbb{R}^{n})\longrightarrow\mathcal{S}^{\prime}(\mathbb{R}^{n})$ with Weyl symbol $a\in\mathcal{S}^{\prime}(\mathbb{R}^{2n})$, we consider the pseudo-differential operators $\widetilde{A}:\mathcal{S}(\mathbb{R}^{n+k})\longrightarrow\mathcal{S}^{\prime}(\mathbb{R}^{n+k})$ associated with the Weyl symbols $\widetilde{a}=(a\otimes1_{2k})\circ{s}$, where $1_{2k}(x)=1$ for all $x\in\mathbb{R}^{2k}$ and ${s}$ is a linear symplectomorphism of $\mathbb{R}^{2(n+k)}$. We call the operators $\widetilde{A}$ symplectic dimensional extensions of $A$. In this paper we study the relation between $A$ and $\widetilde{A}$ in detail, in particular their regularity, invertibility and spectral properties. We obtain an explicit formula allowing to express the eigenfunctions of $\widetilde{A}$ in terms of those of $A$. We use this formalism to construct new classes of pseudo-differential operators, which are extensions of the Shubin classes $HG_{\rho}^{m_{1},m_{0}}$ of globally hypoelliptic operators. We show that the operators in the new classes share the invertibility and spectral properties of the operators in $HG_{\rho }^{m_{1},m_{0}}$ but not the global hypoellipticity property. Finally, we study a few examples of operators that belong to the new classes and which are important in mathematical physics.Comment: 28 pages, new version, accepted for publication in JF

Time dependent transformations in deformation quantization

We study the action of time dependent canonical and coordinate transformations in phase space quantum mechanics. We extend the covariant formulation of the theory by providing a formalism that is fully invariant under both standard and time dependent coordinate transformations. This result considerably enlarges the set of possible phase space representations of quantum mechanics and makes it possible to construct a causal representation for the distributional sector of Wigner quantum mechanics.Comment: 16 pages, to appear in the J. Math. Phy

CFRP group effect and interaction between stirrups and strips on the NSM-shear strengthening of RC beams

Available experimental research shows that the technique based on installing Carbon Fibre Reinforced Polymer (CFRP) strips into slits opened on the cover concrete of the beam’s lateral faces, designated as Near Surface Mounted (NSM), is very effective to increase the shear resistance of reinforced concrete (RC) beams. However, recent research has revealed that, in terms of NSM shear strengthening effectiveness, a detrimental effect can occur between existing steel stirrups and applied strips, as well as amongst the strips when the distance between strips, sf, is lower than a certain limit. In the present work, a test setup was developed and an experimental program was carried out to assess the influence of both sf and interaction between existing steel stirrups and strips on the shear strengthening of RC beams. The experimental program is described and the main results are presented and analyzed.(13-05-04-FDR-00031

Numerical simulation of continuous RC slabs strengthened using NSM technique

The effectiveness of the Near Surface Mounted (NSM) technique for the increase of the flexur-al resistance of reinforced concrete (RC) beams and slabs was already well proved. The NSM technique is es-pecially adapted to increase the negative bending moments of continuous (two or more spans) RC slabs. However, the influence of the NSM strengthening on the moment redistribution capability of RC structures should be investigated. Recently, an exploratory experimental program was conducted to assess the level of moment redistribution that can be obtained in two span RC slabs strengthened with NSM strips for negative bending moments. To help the preparation of an extensive experimental program in this domain, the values of the parameters of a constitutive model, implemented into a FEM-based computer program, were calibrated from the numerical simulation of these tests. The main aspects of the experimental program are presented, the numerical model is briefly described and the numerical simulations are presented and analyzed.The authors wish to acknowledge the support provided by the Empreiteiros Casais, S&P, Secil (Un-ibetao, Braga) Companies. The study reported in this paper forms a part of the research program CUTINEMO Carbon fiber laminates applied according to the near surface mounted technique to increase the flexural resistance to negative moments of continuous reinforced concrete structures supported by FCT, PTDC/ECM/73099/2006.The second author would like to acknowledge the National Council for Scientific and Technological Development(CNPq)Brazil for financial support for scholarship

Shintani descent for standard supercharacters of algebra groups

Let $\mathcal{A}(q)$ be a finite-dimensional nilpotent algebra over a finite field $\mathbb{F}_{q}$ with $q$ elements, and let $G(q) = 1+\mathcal{A}(q)$. On the other hand, let $\Bbbk$ denote the algebraic closure of $\mathbb{F}_{q}$, and let $\mathcal{A} = \mathcal{A}(q) \otimes_{\mathbb{F}_{q}} \Bbbk$. Then $G = 1+\mathcal{A}$ is an algebraic group over $\Bbbk$ equipped with an $\mathbb{F}_{q}$-rational structure given by the usual Frobenius map $F:G\to G$, and $G(q)$ can be regarded as the fixed point subgroup $G^{F}$. For every $n \in \mathbb{N}$, the $n$th power $F^{n}:G\to G$ is also a Frobenius map, and $G^{F^{n}}$ identifies with $G(q^{n}) = 1 + \mathcal{A}(q^{n})$. The Frobenius map restricts to a group automorphism $F:G(q^{n})\to G(q^{n})$, and hence it acts on the set of irreducible characters of $G(q^{n})$. Shintani descent provides a method to compare $F$-invariant irreducible characters of $G(q^{n})$ and irreducible characters of $G(q)$. In this paper, we show that it also provides a uniform way of studying supercharacters of $G(q^{n})$ for $n \in \mathbb{N}$. These groups form an inductive system with respect to the inclusion maps $G(q^{m}) \to G(q^{n})$ whenever $m \mid n$, and this fact allows us to study all supercharacter theories simultaneously, to establish connections between them, and to relate them to the algebraic group $G$. Indeed, we show that Shintani descent permits the definition of a certain ``superdual algebra'' which encodes information about the supercharacters of $G(q^{n})$ for $n \in \mathbb{N}$

Solutions for vehicular communications: a review

Vehicular networks experience a number of unique challenges due to the high mobility of vehicles and highly dynamic network topology, short contact durations, disruption intermittent connectivity, significant loss rates, node density, and frequent network fragmentation. All these issues have a profound impact on routing strategies in these networks. This paper gives an insight about available solutions on related literature for vehicular communications. It overviews and compares the most relevant approaches for data communication in these networks, discussing their influence on routing strategies. It intends to stimulate research and contribute to further advances in this rapidly evolving area where many key open issues that still remain to be addressed are identified.Part of this work has been supported by the Instituto de Telecomunicações, Next Generation Networks and Applications Group (NetGNA), Portugal, in the framework of the Project VDTN@Lab, and by the Euro-NF Network of Excellence of the Seventh Framework Programme of EU, in the framework of the Specific Joint Research Project VDTN