Let $\mathcal{𝒜}$ and $\mathcal{ℬ}$ be two von Neumann algebras. For $A,B\in \mathcal{𝒜}$, define $A•B=A^{\ast }B+B^{\ast }A$ and $A\circ B=A^{\ast }B-B^{\ast }A$ to be the new products of $A$ and $B$. Suppose that a bijective map $\mathrm{\Phi }:\mathcal{𝒜}\to \mathcal{ℬ}$ satisfies $\mathrm{\Phi }(A•B\circ C)=\mathrm{\Phi }(A)•\mathrm{\Phi }(B)\circ \mathrm{\Phi }(C)$ for all $A,B,C\in \mathcal{𝒜}$. In this paper, it is proved that if $\mathcal{𝒜}$ and $\mathcal{ℬ}$ are two von Neumann algebras with no central abelian projections, then the map $\mathrm{\Phi }(I)\mathrm{\Phi }$ is the sum of a $\ast$-isomorphism and a conjugate linear $\ast$-isomorphism, where $\mathrm{\Phi }(I)$ is a self-adjoint central element in $\mathcal{ℬ}$ with $\mathrm{\Phi }{(I)}^2=I$.

We answer in the affirmative a question of Witno (2021) concerning elite primes for Mersenne numbers and repunits.

Let $K=\mathbb{Q}(𝜃)$ be a number field generated by a complex root $𝜃$ of a monic irreducible trinomial $F(x)=x^n+a{x}^m+b\in \mathbb{Z}[x]$. In this paper, we deal with the problem of the nonmonogenity of $K$. More precisely, we provide some explicit conditions on $a$, $b$, $n$, and $m$ for which $K$ is not monogenic. As an application, we show that there are infinite families of nonmonogenic number fields defined by trinomials of degrees $n=s\cdot 2^r\cdot 3^k$, where $s$, $r$, and $k$ are positive integers. Finally, we illustrate our results by giving some examples.

Based on a global continuation theorem by Manásevich and Mawhin, we consider the existence of a positive periodic solution for a generalized Lazer–Solimini equation with indefinite weights. Our new results are applicable to a weak singularity, as well as a strong singularity. We generalize and improve results from the literature. Moreover, we give the existence interval for a positive periodic solution of this equation.

If $(X,{\le }_X)$ is a partially ordered set satisfying certain necessary conditions for $X$ to be order-isomorphic to the spectrum of a Noetherian domain of dimension two, we describe a new poset $(\text{Str\hspace{0.17em}}X,{\le }_{\text{Str\hspace{0.17em}}X})$ that completely determines $X$ up to isomorphism. The order relation ${\le }_{\text{Str\hspace{0.17em}}X}$ imposed on $\text{Str\hspace{0.17em}}X$ is modeled after R. Wiegand’s well-known “P5” condition that can be used to determine when a given partially ordered set $(U,{\le }_U)$ of a certain type is order-isomorphic to $(\text{Spec\hspace{0.17em}}\mathbb{Z}[x],\subseteq )$.

We construct a set of explicit differential operators evaluated at a singular point on a quadratic hypersurface which gives a divisibility criterion. This result is then applied to collect interpolation conditions at distinct points to get new regular Hermite interpolation schemes in ${ℝ}^n$.

Motivated by questions arising in the study of the spectral theory of models of aperiodic order, we investigate sums of functions of semibounded closed subsets of the real line. We show that under suitable thickness assumptions on the sets and growth assumptions on the functions, the sums of such sets contain half-lines. We also give examples to show our criteria are sharp in suitable regimes.

We look at symbolic substitutions and their relationship to Bratteli diagrams and their associated operator algebras. In particular, we consider the equivalence relation on substitutions induced by telescope equivalence of Bratteli diagrams. Such an equivalence preserves pure aperiodicity and primitivity but fails to preserve rank, order, and number of letters. In a similar manner, we consider the equivalence relation on substitutions induced by telescope equivalence of ordered Bratteli diagrams. This results in a finer equivalence but fails to provide a complete invariant. An application to Fibonacci-like substitutions is developed.

We present several new asymptotic expansions of the logarithm of the Barnes $G$-function in terms of the gamma function and the polygamma functions. We also give several explicit formulas and recurrence relations for determining the coefficients in these asymptotic expansions. By using the results obtained, we derive recursion formulas for the Bernoulli numbers and the corresponding results for the Riemann zeta function. Finally, based on the asymptotic expansions which we have presented here, we prove new complete monotonicity properties of some functions involving the Barnes $G$-function, the gamma function and the polygamma functions.

Let $K$ be an algebraically closed field of characteristic 0. For $f,g\in K[x,y]$, when the Jacobian $(\partial f∕\partial x)(\partial g∕\partial y)-(\partial g∕\partial x)(\partial f∕\partial y)$ is a constant, Magnus’ formula describes the relations between the homogeneous degree pieces $f_i$ and $g_i$. We show a more general version of Magnus’ formula, which could provide a potentially useful tool to prove the Jacobian conjecture.

Let $\alpha$ and $\beta$ be real numbers. The corresponding Beatty sequence is defined by

$${\mathcal{ℬ}}_{\alpha ,\beta }=\{\lfloor \alpha n+\beta \rfloor :n\in \mathbb{N}\}.$$

We estimate Kloosterman sums with Beatty sequences of the shape

$$\sum _{\begin{array}{c}1\le m\le N\\ (m,q)=1\\ m\in {\mathcal{ℬ}}_{\alpha ,\beta }\end{array}}{e}_q(rm+s\overline{m}),$$

where $q$ is an integer. When $q=p$ is a prime, bounds for exponential sums with rational functions along Beatty sequences are also obtained. Furthermore, bounds for $\text{max}\{m,\tilde{m}\}$ subject to $m,\tilde{m}\in \mathbb{Z}\cap [1,p)$, $c$ indivisible by $p$, $m\tilde{m}\equiv c(\text{mod\hspace{0.17em}}p)$ and $m$ belonging to some fixed Beatty sequence are improved.

Based on the finite element method, a fully discrete Crank–Nicolson/leapfrog scheme is proposed and studied for the Landau–Lifshitz equation. The scheme avoids stiff time step restrictions and requires that the time step only be bounded by a constant independent of the mesh size. Then, almost unconditional optimal error estimates for the discrete scheme are obtained by using a temporal-spatial error splitting technique. The theoretical results are verified by numerical experiments.

We establish the boundedness of the bilinear $𝜃$-type generalized fractional integral and its commutator on nonhomogeneous metric measure space. Under the assumption that the functions $\mathrm{\Phi }$ and $\lambda$ satisfy certain conditions, we prove that the bilinear $𝜃$-type generalized fractional integral $B{\tilde{T}}_{𝜃,\alpha }$ is bounded from the product of generalized Morrey spaces ${\mathcal{ℒ}}^{p_1,\mathrm{\Phi },\kappa }(\mu )\times {\mathcal{ℒ}}^{p_2,\mathrm{\Phi },\kappa }(\mu )$ into spaces ${\mathcal{ℒ}}^{q,{\mathrm{\Phi }}^{q∕p},\kappa }(\mu )$, and it is also bounded from the product of spaces ${\mathcal{ℒ}}^{p_1,\mathrm{\Phi },\kappa }(\mu )\times {\mathcal{ℒ}}^{p_2,\mathrm{\Phi },\kappa }(\mu )$ into generalized weak Morrey spaces $W{\mathcal{ℒ}}^{1,{\mathrm{\Phi }}^{1∕p},\kappa }(\mu )$. Furthermore, the boundedness of the commutator $[b_1,b_2,B{\tilde{T}}_{𝜃,\alpha }]$ formed by $b_1,b_2\in \widetilde{\text{RBMO\hspace{0.17em}}}(\mu )$ and $B{\tilde{T}}_{𝜃,\alpha }$ on spaces ${\mathcal{ℒ}}^{q,{\mathrm{\Phi }}^{q∕p},\kappa }(\mu )$ and on spaces $W{\mathcal{ℒ}}^{1,{\mathrm{\Phi }}^{1∕p},\kappa }(\mu )$ is also obtained.

Let $n\ge 2$ be an integer and consider the defining ideal of the Fermat configuration of points in ${\mathrm{\mathbb{P}}}^2$: $I_n=\left(x(y^n-z^n),y(z^n-x^n),z(x^n-y^n)\right)\subset R=ℂ[x,y,z]$. We explicitly compute the least degree of generators (the initial degree) of its symbolic powers in all unknown cases. As direct applications, we verify Chudnovsky’s conjecture, Demailly’s conjecture and the Harbourne–Huneke containment problem, as well as calculate the Waldschmidt constant and (asymptotic) resurgence number.

Mixed-norm $\alpha$-modulation spaces were introduced recently by Cleanthous and Georgiadis (Trans. Amer. Math. Soc. 373:5 (2020), 3323–3356). The mixed-norm $\alpha$-modulation spaces $M_{\overrightarrow{p},q}^{s,\alpha }({ℝ}^n)$, $\alpha \in [0,1]$, form a family of smoothness spaces that contains the mixed-norm Besov spaces as special cases. We prove that a pseudodifferential operator $\sigma (x,D)$ with symbol in the Hörmander class $S_{\rho }^b$ extends to a bounded operator $\sigma (x,D):M_{\overrightarrow{p},q}^{s,\alpha }({ℝ}^n)\to M_{\overrightarrow{p},q}^{s-b,\alpha }({ℝ}^n)$, provided , $\overrightarrow{p}\in {(0,\infty )}^n$, and . This extends the known result that pseudodifferential operators, with symbol in the class $S_1^b$, maps the mixed-norm Besov space $B_{\overrightarrow{p},q}^s({ℝ}^n)$ into $B_{\overrightarrow{p},q}^{s-b}({ℝ}^n)$.

Using some recent fixed point theorems of the author (based on Brouwer’s fixed point theorem and continuous selections) we generalize previous results in the literature on the existence of maximal elements. Then, existence results for equilibria are presented for one-person games which will then generate, after some analysis, a variety of existence results for $N$-person games.

We show that an integral domain $R$ is an almost Dedekind domain if and only if $R$ is one-dimensional and every representable $R$-module is strongly representable.

In this article, our interest is to prove the existence of positive solutions to the semilinear elliptic problem

Here $\mathrm{\Omega }\subset {ℝ}^N$ ($N\ge 3$) is a bounded domain with $0\in \mathrm{\Omega }$, and . The function is in suitable Lebesgue spaces. The given problem is notable due to the presence of the variable exponent $\eta (x)$, and it has singularities on the boundary of the domain as well as at the origin.

For each odd prime $p$, we show that the orders of the $KU$-local homotopy groups of the mod $p$ Moore spectrum are equal to denominators of special values of certain quotients of Dedekind zeta-functions of totally real number fields. With this observation in hand, we give a cute topological proof of the Leopoldt conjecture for those number fields, by showing that it is a consequence of periodicity properties of $KU$-local stable homotopy groups.

Let $R$ be a ring with unity 1. We introduce the definition of the (dual) polarity along an element in $R$. Given any $a,d\in R$, an element $a$ is polar along $d$ if there exists some $p\in R$ such that $p^2=p\in \text{comm}(da)$, $pd=d$ and $da+1-p$ is a unit. Also, the dual polarity along an element is given. Simply polar elements, polar elements and well-supported elements are special cases of polar elements along $d$. Moreover, several properties and characterizations of the polarity along an element are investigated.

Several upper estimates for the numerical radius of Hilbert space operators are given. For a bounded linear operator $T$ on a complex Hilbert space $\mathcal{\mathscr{H}}$ and $0\le \alpha \le 1$, with the polar decomposition $T=U\left|T\right|$, it is shown that

$$\omega (T)\le \frac{1}{2}\parallel T\parallel +\frac{1}{4}\left(\parallel {\left|T\right|}^{2\alpha }+{\left|T\right|}^{2(1-\alpha )}\parallel +\omega ({\tilde{T}}_{\alpha })\right),$$

where ${\tilde{T}}_{\alpha }={\left|T\right|}^{\alpha }U{\left|T\right|}^{1-\alpha }$ is the generalized Aluthge transformation of $T$. Related norm inequalities are also obtained.

The generalized central trinomial coefficient $T_n(b,c)$ introduced by Noe is defined as the coefficient of $x^n$ in the expansion of ${(x^2+bx+c)}^n$, where $n\in \mathbb{N}=\{0,1,2,\dots \}$ and $b,c\in \mathbb{Z}$. Let $p$ be an odd prime. In this paper, we determine ${\sum }_{k=0}^{p-1}k{T}_k{(b,c)}^2∕m^k$ modulo $p^2$ for any integer $m$ satisfying the equation ${(m-d)}^2=16mc$, where $d=b^2-4c$. As applications, we prove that for any prime , we have

$$\sum _{n=0}^{p-1}n{\left(\sum _{k=0}^n\frac{1}{2^k}\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)\left(\genfrac{}{}{0.0pt}{}{2k}{k}\right)\right)}^2\equiv \frac{3}{4}\left(\frac{3}{p}\right)p-\left(\frac{-1}{p}\right)(\text{mod}{p}^2)$$

and

$$\sum _{n=0}^{p-1}n{\left(\sum _{k=0}^n\frac{1}{{(-6)}^k}\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)\left(\genfrac{}{}{0.0pt}{}{2k}{k}\right)\right)}^2\equiv -\frac{1}{4}\left(\frac{3}{p}\right)p(\text{mod}{p}^2),$$

as conjectured by Sun (Finite Fields Appl. 46 (2017), 179–216), where $\left(\frac{\cdot }{p}\right)$ stands for the Legendre symbol.

Withers and Nadarajah (2022, to appear) gave solutions to univariate nonlinear recurrence equations. We consider the vector case. Let $\mathcal{𝒞}$ denote the complex numbers. Let $F(z):{\mathcal{𝒞}}^q\to {\mathcal{𝒞}}^q$ be any analytic function. Let $w\in {\mathcal{𝒞}}^q$ be any fixed point of $F(z)$, that is, $F(w)=w$. Set $\dot{F}(z)=dF(z)∕d{z}^{\prime }\in {\mathcal{𝒞}}^{q\times q}$. Then for any eigenvalue $r$ of $\dot{F}(w)$, the recurrence equation

$$z_{n+1}=F(z_n)\in {\mathcal{𝒞}}^q,$$

for $n=0,1,2,\dots$, has a solution of the form

$$z_n=z_n(w,\alpha r^n)=w+\sum _{i=1}^{\infty }{a}_i(w){(\alpha r^n)}^i,$$

where $\alpha \in \mathcal{𝒞}$ is arbitrary and $a_i(w)\in {\mathcal{𝒞}}^q$ are given by recurrence.

The stability problem for the magnetohydrodynamic equations with partial or no dissipation has evoked a considerable interest in recent years. In this paper, we establish the stability near magnetic hydrostatic equilibrium to the three-dimensional magnetic Bénard fluid equations with the mixed partial dissipation involving viscosity, resistivity and heat conduction. Moreover, we obtain the large-time behavior of the corresponding linearized system. Our result mathematically verifies the stabilization of a background magnetic field on the three-dimensional magnetic Bénard fluid.