On compact hypersurfaces in a Riemannian vector bundle with prescribed vertical Gaussian curvature

Let M be a compact Riemannian manifold and E a Riemannian vector bundle on M. We look for hypersurfaces of E with a prescribed vertical Gaussian curvature. In trying to solve this problem fibre-wise, we loose the regularity of the resulting solution. To unsure the smoothness of the solution, we construct it as a radial graph over the unit sphere subbundle of E and prove its existence by solving in this one a nonlinear partial differential equation of Monge-Amp\`ere type.Comment: 30 page

K\"ahler metrics with constant weighted scalar curvature and weighted K-stability

We introduce a notion of a K\"ahler metric with constant weighted scalar curvature on a compact K\"ahler manifold $X$, depending on a fixed real torus $\mathbb{T}$ in the reduced group of automorphisms of $X$, and two smooth (weight) functions $\mathrm{v}>0$ and $\mathrm{w}$, defined on the momentum image (with respect to a given K\"ahler class $\alpha$ on $X$) of $X$ in the dual Lie algebra of $\mathbb{T}$. A number of natural problems in K\"ahler geometry, such as the existence of extremal K\"ahler metrics and conformally K\"ahler, Einstein--Maxwell metrics, or prescribing the scalar curvature on a compact toric manifold reduce to the search of K\"ahler metrics with constant weighted scalar curvature in a given K\"ahler class $\alpha$, for special choices of the weight functions $\mathrm{v}$ and $\mathrm{w}$. We show that a number of known results obstructing the existence of constant scalar curvature K\"ahler (cscK) metrics can be extended to the weighted setting. In particular, we introduce a functional $\mathcal M_{\mathrm{v}, \mathrm{w}}$ on the space of $\mathbb{T}$-invariant K\"ahler metrics in $\alpha$, extending the Mabuchi energy in the cscK case, and show (following the arguments of Li and Sano--Tipler in the cscK and extremal cases) that if $\alpha$ is Hodge, then constant weighted scalar curvature metrics in $\alpha$ are minima of $\mathcal M_{\mathrm{v},\mathrm{w}}$. Motivated by the recent work of Dervan--Ross and Dyrefelt in the cscK and extremal cases, we define a $(\mathrm{v},\mathrm{w})$-weighted Futaki invariant of a $\mathbb{T}$-compatible smooth K\"ahler test configuration associated to $(X, \alpha, \mathbb{T})$, and show that the boundedness from below of the $(\mathrm{v},\mathrm{w})$-weighted Mabuchi functional $\mathcal M_{\mathrm{v}, \mathrm{w}}$ implies a suitable notion of a $(\mathrm{v},\mathrm{w})$-weighted K-semistability.Comment: A link with K\"ahler Ricci solitons and the work of E. Inoue arXiv:1802.08128 added. Presentation improve

Automorphisms and deformations of conformally K\"ahler, Einstein-Maxwell metrics

We obtain a structure theorem for the group of holomorphic automorphisms of a conformally K\"ahler, Einstein-Maxwell metric, extending the classical results of Matsushima, Licherowicz and Calabi in the K\"ahler-Einstein, cscK, and extremal K\"ahler cases. Combined with previous results of LeBrun, Apostolov-Maschler and Futaki-Ono, this completes the classification of the conformally K\"ahler, Einstein--Maxwell metrics on $\mathbb{{CP}}^1 \times \mathbb{{CP}}^1$. We also use our result in order to introduce a (relative) Mabuchi energy in the more general context of $(K, q, a)$-extremal K\"ahler metrics in a given K\"ahler class, and show that the existence of $(K, q, a)$-extremal K\"ahler metrics is stable under small deformation of the K\"ahler class, the Killing vector field $K$ and the normalization constant $a$

New bounds on the Lebesgue constants of Leja sequences on the unit disc and their projections ℜ\Re-Leja sequences

In the papers [6, 7] we have established linear and quadratic bounds, in $k$, on the growth of the Lebesgue constants associated with the $k$-sections of Leja sequences on the unit disc $\mathcal{U}$ and $\Re$-Leja sequences obtained from the latter by projection into $[-1, 1]$. In this paper, we improve these bounds and derive sub-linear and sub-quadratic bounds. The main novelty is the introduction of a "quadratic" Lebesgue function for Leja sequences on $\mathcal{U}$ which exploits perfectly the binary structure of such sequences and can be sharply bounded. This yields new bounds on the Lebesgue constants of such sequences, that are almost of order $\sqrt{k}$ when $k$ has a sparse binary expansion. It also yields an improvement on the Lebesgue constants associated with $\Re$-Leja sequences

Conformally K\"ahler, Einstein--Maxwell metrics and boundedness of the modified Mabuchi-functional

We prove that if a compact smooth polarized complex manifold admits in the corresponding Hodge K\"ahler class a conformally K\"ahler, Einstein--Maxwell metric, or more generally, a K\"ahler metric of constant $(\xi, a, p)$-scalar curvature, then this metric minimizes the $(\xi,a,p)$-Mabuchi functional. Our method of proof extends the approach introduced by Donaldson and developed by Li and Sano--Tipler, via finite dimensional approximations and generalized balanced metrics. As an application of our result and the recent construction of Koca--T{\o}nnesen-Friedman, we describe the K\"ahler classes on a geometrically ruled complex surface of genus greater than 2, which admit conformally K\"ahler, Einstein-Maxwell metrics.Comment: References added, presentation improve

Vector-valued automorphic forms and vector bundles

While vector-valued automorphic forms can be defined for an arbitrary Fuchsian group $\Gamma$ and an arbitrary representation $R$ of $\Gamma$ in GL$(n,{\mathbb C})$, their existence has been established in the literature only when restrictions are imposed on both $\Gamma$ and $R$. In this paper, we prove the existence of $n$ linearly independent vector-valued automorphic forms for any Fuchsian group $\Gamma$ and any $n$-dimensional complex representation $R$ of $\Gamma$. To this end, we realize these automorphic forms as global sections of a special rank $n$ vector bundle built using solutions to the Riemann-Hilbert problem over various noncompact Riemann surfaces and Kodaira's vanishing theorem.Comment: 17 page

Effect of short-range order on transport in one-particle, tight-binding models

We investigate transport properties of topologically disordered, three-dimensional, one-particle, tight binding models, featuring site distance dependent hopping terms. We start from entirely disordered systems into which we gradually introduce some short range order by numerically performing a pertinent structural relaxation using local site-pair interactions. Transport properties of the resulting models within the delocalized regime are analyzed numerically using linear response theory. We find that even though the generated order is very short ranged, transport properties such as conductivity or mean free path scale significantly with the degree of order. Mean free paths may exceed site-pair correlation length. It is furthermore demonstrated that, while the totally disordered model is not in accord with a Drude- or Boltzmann-type description, moderate degrees of order suffice to render such a picture valid.Comment: to appear in Phys. Rev.

New Recurrence Relationships between Orthogonal Polynomials which Lead to New Lanczos-type Algorithms

Lanczos methods for solving $\textit{A}\textbf{x}=\textbf{b}$ consist in constructing a sequence of vectors $(\textbf{x}_k), k=1,...$ such that $\textbf{r}_{k}=\textbf{b}-\textit{A}\textbf{x}_{k}=\textit{P}_{k}(\textit{A})\textbf{r}_{0}$,, where $\textit{P}_{k}$ is the orthogonal polynomial of degree at most $k$ with respect to the linear functional $c$ defined as $c(\xi^i)=(\textbf{y},\textit{A}^i\textbf{r}_{0})$. Let $\textit{P}^{(1)}_{k}$ be the regular monic polynomial of degree $k$ belonging to the family of formal orthogonal polynomials (FOP) with respect to $c^{(1)}$ defined as $c^{(1)}(\xi^{i})=c(\xi^{i+1})$. All Lanczos-type algorithms are characterized by the choice of one or two recurrence relationships, one for $\textit{P}_{k}$ and one for $\textit{P}^{(1)}_{k}$. We shall study some new recurrence relations involving $\textit{P}_{k}$ and $\textit{P}^{(1)}_{k}$ and their possible combination to obtain new Lanczos-type algorithms. We will show that some recurrence relations exist, but cannot be used to derive Lanczos-type algorithms, while others do not exist at all

Enhanced Slotted Aloha Mechanism by Introducing ZigZag Decoding

Various random access mechanisms, such as Aloha protocol and its corresponding variants have been widely studied as efficient methods to coordinate the medium access among competing users. But when two or more wireless users transmit packets at the same time over the same channel a collisions occur. When this happens, the received packets are discarded and retransmissions are required, which is a waste of power and bandwidth. In such a situation one of the most important objectives is to find techniques to improve these protocols to reduce the number of collisions or to avoid them. Several studies have contributed to this problem. In this paper, we propose a new approach named ZigZag decoding to enhance slotted Aloha mechanism by reducing the loss rate of packets colliding. We model the system by a Markov chain witch the number of backlogged packets is taken as the system state. We use a stochastic game to achieve our objective. We evaluate and compare the performances parameters of the proposed approach with those of slotted Aloha mechanism. All found results show that our approach is more efficient than the slotted Aloha mechanism.Comment: 11 pages, 10 figures in Journal of mathematics and computer science 2014. arXiv admin note: substantial text overlap with arXiv:1501.0088

OSCMAC_Duty_Cycle_with_Multi_Helpers_CT_Mode_WILEM_Technology_in_Wireless_Sensor_Networks

Recently, Wireless Sensor Networks (WSNs) grow to be one of the dominant technology trends; new needs are continuously emerging and demanding more complex constraints in a duty cycle, such as extend the life time communication . The MAC layer plays a crucial role in these networks; it controls the communication module and manages the medium sharing. In this work we use OSC-MAC tackles combining with the performance of cooperative transmission (CT) in multi-hop WSN and the Wi-Lem technologyComment: 7 pages , 4 figures, International Journal of wireless and Mobile Networks IJWMN ( 2014