In this paper, we introduce the notion of semi-symmetric structure Jacobi operator for Hopf real hypersufaces in the complex quadric Q^m = SO_m+2/SO_mSO₂. Next we prove that there does not exist any Hopf real hypersurface in the complex quadric Q^m = SO_m+2/SO_mSO₂ with semi-symmetric structure Jacobi operator. As a corollary, we also get a non-existence property of Hopf real hypersurfaces in the complex quadric Q^m with either symmetric (parallel), or recurrent structure Jacobi operator.

Let n ⩾ 2 be an integer, we denote the smallest integer b such that gcd {(ⁿ_k) : b < k < n - b} > 1 as b(n). For any prime p, we denote the highest exponent α such that p^α | n as v_p(n). In this paper, we partially answer a question asked by Hong in 2016. For a composite number n and a prime number p with p | n, let n = a_mp^m + r, 0 ⩽ r < p^m, 0 < a_m < p. Then we have $$v_p\(\text{gcd}\{\(n\\k\)\;:\;b(n)1\}\)=\{\array{1,&&a_m=1\text{ and }r=b(n),\\0,&&\text{otherwise.}}$$

In 2017, Nikiforov proposed the A_𝛼-matrix of a graph G. This novel matrix is defined as A_𝛼(G) = 𝛼D(G) + (1 - 𝛼)A(G), 𝛼 ∈ [0, 1], where D(G) and A(G) are the degree diagonal matrix and adjacency matrix of G, respectively. Recently, Zhai, Xue and Liu [39] considered the Brualdi-Hoffman-type problem for Q-spectra of graphs with given matching number. As a continuance of it, in this contribution we consider the Brualdi-Hoffman-type problem for A_𝛼-spectra of graphs with given matching number. We identify the graphs with given size and matching number having the largest A_𝛼-spectral radius for ${\alpha}{\in}[{\frac{1}{2}},1)$.

In this paper, we give some results on 2-strongly Gorenstein projective modules and related rings. We first investigate the relationship between strongly Gorenstein projective modules and periodic modules and then give the structure of modules over strongly Gorenstein semisimple rings. Furthermore, we prove that a ring R is 2-strongly Gorenstein hereditary if and only if every ideal of R is Gorenstein projective and the class of 2-strongly Gorenstein projective modules is closed under extensions. Finally, we study the relationship between 2-Gorenstein projective hereditary and 2-Gorenstein projective semisimple rings, and we also give an example to show the quotient ring of a 2-Gorenstein projective hereditary ring is not necessarily 2-Gorenstein projective semisimple.

Suppose that M is a strictly convex hypersurface in the (n + 1)-dimensional Euclidean space 𝔼ⁿ⁺¹ with the origin o in its convex side and with the outward unit normal N. For a fixed point p ∈ M and a positive constant t, we put 𝚽_t the hyperplane parallel to the tangent hyperplane 𝚽 at p and passing through the point q = p - tN(p). We consider the region cut from M by the parallel hyperplane 𝚽_t, and denote by I_p(t) the (n + 1)-dimensional volume of the convex hull of the region and the origin o. Then Schneider's characterization theorem for ellipsoids states that among centrally symmetric, strictly convex and closed surfaces in the 3-dimensional Euclidean space 𝔼³, the ellipsoids are the only ones satisfying I_p(t) = 𝜙(p)t, where 𝜙 is a function defined on M. Recently, the characterization theorem was extended to centrally symmetric, strictly convex and closed hypersurfaces in 𝔼ⁿ⁺¹ satisfying for a constant 𝛽, I_p(t) = 𝜙(p)t^𝛽. In this paper, we study the volume I_p(t) of a strictly convex and complete hypersurface in 𝔼ⁿ⁺¹ with the origin o in its convex side. As a result, first of all we extend the characterization theorem to strictly convex and closed (not necessarily centrally symmetric) hypersurfaces in 𝔼ⁿ⁺¹ satisfying I_p(t) = 𝜙(p)t^𝛽. After that we generalize the characterization theorem to strictly convex and complete (not necessarily closed) hypersurfaces in 𝔼ⁿ⁺¹ satisfying I_p(t) = 𝜙(p)t^𝛽.

Let F_n be the Farey sequence of order n. For S ⊆ F_n, let 𝓠(S) be the set of rational numbers x/y with x, y ∈ S, x ≤ y and y ≠ 0. Recently, Wang found all subsets S of F_n with |S| = n + 1 for which 𝓠(S) ⊆ F_n. Motivated by this work, we try to determine the structure of S ⊆ F_n such that |S| = n and 𝓠(S) ⊆ F_n. In this paper, we determine all sets S ⊆ F_n satisfying these conditions for n ∈ {p, 2p}, where p is prime.

For any prime number p, let J(p) be the set of positive integers n such that the numerator of the n^th harmonic number in the lowest terms is divisible by this prime number p. We consider an extension of this set to the generalized harmonic numbers, which are a natural extension of the harmonic numbers. Then, we present an upper bound for the number of elements in this set. Moreover, we state an explicit condition to show the finiteness of our set, together with relations to Bernoulli and Euler numbers.

Let j be a nonnegative integer. We define the Toeplitz-type operators T^(j)_a with symbol a ∈ L^∞(C), which are variants of the traditional Toeplitz operators obtained for j = 0. In this paper, we study the boundedness of these operators and characterize their compactness in terms of its Berezin transform.

In this paper, we prove that a domain R is an FGV-domain if every finitely generated torsion-free R-module is strongly copure projective, and a coherent domain is an FGV-domain if and only if every finitely generated torsion-free R-module is strongly copure projective. To do this, we characterize G-Prüfer domains by G-flat modules, and we prove that a domain is G-Prüfer if and only if every submodule of a projective module is G-flat. Also, we study the D + M construction of G-Prüfer domains. It is seen that there exists a non-integrally closed G-Prüfer domain that is neither Noetherian nor divisorial.

Recently, Alzer and Choi [2] introduced and studied a set of the four linear Euler sums with parameters. These sums are parametric extensions of Flajolet and Salvy's four kinds of linear Euler sums [9]. In this paper, by using the method of residue computations, we will establish two explicit combined formulas involving two parametric linear Euler sums S⁺⁺_p,q (a, b) and S^+-_p,q (a, b) defined by Alzer and Choi, which can be expressed in terms of a linear combinations of products of trigonometric functions, digamma functions and Hurwitz zeta functions.

In this paper, we introduce the idea of twisted product lightlike submanifolds of semi-Riemannian manifolds and provide non-trivial examples of such lightlike submanifolds. Then, we prove the non-existence of proper isotropic or totally lightlike twisted product submanifolds of a semi-Riemannian manifold. We also show that for a twisted product lightlike submanifold of a semi-Riemannian manifold, the induced connection ∇ is not a metric connection. Further, we prove that a totally umbilical SCR-lightlike submanifold of an indefinite Kaehler manifold ${\tilde{M}}$ does not admit any twisted product SCR-lightlike submanifold of the type M_⊥×_ϕM_T, where M_⊥ is a totally real submanifold and MT is a holomorphic submanifold of ${\tilde{M}}$. Consequently, we obtain a geometric inequality for the second fundamental form of twisted product SCR-lightlike submanifolds of the type M_T×_ϕM_⊥ of an indefinite Kaehler manifold ${\tilde{M}}$, in terms of the gradient of ln ϕ, where ϕ stands for the twisting function. Subsequently, the equality case of this inequality is discussed. Finally, we construct a non-trivial example of a twisted product SCR-lightlike submanifold in an indefinite Kaehler manifold.

A celebrated result in the study of integer partitions is the identity due to Lehmer whereby the number of partitions of n with an even number of even parts minus the number of partitions of n with an odd number of even parts equals the number of partitions of n into distinct odd parts. Inspired by Lehmer's identity, we prove explicit formulas for evaluating generating functions for sequences that enumerate integer partitions of fixed width with an even/odd number of even parts. We introduce a technique for decomposing the even entries of a partition in such a way so as to evaluate, using a finite sum over q-binomial coefficients, the generating function for the sequence of partitions with an even number of even parts of fixed, odd width, and similarly for the other families of fixed-width partitions that we introduce.

Let U be the restricted quantized enveloping algebra Ũ_q(𝖘𝖑₂) over an algebraically closed field of characteristic zero, where q is a primitive 𝑙-th root of unity (with 𝑙 being odd and greater than 1). In this paper we show that any indecomposable submodule of U under the adjoint action is generated by finitely many special elements. Using this result we describe all ideals of U. Moreover, we classify annihilator ideals of simple modules of U by generators.

Let p be an odd prime. Let E be an elliptic curve defined over a quadratic imaginary field, where p splits completely. Suppose E has supersingular reduction at primes above p. Under appropriate hypotheses, we extend the results of [17] to ℤ²_p-extensions. We define and study the fine double-signed residual Selmer groups in these settings. We prove that for two residually isomorphic elliptic curves, the vanishing of the signed 𝜇-invariants of one elliptic curve implies the vanishing of the signed 𝜇-invariants of the other. Finally, we show that the Pontryagin dual of the Selmer group and the double-signed Selmer groups have no non-trivial pseudo-null submodules for these extensions.

In this paper, we study products of composition, multiplication and differentiation acting on the fractional Cauchy spaces and mapping into the Zygmund space. Characterizations are provided for boundedness and compactness of these operators.

The main motivation of this paper is to introduce and study the notions of strong Dedekind rings and semi-regular injective modules. Specifically, a ring R is called strong Dedekind if every semi-regular ideal is Q₀-invertible, and an R-module E is called a semi-regular injective module provided Ext¹_R(T, E) = 0 for every 𝓠-torsion module T. In this paper, we first characterize rings over which all semi-regular injective modules are injective, and then study the semi-regular injective envelopes of R-modules. Moreover, we introduce and study the semi-regular global dimensions sr-gl.dim(R) of commutative rings R. Finally, we obtain that a ring R is a DQ-ring if and only if sr-gl.dim(R) = 0, and a ring R is a strong Dedekind ring if and only if sr-gl.dim(R) ≤ 1, if and only if any semi-regular ideal is projective. Besides, we show that the semi-regular dimensions of strong Dedekind rings are at most one.

For any given function f, we focus on the so-called prescribed mean curvature problem for the measure e^-f(|x|2)dx provided thate^-f(|x|2) ∈ L¹(ℝⁿ⁺¹). More precisely, we prove that there exists a smooth hypersurface M whose metric is ds² = dρ² + ρ²d𝜉² and whose mean curvature function is ${\frac{1}{n}}(\frac{u^p}{{\rho}^{\beta}})e^{f({\rho}^2)}{\psi}(\xi)$ for any given real constants p, β and functions f and ψ where u and ρ are the support function and radial function of M, respectively. Equivalently, we get the existence of a smooth solution to the following quasilinear equation on the unit sphere 𝕊ⁿ, $${\sum_{i,j}}({{\delta}_{ij}-{\frac{{\rho}_i{\rho}_j}{{\rho}^2+|{\nabla}{\rho}|^2}})(-{\rho}ji+{\frac{2}{{\rho}}}{\rho}j{\rho}i+{\rho}{\delta}_{ji})={\psi}{\frac{{\rho}^{2p+2-n-{\beta}}e^{f({\rho}^2)}}{({\rho}^2+|{\nabla}{\rho}|^2)^{\frac{p}{2}}}}$$ under some conditions. Our proof is based on the powerful method of continuity. In particular, if we take $f(t)={\frac{t}{2}}$, this may be prescribed mean curvature problem in Gauss measure space and it can be seen as an embedded result in Gauss measure space which will be needed in our forthcoming papers on the differential geometric analysis in Gauss measure space, such as Gauss-Bonnet-Chern theorem and its application on positive mass theorem and the Steiner-Weyl type formula, the Plateau problem and so on.

In this study we propose a model of optimal retirement, consumption and portfolio choice of an individual agent, which encompasses a large class of the models in the literature, and provide a methodology to solve the model. Different from the traditional approach, we consider the problems before and after retirement simultaneously and identify the difference in the dual value functions as the utility value of lifetime labor. The utility value has an option nature, namely, it is the maximized value of choosing the retirement time optimally and we discover it by solving a variational inequality. Then, we discover the dual value functions by using the utility value. We discover the value function and optimal policies by establishing a duality between the value function and the dual value function. The model and approach offer a significant advantage for computation of optimal policies for a large class of problems.

In this paper, we introduce the concept of ω-expansive of random map on compact metric spaces 𝓟. Also we introduce the definitions of positively, negatively shadowing property and shadowing property for two-sided RDS. Then we show that if 𝜑 is ω-expansive and has the shadowing property for ω, then 𝜑 is topologically stable for ω.