Minimal FF-crystals and isomorphism numbers of isosimple FF-crystals

In this paper we generalize minimal $p$-divisible groups defined by Oort to $F$-crystal over an algebraically closed field of positive characteristic. We prove a structural theorem and give an explicit formula of the Frobenius endomorphism of the isosimple minimal $F$-crystals that are the building blocks of minimal $F$-crystals. We then define an invariant called the minimal height for $F$-crystals using minimal $F$-crystals and give an upper bound of the isomorphism numbers of isosimple $F$-crystals in terms of their ranks, Hodge slopes and Newton slopes.Comment: Final accepted version at Math. Nach

Subtle Invariants of FF-crystals

Vasiu proved that the level torsion $\ell_{\mathcal{M}}$ of an $F$-crystal $\mathcal{M}$ over an algebraically closed field of characteristic $p>0$ is a non-negative integer that is an effectively computable upper bound of the isomorphism number $n_{\mathcal{M}}$ of $\mathcal{M}$ and expected that in fact one always has $n_{\mathcal{M}} = \ell_{\mathcal{M}}$. In this paper, we prove that this equality holds.Comment: Final accepted version at J. Ramanujan Math. So

Holographic Representation of Local Operators In De Sitter Space

Assuming the existence of the dS/CFT correspondence, we construct local scalar fields with $m^2>\left( \frac{d}{2} \right)^2$ in de Sitter space by smearing over conformal field theory operators on the future/past boundary. To maintain bulk micro-causality and recover the bulk Wightman function in the Euclidean vacuum, the smearing prescription must involve two sets of single--trace operators with dimensions $\Delta$ and $d-\Delta$. Thus the local operator prescription in de Sitter space differs from the analytic continuation from the prescription in anti--de Sitter space. Pushing a local operator in the global patch to future/past infinity is shown to lead to an operator relation between single--trace operators in conformal field theories at $\mathcal{I}^\pm$, which can be interpreted as a basis transformation, also identified as the relation between an operator in CFT and its shadow operator. Construction of spin$-s$ gauge field operators is discussed, it is shown that the construction of higher spin gauge fields in de Sitter space is equivalent to constructing scalar fields with specific values of mass parameter $m^2<\left( \frac{d}{2} \right)^2$. An acausal higher spin bulk operator which matches onto boundary higher spin current is constructed. Implementation of the scalar operator constructions in AdS and dS with embedding formalism is briefly described.Comment: 35 pages, 3 figures,published in Phys.Rev.D . Comments added, reference fixe

Note on Adiabatic Modes and Ward Identities In A Closed Universe

As statements regarding the soft limit of cosmological correlation functions, consistency relations are known to exist in any flat FRW universe. In this letter we explore the possibility of finding such relations in a spatially closed universe, where the soft limit $\textbf{q}\rightarrow 0$ does not exist in any rigorous sense. Despite the absence of spatial infinity of the spatial slices, we find the adiabatic modes and their associated consistency relations in a toy universe with background topology $R\times S^2$. Flat FRW universe adiabatic modes are recovered via taking the large radius limit $R\gg \mathcal{H}^{-1}$, for which we are living in a small local patch of Hubble size on the sphere. It is shown that both dilation and translation adiabatic modes in the local patch are recovered by a global dilation on the sphere, acting at different places.Comment: 4 page

From Co-prime to the Diophantine Equation Based Sparse Sensing

With a careful design of sample spacings either in temporal and spatial domain, co-prime sensing can reconstruct the autocorrelation at a significantly denser set of points based on Bazout theorem. However, still restricted from Bazout theorem, it is required O(M1 + M2) samples to estimate frequencies in the case of co-prime sampling, where M1 and M2 are co-prime down-sampling rates. Besides, for Direction-of-arrival (DOA) estimation, the sensors can not be arbitrarily sparse in co-prime arrays. In this letter, we restrain our focus on complex waveforms and present a framework under multiple samplers/sensors for both frequency and DOA estimation based on Diophantine equation, which is essentially to estimate the autocorrelation with higher order statistics instead of the second order one. We prove that, given arbitrarily high down-sampling rates, there exist sampling schemes with samples to estimate autocorrelation only proportional to the sum of degrees of freedom (DOF) and the number of snapshots required. In the scenario of DOA estimation, we show there exist arrays of N sensors with O(N^3) DOF and O(N) minimal distance between sensors.Comment: Sparse Sensing; Co-prime Sampling; Co-prime Arra

Multiplicative Iteration for Nonnegative Quadratic Programming

In many applications, it makes sense to solve the least square problems with nonnegative constraints. In this article, we present a new multiplicative iteration that monotonically decreases the value of the nonnegative quadratic programming (NNQP) objective function. This new algorithm has a simple closed form and is easily implemented on a parallel machine. We prove the global convergence of the new algorithm and apply it to solving image super-resolution and color image labelling problems. The experimental results demonstrate the effectiveness and broad applicability of the new algorithm.Comment: 11 pages, 4 figure

On Degree-Based Decentralized Search in Complex Networks

Decentralized search aims to find the target node in a large network by using only local information. The applications of it include peer-to-peer file sharing, web search and anything else that requires locating a specific target in a complex system. In this paper, we examine the degree-based decentralized search method. Specifically, we evaluate the efficiency of the method in different cases with different amounts of available local information. In addition, we propose a simple refinement algorithm for significantly shortening the length of the route that has been found. Some insights useful for the future developments of efficient decentralized search schemes have been achieved.Comment: 6 pages, 3 figs, shortly published by ECCS'0

On Intentional Attacks and Protections in Complex Communication Networks

Being motivated by recent developments in the theory of complex networks, we examine the robustness of communication networks under intentional attack that takes down network nodes in a decreasing order of their nodal degrees. In this paper, we study two different effects that have been largely missed in the existing results: (i) some communication networks, like Internet, are too large for anyone to have global information of their topologies, which makes the accurate intentional attack practically impossible; and (ii) most attacks in communication networks are propagated from one node to its neighborhood node(s), utilizing local network-topology information only. We show that incomplete global information has different impacts to the intentional attack in different circumstances, while local information-based attacks can be actually highly efficient. Such insights would be helpful for the future developments of efficient network attack/protection schemes.Comment: 5 pages, 11 figures, accepted by IEEE Globecom 2006 conferenc

On Solving Ambiguity Resolution with Robust Chinese Remainder Theorem for Multiple Numbers

Chinese Remainder Theorem (CRT) is a powerful approach to solve ambiguity resolution related problems such as undersampling frequency estimation and phase unwrapping which are widely applied in localization. Recently, the deterministic robust CRT for multiple numbers (RCRTMN) was proposed, which can reconstruct multiple integers with unknown relationship of residue correspondence via generalized CRT and achieves robustness to bounded errors simultaneously. Naturally, RCRTMN sheds light on CRT-based estimation for multiple objectives. In this paper, two open problems arising that how to introduce statistical methods into RCRTMN and deal with arbitrary errors introduced in residues are solved. We propose the extended version of RCRTMN assisted with Maximum Likelihood Estimation (MLE), which can tolerate unrestricted errors and bring considerable improvement in robustness

Instability and topological robustness of Weyl semimetals against Coulomb interaction

There is a close connection between various new phenomena in Weyl semimetals and the existence of linear band crossings in the single particle description. We show, by a full self-consistent mean-field calculation, how this picture is modified in the presence of long-range Coulomb interactions. The chiral symmetry breaking occurs at strong enough interactions and the internode interband excitonic pairing channel is found to be significant, which determines the gap-opened band profile varying with interaction strength. Remarkably, in the resultant interacting phase, finite band Chern number jumps in the three-dimensional momentum space are retained, indicating the robustness of the topologically nontrivial features.Comment: 8 pages, 4 figures, accepted by Phys. Rev.