Revista de la Unión Matemática Argentina

Warped product lightlike submanifolds with a slant factor

2022-10-16T01:02:17-03:00

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 Green ring of a family of copointed Hopf algebras

2023-03-02T10:15:41-03:00

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.

Haar wavelet characterization of dyadic Lipschitz regularity

2022-09-14T14:22:06-03:00

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 $h(x)=\mathcal{X}_{[0,1/2)}(x)-\mathcal{X}_{[1/2,1)}(x)$.

Cluster algebras of type $\mathbb{A}_{n-1}$ through the permutation groups $S_{n}$

2022-07-20T18:49:57-03:00

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 $\mathbb{A}$.

Large-scale homogeneity and isotropy versus fine-scale condensation: A model based on Muckenhoupt-type densities

2022-10-27T17:17:23-03:00

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.

Ground State Solutions for Schrodinger Equations in the Presence of a Magnetic Field

2023-06-06T11:14:54-03:00

This paper is dedicated to studying the schrodinger equations in the presence of a magnetic field. Based on variational methods, especially Mountain Pass Theorem, we obtain ground state solutions for the system under certain assumptions.

Clique coloring EPT graphs on bounded degree trees

2023-02-15T21:43:50-03:00

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.

On the wellposedness of a fuel cell problem

2023-04-08T11:20:29-03:00

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.

An improved lopsided shift-splitting preconditioner for three-by-three block saddle point problems

2023-05-03T10:48:51-03:00

In this paper, a improved lopsided shift-splitting (ILSS) preconditioner is considered to solve the three-by-three block saddle point problems, this method is an improvement of the work of Zhang et al. [Comput. Appl. Math. (2022),41:261]. We proved that the iteration method produced by the ILSS preconditioner is unconditionally convergent. In addition, it proved that all eigenvalues of the ILSS preconditioned matrix are real and non-unit eigenvalues are located in a positive interval. Numerical experiments show that the ILSS preconditioner is effective.

The Reconstruction problem for multivalued linear operator’s properties

2023-06-13T09:43:09-03:00

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 M.

On the Maximum Weighted Irredundant Set Problem

2023-05-29T09:23:15-03:00

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.

Covering-based numbers related to the LS-category of finite spaces

2022-09-29T07:58:39-03:00

In this paper, Lusternik-Schinrelmann category and geometric category of finite spaces are considered. We define new numerical invariants of these spaces derived from the geometric category and present an algorithmic approach for its effective computation. The analysis is undertaken by combining homotopic features of the spaces, algorithms and tools from the theory of graphs and hypergraphs. We also provide a number of examples.

Combinatorial formulas for determinant, permanent, and inverse of some circulant matrices with three parameters

2023-02-26T15:54:50-03:00

In this work, 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.

Counterexamples for some results in "On the module intersection graph of ideals of rings"

2023-06-28T02:10:05-03:00

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.

On a fractional Nirenberg equation: compactness and existence results

2023-01-11T14:23:47-03:00

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.

Ricci-Bourguignon solitons on real hypersurfaces in the complex projective space

2023-07-18T10:41:03-03:00

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 $\mathbb{C}P^n$.

Differential graded Brauer groups

2023-05-04T10:01:03-03:00

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 $K$.

Principality by reduced ideals in pure cubic number fields

2023-06-09T08:26:01-03:00

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.

Special Affine Connections on Symmetric Spaces

2023-04-13T13:22:07-03:00

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.

On the Pythagoras number for polynomials of degree 4 in 5 variables

2023-06-20T09:25:13-03:00

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 $p(5,4)$.