For a compact connected Lie group G, the rational cohomology ring of its classifying space BG can be expressed as an invariant ring. Namely, for a maximal torus T^n of G, it is well–known that H^*(BG; ℚ ) ≅ H^*(BT^n; ℚ )^W(G) which is a polynomial ring. If d divides n, the quotient SU(n)/ℤ_d is also a Lie group. Their rational cohomology rings are isomorphic, however, the integral representations of the Weyl groups W(SU(n)/ℤ_d) are not equivalent. So we ask if the modular invariant rings under the actions are polynomial. Our main theorem is a generalization of some results in [4] and [12].
In this paper, we provide a new property for the Smoluchowski-Kramers approximation of stochastic differential equations. We prove the convergence of the derivative of solutions with respect to the initial condition (when the mass of particles tends to zero). We then use the techniques of Malliavin calculus to obtain an explicit Berry-Esseen error bound for the rate of convergence.
In this paper, we establish a non-effective version of Schmidt’s subspace theorem for moving hypersurface targets in weak general position on algebraic varieties over function fields.
This paper deals with certain classes of modules under module-finite extensions. Let φ : R↪ S be a module-finite extension between commutative Noetherian local rings. We investigate the transfer of Artinian module structures and attached primes between R and S. We clarify the behavior of local cohomology modules as well as certain structures of finitely generated S-modules under the restriction of scalars to R via φ . We show that R is a quotient of a Cohen-Macaulay local ring if and only if so is S. As an application, we characterize the structure of Nagata’s idealization. Using Macaulayfication of algebraic varieties and idealization, we give an example to illustrate the results.
In this paper, we investigate the Weyl mean orbital pseudo-metric for Polish dynamical systems and its connections to the properties of the space of invariant measures. We establish equivalent conditions under which the set of invariant measures generated by periodic points is dense in the set of ergodic measures, thereby providing a deeper understanding of the relationship between periodic behavior and measure-theoretic properties in the field of Weyl mean orbital pseudo-metric.
A finite group G is called an m-cyclic group if it has m cyclic subgroups. In this paper, we classify finite 14-cyclic groups. These results involve a subclass of finite m-cyclic groups of order n with |π (n)|≤ 3 , where |π (n)| is the number of prime divisors of n.
We consider the 3D viscous Camassa-Holm equations with damping term in the whole space. Firstly, we prove the existence, uniqueness and regularity of global weak solutions to the equations. Then we prove the existence of a compact global attractor for the associated continuous semigroup. To overcome the essential difficulty when proving the asymptotic compactness of the semigroup, which arises due to the lack of compactness of the Sobolev embeddings, we exploit the energy equation method. Next, by using the regularity of weak solutions and inductive arguments, we study the Sobolev regularity of the global attractor. Finally, we give an explicit upper bound for the fractal dimension of the global attractor.
This paper investigates the solution existence and stability of the semi-affine variational inequality problem in real separable Hilbert spaces. We demonstrate that, under suitable conditions, the solution set of the semi-affine variational inequality problem is nonempty. Additionally, we consider some related stability analyses of the problem. The obtained results contribute to and complement the existing literature.
In this paper, we study the discrete Li–Yau gradient estimates for the positive solutions u to the heat equation on graphs under CDE(n,-K) condition and derive a sharper estimate than Bauer et al. (J. Differ. Geom. 99(3), 359–405 2015) and Wang and Zhang (Comm. Anal. Geom. 27(4), 969–989 2019).
We study the asymptotic behavior of a nonlattice random walk in a general cone of ℝ^d . Following the approach initiated by D. Denisov and V. Wachtel in [8], we use a strong approximation of random walks by the Brownian motion and prove local limit theorems, combining integral theorems for random walks in cones with classical theorems for unrestricted random walks.
In this paper, we investigate global error bounds for nonmonotone affine variational inequalities in finite-dimensional spaces. We establish a characterization of global error bounds on the feasible sets of these problems. Furthermore, we derive several sufficient conditions and necessary conditions for the existence of global error bounds on the feasible sets. Our results complement existing findings for some classes of monotone affine variational inequalities. Several illustrative examples are presented to demonstrate the applicability of the obtained results.
The aim of this short communication is to say about some mistake in the proof of Theorem 2 in my original article “Maximal subgroups of almost subnormal subgroups in division rings” (Bui Xuan Hai, Acta Mathematica Vietnamica 47, 197–209, 2022). Further, some weaker version of this theorem will be provided.
Let D be a division algebra and let n be a positive integer greater than 1. Assume that the commutator width ω (D) of D^* is positive and finite. First, we revisit a question posed by F. S. Cater concerning the decomposition of matrices into a product of reflections over a division ring. Among results, we show that every matrix A in the special linear group SL_n(D) can be expressed as a product of at most rank (A-I_n)+4ω (D) reflections. Next, we study the decomposition of matrices in SL_n(D) into products of commutators of reflections in the general linear group GL_n(D) .
We investigate a space-time finite element method for solving a parabolic advection-diffusion problem. We use the Banach-Nečas-Babuška theorem to show the well-posedness of the continuous Petrov-Galerkin variational formulation for this problem. A fully discrete finite-element scheme is analyzed using the standard Galerkin method and unstructured meshes. An optimal error estimate is established in a discrete energy norm under a globally regularity condition. Some numerical results corroborate our theoretical results.
In this paper, we explore the task of addressing strongly monotone variational inequalities over the solution set of the split common fixed point problem with multiple output sets in real Hilbert spaces. We propose a new iterative algorithm designed specifically for this task, which incorporates dynamic step sizes that adapt based on information from previous iterations. This approach ensures strong convergence without the need for prior knowledge of the norm of the bounded linear operator involved. Additionally, our method does not require information about the Lipschitz constants or the strongly monotone constants of the mappings. We also present several corollaries derived from our main result. Finally, we present an application of the split common fixed point problem with multiple output sets to supply chain planning, and provide numerical experiments to evaluate the performance of the proposed algorithm in comparison with existing methods.
We deal with the Dirichlet problem for nonsymmetric augmented Hessian quotient type equations. First, we look for an admissible solution to the problem for corresponding symmetric augmented Hessian quotient type equations. Then we apply the Banach fixed point theorem to prove the existence of a δ -admissible solution in C^2,α to the problem by assuming that the augmented skew-symmetric matrix is sufficiently small in a certain sense. We also give a necessary condition for the existence and sufficient conditions for uniqueness of this kind of δ -admissible solutions.
We study the initial-boundary value problem for a Keller-Segel-Navier-Stokes model that includes a logistic source term. The system is considered in a two-dimensional bounded domain with a smooth boundary as follows { n_t+u·∇ n=Δ n-∇· (n∇ c)+μ n-κ n^2, c_t+u·∇ c=Δ c -nc, v_t+u·∇ v=Δ v -γ v+n, u_t+(u·∇ ) u+∇π =Δ u-nf, ∇· u=0. . This system characterizes the interaction of chemotactic microorganisms with an incompressible fluid. Under the conditions of positive μ and non-negative κ , sufficiently regular initial data (n_0, c_0, v_0, u_0) with n_0≢0 yield a uniformly bounded global classical solution. Moreover, when μ =0 , the solution obeys n(· ,t)→ 0, c(· ,t)→ 0, v(· ,t)→ 0 and u(· ,t)→ 0 in L^∞(Ω ) as t→∞ .
By using our previous results on Lê modules and an upper-bound on the betti numbers which we proved with Lê, we investigate the cohomology of Milnor fibers and the internal local systems given by the vanishing cycles of hypersurfaces with one-dimensional singular sets and small Lê numbers.
A foundational result by C. Huneke and V. Trivedi provides a formula for the depth of an ideal in terms of height, computed over a finite set of prime ideals, for rings that are homomorphic images of regular rings. Building on a result by the first author for local quotients of Cohen-Macaulay rings, this paper first gives a new proof for the depth formula and derives a similar formula for the finiteness dimension. Our main result then establishes the depth formula for non-local rings that are homomorphic images of a finite-dimensional Gorenstein ring.
