New error bounds for linear complementarity problems of Nekrasov matrices and B-Nekrasov matrices

New error bounds for the linear complementarity problems are given respectively when the involved matrices are Nekrasov matrices and B-Nekrasov matrices. Numerical examples are given to show that new bounds are better respectively than those provided by Garcia-Esnaola and Pena in [15,16] in some cases

Integral almost square-free modular categories

We study integral almost square-free modular categories; i.e., integral modular categories of Frobenius-Perron dimension $p^nm$, where $p$ is a prime number, $m$ is a square-free natural number and ${\rm gcd}(p,m)=1$. We prove that if $n\leq 5$ or $m$ is prime with $m<p$ then they are group-theoretical. This generalizes several results in the literature and gives a partial answer to the question posed by the first author and H. Tucker. As an application, we prove that an integral modular category whose Frobenius-Perron dimensions is odd and less than $1125$ is group-theoretical.Comment: Some typos are corrected. Section 4 in previous version is moved. The current version is revised by Libin Li and Li Da

Non-uniform continuous dependence on initial data of solutions to the Euler-Poincar\'{e} system

In this paper, we investigate the continuous dependence on initial data of solutions to the Euler-Poincar\'{e} system. By constructing a sequence approximate solutions and calculating the error terms, we show that the data-to-solution map is not uniformly continuous in Sobolev space $H^s(\mathbb{R}^d)$ for $s>1+\frac d2$.Comment: 10 pages. arXiv admin note: text overlap with arXiv:1505.00086 by other author

Existence of Tannakian subcategories and its applications

We study several classes of braided fusion categories, and prove that they all contain nontrivial Tannakian subcategories. As applications, we classify some fusion categories in terms of solvability and group-theoreticality.Comment: Title is changed; Section 5 is new; one result is correcte

On semisimple quasitriangular Hopf algebras of dimension dqndq^n

Let $q>2$ be a prime number, $d$ be an odd square-free natural number, and $n$ be a non-negative integer. We prove that a semisimple quasitriangular Hopf algebra of dimension $dq^n$ is solvable in the sense of Etingof, Nikshych and Ostrik. In particular, if $n\leq 3$ then it is either isomorphic to $k^G$ for some abelian group $G$, or twist equivalent to a Hopf algebra which fits into a cocentral abelian exact sequence.Comment: 9 pages, first version, comments are welcom

Special Relativity for the Full Speed Range -- speed slower than CRC_R also equal to and faster than CRC_R

In this paper, we establish a theory of Special Relativity valid for the entire speed range without the assumption of constant speed of light. Two particles species are defined, one species of particles have rest frames with rest mass, and another species of particles do not have rest frame and can not define rest mass. We prove that for the particles which have rest frames, the Galilean transformation is the only linear transformation of space-time that allows infinite speed of particle motion. Hence without any assumption, an upper bound of speed is required for all non-Galilean linear transformations. We then present a novel derivation of the mass-velocity and the mass-energy relations in the framework of relativistic dynamics, which is solely based on the principle of relativity and basic definitions of relativistic momentum and energy. The generalized Lorentz transformation is then determined. The new relativistic formulas are not related directly to the speed of light $c$, but are replaced by a Relativity Constant $C_R$ which is an universal speed constant of the Nature introduced in relativistic dynamics. Particles having rest mass are called tardyons moving slower than $C_R$. Particles having neither rest frames nor rest mass are called $tachyons$ moving faster than $C_R$, and with the real mass-velocity relation $m=|\vec{p}_\infty |(v^2-C_R^2)^{-1/2}$ where $\vec{p}_\infty$ is the finite momentum of tachyon at infinite speed. Moreover, the particles with constant-speed $C_R$, also having neither rest frames nor rest mass, are called $constons$. For all particles, $p^2=\vec{p}^2-({E^2}/{C_R^2})$ remains invariant under transformation between inertia frames. The invariant reads $p^2=-m_0^2C_R^2 <0$ for tardyons, $p^2 =0$ for constons and $p^2=|\vec{p}_\infty|^2 >0$ for tachyons, respectively.Comment: 22 pages, 10 figure

Mass under the Ricci flow

In this paper, we study the change of the ADM mass of an ALE space along the Ricci flow. Thus we first show that the ALE property is preserved under the Ricci flow. Then, we show that the mass is invariant under the flow in dimension three (similar results hold in higher dimension with more assumptions). A consequence of this result is the following. Let $(M,g)$ be an ALE manifold of dimension $n=3$. If $m(g)\neq 0$, then the Ricci flow starting at $g$ can not have Euclidean space as its (uniform) limit

New implementation of hybridization expansion quantum impurity solver based on Newton-Leja interpolation polynomial

We introduce a new implementation of hybridization expansion continuous time quantum impurity solver which is relevant to dynamical mean-field theory. It employs Newton interpolation at a sequence of real Leja points to compute the time evolution of the local Hamiltonian efficiently. Since the new algorithm avoids not only computationally expansive matrix-matrix multiplications in conventional implementations but also huge memory consumptions required by Lanczos/Arnoldi iterations in recently developed Krylov subspace approach, it becomes advantageous over the previous algorithms for quantum impurity models with five or more bands. In order to illustrate the great superiority and usefulness of our algorithm, we present realistic dynamical mean-field results for the electronic structures of representative correlated metal SrVO$_{3}$.Comment: 10 pages, 6 figure

On Kaplansky's sixth conjecture

About $39$ years ago, Kaplansky conjectured that the dimension of a semisimple Hopf algebra over an algebraically closed field of characteristic zero is divisible by the dimensions of its simple modules. Although it still remains open, some partial answers to this conjecture play an important role in classifying semisimple Hopf algebras. This paper focuses on the recent development of Kaplansky's sixth conjecture and its applications in classifying semisimple Hopf algebras.Comment: 17 pages, final version was accepted for publication in Rendiconti del Seminario Matematico della Universita di Padova (European Mathematical Society). arXiv admin note: text overlap with arXiv:0809.3031 by other author

On cyclic Higgs bundles

In this paper, we derive a maximum principle for a type of elliptic systems and apply it to analyze the Hitchin equation for cyclic Higgs bundles. We show several domination results on the pullback metric of the (possibly branched) minimal immersion $f$ associated to cyclic Higgs bundles. Also, we obtain a lower and upper bound of the extrinsic curvature of the image of $f$. As an application, we give a complete picture for maximal $Sp(4,\mathbb{R})$-representations in the $2g-3$ Gothen components and the Hitchin components.Comment: 27 pages, comments are welcom