arXiv subjects
Publications and source records attributed to José M. Rodríguez.
To our knowledge, there is no rigorous mathematical justification of the Reynolds equation for a spherical bearing. In this article, we demonstrate that the solution of the Stokes problem in a domain between two closely spaced spheres converges, as the distance between the spheres approaches zero, to the solution of a "Reynolds equation".
We propose a two-dimensional flow model of a viscous fluid between two close moving surfaces. We show, using a formal asymptotic expansion of the solution, that its asymptotic behavior, when the distance between the two surfaces tends to zero, is the same as that of the the Navier-Stokes equations. The leading term of the formal asymptotic expansions of the solutions to the new model and Navier-Stokes equations are solution of the same limit problem, and the type of the limit problem depends on the boundary conditions. If slip velocity boundary conditions are imposed on the upper and lower bound surfaces, the limit is a solution of a lubrication model, but if the tractions and friction forces are known on both bound surfaces, the limit is a solution of a thin fluid layer model. The model proposed has been obtained to be a valuable tool for computing viscous fluid flow between two nearby moving surfaces, without the need to decide a priori whether the flow is typical of a lubrication or a thin fluid layer problem, and without the enormous computational effort that would be required to solve the Navier-Stokes equations in such a thin domain.
We propose a two-dimensional flow model of a viscous fluid between two close moving surfaces. We show that its asymptotic behavior, when the distance between the two surfaces tends to zero, is the same as that of the the Navier-Stokes equations. The solutions of the new model and Navier-Stokes equations converge to the same limit problem, that depends on the boundary conditions. If slip velocity boundary conditions are imposed on the upper and lower bound surfaces, the limit is solution of a lubrication model, but if the tractions and friction forces are known on both bound surfaces, the limit is solution of a shallow water model. The model proposed has been obtained to be a valuable tool for computing viscous fluid flow between two close moving surfaces, without the need to decide a priori whether the flow is typical of a lubrication problem or it is of shallow water type, and without the enormous computational effort that would be required to solve the Navier-Stokes equations in such a thin domain.
Inequalities play an important role in pure and applied mathematics. In particular, Opial inequality plays a main role in the study of the existence and uniqueness of initial and boundary value problems for differential equations. It has several interesting generalizations. In this work we prove some new Opial-type inequalities, and we apply them to generalized Riemann-Liouville-type integral operators.
Inequalities play an important role in pure and applied mathematics. In particular, Jensen's inequality, one of the most famous inequalities, plays a main role in the study of the existence and uniqueness of initial and boundary value problems for differential equations. In this work we prove some new Jensen-type inequalities for $m$-convex functions, and we apply them to generalized Riemann-Liouville-type integral operators. It is remarkable that, if we consider $m=1$, we obtain new inequalities for convex functions.
We introduce a degree-based variable topological index inspired on the power (or generalized) mean. We name this new index as the mean Sombor index: $mSO_α(G) = \sum_{uv \in E(G)} \left[\left( d_u^α+d_v^α\right) /2 \right]^{1/α}$. Here, $uv$ denotes the edge of the graph $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $α\in \mathbb{R} \backslash \{0\}$. We also consider the limit cases $mSO_{α\to 0}(G)$ and $mSO_{α\to\pm\infty}(G)$. Indeed, for given values of $α$, the mean Sombor index is related to well-known topological indices such as the inverse sum indeg index, the reciprocal Randic index, the first Zagreb index, the Stolarsky--Puebla index and several Sombor indices. Moreover, through a quantitative structure property relationship (QSPR) analysis we show that $mSO_α(G)$ correlates well with several physicochemical properties of octane isomers. Some mathematical properties of mean Sombor indices as well as bounds and new relationships with known topological indices are also discussed.
The aim of this paper is to obtain new inequalities for a large family of topological indices restricted to unicyclic graphs and to characterize the set of extremal unicyclic graphs with respect to them. This family includes variable first Zagreb, variable sum exdeg, multiplicative second Zagreb and Narumi-Katayama indices. Our main results provide upper and lower bounds for these topological indices on unicyclic graphs, fixing or not the maximum degree or the number of pendant vertices.
A celebrated theorem of Kanai states that quasi-isometries preserve isoperimetric inequalities between uniform Riemannian manifolds (with positive injectivity radius) and graphs. Our main result states that we can study the (Cheeger) isoperimetric inequality in a Riemann surface by using a graph related to it, even if the surface has injectivity radius zero (this graph is inspired in Kanai's graph, but it is different from it). We also present an application relating Gromov boundary and isoperimetric inequality.
The aim of this paper is to obtain new sharp inequalities for a large family of topological indices, including the first variable Zagreb index $M_1^α$, and to characterize the set of extremal graphs with respect to them. Our main results provide lower bounds on this family of topological indices involving just the minimum and the maximum degree of the graph. These inequalities are new even for the first Zagreb, the inverse and the forgotten indices.
The concept of geometric-arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. The aim of this paper is to obtain new inequalities involving the geometric-arithmetic index $GA_1$ and characterize graphs extremal with respect to them. Our main results provide lower bounds $GA_1(G)$ involving just the minimum and the maximum degree of the graph $G$.
If $X$ is a geodesic metric space and $x_1,x_2,x_3\in X$, a {\it geodesic triangle} $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $δ$-\emph{hyperbolic} $($in the Gromov sense$)$ if any side of $T$ is contained in a $δ$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$. If $X$ is hyperbolic, we denote by $δ(X)$ the sharp hyperbolicity constant of $X$, i.e., $δ(X)=\inf\{δ\ge 0: \, X \, \text{ is $δ$-hyperbolic}\,\}.$ Some previous works characterize the hyperbolic product graphs (for the Cartesian, strong, join, corona and lexicographic products) in terms of properties of the factor graphs. However, the problem with the direct product is more complicated. In this paper, we prove that if the direct product $G_1\times G_2$ is hyperbolic, then one factor is hyperbolic and the other one is bounded. Also, we prove that this necessary condition is, in fact, a characterization in many cases. In other cases, we find characterizations which are not so simple. Furthermore, we obtain formulae or good bounds for the hyperbolicity constant of the direct product of some important graphs.
In this paper we study the relationship of hyperbolicity and (Cheeger) isoperimetric inequality in the context of Riemannian manifolds and graphs. We characterize the hyperbolic manifolds and graphs (with bounded local geometry) verifying this isoperimetric inequality, in terms of their Gromov boundary. Furthermore, we characterize the trees with isoperimetric inequality (without any hypothesis). As an application of our results, we obtain the solvability of the Dirichlet problem at infinity for these Riemannian manifolds and graphs, and that the Martin boundary is homeomorphic to the Gromov boundary.
Large Eddy Simulation (LES) is a very useful tool when simulating turbulent flows if we are only interested in its "larger" scales. One of the possible ways to derive the LES equations is to apply a filter operator to the Navier-Stokes equations, obtaining a new equation governing the behavior of the filtered velocity. This approach introduces in the equations the so called subgrid-scale tensor, that must be expressed in terms of the filtered velocity to close the problem. One of the most popular models is that proposed by Smagorinsky, where the subgrid-scale tensor is modeled by introducing an eddy viscosity. In this work, we shall propose a new approximation to this problem by applying the filter, not to the Navier-Stokes equations, but to a generalized version of them with nonlinear viscosity. That is, we shall introduce a nonlinear viscosity, not as a procedure to close the subgrid-scale tensor, but as part of the model itself (see below). Consequently, we shall need a different method to close the subgrid-scale tensor, and we shall use the Clark approximation, where the Taylor expansion of the subgrid-scale tensor is computed.
The alliance polynomial of a graph $G$ with order $n$ and maximum degree $Δ$ is the polynomial $A(G; x) = \sum_{k=-Δ}^Δ A_{k}(G) \, x^{n+k}$, where $A_{k}(G)$ is the number of exact defensive $k$-alliances in $G$. We obtain some properties of $A(G; x)$ and its coefficients for regular graphs. In particular, we characterize the degree of regular graphs by the number of non-zero coefficients of their alliance polynomial. Besides, we prove that the family of alliance polynomials of $Δ$-regular graphs with small degree is a very special one, since it does not contain alliance polynomials of graphs which are not $Δ$-regular. By using this last result and direct computation we find that the alliance polynomial determines uniquely each cubic graph of order less than or equal to $10$.
If $X$ is a geodesic metric space and $x_1,x_2,x_3\in X$, a {\it geodesic triangle} $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $δ$-\emph{hyperbolic} $($in the Gromov sense$)$ if any side of $T$ is contained in a $δ$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$. If $X$ is hyperbolic, we denote by $δ(X)$ the sharp hyperbolicity constant of $X$, i.e. $δ(X)=\inf\{δ\ge 0: \, X \, \text{ is $δ$-hyperbolic}\}.$ In this paper we characterize the lexicographic product of two graphs $G_1\circ G_2$ which are hyperbolic, in terms of $G_1$ and $G_2$: the lexicographic product graph $G_1\circ G_2$ is hyperbolic if and only if $G_1$ is hyperbolic, unless if $G_1$ is a trivial graph (the graph with a single vertex); if $G_1$ is trivial, then $G_1\circ G_2$ is hyperbolic if and only if $G_2$ is hyperbolic. In particular, we obtain the sharp inequalities $δ(G_1)\le δ(G_1\circ G_2) \le δ(G_1) + 3/2$ if $G_1$ is not a trivial graph, and we characterize the graphs for which the second inequality is attained.
If $X$ is a geodesic metric space and $x_1,x_2,x_3\in X$, a geodesic triangle $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $δ$-hyperbolic (in the Gromov sense) if any side of $T$ is contained in a $δ$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. In the context of graphs, to remove and to contract an edge of a graph are natural transformations. The main aim in this work is to obtain quantitative information about the distortion of the hyperbolicity constant of the graph $G \setminus e$ (respectively, $\,G/e\,$) obtained from the graph $G$ by deleting (respectively, contracting) an arbitrary edge $e$ from it. This work provides information about the hyperbolicity constant of minor graphs.
The aim of this paper is to provide Markov-type inequalities in the setting of weighted Sobolev spaces when the considered weights are generalized classical weights. Also, as results of independent interest, some basic facts about Sobolev spaces with respect to certain vector measures are stated.
If X is a geodesic metric space and $x_1,x_2,x_3\in X$, a {\it geodesic triangle} $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $δ$-\emph{hyperbolic} $($in the Gromov sense$)$ if any side of $T$ is contained in a $δ$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$. If $X$ is hyperbolic, we denote by $δ(X)$ the sharp hyperbolicity constant of $X$, i.e. $δ(X)=\inf\{δ\ge 0: \, X \, \text{ is $δ$-hyperbolic}\,\}\,.$ Some previous works characterize the hyperbolic product graphs (for the Cartesian product, strong product and lexicographic product) in terms of properties of the factor graphs. In this paper we characterize the hyperbolic product graphs for graph join $G_1\uplus G_2$ and the corona $G_1\diamond G_2$: $G_1\uplus G_2$ is always hyperbolic, and $G_1\diamond G_2$ is hyperbolic if and only if $G_1$ is hyperbolic. Furthermore, we obtain simple formulae for the hyperbolicity constant of the graph join $G_1\uplus G_2$ and the corona $G_1\diamond G_2$.