Cheng-Hu-Moruz (2017) completely classified the locally strongly convex centroaffine hypersurfaces with parallel cubic form based on the Calabi product (called the type I Calabi product for short) proposed by Li-Wang (1991).

In the present paper, the authors introduce the type II Calabi product (in case λ₁ = 2λ₂), complementing the type I Calabi product (in case λ₁ ≠ 2λ₂), and achieve a classification of the locally strongly convex centroaffine hypersurfaces in ℝⁿ⁺¹ with vanishing centroaffine shape operator and Weyl curvature tensor by virtue of the types I and II Calabi product.

As a corollary, 3-dimensional complete locally strongly convex centroaffine hypersurfaces with vanishing centroaffine shape operator are completely classified, which positively answers the centroaffine Bernstein problems III and V by Li-Li-Simon (2004).

The results of this work deal with the existence and blow up of solutions for the following damped extensible beam with degenerate nonlocal damping and source term $u_{tt}+\Delta^{2}u-M(\Vert\nabla u \Vert^{2})\Delta u+\Vert \Delta u\Vert^{2\alpha}\vert u_{t}\vert^{\gamma}u_{t}=\vert u\vert^{\rho}u$. It is regarded as the second part of the paper by Narciso et al. (in 2023), where global existence, uniqueness and asymptotic stability of strong solutions were obtained for regular initial data in the case ∣u∣^ρu ≡ 0.

Let

be a space of homogeneous type, in the sense of Coifman and Weiss, and

\phi :\mathcal{X}\times [0,\mathrm{\infty })\to [0,\mathrm{\infty })

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varphi:\cal{X}\times[0,\infty)\rightarrow[0,\infty)$$\end{document}

satisfy that, for almost every

x\in \mathcal{X},\phi (x,\cdot )

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x\in\cal{X},\varphi(x,\cdot)$$\end{document}

is an Orlicz function and that φ(·, t) is a Muckenhoupt

{\mathbb{A}}_{\mathrm{\infty }}(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{A}_{\infty}(\cal{X})$$\end{document}

weight uniformly in t ∈ [0, ∞). In this article, the authors first establish a new molecular characterization, associated with admissible sequences of balls on

\mathcal{X}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\cal{X}$$\end{document}

, of the Musielak-Orlicz Hardy space

H^{\phi }(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^{\varphi}(\cal{X})$$\end{document}

. As an application, the authors also obtain the boundedness of Calderón-Zygmund operators from

H^{\phi }(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^{\varphi}(\cal{X})$$\end{document}

to

H^{\phi }(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^{\varphi}(\cal{X})$$\end{document}

or to the Musielak-Orlicz space

L^{\phi }(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^{\varphi}(\cal{X})$$\end{document}

. The main novelty of these results is that, in the proof of the boundedness of Calderón-Zygmund operators on

H^{\phi }(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^{\varphi}(\cal{X})$$\end{document}

, the authors get rid of the dependence on the reverse doubling property of μ by using this new molecular characterization of

H^{\phi }(\mathcal{X})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^{\varphi}(\cal{X})$$\end{document}

.

For any positive integer m, let ℤ_m be the additive group of residue classes modulo m. For A ⊆ ℤ_m and

, let the representation function

R_A(\overline{n})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R_{A}(\overline{n})$$\end{document}

denote the number of solutions of the equation

\overline{n}=\overline{a}+\overline{a^{\mathrm{\prime }}}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\overline{n}=\overline{a}+\overline{a^{\prime}}$$\end{document}

with unordered pairs

(\overline{a},\overline{a^{\mathrm{\prime }}})\in A\times A

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(\overline{a},\overline{a^{\prime}})\in A\times A$$\end{document}

. Let m = 2^αM > 2, where α is a positive integer and M is a positive odd integer. In this paper, the author proves that if M ≥ 3, then there exist two distinct sets A, B ⊆ ℤ_m with ∣A ∪ B∣ = m − 2, A ∩ B = ∅ and

B\ne \frac{\overline{m}}{2}+A

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B\ne{\overline{m}\over{2}}+A$$\end{document}

such that

R_A(\overline{n})=R_B(\overline{n})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R_{A}(\overline{n})=R_{B}(\overline{n})$$\end{document}

for all

\overline{n}\in {\mathbb{Z}}_m

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\overline{n}\in\mathbb{Z}_{m}$$\end{document}

. The author also proves that if M = 1 and A, B ⊆ ℤ_m with ∣A ∪ B∣ = m − 2 and A ∩ B = ∅, then

R_A(\overline{n})=R_B(\overline{n})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R_{A}(\overline{n})=R_{B}(\overline{n})$$\end{document}

for all

\overline{n}\in {\mathbb{Z}}_m

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\overline{n}\in\mathbb{Z}_{m}$$\end{document}

if and only if

B=\frac{\overline{m}}{2}+A

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B={\overline{m}\over{2}}+A$$\end{document}

.

The authors carry out numerical experiments with regard to the Monte Carlo integration method, using as input the pseudorandom vectors that are generated by the algorithm proposed in [Mok, C. P., Pseudorandom Vector Generation Using Elliptic Curves and Applications to Wiener Processes, Finite Fields and Their Applications, 85, 2023, 102129], which is based on the arithmetic theory of elliptic curves over finite fields. They consider integration in the following two cases: The case of Lebesgue measure on the unit hypercube [0, 1]^d, and as well as the case of Wiener measure. In the case of Wiener measure, the construction gives discrete time simulation of an independent sequence of standard Wiener processes, which is then used for the numerical evaluation of Feynman-Kac formulas.

The authors exhibit some new families of cyclotomic fields which have non-trivial plus parts of their class numbers. They also prove the 3-divisibility of the plus part of the class number of another family consisting of infinitely many cyclotomic fields. At the end, they provide some numerical examples supporting our results.

In this paper, the author proves that if the dual X* of X is weakly locally uniformly convex and the convex function f is continuous on X, then there exist two sequences

and

\{g_n{\}}_{n=1}^{\mathrm{\infty }}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\{g_{n}\}_{n=1}^{\infty}$$\end{document}

of continuous functions on X** such that (1) f_n(x) ≤ f_n+1(x) ≤ f(x) ≤ g_n+1(x) ≤ g_n(x) whenever x ∈ X; (2) the two convex functions f_n and g_n are Gâteaux differentiable on X; (3) f_n → f and g_n → f uniformly on X. Moreover, if the function f is coercive on X, then (1) f_n and g_n are two w*-lower semicontinuous convex functions on X*; (2)

\text{epi}{f}_n={\overline{\text{epi}{f}_n\cap (X\times R)}}^{w^{\ast }}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\text{epi}\;f_{n}=\overline{\text{epi}\;f_{n}\cap(X\times R)}^{w^{\ast}}$$\end{document}

and

\text{epi}{g}_n={\overline{\text{epi}{g}_n\cap (X\times R)}}^{w^{\ast }}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\text{epi}\;g_{n}=\overline{\text{epi}\;g_{n}\cap(X\times R)}^{w^{\ast}}$$\end{document}

.

A book embedding of a graph G is a placement of its vertices along the spine of a book, and an assignment of its edges to the pages such that no two edges on the same page cross. The pagenumber of a graph is the minimum number of pages in which it can be embedded. Determining the pagenumber of a graph is NP-hard. A graph is said to be 1-planar if it can be drawn in the plane so that each edge is crossed at most once. The anthors prove that the pagenumber of 1-planar graphs is at most 10.

In this paper, the authors investigate exceptional sets in the Waring-Goldbach problem for unlike powers. For example, estimates are obtained for sufficiently large integers below a parameter subject to the necessary local conditions that do not have a representation as the sum of a square of prime, a cube of prime and a sixth power of prime and a k-th power of prime. These results improve the recent result due to Brüdern in the order of magnitude. Furthermore, the method can be also applied to the similar estimates for the exceptional sets for Waring-Goldbach problem for unlike powers.