On the peakon inverse problem for the Degasperis-Procesi equation

The peakon inverse problem for the Degasperis-Procesi equation is solved directly on the real line, using Cauchy biorthogonal polynomials, without any additional transformation to a "string" type boundary value problem known from prior works

A New Interpretation for Orthofermions

In this article we introduce a simple physical model which realizes the algebra of orthofermions. The model is constructed from a cylinder which can be filled with some balls. The creation and annihilation operators of orthofermions are related to the creation and annihilation operators of balls in certain positions in the cylinder. Relationship between this model and topological symmetries in quantum mechanics is investigated.Comment: To appear in Rep. Math. Phy

Existence of a Not Necessarily Symmetric Matrix with Given Distinct Eigenvalues and Graph

For given k distinct complex conjugate pairs, l distinct real numbers, and a given graph G on 2k+l vertices with a matching of size at least k, we will show that there is a real matrix whose eigenvalues are the given numbers and its graph is G. In particular, this implies that any real matrix with distinct eigenvalues is similar to a real, irreducible, tridiagonal matrix

Online SLAM with Any-time Self-calibration and Automatic Change Detection

A framework for online simultaneous localization, mapping and self-calibration is presented which can detect and handle significant change in the calibration parameters. Estimates are computed in constant-time by factoring the problem and focusing on segments of the trajectory that are most informative for the purposes of calibration. A novel technique is presented to detect the probability that a significant change is present in the calibration parameters. The system is then able to re-calibrate. Maximum likelihood trajectory and map estimates are computed using an asynchronous and adaptive optimization. The system requires no prior information and is able to initialize without any special motions or routines, or in the case where observability over calibration parameters is delayed. The system is experimentally validated to calibrate camera intrinsic parameters for a nonlinear camera model on a monocular dataset featuring a significant zoom event partway through, and achieves high accuracy despite unknown initial calibration parameters. Self-calibration and re-calibration parameters are shown to closely match estimates computed using a calibration target. The accuracy of the system is demonstrated with SLAM results that achieve sub-1% distance-travel error even in the presence of significant re-calibration events.Comment: 8 pages, 6 figure

Analytical results in coherent quantum transport for periodic quantum dot

In this paper we have calculated electron transport coefficient in ballistic regime through a periodic dot sandwiched between uniform leads. We have calculated the Green's function (GF), density of states (Dos) and the coherent transmission coefficient (conductance) fully analytically. The quasi gap, bound states energies, the energy and wire-length dependence of the GF and conductance for this system are also derived.Comment: 7 Pages, 3 figure

Some Remarks on Ideals with large Regularity and Regularity Jumps

This paper exhibits some new examples of the behavior of the Castelnuovo-Mumford regularity of homogeneous ideals in polynomial rings. More precisely, we present new examples of homogenous ideals with large regularity compared to the generating degree. Then we consider the regularity jumps of ideals. In particular we provide an infinite family of ideals having regularity jumps at a certain power.Comment: 11 page

On a generalization of the Hadwiger-Nelson problem

For a field $F$ and a quadratic form $Q$ defined on an $n$-dimensional vector space $V$ over $F$, let $\mathrm{QG}_Q$, called the quadratic graph associated to $Q$, be the graph with the vertex set $V$ where vertices $u,w \in V$ form an edge if and only if $Q(v-w)=1$. Quadratic graphs can be viewed as natural generalizations of the unit-distance graph featuring in the famous Hadwiger-Nelson problem. In the present paper, we will prove that for a local field $F$ of characteristic zero, the Borel chromatic number of $\mathrm{QG}_Q$ is infinite if and only if $Q$ represents zero non-trivially over $F$. The proof employs a recent spectral bound for the Borel chromatic number of Cayley graphs, combined with an analysis of certain oscillatory integrals over local fields. As an application, we will also answer a variant of question 525 proposed in the 22nd British Combinatorics Conference 2009.Comment: This is the final version. Accepted in Israel Journal of Mathematic

On the chromatic number of structured Cayley graphs

In this paper, we will study the chromatic number of Cayley graphs of algebraic groups that arise from algebraic constructions. Using Lang-Weil bound and representation theory of finite simple groups of Lie type, we will establish lower bounds on the chromatic number of these graphs. This provides a lower bound for the chromatic number of Cayley graphs of the regular graphs associated to the ring of $n\times n$ matrices over finite fields. Using Weil's bound for Kloosterman sums we will also prove an analogous result for $\mathrm{SL}_2$ over finite rings.Comment: arXiv admin note: text overlap with arXiv:1507.0530

Divergence of effective mass in 'Uncorrelated State Percolation' Model

We want to answer the question of whether the divergence in the effective mass in metal-insulator transition (MIT) in 2DEG is in the same universality class as percolation. We use a model to make Percolated state in 2D and then calculate the effective mass in a super-cell and the Bloch Theorem. It is seen that the effective mass, m*, scales as m*~(P-Pc)^a with a=1 and Pc being the (classical) percolation threshold.Comment: 5 pages, 5 figure

Quasi-Random profinite groups

We will investigate quasi-randomness for profinite groups. We will obtain bounds for the mininal degree of non-trivial representations of $\mathrm{SL}_k(\mathbb{Z}/(p^n\mathbb{Z}))$ and $\mathrm{Sp}_{2k}(\mathbb{Z}/(p^n\mathbb{Z}))$. Our method also delivers a lower bound for the minimal degree of a faithful representation for these groups. Using the suitable machinery from functional analysis, we establish exponential lower and upper bounds for the supremal measure of a product-free measurable subset of the profinite groups $\mathrm{SL}_{k}({\mathbb{Z}_p})$ and $\mathrm{Sp}_{2k}(\mathbb{Z}_p)$. We also obtain analogous bounds for a special subgroup of the automorphism group of a regular tree.Comment: This is the final version. To appear in Glasgow Mathematical Journa