The set of factorizations of permutations into $m$ transpositions of some symmetric group ${\mathcal{𝒮}}_n$ is naturally in bijection with the set of graphs of order $n$ and size $m$ with both edges and vertices labeled. We define a notion of duality (the mind-body duality) for factorizations and such labeled graphs and interpret it in terms of properly embedded graphs, a class of graphs embedded in a bounded compact oriented surface with all the vertices lying in the boundary, and show a close connection of this duality with the Hurwitz action of the braid group. Connections with the theory of cellularly embedded graphs are highlighted and hints of possible applications are given. In this paper we focus on developing the necessary theory, leaving specific applications and further developments for future projects.

Let $k\ge 2$ be an integer and let $P(X)$ be a monic polynomial of $\mathbb{Z}[X]$ with degree $k-1$. We say that $P(X)$ is a $G$-polynomial if for each $n\in \{1,2,\dots ,k\}$, $P(X)$ divides $P(X^n)$ in the ring $\mathbb{Z}[X]$ if and only if $\text{gcd }(n,k)=1$. We present several approaches on finding necessary and sufficient conditions so that $P(X)$ is a $G$-polynomial. Among other interesting results, we show that $A_k(X):=X^{k-1}+X^{k-2}+\cdots X+1$ is the unique $G$-polynomial of degree $k-1$ if and only if $k$ is a prime number.

Let $n_0,n_1,\dots ,n_p$ be a sequence of positive integers such that , $\text{gcd }(n_0,n_1,\text{… },n_p)=1$. Let $S=⟨(0,n_p),(n_0,n_p-n_0),\dots ,(n_{p-1},n_p-n_{p-1}),(n_p,0)⟩$ be an affine semigroup in ${\mathbb{N}}^2$. The semigroup ring $k[S]$ is the coordinate ring of the projective monomial curve in the projective space ${\mathbb{P}}_k^{p+1}$, which is defined parametrically by

$$x_0=v^{n_p},x_1=u^{n_0}{v}^{n_p-n_0},\dots ,x_p=u^{n_{p-1}}{v}^{n_p-n_{p-1}},x_{p+1}=u^{n_p}.$$

In this article, we consider the case when $n_0,n_1,\dots ,n_p$ forms an arithmetic sequence, and give an explicit set of minimal generators for the derivation module ${\text{Der }}_k(k[S])$. Further, we give an explicit formula for the Hilbert–Kunz multiplicity of the coordinate ring of a projective monomial curve.

Given an $m$-isometric Hilbert space operator $A\in \mathcal{ℬ}(\mathcal{\mathscr{H}})$, ${△}_{A^{\ast },A}^m(I)={\sum }_{j=0}^m{(-1)}^j\left(\genfrac{}{}{0.0pt}{}{m}{j}\right)A^{\ast j}{A}^j=0$, with polar decomposition $A=U\left|A\right|$, the Aluthge transform $Ã=\left|A{|}^{1∕2}U\right|A{|}^{1∕2}$ preserves almost all the spectral properties of $A$. However, the $m$-isometric property of an operator neither implies nor is implied by the $m$-isometric property of its Aluthge transform. The operator $A$ has an $\left|A\right|$-adjoint $\mathcal{𝒜}$, ${\mathcal{𝒜}}^{\ast }={[A]}^{\ast }=U^{\ast }\left|A\right|$. If $A_i$, $i=1,2$, doubly commute and ${Ã}_i$ (resp. ${\mathcal{𝒜}}_i$) is strict $m_i$-isometric, then $\widetilde{A_1{A}_2}$ (resp. ${\mathcal{𝒜}}_1{\mathcal{𝒜}}_2$) is strict $(m_1+m_2-1)$-isometric. The converse fails for products $A_1{A}_2$, ${Ã}_1{Ã}_2$ and ${\mathcal{𝒜}}_1{\mathcal{𝒜}}_2$, but has a positive answer for tensor products $A_1\otimes A_2$, ${Ã}_1\otimes {Ã}_2$, ${\mathcal{𝒜}}_1\otimes {\mathcal{𝒜}}_2$ (and their Hilbert–Schmidt class identifications with the elementary operators $L_{A_1}{R}_{A_2^{\ast }}$, $L_{{Ã}_1}{R}_{{Ã}_2^{\ast }}$ and $L_{{\mathcal{𝒜}}_1}{R}_{A_2^{\ast }}$); if $S\otimes T$, where $S\otimes T$ stands for either of the three tensor products above, is strict $m$-isometric, then there exist scalars $c$ and $d$, $\left|cd\right|=1$, and positive integers $m_1$ and $m_2$, $m=m_1+m_2-1$, such that $cS$ is strict $m_1$-isometric and $dT$ is strict $m_2$-isometric.

The existence of the Gorenstein projective precovers over arbitrary rings is an open question. It is known that if the ring has finite Gorenstein global dimension, then every module has a Gorenstein projective precover. We prove here a “reduction” property—we show that, over any ring, it suffices to consider finitely presented modules: if there exists a nonnegative integer $n$ such that every finitely presented module has Gorenstein projective dimension $\le n$, then the class of Gorenstein projective modules is special precovering.

A positive integer is a $b$-Niven number (or $b$-harshad number) if it is a multiple of the sum of the digits of its base-$b$ representation. For each base $b\ge 2$, the maximum length of a sequence of consecutive $b$-Niven numbers is known to be $2b$. We investigate the maximum lengths of arithmetic progressions of $b$-Niven numbers.

Let $X$ be a Tychonoff space, $Y$ be a metrizable one and $C(X,Y)$ be the space of continuous functions from $X$ to $Y$. It is a classical result that two compatible metrics on $Y$ generate the same topologies of uniform convergence on compacta on $C(X,Y)$. We extend the result for the space $U(X,Y)$ of upper semicontinuous nonempty compact-valued maps from $X$ to $Y$. We also present characterizations of uniform equivalence of metrics via uniform convergence on functional spaces, as well as functional characterizations of locally compact and $k$-spaces. We give a partial answer to a question posed in Topological properties of spaces of continuous functions (1988), after Example 1.2.7, when two compatible metrics on $Y$ generate the same topologies of uniform convergence on $C(X,Y)$.

A. L. Shields (1974) proved a well-known theorem for the similarity of unilateral weighted shift operators. By using the generalization of this theorem for multivariable weighted shifts and the curvature of holomorphic bundles, we give a necessary and sufficient condition for the similarity of $m$-tuples in the Cowen–Douglas class. We also present a necessary condition for commuting $m$-tuples of backward weighted shift operators to be $n$-hypercontractive in terms of the weight sequences.

Given a continuum $X$ and $n\in \mathbb{N}$, let $C_n(X)$ (resp., $F_n(X)$) be the hyperspace of nonempty closed sets with at most $n$ components (resp., $n$ points). Let $S^1$ denote the unit circle in the plane. Given $1\le m\le n$, we consider the quotient space $C_n(S^1)∕F_m(S^1)$. The homogeneity degree of $X$, $\text{hd}(X)$, is the number of orbits of the group of homeomorphisms of $X$. We discuss the known models for the hyperspaces of $S^1$, we construct a new model for a hyperspace of $S^1$ by proving that $C_2(S^1)∕F_2(S^1)$ is homeomorphic to the topological suspension of a solid torus, and we show that $\text{hd}(C_2(X)∕F_2(X))=3$ and $\text{hd}(C_2(X)∕F_1(X))=4$.

We consider the two Hecke groups $G_4$ and $G_6$ generated by the transformations $S$ and $T$ defined by $S(z)=z+{\lambda }_m$ and $T(z)=-1∕z$ where ${\lambda }_m=2\text{ cos}(\pi ∕m)$ with $m\in \{4,6\}$. We give a full characterization of purely periodic Rosen continued fractions over $G_4$ and $G_6$. Finally, we end by finding a family of examples of purely periodic Rosen expansions of period length two and some related examples.

In a ring with involution, we first investigate some necessary and sufficient conditions under which Jacobson’s lemma for weak core inverses holds true. Then, we present reverse order laws of weak core inverses and some equivalent conditions under which absorption laws of weak core inverses hold true. Finally, some equivalent characterizations of $a^{\ast }$ commuting with the weak core inverse of $a$ are shown, which improve the relevant result of Zhou et al. (Weak group inverses and partial isometries in proper $\ast$-rings, Linear Multilinear Algebra 70:21 (2022), 6876–6890).

In a recent paper Garoufalidis and Reid constructed pairs of $1$-cusped hyperbolic $3$-manifolds which are isospectral but not isometric. We extend this work to the multicusped setting by constructing isospectral but not isometric hyperbolic $3$-manifolds with arbitrarily many cusps. The manifolds we construct have the same Eisenstein series, the same infinite discrete spectrum and the same complex length spectrum. Our construction makes crucial use of Sunada’s method and the strong approximation theorem of Nori and Weisfeiler.

Let $n\in \mathbb{N}$ and $\alpha \in (n-1,n)$. Under uniqueness and continuity conditions, we show that solutions to Caputo fractional differential equations depend continuously upon the boundary conditions. We also provide sufficient conditions on interval length between the boundary points in a specific case for $\alpha \in (1,2)$.

Our work is devoted to the construction of real valued noncommutative $L_p$ spaces associated with real $W^{\ast }$-algebras of different types. We construct such real noncommutative $L_p$-spaces, , and we prove the theorem of isomorphism for real noncommutative $L_p$-spaces. Finally, we build an approximation of $L_p$-spaces associated with real $W^{\ast }$-algebras of type $III$ via $L_p$-spaces associated with finite real $W^{\ast }$-algebras, analogously to the work of U. Haagerup, M. Junge and Q. Xu (Trans. Amer. Math. Soc. 362:4 (2010), 2125–2165).

An ideal $I$ in a Noetherian ring $R$ is called of linear type if the Rees algebra of $I$ is isomorphic to the symmetric algebra of $I$. We prove that the Jacobian ideal of any reduced plane curve with singular points of multiplicity $2$ is of linear type. We characterize plane curves with singular points of multiplicity $3$ whose Jacobian ideal is of linear type.

We focus on the nonlinear Lane–Emden-type delay problems connected with $g$-Caputo fractional derivatives. Here, we investigate the problems under the assumption that the source functions have time-singular coefficients. Via the Bielecki-type norm, we derive the existence and uniqueness of mild solutions for the problem. In a special case, by subdividing the interval of time, we obtain a unique mild solution for the problem under a weaker condition than the previous one. Besides, we propose and discuss a new Ulam–Hyers-type stability for the main equation. Meanwhile, a new inequality is established to prove the main results of the paper. Some examples are provided to illustrate the theoretical results.

We prove the existence and uniqueness results for a class of implicit fractional differential equations involving the combined Caputo fractional derivative with advanced arguments by using the fixed point theorems of Banach and nonlinear alternative of Leray and Schauder. We will also establish the Ulam stability and give some examples to show the applicability of our results.

Bremner and Macleod (Ann. Math. Inform. 43 (2014), 29–41) showed that for all odd positive integers $n$, the equation

$$n=\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}$$

has no solutions in the positive integers. We extend this theorem to the equation

$$2n{a}^2{b}^2{c}^2-a^2-b^2-c^2=\frac{a^{2}x}{y+z}+\frac{b^{2}y}{z+x}+\frac{c^{2}z}{x+y},$$

where $a,b,c\in \mathbb{Z}-\{0\}$ and $n,x,y,z\in {\mathbb{Z}}^{+}$. Furthermore, we show that the insolubility of this equation (under some conditions on $a,b,c,n$) can be explained by a Brauer–Manin obstruction to weak approximation for an elliptic curve model of the defining equation.

We investigate the right Hadamard fractional differences. First, a $Q$-operator is defined for continuous Hadamard fractional calculus. It shows that the left and right Hadamard operators satisfy a dual identity. Then, the $Q$-operator is extended to define the right Hadamard fractional sum and difference and present their properties. A terminal value problem is discussed by use of the right fractional difference. Finally, the related summation by parts formulas are obtained. It can be concluded that the right Hadamard fractional sum and difference are well defined.

We focus on the energy equality for axisymmetric weak solutions of the 3D Navier–Stokes equations. The classical Shinbrot condition says that if the weak solution $u$ of the Navier–Stokes equations belongs to $L^q(0,T;L^p({ℝ}^3))$ with $\frac{1}{q}+\frac{1}{p}=\frac{1}{2}$ and $p\ge 4$, then $u$ must satisfy the energy equality. A novel point is that, for the axisymmetric Navier–Stokes equations, the Shinbrot condition can be relaxed as follows: if $ũ=u^r{e}_r+u^z{e}_z\in L^q(0,T;L^p({ℝ}^3))$ with $\frac{1}{q}+\frac{1}{p}=\frac{1}{2}$ and $p\ge 4$, then $u$ must satisfy the energy equality. Some other interesting results will also be obtained.

Let $\mathcal{𝒜}$ be a unital $\ast$-algebra. It is shown, under some mild conditions on $\mathcal{𝒜}$, that a map $\mathrm{\Phi }:\mathcal{𝒜}\to \mathcal{𝒜}$ is the second nonlinear mixed Jordan triple $\ast$-derivation if and only if $\mathrm{\Phi }$ is an additive $\ast$-derivation. In particular, we apply the above result to prime $\ast$-algebras, von Neumann algebras with no central summands of type $I_1$,factor von Neumann algebras and standard operator algebras.

We provide a variant of Baer’s theorem about isomorphism of endomorphism rings of vector spaces of division rings, where the full endomorphism rings are replaced by some subrings of finitary maps.