Infinite Branches Research Articles

Hyperbolic radial spanning tree

We define and analyze an extension to the d-dimensional hyperbolic space of the Radial Spanning Tree (RST) introduced by Baccelli and Bordenave in the two-dimensional Euclidean space (2007). In particular, we will focus on the description of the infinite branches of the tree. The properties of the two-dimensional Euclidean RST are extended to the hyperbolic case in every dimension: almost surely, every infinite branch admits an asymptotic direction and each asymptotic direction is reached by at least one infinite branch. Moreover, the branch converging to any deterministic asymptotic direction is unique almost surely. To obtain results for any dimension, a completely new approach is considered here. Our strategy mainly involves the two following ingredients, that rely on the hyperbolic Directed Spanning Forest (DSF) introduced and studied in Flammant (2019). First, the hyperbolic metric allows us to obtain fine control of the branches’ fluctuations in the hyperbolic DSF without using planarity arguments. Then, we couple the hyperbolic RST with the hyperbolic DSF and conclude thanks to the precise estimates mentioned before.

Asymptotic Behavior of a Parametric Algebraic Surface

Starting from the concept of infinite branches and approximation surfaces, we present a method to compute infinite branches and surfaces having the same asymptotic behavior as an input parametric surface. The results obtained in this work represent a breakthrough for the study of surfaces and their applications.

Deciphering the intricate dielectric relaxation processes of cellulose paper: Extraction of distribution of relaxation time and analysis of degradation characteristics

Cellulose material is a dielectric with intricate microscopic relaxation processes due to its complex structure. However, conventional models and curve fitting methods used for tracing and analyzing these processes often fail to capture crucial dielectric information. This paper aimed to extract the Distribution of Relaxation Time (DRT), the most fundamental and effective dielectric information providing the time scale and relative contribution of all microscopic relaxation processes. First, a distributed extended Debye model with infinite branches was constructed based on the microscopic nature of dielectric relaxation. Then, an implicit equation of the DRT function was established, inspired by the mathematical principles of infinite subdivision and summation. To obtain the numeral solution of the DRT function, a regularization method was proposed and validated. Finally, the approach was applied to cellulose insulating paper with varying degradation degrees. The relaxation process with a long time constant played a significant role, and variations during the degradation process were attributed to reduced activation energy. With clear physical interpretation and robust mathematical foundation, our method sheds light on the intricate dielectric relaxation processes in cellulose. This not only enhances the theoretical understanding and practical application of cellulose materials but also provides valuable insights for the analysis and application of other materials.

Note on the Generalized Branching Random Walk on the Galton–Watson Tree

Let ∂T be a super-critical Galton–Watson tree. Recently, the first author computed almost surely and simultaneously the Hausdorff dimensions of the sets of infinite branches of the boundary of ∂T along which the sequence SnX(t)/SnX˜(t) has a given set of limit points, where SnX(t) and SnX˜(t) are two branching random walks defined on ∂T. In this study, we are interested in the study of the speed of convergence of this sequence. More precisely, for a given sequence s=(sn), we consider Eα,s=t∈∂T:SnX(t)−αSnX˜(t)∼snasn→+∞. We will give a sufficient condition on (sn) so that Eα,s has a maximal Hausdorff and packing dimension.

Ultradiscrete hard-spring equation and its phase plane analysis

Ultradiscretization enables us to construct a piecewise linear equation which approximates a given subtraction-free difference equation. Recently proposed “ultradiscretization with parity variables” (pUD) can treat an equation with subtraction. However, its solution may have an infinite number of branches under some specific conditions. In this paper, solutions of pUD for the hard-spring equation is investigated. The pUD equation is reinterpreted as the mapping which maps a set on the phase plane to another one, and the behaviour of the solutions are analyzed through approximative transition diagrams.As a result, the infinite branches are translated into a few finite branches and allowed easier understanding.

Symmetry-breaking longitude bifurcations for a free boundary problem modeling small plaques in three dimensions.

Atherosclerosis, one of the leading causes of death in USA and worldwide, begins with a lesion in the intima of the arterial wall, allowing LDL to penetrate into the intima where they are oxidized. The immune system considers these oxidized LDL as a dangerous substance and tasks the macrophages to attack them; incapacitated macrophages become foam cells and leads to the formation of a plaque. As the plaque continues to grow, it progressively restricts the blood flow, possibly triggering heart attack or stroke. Because the blood vessels tend to be circular, two-space dimensional cross section model is a good approximation, and the two-space dimensional models are studied in Friedman et al. (J Differ Equ 259(4):1227-1255, 2015) and Zhao and Hu (J Differ Equ 288:250-287, 2021). It is interesting to see whether a true three-space dimensional stationary solution can be developed. We shall establish a three-space dimensional stationary solution for the mathematical model of the initiation and development of atherosclerosis which involves LDL and HDL cholesterols, macrophages and foam cells. The model is a highly nonlinear and coupled system of PDEs with a free boundary, the interface between the plaque and the blood flow. We establish infinite branches of symmetry-breaking stationary solutions which bifurcate from the annular stationary solution in the longitude direction.

On the multifractal analysis of a non-standard branching random walk

Consider two branching random walks SnX and \(S_{n}\widetilde{X}\) on a supercritical random Galton–Watson tree. We compute, almost surely and simultaneously, the Hausdorff and packing dimensions of the level set \(E_{X,\widetilde{X}}(a,b)\) of infinite branches in the boundary of the tree along which the averages \(S_{n}X/S_{n}\widetilde{X}\) have a given set of limit points [a,b].

GAMES CHARACTERIZING LIMSUP FUNCTIONS AND BAIRE CLASS 1 FUNCTIONS

AbstractWe consider a real-valued function f defined on the set of infinite branches X of a countably branching pruned tree T. The function f is said to be a limsup function if there is a function $u \colon T \to \mathbb {R}$ such that $f(x) = \limsup _{t \to \infty } u(x_{0},\dots ,x_{t})$ for each $x \in X$ . We study a game characterization of limsup functions, as well as a novel game characterization of functions of Baire class 1.

Radial perturbations of the scalarized black holes in Einstein-Maxwell-conformally coupled scalar theory

We perform the stability analysis for the scalarized charged black holes obtained from Einstein-Maxwell-conformally coupled scalar (EMCS) theory by employing the radial perturbations. The targeting black holes include a single branch of scalarized charged black hole with the coupling parameter $\alpha>0$ inspired by the constant scalar hairy black hole as well as infinite branches of $n=0(\alpha \ge 8.019),1(\alpha \ge 40.84),2(\alpha \ge 99.89),\cdots$ scalarized charged black holes found through the spontaneous scalarization on the Reissner-Nordstr\"om black hole. It turns out that the black hole in the single branch and the $n = 0$ black hole are stable against radial perturbations, while the $n = 1, 2$ excited black holes are unstable in the EMCS theory with both quadratic and exponential couplings.

Remarks on the Interpolation Method

We discuss a generalization of the classic condition of validity of the interpolation method for the density of quenched free energy of mean field spin glasses. The condition is written just in terms of the L^2 metric structure of the Gaussian random variables. As an example of application we deduce the existence of the thermodynamic limit for a GREM model with infinite branches for which the classic conditions of validity fail. We underline the dependence of the density of quenched free energy just on the metric structure and discuss the models from a metric viewpoint.

Magnetic quivers from brane webs with O5 planes

Magnetic quivers have led to significant progress in the understanding of gauge theories with 8 supercharges at UV fixed points. For a given low-energy gauge theory realised via a Type II brane construction, there exist magnetic quivers for the Higgs branches at finite and infinite gauge coupling. Comparing these moduli spaces allows one to study the non-perturbative effects when transitioning to the fixed point. For 5d mathcal{N} = 1 SQCD, 5-brane webs have been an important tool for deriving magnetic quivers. In this work, the emphasis is placed on 5-brane webs with orientifold 5-planes which give rise to 5d theories with orthogonal or symplectic gauge groups. For this set-up, the magnetic quiver prescription is derived and contrasted against a unitary magnetic quiver description extracted from an O7− construction. Further validation is achieved by a derivation of the associated Hasse diagrams. An important class of families considered are the orthogonal exceptional En families (−∞ < n ≤ 8), realised as infinite coupling Higgs branches of Sp(k) gauge theories with fundamental matter. In particular, the moduli spaces are realised by a novel type of magnetic quivers, called unitary-orthosymplectic quivers.

Scalar hairy black holes in Einstein-Maxwell-conformally coupled scalar theory

We obtain scalar hairy black holes from Einstein-Maxwell-conformally coupled scalar (EMCS) theory with the scalar coupling parameter α to the Maxwell term. In case of α=0, the α=0 EMCS theory provides constant (charged) scalar hairy black hole and charged BBMB (Bocharova-Bronnikov-Melnikov-Bekenstein) black hole where the former is stable against full perturbations, while the latter remains unstable because it belongs to an extremal black hole. It is noted that for α≠0, the unstable Reissner-Nordström black holes without scalar hair imply infinite branches of n=0(α≥8.019),1(α≥40.84),2(α≥99.89),⋯ scalarized charged black holes. In addition, for α>0, we develop a single branch of scalarized charged black hole solutions inspired by the constant scalar hairy black hole. Finally, we obtain the numerical charged BBMB black hole solution from the α=0 EMCS theory.

On the Multifractal Analysis of Branching Random Walk on Galton–Watson Tree with Random Metric

We consider a branching random walk $$S_nX(t)$$ on a supercritical random Galton–Watson tree. We compute the Hausdorff and packing dimensions of the level set $$E(\alpha )$$ of infinite branches in the boundary of tree endowed with random metric along which the average of $$S_n X(t)/n$$ have a given limit point.

Bifurcation for a free boundary problem modeling the growth of multilayer tumors with ECM and MDE interactions

We study a free boundary problem modeling the growth of multilayer tumors. This model describes the invasion of tumors: the tumor cells produce matrix degrading enzymes (MDEs) to degrade the extracellular matrix (ECM) which provides structural support of the surrounding tissue. As in Pan and Xing, the influence of ECM and MDE interactions is considered in this paper. The model equations include two diffusion equations for the nutrient concentration and MDE concentration and an ordinary differential equation for ECM concentration. The tumor cell proliferation is at the rate , which characterizes the aggressiveness of the tumor. In contrast to Pan and Xing, we consider “flat stationary solution” in this paper. We first show that there exists a unique flat stationary solution for any . Then, we prove that there are infinite branches of bifurcation solutions from the flat stationary solutions at the bifurcation points ( ).

On the exact dimension of Mandelbrot measure

We develop, in the context of the boundary of a supercritical Galton–Watson tree, a uniform version of the argument used by Kahane 1987 on homogeneous trees to estimate almost surely and simultaneously the Hausdorff and packing dimensions of the Mandelbrot measure over a suitable set J . As an application, we compute, almost surely and simultaneously, the Hausdorff and packing dimensions of the level sets Eα of infinite branches of the boundary of the tree along which the averages of the branching random walk have a given limit point.

On the stabilizers of finite sets of numbers in the R. Thompson group $F$

We study subgroups $H_U$ of the R. Thompson group $F$ which are stabilizers of finite sets $U$ of numbers in the interval $(0,1)$. We describe the algebraic structure of $H_U$ and prove that the stabilizer $H_U$ is finitely generated if and only if $U$ consists of rational numbers. We also show that such subgroups are isomorphic surprisingly often. In particular, we prove that if finite sets $U\subset [0,1]$ and $V\subset [0,1]$ consist of rational numbers which are not finite binary fractions, and $|U|=|V|$, then the stabilizers of $U$ and $V$ are isomorphic. In fact these subgroups are conjugate inside a subgroup $\bar F<\Homeo([0,1])$ which is the completion of $F$ with respect to what we call the Hamming metric on $F$. Moreover the conjugator can be found in a certain subgroup $\F < \bar F$ which consists of possibly infinite tree-diagrams with finitely many infinite branches. We also show that the group $\F$ is non-amenable.

Existence Condition of Strong Stationary Times for Continuous Time Markov Chains on Discrete Graphs

We consider a random walk on a discrete connected graph having some infinite branches plus finitely many vertices with finite degrees. We find the generator of a strong stationary dual in the sense of Fill, and use it to find some equivalent condition to the existence of a strong stationary time. This strong stationary dual process lies in the set of connected compact sets of the compactification of the graph. When the graph is $$\mathbb Z$$ , the set here is simply the set of (possibly infinite) segments of $$\mathbb Z$$ .

Design of compact Marx module with square pulse output

Compact pulsed power system based on compact Marx generator is widely used in terms of drive resistance and capacitive loads. This system usually adopts high performance components such as high energy density capacitors, compact switches, and integrated structure. Traditional compact Marx generator can only output double-exponential pulse profile. In this paper a compact, low-impedance Marx module which can output rectangular pulse profile is design and tested. This module has multiple circuits of different discharge frequencies in parallel to generate quasi-rectangular pulse. Discharge characteristic of an ideal module with infinite branches is calculated theoretically. A module with two branches has been designed and tested. Test results show that the impedance of the module is 1.2 Ω. When charging voltage is 100.6 kV and load resistance is 1 Ω, the peak output pulse is 45.2 kV voltage, the peak power is about 2 GW, the pulse width is about 130 ns, and the rise time is about 35 ns. The energy density and power density of the module are 15 kJ/m(3) and 140 GW/m(3), respectively.

Stationary solutions of a free boundary problem modeling the growth of tumors with Gibbs–Thomson relation

In this paper we study a free boundary problem modeling tumor growth. The model consists of two elliptic equations describing nutrient diffusion and pressure distribution within tumors, respectively, and a first-order partial differential equation governing the free boundary, on which a Gibbs–Thomson relation is taken into account. We first show that the problem may have none, one or two radial stationary solutions depending on model parameters. Then by bifurcation analysis we show that there exist infinite many branches of non-radial stationary solutions bifurcating from given radial stationary solution. The result implies that cell-to-cell adhesiveness is the key parameter which plays a crucial role on tumor invasion.

Does ultra-slow diffusion survive in a three dimensional cylindrical comb?

We present an exact analytical result on ultra-slow diffusion by solving a Fokker–Planck equation, which describes anomalous transport in a three dimensional (3D) comb. This 3D cylindrical comb consists of a cylinder of discs of either infinite or finite radius, threaded on a backbone. It is shown that the ultra-slow particle spreading along the backbone is described by the mean squared displacement (MSD) of the order of ln (t). This phenomenon takes place only for normal two dimensional diffusion inside the infinite secondary branches (discs). When the secondary branches have finite boundaries, the ultra-slow motion is a transient process and the asymptotic behavior is normal diffusion. In another example, when anomalous diffusion takes place in the secondary branches, a destruction of ultra-slow (logarithmic) diffusion takes place as well. As the result, one observes “enhanced” subdiffusion with the MSD ∼t1−αln(t), where 0 < α < 1.