Volume 35, Issue 6

We complete the work done by James A. Ward in the mid-twentieth century on a system of partial differential equations that defines an algebra 𝔸 {\mathbb{A}} for which this system is the generalized Cauchy–Riemann equations for the derivative introduced by Sheffers at the end of the nineteenth century with respect to 𝔸 {\mathbb{A}} , which is also known as the Lorch derivative with respect to 𝔸 {\mathbb{A}} , and recently simply called 𝔸 {\mathbb{A}} -differentiability. We get a characterization of finite-dimensional algebras, which are associative commutative with unity.

We carry on the investigation started in [P. Ambrosio and A. Passarelli di Napoli, Regularity results for a class of widely degenerate parabolic equations, preprint 2022, version 3, https://arxiv.org/abs/2204.05966v3] about the regularity of weak solutions to the strongly degenerate parabolic equation u t - div ⁡ [ ( | D ⁢ u | - 1 ) + p - 1 ⁢ D ⁢ u | D ⁢ u | ] = f   in  ⁢ Ω T = Ω × ( 0 , T ) , u_{t}-\operatorname{div}\Bigl{[}(\lvert Du\rvert-1)_{+}^{p-1}\frac{Du}{\lvert Du% \rvert}\Bigr{]}=f\quad\text{in }\Omega_{T}=\Omega\times(0,T), where Ω is a bounded domain in ℝ n {\mathbb{R}^{n}} for n ≥ 2 {n\geq 2} , p ≥ 2 {p\geq 2} and ( ⋅ ) + {(\,\cdot\,)_{+}} stands for the positive part. Here, we weaken the assumption on the right-hand side, by assuming that f ∈ L loc p ′ ⁢ ( 0 , T ; B p ′ , ∞ , loc α ⁢ ( Ω ) ) , f\in L_{\mathrm{loc}}^{p^{\prime}}(0,T;B_{p^{\prime},\infty,\mathrm{loc}}^{% \alpha}(\Omega)), with α ∈ ( 0 , 1 ) {\alpha\in(0,1)} and p ′ = p / ( p - 1 ) {p^{\prime}=p/(p-1)} . This leads us to obtain higher fractional differentiability results for a function of the spatial gradient Du of the solutions. Moreover, we establish the higher summability of Du with respect to the spatial variable. The main novelty of the above equation is that the structure function satisfies standard ellipticity and growth conditions only outside the unit ball centered at the origin. We would like to point out that the main result of this paper can be considered, on the one hand, as the parabolic counterpart of an elliptic result contained in [P. Ambrosio, Besov regularity for a class of singular or degenerate elliptic equations, J. Math. Anal. Appl. 505 2022, 2, Paper No. 125636], and on the other hand as the fractional version of some results established in [P. Ambrosio and A. Passarelli di Napoli, Regularity results for a class of widely degenerate parabolic equations, preprint 2022, version 3, https://arxiv.org/abs/2204.05966v3].

In this article, we study minimal isometric immersions of Kähler manifolds into product of two real space forms. We analyse the obstruction conditions to the existence of pluriharmonic isometric immersions of a Kähler manifold into those spaces and we prove that the only ones into 𝕊 m - 1 × ℝ {\mathbb{S}^{m-1}\times\mathbb{R}} and ℍ m - 1 × ℝ {\mathbb{H}^{m-1}\times\mathbb{R}} are the minimal isometric immersions of Riemannian surfaces. Furthermore, we show that the existence of a minimal isometric immersion of a Kähler manifold M 2 ⁢ n {M^{2n}} into 𝕊 m - 1 × ℝ {\mathbb{S}^{m-1}\times\mathbb{R}} and 𝕊 m - k × ℍ k {\mathbb{S}^{m-k}\times\mathbb{H}^{k}} imposes strong restrictions on the Ricci and scalar curvatures of M 2 ⁢ n {M^{2n}} . In this direction, we characterise some cases as either isometric immersions with parallel second fundamental form or anti-pluriharmonic isometric immersions.

Let 𝔽 q {\mathbb{F}_{q}} be the finite field of odd characteristic p with q elements ( q = p n {q=p^{n}} , n ∈ ℕ {n\in\mathbb{N}} ) and let 𝔽 q * {\mathbb{F}_{q}^{*}} represent the set of nonzero elements of 𝔽 q {\mathbb{F}_{q}} . By making use of the Smith normal form of exponent matrices, we obtain an explicit formula for the number of rational points on the variety defined by the following system of equations over 𝔽 q {\mathbb{F}_{q}} : { ∑ i = 1 r a i ( 1 ) ⁢ x 1 e i ⁢ 1 ( 1 ) ⁢ ⋯ ⁢ x n e i ⁢ n ( 1 ) = b 1 , ∑ j ′ = 0 t - 1 ∑ i ′ = 1 r j ′ + 1 - r j ′ a r j ′ + i ′ ( 2 ) ⁢ x 1 e r j ′ + i ′ , 1 ( 2 ) ⁢ ⋯ ⁢ x n j ′ + 1 e r j ′ + i ′ , n j ′ + 1 ( 2 ) = b 2 , \left\{\begin{aligned} &\displaystyle\sum_{i=1}^{r}a^{(1)}_{i}x_{1}^{e_{i1}^{(% 1)}}\cdots x_{n}^{e_{in}^{(1)}}=b_{1},\\ &\displaystyle\sum^{t-1}_{j^{\prime}=0}\sum^{r_{j^{\prime}+1}-r_{j^{\prime}}}_% {i^{\prime}=1}a^{(2)}_{r_{j^{\prime}}+i^{\prime}}x_{1}^{e_{r_{j^{\prime}}+i^{% \prime},1}^{(2)}}\cdots x_{n_{{j^{\prime}}+1}}^{e_{r_{j^{\prime}}+i^{\prime},n% _{{j^{\prime}}+1}}^{(2)}}=b_{2},\end{aligned}\right.\vspace*{1mm} where b i ∈ 𝔽 q {b_{i}\in\mathbb{F}_{q}} ( i = 1 , 2 {i=1,2} ), t ∈ ℕ {t\in\mathbb{N}} , 0 = n 0 < n 1 < n 2 < ⋯ < n t , 0=n_{0}<n_{1}<n_{2}<\cdots<n_{t},\vspace*{1mm} n k - 1 < n ≤ n k {n_{k-1}<n\leq n_{k}} for some 1 ≤ k ≤ t {1\leq k\leq t} , 0 = r 0 < r 1 < r 2 < ⋯ < r t , 0=r_{0}<r_{1}<r_{2}<\cdots<r_{t},\vspace*{1mm} a i ( 1 ) ∈ 𝔽 q * {a^{(1)}_{i}\in\mathbb{F}_{q}^{*}} for i ∈ { 1 , … , r } {i\in\{1,\ldots,r\}} , a i ′ ( 2 ) ∈ 𝔽 q * {a^{(2)}_{i^{\prime}}\in\mathbb{F}_{q}^{*}} for i ′ ∈ { 1 , … , r t } {{i^{\prime}}\in\{1,\ldots,r_{t}\}} , and the exponent of each variable is a positive integer. This generalizes the results obtained previously by Wolfmann, Sun, Cao, and others. Our result also gives a partial answer to an open problem raised by Hu, Hong and Zhao [S. N. Hu, S. F. Hong and W. Zhao, The number of rational points of a family of hypersurfaces over finite fields, J. Number Theory 156 2015, 135–153].

We prove two conjectural identities of Z.-W. Sun concerning Apéry-like series. One of the series is alternating, whereas the other one is not. Our main strategy is to convert the series and the alternating series to log-sine-cosine and log-sinh-cosh integrals, respectively. Then we express all these integrals using single-valued Bloch–Wigner–Ramakrishnan–Wojtkowiak–Zagier polylogarithms. The conjectures then follow from a few rather non-trivial functional equations of those polylogarithms in weights 3 and 4.

Let Ω ⊂ R n \Omega\subset\mathbb{R}^{n} be any open set and 𝑢 a weak supersolution of L ⁢ u = c ⁢ ( x ) ⁢ g ⁢ ( | u | ) ⁢ u | u | \mathcal{L}u=c(x)g(\lvert u\rvert)\frac{u}{\lvert u\rvert} , where L ⁢ u ⁢ ( x ) = p.v. ⁢ ∫ R n g ⁢ ( | u ⁢ ( x ) − u ⁢ ( y ) | | x − y | s ) ⁢ u ⁢ ( x ) − u ⁢ ( y ) | u ⁢ ( x ) − u ⁢ ( y ) | ⁢ K ⁢ ( x , y ) ⁢ d ⁢ y | x − y | s \mathcal{L}u(x)=\textup{p.v.}\int_{\mathbb{R}^{n}}g\biggl{(}\frac{\lvert u(x)-u(y)\rvert}{\lvert x-y\rvert^{s}}\biggr{)}\frac{u(x)-u(y)}{\lvert u(x)-u(y)\rvert}K(x,y)\frac{dy}{\lvert x-y\rvert^{s}} and g = G ′ g=G^{\prime} for some Young function 𝐺. This note imparts a Hopf type lemma and strong minimum principle for 𝑢 when c ⁢ ( x ) c(x) is continuous in Ω ¯ \overline{\Omega} that extend the results of Del Pezzo and Quaas [A Hopf’s lemma and a strong minimum principle for the fractional 𝑝-Laplacian, J. Differential Equations 263 (2017), 1, 765–778] in fractional Orlicz–Sobolev setting.

In this article, we study the asymptotic behavior and decay of the solution of the fully discrete heat problem. We show basic properties of its solutions, such as the mass conservation principle and their moments, and we compare them to the known ones for the continuous analogue problems. We present the fundamental solution, which is given in terms of spherical harmonics, and we state pointwise and ℓ p {\ell^{p}} estimates for that. Such considerations allow to prove decay and large-time behavior results for the solutions of the fully discrete heat problem, giving the corresponding rates of convergence on ℓ p {\ell^{p}} spaces.

Let W ⁢ ( n ) {W(n)} be the Jacobson–Witt algebra over algebraic closed field 𝕂 {\mathbb{K}} with characteristic p > 2 {p>2} . In [K. Ou and B. Shu, Borel subalgebras of restricted Cartan-type Lie algebras, J. Algebra Appl. 21 2022, 11, Paper No. 2250210], we introduced the so-called B-subalgebra of W ⁢ ( n ) {W(n)} , which serves as an analog of the Borel subalgebra of classical Lie algebras. As a sequel, we describe the structure of the variety ℬ {\mathcal{B}} consisting of all B-subalgebras of W ⁢ ( n ) {W(n)} in this paper. This variety presents an analog of the flag variety for classical Lie algebras. It is shown that ℬ {\mathcal{B}} is related to the variety of all full flags in 𝕂 n + 1 {\mathbb{K}^{n+1}} . Additionally, we provide a detailed description of the varieties for W ⁢ ( 1 ) {W(1)} as an illustrative example. With the above setting-up, one may establish the Springer theory and geometric representations for the Jacobson–Witt algebras.

In the first part of the paper, we characterize certain systems of first-order nonlinear differential equations whose space of solutions is an s ⁢ l 2 ⁢ ( C ) \mathfrak{sl}_{2}(\mathbb{C}) -module. We prove that such systems, called Ramanujan systems of Rankin–Cohen type , have a special shape and are precisely the ones whose solution space admits a Rankin–Cohen structure. In the second part of the paper, we consider triangle groups Δ ⁢ ( n , m , ∞ ) \Delta(n,m,\infty) . By means of modular embeddings, we associate to every such group a number of systems of nonlinear ODEs whose solutions are algebraically independent twisted modular forms. In particular, all rational weight modular forms on Δ ⁢ ( n , m , ∞ ) \Delta(n,m,\infty) are generated by the solutions of one such system (which is of Rankin–Cohen type). As a corollary, we find new relations for the Gauss hypergeometric function evaluated at functions on the upper half-plane. To demonstrate the power of our approach in the non-classical setting, we construct the space of integral weight twisted modular form on Δ ⁢ ( 2 , 5 , ∞ ) \Delta(2,5,\infty) from solutions of systems of nonlinear ODEs.

Skew-braces are ring-like objects arising in connection with Hopf–Galois theory and set-theoretic solutions 𝑆 to the Yang–Baxter equation. Interactions between skew-braces are often related to 𝑞-braces. For example, every 𝑞-brace 𝐴 is given by a pair of skew-braces which induces a ℤ-indexed sequence of skew-braces. The sequence collapses if 𝐴 itself is a skew-brace. The free group over the underlying set of a solution 𝑆 is a 𝑞-brace. Bi-crossed products of skew-braces are shown to be 𝑞-braces, and criteria are developed when they are skew-braces. Two classes of skew-braces are put into a bijective correspondence with special 𝑞-braces, which themselves need not be skew-braces. Characterizations of 𝑞-braces, including one in terms of a graph and one as special modules over skew-braces, are given.

Let F be an algebraically closed field and let R be a locally finite algebra over F . This paper aims to show that any element of R is a product of at most three unipotent elements from R if and only if the element lies in the first derived subgroup of the unit group of R . In addition, this necessary and sufficient condition is applied to twisted group algebras of locally finite groups over a field of either zero characteristic or characteristic p ≠ 0 {p\not=0} for some prime p . Moreover, we explore some crucial properties satisfied by certain algebras like the conditions concerned with the connections between unipotent elements of index 2 and commutators, as well as we investigate the unipotent radical of some subgroups of a finite-dimensional algebra R over a field F with at least four elements. In particular, we again apply these results to twisted group algebras.

Let μ M , D {\mu_{M,D}} be a self-similar measure generated by an n × n {n\times n} expanding real matrix M = ρ - 1 ⁢ I {M=\rho^{-1}I} and a finite digit set D ⊂ ℤ n {D\subset{\mathbb{Z}}^{n}} , where 0 < | ρ | < 1 {0<\lvert\rho\rvert<1} and I is an n × n {n\times n} unit matrix. In this paper, we study the existence of a Fourier basis for L 2 ⁢ ( μ M , D ) {L^{2}(\mu_{M,D})} , i.e., we find a discrete set Λ such that E Λ = { e 2 ⁢ π ⁢ i ⁢ 〈 λ , x 〉 : λ ∈ Λ } {E_{\Lambda}=\{e^{2\pi i\langle\lambda,x\rangle}:\lambda\in\Lambda\}} is an orthonormal basis for L 2 ⁢ ( μ M , D ) {L^{2}(\mu_{M,D})} . Under some suitable conditions for D , some necessary and sufficient conditions for L 2 ⁢ ( μ M , D ) {L^{2}(\mu_{M,D})} to admit infinite orthogonal exponential functions are given. Then we set up a framework to obtain necessary and sufficient conditions for L 2 ⁢ ( μ M , D ) {L^{2}(\mu_{M,D})} to have a Fourier basis. Finally, we demonstrate how these results can be applied to self-similar measures.

In this paper, we consider the solvability over non-negative integers of certain Diophantine equations coming from representations of integers as sums of pentagonal numbers (counting the number of dots in a regular pentagon). We study a general method to obtain generalized versions of Cauchy’s lemma. Using this, we show the “pentagonal theorem of 63”, which states that a sum of pentagonal numbers represents every non-negative integer if and only if it represents the integers 1 , 2 , 3 , 4 , 6 , 7 , 8 , 9 , 11 , 13 , 14 , 17 , 18 , 19 , 23 , 28 , 31 , 33 , 34 , 39 , 42 , 63 . 1,~{}2,~{}3,~{}4,~{}6,~{}7,~{}8,~{}9,~{}11,~{}13,~{}14,~{}17,~{}18,~{}19,~{}23% ,~{}28,~{}31,~{}33,~{}34,~{}39,~{}42,~{}63. We further show that these integers form a unique minimal universality criterion set.

This paper continues our previous program to study the restriction estimates in a class of conical singular spaces X = C ⁢ ( Y ) = ( 0 , ∞ ) r × Y {X=C(Y)=(0,\infty)_{r}\times Y} equipped with the metric g = d ⁢ r 2 + r 2 ⁢ h {g=\mathrm{d}r^{2}+r^{2}h} , where the cross section Y is a compact ( n - 1 ) {(n-1)} -dimensional closed Riemannian manifold ( Y , h ) {(Y,h)} . Assuming the initial data possesses additional regularity in the angular variable θ ∈ Y {\theta\in Y} , we prove some linear restriction estimates for the solutions of Schrödinger equations on the cone X . The smallest positive eigenvalue of the operator Δ h + V 0 + ( n - 2 ) 2 / 4 {\Delta_{h}+V_{0}+(n-2)^{2}/4} plays an important role in the result. As applications, we prove local energy estimates and Keel–Smith–Sogge estimates for the Schrödinger equation in this setting.

We obtain sharp L p {L^{p}} - L q {L^{q}} estimates for fractional integrals generated by Radon transforms of the following three types: The classical Radon transform over the set of all hyperplanes in ℝ n {\mathbb{R}^{n}} , the Strichartz transversal transform over only those hyperplanes which meet the last coordinate axis, and the Radon transform associated with paraboloids. The method relies on a version of Stein’s interpolation theorem for analytic families of operators communicated by L. Grafakos.