In this paper, we derive the sub-Riemannian version of the Kastler-Kalau-Walze type theorem and the Dabrowski-Sitarz-Zalecki type theorem for the twisted BCV spaces. We also compute the Connes conformal invariants for the twisted product, as well as the sub-Riemannian limits of the Connes conformal invariants for the twisted BCV spaces.
We evaluate some series with summands involving a single binomial coefficient(6k3k).For example,we prove that∞∑k=0(63k2+78k+22)8k/(2k+1)(6k+1)(6k+5)(6k3k)=3π/2.Motivated by Galois theory,we introduce the so-called Duality Principle for irrational series of Ramanujan's type or Zeilberger's type,and apply it to find 26 new irrational series identities.For example,we conjecture that∞∑k=1(32(91√33-523))k/k3(2kk)2(3kk)((91√33+891)k-33√33-225)=320(11/3√33L-11(2)-27L-3(2)),where Ld(2)=∑∞k=1(d/k)/k2 for any integer d≡0,1(mod 4)with(d/k)the Kronecker symbol.
Fibonacci sequence,generated by summing the preceding two terms,is a classical sequence renowned for its elegant properties.In this paper,leveraging properties of generalized Fibonacci sequences and formulas for consecutive sums of equidistant sub-sequences,we investigate the ratio of the sum of numbers along main-diagonal and sub-diagonal of odd-order grids containing generalized Fibonacci sequences.We show that this ratio is solely dependent on the order of the grid,providing a concise and splendid identity.
We prove the equivalence between two explicit expressions for two-point Witten-Kontsevich correlators obtained by M. Bertola, B. Dubrovin, D. Yang and by P. Zograf.
In this paper,a discrete predator-prey model with prey refuge is investigated.It is proved that the model undergoes codimension-2 bifurcations associated with 1∶2 and 1∶3 resonances.The bifurcation diagrams and phase portraits show that the model has some interesting complex dynamical behaviors,such as limit cycle,periodic solutions,chaos and codimension-1 bifurcations.
Let $a,b$ be two positive integers such that $a \le b$ and $a \equiv b$ (mod $2$). We say that a graph $G$ has an $(a,b)$-parity factor if $G$ has a spanning subgraph $F$ such that $d_{F}(v) \equiv b$ (mod $2$) and $a \le d_{F}(v) \le b$ for all $v \in V (G)$. In this paper, we provide a tight spectral radius condition for a graph to have $(a,b)$-parity factors.
We prove a generalization of the classical Gauss-Bonnet formula for a conical metric on a compact Riemann surface provided that the Gaussian curvature is Lebesgue integrable with respect to the area form of the metric. We also construct explicitly some conical metrics whose curvature is not integrable.
Based on the theory of semi-global classical solutions to quasilinear hyperbolic systems, the local exact boundary observability for a kind of second-order quasilinear hyperbolic systems is obtained by a constructive method.
We aim to find the eigenvalues and eigenfunctions of the barrier potential case for Strum-Liouville operator on the finite interval[0,-rr]when λ＞0.Generally,the eigenvalue problem for the Sturm-Liouville operator is often solved by using integral equations,which are sometimes complex to solve,and difficulties may arise in computing the boundary values.Considering the said complexity,we have successfully developed a technique to give the asymptotic formulae of the eigenvalue and the eigenfunction for Sturm-Liouville operator with barrier potential.The results are of significant interest in the field of quantum mechanics and atomic systems to observe discrete energy levels.
In this article,we give a further survey of some progress of the applications of group actions in the complex geometry after my earlier survey around 2020,mostly related to my own interests.
In this paper,we first survey existed theorems and propose all 46 related open problems of the existence of global attractors for autonomous dynamical systems,then establish a new existence theorem of global attractors which will be applied to a nonclassical diffusion equation for the norm-to-weak continuous,weakly compact semigroup on H01(Ω)and H2(Ω)∩H10(Ω)respectively.As an application of this new existence theorem of global attractors,we obtain the existence of the global attractors on H10(Ω)and H2(Ω)∩ H01(Ω)respectively for a nonclassical-diffusion equation.
This is a survey paper that lists our research works in the study of Stokes phenomenon of meromorphic ordinary differential equations and its relation with repre-sentation theory of quantum groups.
In this survey article, we present two applications of surface curvatures in theoretical physics. The first application arises from biophysics in the study of the shape of cell vesicles involving the minimization of a mean curvature type energy called the Helfrich bending energy. In this formalism, the equilibrium shape of a cell vesicle may present itself in a rich variety of geometric and topological characteristics. We first show that there is an obstruction, arising from the spontaneous curvature, to the existence of a minimizer of the Helfrich energy over the set of embedded ring tori. We then propose a scale-invariant anisotropic bending energy, which extends the Canham energy, and show that it possesses a unique toroidal energy minimizer, up to rescaling, in all parameter regime. Furthermore, we establish some genus-dependent topological lower and upper bounds, which are known to be lacking with the Helfrich energy, for the proposed energy. We also present the shape equation in our context, which extends the Helfrich shape equation. The second application arises from astrophysics in the search for a mechanism for matter accretion in the early universe in the context of cosmic strings. In this formalism, gravitation may simply be stored over a two-surface so that the Einstein tensor is given in terms of the Gauss curvature of the surface which relates itself directly to the Hamiltonian energy density of the matter sector. This setting provides a lucid exhibition of the interplay of the underlying geometry, matter energy, and topological characterization of the system. In both areas of applications, we encounter highly challenging nonlinear partial differential equation problems. We demonstrate that studies on these equations help us to gain understanding of the theoretical physics problems considered.
Recently,some authors(Shen and Shi,2016)studied the generalized shift-splitting(GSS)iteration method for singular saddle point problem with nonsymmetric positive definite(1,1)-block and symmetric positive semidefinite(2,2)-block.In this paper,we further apply the GSS iteration method to solve singular saddle point problem with nonsymmetric positive semidefinite(1,1)-block and symmetric positive semidefinite(2,2)-block,prove the semi-convergence of the GSS iteration method and analyze the spectral properties of the corresponding preconditioned matrix.Numerical experiment is given to indicate that the GSS iteration method with appropriate iteration parameters is effective and competitive for practical use.
H-tensor plays an important role in identifying positive definiteness of even order real symmetric tensors.In this paper,some definitions and theorems related to H-tensors are introduced firstly.Secondly,some new criteria for identifying nonsingular H-tensors are proposed,moreover,a new theorem for identifying positive definiteness of even order real symmetric tensors is obtained.Finally,some numerical examples are given to illustrate our results.
In this paper we discuss the following Kirchhoff equation{-(a+b∫R3 |▽u|2dx)Δu+V(x)u+λu=μ|u|q-2u+|u|p-2u in R3,{∫R3u2dx=c2,where a,b,μ and c are positive numbers,λ is unknown and appears as a Lagrange multiplier,14/3＜q＜p＜6 and V is a continuous non-positive function vanishing at in-finity.Under some mild assumptions on V,we prove the existence of a mountain pass normalized solution.Here we study the existence of normalized solution to mass super-critical Kirchhoff equation with potential via the minimax principle and Nehari-Pohozaev manifold.
In this paper,we obtain the boundedness of multilinear Calderón-Zygmund operators with kernels of Dini type and commutators with variable exponent λ-central BMO functions in variable exponent central Morrey spaces.
Let P∈Cm×m and Q∈Cn×n be Hermitian and {k+1}-potent matrices,i.e.,Pk+1=P=P*,Qk+1=Q=Q*,where(·)*stands for the conjugate transpose of a matrix.A matrix X∈Cm×n is called {P,Q,k+1}-reflexive(anti-reflexive)if PXQ=X(PXQ=-X).In this paper,the least squares solution of the matrix equation AXB=C subject to {P,Q,k+1}-reflexive and anti-reflexive constraints are studied by converting into two simpler cases:k=1 and k=2.
