arXiv subjects

J. Toledo

Publications and source records attributed to J. Toledo.

11 recordsLinked to original sources

arXiv2026

Doubly Nonlinear Diffusion Equations on Metric Graphs

In this paper we study existence and uniqueness of solutions for a very general class of doubly nonlinear diffusion equations on metric graphs, which provide the appropriate mathematical framework to describe complex tubular networks in which axial diffusion is the main focus. Some important particular cases covered in our study are the Porous Medium Equation and the evolution equation for the $p$-Laplacian, but we also consider the case in that diffusion changes from one edge to another, which takes into account the influence of the properties of the tubules forming the network on axial diffusion. Furthermore, the problem is studied under non-homogeneous Neumann-Kirchhoff conditions on the vertices of the graph.

math.AP↗

arXiv2025

Evolution problem for the $1$-Laplacian with mixed boundary conditions

This paper deals with evolution problem for the $1$-Laplacian with mixed boundary conditions on a bounded open set $\Omega$ of $\R^N$. We prove existence and uniqueness of strong solutions for data in $L^2(\Omega)$ by mean of the theory of maximal monotone operator. We also see that if the flux on the boundary is~$1$ (that is, the maximum possible) then these strong solutions can be seen as the large solutions introduced in \cite{MP}. We give explicit examples of solutions.

math.AP↗

arXiv2025

CRYSP: a Total-Body PET based on cryogenic cesium iodide crystals

Total Body PET (TBPET) scanners have the potential to substantially reduce both acquisition time and administered radiation dose, owing to their high sensitivity. However, their widespread clinical adoption is hindered by the high cost of currently available systems. This work explores the use of pure cesium iodide (CsI) monolithic crystals operated at cryogenic temperatures as a cost-effective alternative to rare-earth scintillators for TBPET. We investigate the performance of pure CsI crystals operated at cryogenic temperatures ($\sim$100 K), where they achieve a light yield of approximately $10^5$ photons/MeV. The implications for energy resolution, spatial resolution (including depth-of-interaction capability), and timing performance are assessed, with a view toward their integration into a TBPET system. Cryogenic CsI crystals demonstrated energy resolution below 7% and coincidence time resolution at the nanosecond level, despite their relatively slow scintillation decay time. A Monte Carlo simulation of monolithic CsI crystals shows that a millimeter-scale spatial resolution in all three dimensions can be obtained. These characteristics indicate that high-performance PET imaging is achievable with this technology. A TBPET scanner based on cryogenic CsI monolithic crystals could combine excellent imaging performance with significantly reduced detector costs, enabling broader accessibility and accelerating the adoption of TBPET in both clinical and research settings.

physics.ins-det↗

arXiv2024

Gelfand-Type problems in Random Walk Spaces

This paper deals with Gelfand-type problems \begin{equation}\label{Gelfand10} \qquad\qquad\left\{\begin{array}{ll} - \Delta_m u = \lambda f(u), \quad&\hbox{in} \ \Omega, \ \lambda >0, \\[10pt] u =0, \quad&\hbox{on} \ \partial_m\Omega, \end{array} \right. \end{equation} in the framework of Random Walk Spaces, which includes as particular cases: Gelfand-type problems posed on locally finite weighted connected graphs and Gelfand-type problems driven by convolution integrable kernels. Under the same assumption on the nonlinearity $f$ as in the local case, we show there exists an extremal parameter $\lambda^* \in (0, \infty)$ such that, for $0 \leq \lambda < \lambda^*$, problem \eqref{Gelfand10} admits a minimal bounded solution $u_\lambda$ and there are not solution for $\lambda > \lambda^*$. Moreover, assuming $f$ is convex, we show that Problem \eqref{Gelfand10} admits a minimal bounded solution for $\lambda = \lambda^*$. We also show that $u_\lambda$ are stable, and, for $f$ strictly convex, we show that they are the unique stable solutions. We give simple examples that illustrate the many situations that can occur when solving Gelfand-type problems on weighted graphs.

math.AP↗

arXiv2024

Evolution problems with perturbed $1$-Laplacian type operators on random walk spaces

Random walk spaces are a general framework for the study of PDEs. They include as particular cases locally finite weighted connected graphs and nonlocal settings involving symmetric integrable kernels on $\mathbb{R}^N$. We are interested in the study of evolution problems involving two random walk structures so that the associated functionals have different growth on each structure. We also deal with the case of a functional with different growth on a partition of the random walk.

math.AP↗

arXiv2024

Normally torsion-freeness and normality criteria for monomial ideals

In this paper, we focus on the associated primes of powers of monomial ideals and asymptotic behavior properties such as normally torsion-freeness, normality, the strong persistence property, and the persistence property. In particular, we introduce the concept of monomial ideals of well-nearly normally torsion-free type, and show that these ideals are normal. After that, we present some results on the existence of embedded associated prime ideals in the associated primes set of powers of monomial ideals. Further, we employ them in investigating the edge and cover ideals of cones of graphs. Next, we present counterexamples to several questions concerning the relations between relevant algebraic properties of the edge ideals of clutters and complement clutters. We conclude by providing counterexamples to questions on the possible connections between normally torsion-freeness and normality of monomial ideals under polarization.

math.AC↗

arXiv2024

Two models forsandpile growth in weighted graphs

In this paper we study $\infty$-Laplacian type diffusion equations in weighted graphs obtained as limit as $p\to \infty$ to two types of $p$-Laplacian evolution equations in such graphs. We propose these diffusion equations, that are governed by the subdifferential of a convex energy functionals associated to the indicator function of the set $$K^G_{\infty}:= \left\{ u \in L^2(V, ν_G) \ : \ \vert u(y) - u(x) \vert \leq 1 \ \ \hbox{if} \ \ x \sim y \right\}$$ and the set $$K^w_{\infty}:= \left\{ u \in L^2(V, ν_G) \ : \ \vert u(y) - u(x) \vert \leq \sqrt{w_{xy}} \ \ \hbox{if} \ \ x \sim y \right\}$$ as models for sandpile growth in weighted graphs. Moreover, we also analyse the collapse of the initial condition when it does not belong to the stable sets $K^G_{\infty}$ or $K^w_{\infty}$ by means of an abstract result given in~\cite{BEG}. We give an interpretation of the limit problems in terms of Monge-Kantorovich mass transport theory. Finally, we give some explicit solutions of simple examples that illustrate the dynamics of the sandpile growing or collapsing.

math.AP↗

arXiv2020

The Total Variation Flow in Metric Random Walk Spaces

In this paper we study the Total Variation Flow (TVF) in metric random walk spaces, which unifies into a broad framework the TVF on locally finite weighted connected graphs, the TVF determined by finite Markov chains and some nonlocal evolution problems. Once the existence and uniqueness of solutions of the TVF has been proved, we study the asymptotic behaviour of those solutions and, with that aim in view, we establish some inequalities of Poincaré type. In particular, for finite weighted connected graphs, we show that the solutions reach the average of the initial data in finite time. Furthermore, we introduce the concepts of perimeter and mean curvature for subsets of a metric random walk space and we study the relation between isoperimetric inequalities and Sobolev inequalities. Moreover, we introduce the concepts of Cheeger and calibrable sets in metric random walk spaces and characterize calibrability by using the $1$-Laplacian operator. Finally, we study the eigenvalue problem whereby we give a method to solve the optimal Cheeger cut problem.

math.AP↗

arXiv2020

$(BV,L^p)$-decomposition, $p=1,2$, of Functions in Metric Random Walk Spaces

In this paper we study the $(BV,L^p)$-decomposition, $p=1,2$, of functions in metric random walk spaces, a general workspace that includes weighted graphs and nonlocal models used in image processing. We obtain the Euler-Lagrange equations of the corresponding variational problems and their gradient flows. In the case $p=1$ we also study the associated geometric problem and the thresholding parameters.

math.AP↗

arXiv2012

χ-Systems for Correlation Functions

We consider the strong coupling limit of 4-point functions of heavy operators in N=4 SYM dual to strings with no spin in AdS. We restrict our discussion for operators inserted on a line. The string computation factorizes into a state-dependent sphere part and a universal AdS contribution which depends only on the dimensions of the operators and the cross ratios. We use the integrability of the AdS string equations to compute the AdS part for operators of arbitrary conformal dimensions. The solution takes the form of TBA-like integral equations with the minimal AdS string-action computed by a corresponding free-energy-like functional. These TBA-like equations stem from a peculiar system of functional equations which we call a χ-system. In principle one could use the same method to solve for the AdS contribution in the N-point function. An interesting feature of the solution is that it encodes multiple string configurations corresponding to different classical saddle-points. The discrete data that parameterizes these solutions enters through the analog of the chemical-potentials in the TBA-like equations. Finally, for operators dual to strings spinning in the same equator in S^5 (i.e. BPS operators of the same type) the sphere part is simple to compute. In this case (which is generically neither extremal nor protected) we can construct the complete, strong-coupling 4-point function.

hep-th↗

arXiv2010

Detection and Imaging of High-Z Materials with a Muon Tomography Station Using GEM Detectors

Muon tomography based on the measurement of multiple scattering of atmospheric cosmic ray muons is a promising technique for detecting and imaging heavily shielded high-Z nuclear materials such as enriched uranium. This technique could complement standard radiation detection portals currently deployed at international borders and ports, which are not very sensitive to heavily shielded nuclear materials. We image small targets in 3D using $2\times 2 \times 2 mm^3$ voxels with a minimal muon tomography station prototype that tracks muons with Gas Electron Multiplier (GEM) detectors read out in 2D with x-y microstrips of 400 micron pitch. With preliminary electronics, the GEM detectors achieve a spatial resolution of 130 microns in both dimensions. With the next GEM-based prototype station we plan to probe an active volume of ~27 liters. We present first results on reading out all 1536 microstrips of a $30 \times 30 cm^2$ GEM detector for the next muon tomography prototype with final frontend electronics and DAQ system. This constitutes the first full-size implementation of the Scalable Readout System (SRS) recently developed specifically for Micropattern Gas Detectors by the RD51 collaboration. Design of the SRS and first performance results when reading out GEM detectors are presented.

physics.ins-det↗