In this study, we give weighted mean and weighted Gaussian curvatures of two types of timelike general rotational surfaces with non-null plane meridian curves in four-dimensional Minkowski space 𝔼⁴₁ with density e^{λ₁x2+λ₂y2+λ₃z2+λ₄t2}, where λ_i (i = 1, 2, 3, 4) are not all zero. We give some results about weighted minimal and weighted flat timelike general rotational surfaces in 𝔼⁴₁ with density. Also, we construct some examples for these surfaces.

We identify stable pairs and stable framed sheaves as epimorphisms and monomorphisms in the hearts of tilted t-structures under appropriate conditions. We then identify the moduli spaces with the corresponding Quot spaces. As a result, we obtain the projectivity of the Quot spaces in absolute cases. In addition, we prove a formula in a motivic Hall algebra, which relates together the Quot spaces under a tilt.

We show that under a mild thickness condition, every Tits pentagon arises through a folding process from a free module of rank 5 over a unitary associative ring of stable range 1.

Let Q be an integral positive definite quadratic form of level N in 2k(≥ 4) variables. We assume that (-1)^kN is a fundamental discriminant and the associated character χ of Q is primitive of conductor N. Under our assumption, we find the pairs (k, N) such that the dimension of spaces of cusp forms of weight k and level N with Nebentypus χ is one. Furthermore, we explicitly construct their bases by using Eisenstein series of lower weights. For the above pairs (k, N), we use these cusp forms to provide closed formulas for the representation numbers by quadratic forms of level N in 2k variables, which are expressed in terms of divisor functions and their convolution sums.

In this paper, in one spatial dimension, we study the convergence rate for Schrödinger operators with complex time P^t_a,𝛾, which is defined by $${P^t_{a,{\gamma}}}f(x)={S^{t+it^{\gamma}}_a}f(x)=\int_{\mathbb{R}}{e^{ix{\xi}}\,e^{ix{\mid}{\xi}{\mid}^a}\,e^{-t^{\gamma}{\mid}{\xi}{\mid}^a}}{\hat{f}}({\xi})d{\xi},$$ where 𝛾 > 0 and a > 0. The convergence rate for Schrödinger operators with complex time is different from that of classical Schrödinger operators in Cao-Fan-Wang (Illinois J. Math. 62: 365-380, 2018).

For a domain D ⊂ ℂⁿ and an admissible weight µ on it, we consider the weighted Bergman kernel K_D,µ and the corresponding weighted Bergman metric on D. In particular, motivated by work of Mok, Ng, Chan-Yuan and Chan-Xiao-Yuan among others, we study the nature of holomorphic isometries from the disc 𝔻 ⊂ ℂ with respect to the weighted Bergman metrics arising from weights of the form µ = K^-d_𝔻 for some integer d ≥ 0. These metrics provide a natural class of examples that give rise to positive conformal constants that have been considered in various recent works on isometries. Specific examples of isometries that are studied in detail include those in which the isometry takes values in 𝔻ⁿ and 𝔻 × 𝔹ⁿ where each factor admits a weighted Bergman metric as above for possibly different non-negative integers d. Finally, the case of isometries between polydisks in possibly different dimensions, in which each factor has a different weighted Bergman metric as above, is also presented.

Let 𝓔 be an injectively resolving class of left R-modules and n≥1 be an integer. In this paper, we introduce and study n-strongly 𝓔-Gorenstein injective and flat modules, which are common generalizations of strongly Gorenstein injective and flat modules [2, 28], n-strongly Gorenstein injective and flat modules [3, 29], respectively. Then we discuss some properties of these modules and show how the property of being an n-strongly 𝓔-Gorenstein injective and flat module can be inherited by its direct summands under certain condition. Finally, we consider the connections between n-strongly 𝓔-Gorenstein injective and flat modules. As applications, some known results are obtained as corollaries.

Let R be an associative ring with an identity, 𝜎 be an automorphism, and 𝛿 be a 𝜎-derivation of R. In this article, we describe all (nilpotent) associated primes of the skew inverse Laurent series ring R((x^-1; 𝜎, 𝛿)) in terms of the (nilpotent) associated primes of R.

The objective which drives the writing of this article is to study the behavior of the algebraic transfer for ranks h ∈ {6, 7, 8} across various internal degrees. More precisely, we prove that the algebraic transfer is an isomorphism in certain bidegrees. A noteworthy aspect of our research is the rectification of the results outlined by M. Moetele and M. F. Mothebe in [East-West J. of Mathematics 18 (2016), 151-170]. This correction focuses on the 𝒜-generators for the polynomial algebra ℤ/2[t₁, t₂, . . . , t_h] in degree thirteen and the ranks h mentioned above. As direct consequences, we are able to confirm the Singer conjecture for the algebraic transfer in the cases under consideration. Especially, we affirm that the decomposable element h₆Ph₂ ∈ Ext^6,80_𝒜 (ℤ/2, ℤ/2) does not reside within the image of the sixth algebraic transfer. This event carries significance as it enables us to either strengthen or refute the Singer conjecture, which is relevant to the behavior of the algebraic transfer. Additionally, we also show that the indecomposable element q ∈ Ext^6,38_𝒜 (ℤ/2, ℤ/2) is not detected by the sixth algebraic transfer. Prior to this research, no other authors had delved into the Singer conjecture for these cases. The significant and remarkable advancement made in this paper regarding the investigation of Singer's conjecture for ranks h, 6 ≤ h ≤ 8, highlights a deeper understanding of the enigmatic nature of Ext^h,h+•_𝒜 (ℤ/2, ℤ/2).