Third order three-dimensional symmetric and traceless tensors play an important role in physics and tensor representation theory. A minimal integrity basis of a third order three-dimensional symmetric and traceless tensor has four invariants with degrees two, four, six, and ten, respectively. In this paper, we show that any minimal integrity basis of a third order three-dimensional symmetric and traceless tensor is also an irreducible function basis of that tensor, and there is no syzygy relation among the four invariants of that basis, i.e., these four invariants are algebraically independent.

Let \mathcal{H} be a k-uniform hypergraph on n vertices with degree sequence \Delta ={{\displaystyle d}}_{\mathrm{1}}\ge \mathrm{...}\ge {{\displaystyle d}}_n=\delta. In this paper, in terms of degree d_i, we give some upper bounds for the Z-spectral radius of the signless Laplacian tensor (\mathcal{Q}(\mathcal{H})) of \mathcal{H}. Some examples are given to show the eciency of these bounds.

Let \mathcal{E} be a proper class of triangles in a triangulated category \mathcal{C}, and let (\mathcal{A},\mathcal{B},\mathcal{C}) be a recollement of triangulated categories. Based on Beligiannis's work, we prove that \mathcal{A} and \mathcal{C} have enough \mathcal{E} -projective objects whenever \mathcal{B} does. Moreover, in this paper, we give the bounds for the \mathcal{E} -global dimension of \mathcal{B} in a recollement (\mathcal{A},\mathcal{B},\mathcal{C}) by controlling the behavior of the \mathcal{E} -global dimensions of the triangulated categories A and \mathcal{C} : In particular, we show that the niteness of the \mathcal{E} -global dimensions of triangulated categories is invariant with respect to the recollements of triangulated categories.

We bring in Landau-Lifshitz-Bloch equation on m-dimensional closed Riemannian manifold and prove that it admits a unique local solution. When m\ge \mathrm{3} and the initial data in L^{\infty }-norm is suciently small, the solution can be extended globally. Moreover, for m\ge \mathrm{2}, we can prove that the unique solution is global without assuming small initial data.

A non-local abstract Cauchy problem with a singular integral is studied, which is a closed system of two evolution equations for a real-valued function and a function-valued function. By proposing an appropriate Banach space, the well-posedness of the evolution system is proved under some bounded-ness and smoothness conditions on the coeffcient functions. Furthermore, an isomorphism is established to extend the result to a partial integro-differential equation with a singular convolution kernel, which is a generalized form of the stationary Wigner equation. Our investigation considerably improves the understanding of the open problem concerning the well-posedness of the stationary Wigner equation with inflow boundary conditions.

We establish the boundedness and continuity of parametric Marcinkiewicz integrals associated to homogeneous compound mappings on Triebel-Lizorkin spaces and Besov spaces. Here the integral kernels are provided with some rather weak size conditions on the unit sphere and in the radial direction. Some known results are naturally improved and extended to the rough case.

Let f be a full-level cusp form for GL_m(\mathbb{Z} ) with Fourier coeffcients {{\displaystyle A}}_f({{\displaystyle c}}_{m-\mathrm{2}},\cdots ,{{\displaystyle c}}_{\mathrm{1}},n). Let \lambda (n) be either the von Mangoldt function \wedge (n) or the k-th divisor function {{\displaystyle \tau }}_k(n) : We consider averages of shifted convolution sums of the type : We succeed in obtaining a saving of an arbitrary power of the logarithm, provided that X^{\frac{\mathrm{8}}{\mathrm{33}}+\epsilon }\le H\le X^{\mathrm{1}-\epsilon } .

The path independence of additive functionals for stochastic differential equations (SDEs) driven by the G-Brownian motion is characterized by the nonlinear partial differential equations. The main result generalizes the existing ones for SDEs driven by the standard Brownian motion.

We investigate the Cauchy problem for the 3D magnetohydrodynamics equations with only horizontal dissipation for the small initial data. With the help of the dissipation in the horizontal direction and the structure of the system, we analyze the properties of the decay of the solution and apply these decay properties to get the global regularity of the solution. In the process, we mainly use the frequency decomposition in Green's function method and energy method.

We establish a new characterization of the Musielak–Orlicz–Sobolev space on {ℝ}^n; which includes the classical Orlicz–Sobolev space, the weighted Sobolev space, and the variable exponent Sobolev space as special cases, in terms of sharp ball averaging functions. Even in a special case, namely, the variable exponent Sobolev space, the obtained result in this article improves the corresponding result obtained by P. Hästö and A. M. Ribeiro [Commun. Contemp. Math., 2017, 19: 1650022] via weakening the assumption f\in L^{\mathrm{1}}({ℝ}^n) into f\in L_{}^{\mathrm{1}}({ℝ}^n), which was conjectured to be true by Hästö and Ribeiro in the aforementioned same article.

Addition formulae of trigonometric and elliptic functions are used to generate Backlund transformations together with their connecting quadrilateral equations. As a result, we obtain the periodic solutions for a number of multidimensionally consistent affine linear and multiquadratic quadrilateral equations.

Eigenvalues of tensors play an increasingly important role in many aspects of applied mathematics. The characteristic polynomial provides one of a very few ways that shed lights on intrinsic understanding of the eigenvalues. It is known that the characteristic polynomial of a third order three dimensional tensor has a stunning expression with more than 20000 terms, thus prohibits an effective analysis. In this article, we are trying to make a concise representation of this characteristic polynomial in terms of certain basic determinants. With this, we can successfully write out explicitly the characteristic polynomial of a third order three dimensional tensor in a reasonable length. An immediate benefit is that we can compute out the third and fourth order traces of a third order three dimensional tensor symbolically, which is impossible in the literature.