A {(3,4),4}-fullerene graph S is a 4-regular map on the sphere whose faces
are of length 3 or 4. It follows from Euler’s formula that the number of triangular faces
is eight. A set H of disjoint quadrangular faces of S is called resonant pattern if S
has a perfect matching M such that every quadrangular face in H is M-alternating.
Let k be a positive integer, S is k-resonant if any i≤k disjoint quadrangular faces of
S form a resonant pattern. Moreover, if graph S is k-resonant for any integer k, then
S is called maximally resonant.

In this paper, an efficient numerical method is proposed to solve the CaputoRiesz fractional diffusion equation with fractional Robin boundary conditions. We
approximate the Riesz space fractional derivatives using the fractional central difference
scheme with second-order accurate. A priori estimation of the solution of the numerical
scheme is given, and the stability and convergence of the numerical scheme are analyzed.
Finally, a numerical example is used to verify the accuracy and efficiency of the numerical
method.

Suppose (M,F) is a convex complex Finsler manifold. We prove that geodesics
of (M,F) are locally minimizing. Hence, F introduces a distance function d such that
(M,d) is a metric space from topology. Next, we prove the classical Hopf-Rinow Theorem
holds on (M,F).

Using the weight coefficient method, we first discuss semi-discrete Hilberttype inequalities, and then discuss boundedness of integral and discrete operators and
operator norm estimates based on Hilbert-type inequalities in weighted Lebesgue space
and weighted normed sequence space.

In this paper, we study the regularity criterion for the three-dimensional
Boussinesq equations in Besov spaces. We show that the smooth solution (u,θ) is regular
if the horizonal velocity uh holds

We prove the equivalence between two explicit expressions for two-point
Witten-Kontsevich correlators obtained by Bertola-Dubrovin-Yang and by Zograf, respectively.

Magnetic monopoles stand for the static solution arising from a (1+ 3)–
dimensional theory describing the interaction between a real scalar triplet and non–Abelian
gauge field. In this paper, we obtain a two–point boundary value problem of a first–order
ordinary differential equations from the self–dual monopole model. Then we establish
the existence and uniqueness theorem for the problem by using a dynamical shooting
method, we also obtain sharp asymptotic estimates for the solutions at infinity.

We consider the initial-boundary value problem for finitely degenerate
parabolic equation. We first give sufficient conditions for the blow-up and global existence
of the parabolic equation at high initial energy level. Then, we establish the existence of
solutions blowing up in finite time with initial data at arbitrary energy level. Finally, we
estimate the upper bound of the blow-up time under certain conditions.