The confirmation of a conjecture on disjoint cycles in a graph

In this paper, we prove the following conjecture proposed by Gould, Hirohata and Keller [Discrete Math. submitted]: Let $G$ be a graph of sufficiently large order. If $\sigma_t(G) \geq 2kt - t + 1$ for any two integers $k \geq 2$ and $t \geq 5$, then $G$ contains $k$ disjoint cycles

Symmetry of Solutions for a Fractional System

We consider the following equations: \begin{equation*} \left\{\begin{array}{ll} (-\triangle)^{\alpha/2}u(x)=f(v(x)), \\ (-\triangle)^{\beta/2}v(x)=g(u(x)), &x \in R^{n},\\ u,v\geq 0, &x \in R^{n}, \end{array} \right. \end{equation*} for continuous $f, g$ and $\alpha, \beta \in (0,2)$. Under some natural assumptions on $f$ and $g$, by applying the \emph{method of moving planes} directly to the system, we obtain symmetry on non-negative solutions without any decay assumption on the solutions at infinity

Determine Arbitrary Feynman Integrals by Vacuum Integrals

By introducing an auxiliary parameter, we find a new representation for Feynman integrals, which defines a Feynman integral by analytical continuation of a series containing only vacuum integrals. The new representation therefore conceptually translates the problem of computing Feynman integrals to the problem of performing analytical continuations. As an application of the new representation, we use it to construct a novel reduction method for multi-loop Feynman integrals, which is expected to be more efficient than known integration-by-parts reduction method. Using the new method, we successfully reduced all complicated two-loop integrals in $gg\to HH$ process and $gg\to ggg$ process.Comment: Version published in PRD (Rapid Communication

Hom-Nijienhuis operator and TT*-extension of Hom-Lie Superalgebras

In this paper, we study hom-Lie superalgebras. We give the definition of hom-Nijienhuis operators of regualr hom-Lie superalgebras and show that the deformation generated by a hom-Nijienhuis operator is trivial. Moreover, we introduce the definition of $T^*$-extensions of Hom-Lie superalgebras and show that $T^*$-extensions preserve many properties such as nilpotency, solvability and decomposition in some sense. We also investigate the equivalence of $T^*$-extensions.Comment: arXiv admin note: text overlap with arXiv:1005.0140 by other author

Symmetry and nonexistence of positive solutions for fractional systems

This paper is devoted to study the nonexistence results of positive solutions for the following fractional H$\acute{e}$non system \begin{eqnarray*}\left\{ \begin{array}{lll} &(-\triangle)^{\alpha/2}u=|x|^av^p,~~~&x\in R^n, &(-\triangle)^{\alpha/2}v=|x|^bu^q,~~~ &x\in R^n, &u\geq0, v\geq 0, \end{array} \right. \end{eqnarray*} where $0<\alpha<2$, $0<p,q<\infty$, $a$, $b$ $\geq0$, $n\geq2$. Using a direct method of moving planes, we prove non-existence of positive solution in the subcritical case

The rotational invariants constructed by the products of three spherical harmonic polynomials

The rotational invariants constructed by the products of three spherical harmonic polynomials are expressed generally as homogeneous polynomials with respect to the three coordinate vectors, where the coefficients are calculated explicitly in this paper

Representations and module-extensions of hom 3-Lie algebras

In this paper, we study the representations and module-extensions of hom 3-Lie algebras. We show that a linear map between hom 3-Lie algebras is a morphism if and only if its graph is a hom 3-Lie subalgebra and show that the derivations of a hom 3-Lie algebra is a Lie algebra. Derivation extension of hom 3-Lie algebras are also studied as an application. Moreover, we introduce the definition of $T_{\theta}$-extensions and $T^{*}_{\theta}$-extensions of hom 3-Lie sub-algebras in terms of modules, provide the necessary and sufficient conditions for $2k$-dimensional metric hom 3-Lie algebra to be isomorphic to a $T^{*}_{\theta}$-extensions.Comment: arXiv admin note: substantial text overlap with arXiv:1005.0140 by other author

A Pohozaev Identity for the Fractional Heˊ\acute{e}non System

In this paper, we study the Pohozaev identity associated with a H$\acute{e}$non-Lane-Emden system involving the fractional Laplacian: \begin{equation} \left\{\begin{array}{ll} (-\triangle)^su=|x|^av^p,&x\in\Omega, (-\triangle)^sv=|x|^bu^q,&x\in\Omega, u=v=0,&x\in R^n\backslash\Omega, \end{array} \right. \end{equation} in a star-shaped and bounded domain $\Omega$ for $s\in(0,1)$. As an application of our identity, we deduce the nonexistence of positive solutions in the critical and supercritical cases

Theory for quarkonium: from NRQCD factorization to soft gluon factorization

We demonstrate that the recently proposed soft gluon factorization (SGF) is equivalent to the nonrelativistic QCD (NRQCD) factorization for heavy quarkonium production or decay, which means that for any given process these two factorization theories are either both valid or both violated. We use two methods to achieve this conclusion. In the first method, we apply the two factorization theories to the physical process $J/\psi \to e^+e^-$. Our explicit calculation shows that both SGF and NRQCD can correctly reproduce low energy physics of full QCD, and thus the two factorizations are equivalent. In the second method, by using equations of motion we successfully deduce SGF from NRQCD effective field theory. By identifying SGF with NRQCD factorization, we establish relations between the two factorization theories and prove the generalized Gremm-Kapustin relations as a by product. Comparing with the NRQCD factorization, the advantage of SGF is that it resums the series of relativistic corrections originated from kinematic effects to all powers, which gives rise to a better convergence in relativistic expansion.Comment: 15 pages, 1 figur

Formation and local symmetry of Holstein polaron in t-J model

The formation and local symmetry of spin-lattice polaron has been investigated semiclassically in the planar Holstein t-J-like models within the exact diagonalization method. Due to the interplay of strong correlations and electron-lattice interaction, the doped hole may either move freely or lead to the localized spin-lattice distortion and form a Holstein polaron. The formation of polaron breaks the translational symmetry by suppression of antiferromagnetic correlations and inducement of ferromagnetic correlations locally. Moreover, the breaking of local rotational symmetry around the polaron has been shown. The ground state is generically a parity singlet and the first excited state maybe a parity doublet. Further consequences of the density of states spectra for comparison with future STM experiments are discussed.Comment: 6 pages, 5 figure