In the present study, we investigate a new type of warped products on manifolds with indefinite metrics, namely, warped product lightlike submanifolds of indefinite Kaehler manifolds with a slant factor. First, we show that indefinite Kaehler manifolds do not admit any proper warped product semi-slant lightlike submanifolds of the type $N_{T}\times_{\lambda}N_{\theta}$, $N_{\theta}\times_{\lambda}N_{T}$, $N_{\perp}\times_{\lambda}N_{\theta}$ and $N_{\theta}\times_{\lambda}N_{\perp}$, where $N_{T}$ is a holomorphic submanifold, $N_{\perp}$ is a totally real submanifold and $N_{\theta}$ is a proper slant submanifold. Then, we study warped product semi-slant lightlike submanifolds of the type $B\times_{\lambda}N_{\theta}$, where $B = N_{T}\times N_{\perp}$, of an indefinite Kaehler manifold. Following this, we give one non-trivial example for this kind of warped products of indefinite Kaehler manifolds. Then, we establish a geometric estimate for the squared norm of the second fundamental form involving the Hessian of warping function $\lambda$ for this class of warped products. Finally, we present a sharp geometric inequality for the squared norm of second fundamental form of warped product semi-slant lightlike submanifolds of the type $B\times_{\lambda}N_{\theta}$.

The copointed liftings of the Fomin–Kirillov algebra $\mathcal{FK}_3$ over the algebra of functions on the symmetric group $\mathbb{S}_3$ were classified by Andruskiewitsch and the author. We demonstrate here that those associated to a generic parameter are Morita equivalent to the regular blocks of well-known Hopf algebras: the Drinfeld doubles of the Taft algebras and the small quantum groups $u_{q}(\mathfrak{sl}_2)$. The indecomposable modules over these were classified independently by Chen, Chari–Premet and Suter. Consequently, we obtain the indecomposable modules over the generic liftings of $\mathcal{FK}_3$. We decompose the tensor products between them into the direct sum of indecomposable modules. We then deduce a presentation by generators and relations of the Green ring.

We obtain a necessary and sufficient condition on the Haar coefficients of a real function $f$ defined on $\mathbb{R}^+$ for the Lipschitz $\alpha$ regularity of $f$ with respect to the ultrametric $\delta(x,y)=\inf \{|I| : x, y\in I; I\in\mathcal{D}\}$, where $\mathcal{D}$ is the family of all dyadic intervals in $\mathbb{R}^+$ and $\alpha$ is positive. Precisely, $f\in \mathrm{Lip}_\delta(\alpha)$ if and only if ${\vert\langle{f}{h^j_k}\rangle\vert}\leq C 2^{-(\alpha + 1/2)j}$ for some constant $C$, every $j\in\mathbb{Z}$ and every $k=0,1,2,\ldots$ Here, as usual, $h^j_k(x)= 2^{j/2}h(2^jx-k)$ and $h(x)=\mathcal{X}_{[0,1/2)}(x)-\mathcal{X}_{[1/2,1)}(x)$.

Flips of triangulations appear in the definition of cluster algebras by Fomin and Zelevinsky. In this article we give an interpretation of mutation in the sense of permutation using triangulations of a convex polygon. We thus establish a link between cluster variables and permutation mutations in the case of cluster algebras of type $\mathbb{A}$.

In this brief note we aim to provide, through a well-known class of singular densities in harmonic analysis, a simple approach to the fact that the homogeneity of the universe on scales of the order of a hundred million light years is entirely compatible with the fine- scale condensation of matter and energy. We give precise and quantitative definitions of homogeneity and isotropy on large scales. Then we show that Muckenhoupt densities have the ingredients required for a model of the large-scale homogeneity and the fine-scale condensation of the universe. In particular, these densities can take locally infinitely large values (black holes) and, at the same time, they are independent of location at large scales. We also show some locally singular densities that satisfy the large-scale isotropy property.

The edge-intersection graph of a family of paths on a host tree is called an EPT graph. When the host tree has maximum degree $h$, we say that the graph is $[h,2,2]$. If the host tree also satisfies being a star, we have the corresponding classes of EPT-star and $[h,2,2]$-star graphs. In this paper, we prove that $[4,2,2]$-star graphs are $2$-clique colorable, we find other classes of EPT-star graphs that are also $2$-clique colorable, and we study the values of $h$ such that the class $[h,2,2]$-star is $3$-clique colorable. If a graph belongs to $[4,2,2]$ or $[5,2,2]$, we prove that it is $3$-clique colorable, even when the host tree is not a star. Moreover, we study some restrictions on the host trees to obtain subclasses that are $2$-clique colorable.

This paper investigates the existence of weak solutions to a fuel cell problem modeled by a boundary value problem (BVP) in the multiregion domain. The BVP consists of the coupled Stokes/Darcy-TEC (thermoelectrochemical) system of elliptic equations, with Beavers–Joseph–Saffman and regularized Butler–Volmer boundary conditions being prescribed on the interfaces, porous-fluid and membrane, respectively. The present model includes macrohomogeneous models for both hydrogen and methanol crossover. The novelty in the coupled Stokes/Darcy-TEC system lies in the presence of the Joule effect together with the quasilinear character given by (1) temperature dependence of the viscosities and the diffusion coefficients; (2) the concentration-temperature dependence of Dufour–Soret and Peltier–Seebeck cross-effect coefficients, and (3) the pressure dependence of the permeability. We derive quantitative estimates of the solutions to clarify smallness conditions on the data. We use fixed-point and compactness arguments based on the quantitative estimates of approximated solutions.

We consider an improved lopsided shift-splitting (ILSS) preconditioner for solving three- by-three block saddle point problems. This method enhances the work of Zhang et al. [Comput. Appl. Math. 41 (2022), 261]. We prove that the iteration method produced by the ILSS preconditioner is unconditionally convergent. Additionally, we show that all eigenvalues of the ILSS preconditioned matrix are real, with non-unit eigenvalues located in a positive interval. Numerical experiments demonstrate the effectiveness of the ILSS preconditioner.

The reconstruction problem for a multivalued linear operator (linear relation) $T$ is viewed as the exploration of some properties of $T$ from those of a restriction of $T$ on an invariant linear subspace.

In this paper, we are dedicated to studying the Schrödinger equations in the presence of a magnetic field. Based on variational methods, especially the mountain pass theorem, we obtain ground state solutions for the system under certain assumptions.

We present a generalization of a well-known domination parameter, the upper irredundance number, and address its associated optimization problem, namely the maximum weighted irredundant set (MWIS) problem, which models some service allocation problems. We establish a polynomial-time reduction to the maximum weighted stable set (MWSS) problem that we use to find graph classes for which the MWIS problem is polynomial, among other results. We formalize these results in the proof assistant Coq. This is mainly convenient in the case of some of them due to the structure of their proofs. We also present a heuristic and an integer programming formulation for the MWIS problem and check that the heuristic delivers good quality solutions through experimentation.

In this paper, we consider the Lusternik–Schnirelmann and geometric categories of finite spaces. We define new numerical invariants for these spaces derived from the geometric category and present an algorithmic approach for their effective computation. Our analysis combines homotopy-theoretic properties of these spaces with algorithms and tools from graph and hypergraph theory. We also provide several examples to illustrate our results.

We give closed formulas for determinant, permanent, and inverse of circulant matrices with three non-zero coefficients. The techniques that we use are related to digraphs associated with these matrices.

Let $R$ be a commutative ring and $M$ be an $R$-module, and let $I(R)^*$ be the set of all nontrivial ideals of $R$. The $M$-intersection graph of ideals of $R$, denoted by $G_M(R)$, is a graph with the vertex set $I(R)^*$, and two distinct vertices $I$ and $J$ are adjacent if and only if $IM\cap JM\neq 0$. In this note, we provide counterexamples for some results proved in a paper by Asir, Kumar, and Mehdi [Rev. Un. Mat. Argentina 63 (2022), no. 1, 93–107]. Also, we determine the girth of $G_M(R)$ and derive a necessary and sufficient condition for $G_M(R)$ to be weakly triangulated.

This paper deals with a fractional Nirenberg equation of order $\sigma\in (0, n/2)$, $n\geq2$. We study the compactness defect of the associated variational problem. We determine precise characterizations of critical points at infinity of the problem, through the construction of a suitable pseudo-gradient at infinity. Such a construction requires detailed asymptotic expansions of the associated energy functional and its gradient. This study will then be used to derive new existence results for the equation.

We give a complete classification of Ricci–Bourguignon solitons on real hypersurfaces in the complex projective space $\mathbb{C}P^n=SU_{n+1}/S(U_1 \cdot U_n)$. Next, as an application, we give some non-existence properties for gradient Ricci–Bourguignon solitons on real hypersurfaces with isometric Reeb flow and contact real hypersurfaces in the complex projective space $\mathbb{C}P^n$.

We consider central simple $K$-algebras which happen to be differential graded $K$-algebras. Two such algebras $A$ and $B$ are considered equivalent if there are bounded complexes of finite-dimensional $K$-vector spaces $C_A$ and $C_B$ such that the differential graded algebras $A\otimes_K \mathrm{End}_K^\bullet(C_A)$ and $B\otimes_K \mathrm{End}_K^\bullet(C_B)$ are isomorphic. Equivalence classes form an abelian group, which we call the dg Brauer group. We prove that this group is isomorphic to the ordinary Brauer group of the field $K$.

This paper describes a method for determining the list of reduced ideals of any pure cubic number field, which we can use for testing the principality of these fields and give a generator for a principal ideal.

Let $(G,H,\sigma)$ be a symmetric pair and $\mathfrak{g}=\mathfrak{m}\oplus\mathfrak{h}$ the canonical decomposition of the Lie algebra $\mathfrak{g}$ of $G$. We denote by $\nabla^0$ the canonical affine connection on the symmetric space $G/H$. A torsion-free $G$-invariant affine connection on $G/H$ is called special if it has the same curvature as $\nabla^0$. A special product on $\mathfrak{m}$ is a commutative, associative, and $\operatorname{Ad}(H)$-invariant product. We show that there is a one-to-one correspondence between the set of special affine connections on $G/H$ and the set of special products on $\mathfrak{m}$. We introduce a subclass of symmetric pairs, called strongly semi-simple, for which we prove that the canonical affine connection on $G/H$ is the only special affine connection, and we give many examples. We study a subclass of commutative, associative algebra which allows us to give examples of symmetric spaces with special affine connections. Finally, we compute the holonomy Lie algebra of special affine connections.

We give an example of a polynomial of degree 4 in 5 variables that is the sum of squares of 8 polynomials and cannot be decomposed as the sum of 7 squares. This improves the current existing lower bound of 7 polynomials for the Pythagoras number $p(5,4)$.

Let $\mathcal{K}(1)$ be the Lie superalgebra of contact vector fields on the $(1,1)$-dimensional complex superspace; it contains the Möbius superalgebra $\mathfrak{osp}(1|2)$. We classify $\mathfrak{osp}(1|2)$-invariant superanti-symmetric binary differential operators from $\mathcal{K}(1)\wedge\mathcal{K}(1)$ to $\mathfrak{D}_{\lambda,\mu}$ vanishing on $\mathfrak{osp}(1|2)$, where $\mathfrak{D}_{\lambda,\mu}$ is the superspace of linear differential operators acting on the superspaces of weighted densities. This result allows us to compute the second differential $\mathfrak{osp}(1|2)$-relative cohomology of $\mathcal{K}(1)$ with coefficients in $\mathfrak{D}_{\lambda,\mu}$.

We revisit the fact that the cocommutative comonoids in a symmetric monoidal category form the best possible approximation by a cartesian category, now considering the case where the original category is only braided monoidal. This leads to the question of when the endomorphism operad of a comonoid is a clone (a Lawvere theory). By giving an explicit example, we prove that this does not imply that the comonoid is cocommutative.

We prove that fusion categories of Frobenius–Perron dimensions 120 are of Frobenius type. Combining this with known results in the literature, we get that all weakly integral fusion categories of Frobenius–Perron dimension less than 126 are of Frobenius type.

We classify all the $\Delta$-coherent pairs of measures of the second kind on the real line. We obtain five cases, corresponding to all the families of discrete semiclassical orthogonal polynomials of class $s\leq1$.

A finite group $G$ is called an MNP-group if all maximal subgroups of the Sylow subgroups of $G$ are normal in $G$. The aim of this paper is to give a necessary and sufficient condition for a group to be an MNP-group, characterize the structure of finite groups whose maximal subgroups (respectively, maximal subgroups of even order) are all MNP-groups, and determine finite non-abelian simple groups whose second maximal subgroups (respectively, maximal subgroups of even order) are all MNP-groups.

The present paper is devoted to showing that on every compact, connected homogeneous isoparametric submanifold $M=G/K$ of codimension $h\geq2$ in a Euclidean space, there exist canonical distributions which are generated by the compact symmetric spaces associated to $M$ (i.e., those corresponding to the group $G$). The central objective is to show that all these distributions are bracket generating of step 2. To that end, formulae that complement those in the first article of this series (Rev. Un. Mat. Argentina 61, no. 1 (2020), 113–130) are obtained.

Consider a Brownian particle released from a fixed point $z_0$ in a region $\Omega$. The harmonic-measure distribution function, or $h$-function, $h(r)$, expresses the probability that the Brownian particle first hits the boundary $\partial\Omega$ of the region $\Omega$ within distance $r$ of $z_0$. In this paper, we compute the $h$-function of several new planar simply connected two-dimensional regions by using two different methods, both involving conformal maps. We also explain the asymptotic behaviour at certain values of $r$ where two different regimes meet. Moreover, for some regions, we examine how the behaviour of $h(r)$ changes when part of the boundary changes.

The focus of the paper is on the study of the moduli space of left-invariant pseudo- Riemannian metrics on the cotangent bundle of the Heisenberg group. We use algebraic approach to obtain orbits of the automorphism group acting in a natural way on the space of left invariant metrics. However, geometric tools such as the classification of hyperbolic plane conics are often needed. For the metrics obtained by the classification, we study geometric properties: curvature, Ricci tensor, sectional curvature, holonomy, and parallel vector fields. The classification of algebraic Ricci solitons is also presented, as well as the classification of pseudo-Kähler and pp-wave metrics. We obtain description of parallel symmetric tensors for each metric and show that they are derived from parallel vector fields. Finally, we study the totally geodesic subalgebras and show that for each subalgebra of the observed algebra there is a metric which makes it totally geodesic.

We study the subvariety of fixed points of an automorphism of a Calogero–Moser space induced by a regular element of finite order of the normalizer of the associated complex reflection group $W$. We determine some of (and conjecturally all) the ${\mathbb{C}}^\times$-fixed points of its unique irreducible component of maximal dimension in terms of the character table of $W$. This is inspired by the mysterious relations between the geometry of Calogero–Moser spaces and unipotent representations of finite reductive groups, which is the theme of another paper [Pure Appl. Math. Q. 21 no. 1 (2025), 131–200].

Let $R=\bigoplus_{\alpha\in\Gamma}R_{\alpha}$ be a commutative ring graded by an arbitrary torsionless monoid $\Gamma$. We say that $R$ is a graded almost valuation ring (gr AV- ring) if for every two homogeneous elements $a,b$ of $R$, there exists a positive integer $n$ such that either $a^{n}$ divides $b^{n}$ (in $R$) or $b^{n}$ divides $a^{n}$. In this paper, we introduce and study the graded version of the almost valuation ring which is a generalization of gr-AVD to the context of arbitrary $\Gamma$-graded rings (with zero- divisors). Next, we study the possible transfer of this property to the graded trivial ring extension $A\ltimes E$. Our aim is to provide examples of new classes of $\Gamma$-graded rings satisfying the above mentioned property.

We study a generalized form of the Bernoulli differential equation, employing a generalized conformable derivative. We first establish a generalized variant of Gronwall's inequality, which is essential for assessing the stability of generalized differential equation systems, and offer insights into the qualitative behavior of the trivial solution of the proposed equation. We then present and prove the main results concerning the solution of the generalized Bernoulli differential equation, complemented by illustrative examples that highlight the advantages of this generalized derivative approach. Furthermore, we introduce a finite difference method as an alternative technique to approximate the solution of the generalized Bernoulli equation and demonstrate its validity through practical examples.

I extend the pointillist principles of Moon and Carrillo–de Guzmán to variational operators and jump functions.

We propose a compact finite difference (CFD) scheme for the solution of time-fractional diffusion equations (TFDE) with the Caputo–Fabrizio derivative. The Caputo–Fabrizio derivative is discussed in the time direction and is discretized by a special discrete scheme. The compact difference operator is introduced in the space direction. We prove the unconditional stability and convergence of the proposed scheme. We show that the convergence order is $O(\tau^3+h^4)$, where $\tau$ and $h$ are the temporal stepsize and spatial stepsize, respectively. Our main purpose is to show that the Caputo–Fabrizio derivative without singular term can improve the accuracy of the discrete scheme. Numerical examples demonstrate the efficiency of the proposed method, and the numerical results agree well with the theoretical predictions.

Finding the Hamiltonian cycles in graphs is a well-known problem. Although the Hamiltonicity of grid graphs has been studied in the literature, there are few results on Hamiltonicity of grid graphs with holes. In this paper, we study the Hamiltonicity of rectangular grid graphs (meshes) with an L-shaped hole, and give a linear-time algorithm. The holes in meshes correspond to the faulty nodes.

Motivated by financial considerations, we develop a non-classical integration theory that is not necessarily associated with a measure. The base space consists of stock price trajectories and embodies a natural no-arbitrage condition. Conditional integrals are introduced, representing the investment required to hedge an option payoff when entering the market at any later time. Here, the investment may depend on the stock price history, and hedging takes place almost everywhere and as a limit over an increasing number of portfolios. In our setting, the space of elementary integrands fails to satisfy the lattice property and the notion of null sets is financially motivated and not measure- theoretic. Therefore, option prices arise from conditional non-lattice integrals rather than expectations, with no need to impose measurability assumptions.

Let $J_{n,m}:=(x_1x_2\cdots x_m, x_2x_3\cdots x_{m+1}, \ldots, x_{n-m+1}\cdots x_n, x_{n-m+2}\cdots x_nx_1, \ldots, x_nx_1\cdots x_{m-1})$ be the $m$-path ideal of the cycle graph of length $n$ in the ring $S=K[x_1,\ldots,x_n]$. Let $d=\gcd(n,m)$. We prove that $\operatorname{depth}(S/J_{n,m}^t)\leq d-1$ for all $t\geq n-1$. We show that $\operatorname{sdepth}(S/J_{n,n-1}^t)=\operatorname{depth}(S/J_{n,n-1}^t)=\max\{n-t-1,0\}$ for all $t\geq 1$. Also, we give some bounds for $\operatorname{depth}(S/J_{n,m}^t)$ and $\operatorname{sdepth}(S/J_{n,m}^t)$, where $t\geq 1$.

The classical Muckenhoupt $A_p$ condition is necessary and sufficient for the boundedness of the maximal operator $M$ on $L^p(w)$ spaces. In this paper we obtain another characterization of the $A_p$ condition. As a result, we show that some strong versions of the weighted $L^p(w)$ Coifman–Fefferman and Fefferman–Stein inequalities hold if and only if $w\in A_p$. We also give new examples of Banach function spaces $X$ such that $M$ is bounded on $X$ but not bounded on the associate space $X'$.

A connected graph $G$ with $\mathrm{diam}(G) \ge \ell$ is $\ell$-distance-balanced if $|W_{xy}|=|W_{yx}|$ for every $x,y\in V(G)$ with $d_{G}(x,y)=\ell$, where $W_{xy}$ is the set of vertices of $G$ that are closer to $x$ than to $y$. Miklavič and Šparl [Discrete Appl. Math. 244 (2018), 143–154] conjectured that if $n>n_k$, where $n_k=11$ if $k=2$, $n_k=(k+1)^2$ if $k$ is odd, and $n_k=k(k+2)$ if $k\ge 4$ is even, then the generalized Petersen graph $\mathrm{GP}(n,k)$ is not $\ell$-distance-balanced for any $1\le \ell < \mathrm{diam}(\mathrm{GP}(n,k))$. In the seminal paper, the conjecture was verified for $k=2$. In this paper we prove that the conjecture holds for $k=3$ and for $k=4$.

We establish an extension of Ky Fan's determinant inequality when the usual matrix addition is replaced by the power mean of positive definite matrices. We further explore variants of this newly derived $L_p$ Ky Fan inequality, extending a determinant difference inequality formulated by Yuan and Leng [J. Aust. Math. Soc. 83 no. 1 (2007)].

Under certain natural sufficient conditions on the sequence of uniformly bounded closed sets $E_k\subset\mathbb{R}$ of admissible coefficients, we construct a polynomial $P_n(x)=1+\sum_{k=1}^n\varepsilon_k x^k$, $\varepsilon_k\in E_k$, with at least $c\sqrt n$ distinct roots in $[0,1]$, which matches the classical upper bound up to the value of the constant $c>0$. Our sufficient conditions cover the Littlewood ($E_k=\{-1,1\}$) and Newman ($E_k=\{0,(-1)^k\}$) polynomials and are also necessary for the existence of such polynomials with arbitrarily many roots in the case when the sequence $E_k$ is periodic.

For a finite group $G$, the superpower graph $S(G)$ is a simple undirected graph with vertex set $G$, where two distinct vertices are adjacent if and only if the order of one divides that of the other. The aim of this paper is to provide tight bounds for the vertex connectivity of $S(G)$, together with some structural properties such as maximal domination sets, Hamiltonicity, and its variations for superpower graphs of finite abelian groups. The paper concludes with some open problems.

We provide a characterization of multiply recurrent operators that act on a Fréchet space. As an application, we extend the weighted shift results established by Costakis and Parissis (2012). We achieve this by characterizing topologically multiply recurrent pseudo-shifts acting on an $F$-sequence space indexed by an arbitrary countable infinite set. This characterization is in terms of the weights, the OP-basis and the shift mapping. Additionally, we establish that the recurrence and the hypercyclicity of pseudo-shifts are equivalent.