Continuous Branching Research Articles

Directed random walks on hierarchical trees with continuous branching: A renormalization group approach

We investigate the directed random walk on hierarchic trees. Two cases are investigated: random variables on deterministic trees with a continuous branching, and random variables on the trees constructed through the random branching process. We derive renormalization group (partial differential) equations for the branching models with binomial, Poisson, and compound Poisson distributions of random variables on the links of a tree. These renormalization group equations are a new class of reaction-diffusion equations in one dimension.

Invariance Principle for a Critical Galton–Watson Process Attaining a High Level

An invariance principle is established for a critical Galton–Watson process attaining a high level. The limiting process is the limit, as $\varepsilon \to 0$ of the continuous Feller branching process given it starts from the level $\varepsilon>0$ and attains level 1. An iterative procedure is outlined for calculating finite-dimensional Laplace transforms of the limiting process.

ℕ-measure for continuous state branching processes and its application

In this paper, we first give a direct construction of the ℕ-measure of a continuous state branching process. Then we prove, with the help of this ℕ-measure, that any continuous state branching process with immigration can be constructed as the independent sum of a continuous state branching process (without immigration), and two immigration parts (jump immigration and continuum immigration). As an application of this construction of a continuous state branching process with immigration, we give a proof of a necessary and sufficient condition, first stated without proof by M. A. Pinsky [Bull. Amer. Math. Soc., 1972, 78: 242–244], for a continuous state branching process with immigration to a proper almost sure limit. As another application of the ℕ-measure, we give a “conceptual” proof of an L log L criterion for a continuous state branching process without immigration to have an L 1-limit first proved by D. R. Grey [J. Appl. Prob., 1974, 11: 669–677].

Robert Bentley Todd's Contribution to Cell Theory and The Neuron Doctrine

Robert Bentley Todd, who is best remembered for “Todd's paralysis,” made many more important contributions to neurology and neuroscience, including the concept of brain electricity and electrical discharges in epilepsy. He was also a pioneering microscopist and we here review his neurohistological studies and his contributions to the application of Schwann's (1839) cell theory to the nervous system and the later neuron doctrine, as described in his textbook The Descriptive and Physiological Anatomy of the Brain, Spinal Cord and Ganglions (Todd, 1845), his Cyclopaedia of Anatomy and Physiology (1847) and his joint textbook with William Bowman The Physiological Anatomy and Physiology of Man (1845). Writing in the mid-1840s, Todd acknowledged that the “vesicles” he observed corresponded to the earlier descriptions of “globules” or “kugeln” by Valentin and which Schwann first interpreted as cell bodies. Todd was among the first to recognize that nerve cell bodies were in continuity with axons (“axis cylinders”), sometimes associated with “the white substance of Schwann” (“tubular” fibers), or sometimes without (“gelatinous” fibers). He also described continuous nerve cell branching processes, later called dendrites. He was the first to recognize the insulating properties of Schwann's “white substance” (myelin) to facilitate conduction. Influenced by his contemporary, Faraday, Todd was also the first to develop the functional concept of dynamic polarization (“nervous polarity”) to explain nerve cell conduction.

On continuous branches of very singular similarity solutions of a stable thin film equation. I – The Cauchy problem

We consider the fourth-order thin film equation, with a stable second-order diffusion term. For the first critical exponent, where N ≥ 1 is the space dimension, the Cauchy problem is shown to admit countable continuous branches of source-type self-similar very singular solutions of the form These solutions are inherently oscillatory in nature and will be shown in Part II to be the limit of appropriate free-boundary problem solutions. For p ≠ p0, the set of very singular solutions is shown to be finite and to be consisting of a countable family of branches (in the parameter p) of similarity profiles that originate at a sequence of critical exponents {pl, l ≥ 0}. At p = pl, these branches appear via a non-linear bifurcation mechanism from a countable set of similarity solutions of the second kind of the pure thin film equation Such solutions are detected by the ‘Hermitian spectral theory’, which allows an analytical n-branching approach. As such, a continuous path as n → 0+ can be constructed from the eigenfunctions of the linear rescaled operator for n = 0, i.e. for the bi-harmonic equation ut = −Δ2u. Numerics are used, wherever appropriate, to support the analysis.

On continuous branches of very singular similarity solutions of the stable thin film equation. II – Free-boundary problems

We discuss the fourth-order thin film equation with a stable second-order diffusion term, in the context of a standard free-boundary problem with zero height, zero contact angle and zero-flux conditions imposed at an interface. For the first critical exponent where N ≥ 1 is the space dimension, there are continuous sets (branches) of source-type very singular self-similar solutions of the form For p ≠ p0, the set of very singular self-similar solutions is shown to be finite and consists of a countable family of branches (in the parameter p) of similarity profiles that originate at a sequence of critical exponents {pl, l ≥ 0}. At p = pl, these branches appear via a non-linear bifurcation mechanism from a countable set of second kind similarity solutions of the pure thin film equation Such solutions are detected by a combination of linear and non-linear ‘Hermitian spectral theory’, which allows the application of an analytical n-branching approach. In order to connect with the Cauchy problem in Part I, we identify the cauchy problem solutions as suitable ‘limits’ of the free-boundary problem solutions.

Stochastic continuous time neurite branching models with tree and segment dependent rates

In this paper we introduce a continuous time stochastic neurite branching model closely related to the discrete time stochastic BES-model. The discrete time BES-model is underlying current attempts to simulate cortical development, but is difficult to analyze. The new continuous time formulation facilitates analytical treatment thus allowing us to examine the structure of the model more closely. We derive explicit expressions for the time dependent probabilities p ( γ , t ) for finding a tree γ at time t, valid for arbitrary continuous time branching models with tree and segment dependent branching rates. We show, for the specific case of the continuous time BES-model, that as expected from our model formulation, the sums needed to evaluate expectation values of functions of the terminal segment number μ ( f ( n ) , t ) do not depend on the distribution of the total branching probability over the terminal segments. In addition, we derive a system of differential equations for the probabilities p( n, t) of finding n terminal segments at time t. For the continuous BES-model, this system of differential equations gives direct numerical access to functions only depending on the number of terminal segments, and we use this to evaluate the development of the mean and standard deviation of the number of terminal segments at a time t. For comparison we discuss two cases where mean and variance of the number of terminal segments are exactly solvable. Then we discuss the numerical evaluation of the S-dependence of the solutions for the continuous time BES-model. The numerical results show clearly that higher S values, i.e. values such that more proximal terminal segments have higher branching rates than more distal terminal segments, lead to more symmetrical trees as measured by three tree symmetry indicators.

High-Frequency Noise in Modern FET/HEMT Channels Caused by the Excitation of 2D-Plasma Waves

The problems related with the intrinsic noise in FET/HEMT channels induced by continuous branching of the total current between channel and gate are considered in the framework of a simple analytical model and its predictions on the current-noise spectra. Main branching-induced effects such as the appearance of an additional noise related to the excitation of plasma waves, its dependence on FET/HEMT embedding circuits, interference properties, etc. are analysed.

Catalytic Branching Random Walk on a Two-Dimensional Lattice

We study a critical continuous time catalytic branching random walk on two-dimensional lattice under the assumption that the particles' birth and death occur at a single source of branching located...

Advanced Bifurcation Systems’ mother-daughter platform

Description of the device and methodology The Advanced Bifurcation Systems (ABS Los Angeles, CA, USA) line of modular bifurcation systems is a novel stent platform designed to uniformly mould all coronary bifurcations regardless of angles and sizes in a simple, intuitive procedure. Orientation and alignment of main branch (mother) and side branch (daughter) segments is automatic with complete tissue coverage, evenly conforming to the anatomy without overlapping stent struts, while providing continuous side branch access. The ABS line of devices consist of a modular, dual-catheter, independently movable positioning system (currently 7 Fr) for complete multi-branch as well as provisional side branch stenting. The full bifurcation stenting system (MD-BI) consists of two balloon catheters, mother-daughter balloon catheter and daughter balloon catheter and two stents, mother-daughter stent and daughter stent. The mother-daughter balloon catheter (MDC) has a mother-daughter stent (MDS) with an aperture through which the daughter balloon catheter (DC) extrudes and leads the system. The MDS is fully circumferentially (360 degrees) crimped on the MDC, only distal to this aperture. The MDS is only partially crimped proximal to the aperture in such a fashion that the shaft of the DC can slide back and forth. In other words, the MDS is only partially longitudinally and circumferentially crimped on its delivery balloon in different segments. The daughter balloon catheter has a stent mounted only on the distal half. Additionally, the mother balloon catheter has a “sleeve” proximally through which the daughter catheter is loaded. This sleeve ensures the coaxial movement of the mother and daughter catheters. The daughter catheter leads the system by a few centimetres. A clip placed at the proximal hub of the two catheters allows for simultaneous or independent movement of the catheters. After predilatation, the system is loaded on the two wires (main and side branch wires) and advanced until it reaches the carina at which point it can not be advanced any further. The DC balloon, which now is in the daughter vessel, is pulled back, while the MDC balloon is held in place. Pull-back is completed when the proximal half of the DC is in the proximal segment of the MDS and the two proximal balloon markers are aligned next to each other. The DS, which is on the distal half of the DC balloon, is now abutting the mother stent, and a bifurcating stent is assembled on the site. The DC balloon is then inflated first, which simultaneously deploys the proximal segment of the MDS, as well as the DS, thus ensuring perfect alignment of the two stents and proper orientation of the proximal segment of the MDS. This automatic alignment is accomplished using the only fixed (non-variable) structure in any bifurcation, namely the carina. This simultaneous deployment and alignment also ensures full tissue coverage. The MDC balloon is inflated next followed by kissing inflation. The sequences of system deployment is shown in Figures 1A, 1B. The diameter sizing of the system is based on the equation governing the sizes of branching arteries according to the concept of pseudo-fractals known as the Finet’s law. This states that the diameter of a mother vessel is 0.678 times of the sum of the two daughter branches: M = 0.678 (D 1 +D 2 ). The provisional side branch stenting (MD-P) device has a shorter balloon on the DC and does not have a stent mounted on it. The deployment method is otherwise similar.

Branching asymptotics on manifolds with edge

We study pseudo-differential operators on a wedge with continuous and variable discrete branching asymptotics. Our approach relies on the fact that such asymptotics for r → 0 in the image under the Mellin transform turn to poles of corresponding meromorphic functions, depending on the edge variable y. Those represent families of analytic functionals that reproduce the asymptotics as values on the holomorphic function r−z. Variable and branching means that we admit variable multiplicities of these poles. We show that the corresponding y-wise discrete asymptotics are preserved under the action of Green and Mellin operators of the edge calculus.

Exact multiplicity of nematic states for an Onsager model

Isotropic–nematic phase transition due to excluded volume effects is considered for the 2D Onsager model of rigid rod-like polymers with a general interaction kernel involving a finite number of Fourier modes. Nematic phases appear above a concentration threshold. We prove the existence of continuous branches of nematic states bifurcating from the isotropic phase at concentrations proportional to the Fourier coefficients of the interaction kernel. The bifurcation structure is shown to depend on the size of the spectral gaps of the interaction operator. Exact multiplicity of nematic phases is proved for a class of two-mode trigonometric kernels. Our arguments use simple bifurcation and variational tools applied to an equivalent finite dimensional problem that involves multi-variable Bessel functions.

Coupled dielectric waveguides with photonic crystal properties

A system of planar waveguides coupled through a periodic set of small windows is considered. It is shown that the weak periodic coupling of the waveguides leads to an additional eigenvalue band separated by a gap from the threshold of a continuous spectrum branch. Thus, the system has photonic crystal properties, which can be used to construct optical fiber devices. This system can also play the role of a SCISSOR device for reducing the group velocity of light, in optical delay lines, etc.

Nonlinear PDEs and measure-valued branching type processes

We deal with the probabilistic approach to a nonlinear operator Λ of the form Λ u = Δ u + ∑ k = 1 ∞ q k u k , in connection with the works of M. Nagasawa, N. Ikeda, S. Watanabe, and M.L. Silverstein on the discrete branching processes. Instead of the Laplace operator we may consider the generator of a right (Markov) process, called base process, with a general (not necessarily locally compact) state space. It turns out that solutions of the nonlinear equation Λ u = 0 are produced by the harmonic functions with respect to the (linear) generator of a discrete branching type process. The consideration of the general state space allows to take as base process a measure-valued superprocess (in the sense of E.B. Dynkin). The probabilistic counterpart is a Markov process which is a combination between a continuous branching process (e.g., associated with a nonlinear operator of the form Δ u − u α , 1 < α ⩽ 2 ) and a discrete branching type one, on a space of configurations of finite measures. Our approach uses probabilistic and analytic potential theoretical tools, like the potential kernel of a continuous additive functional and the subordination operators.

Qualitative analysis of solutions to discrete static contact problems with Coulomb friction

We analyze properties of solutions to discrete contact problems with Coulomb friction which are parametrized by the coefficient of friction F . Using a generalized variant of the implicit-function theorem we establish conditions under which there exists a local Lipschitz continuous branch of solutions around a reference point. Finally, a piecewise smooth continuation algorithm which allows to follow such branches of solutions is proposed.

A hybrid continuum–discrete modelling approach to predict and control angiogenesis: analysis of combinatorial growth factor and matrix effects on vessel-sprouting morphology

Angiogenesis is crucial during many physiological processes and is influenced by various biochemical and biomechanical factors. Models have proved useful in understanding the mechanisms of angiogenesis and also the characteristics of the capillaries formed as part of the process. We have developed a three-dimensional hybrid, agent-field model where individual cells are modelled as sprout-forming agents in a matrix field. Cell independence, cell-cell communication and stochastic cell response are integral parts of the model. The model simulations incorporate probabilities of an individual cell to transition into one of four stages--quiescence, proliferation, migration and apoptosis. We demonstrate that several features, such as continuous sprouts, cell clustering and branching, that are observed in microfluidic experiments conducted under controlled conditions using few angiogenic factors can be reproduced by this model. We also identify the transition probabilities that result in specific sprout characteristics such as long continuous sprouts and specific branching patterns. Thus, this model can be used to cluster sprout morphology as a function of various influencing factors.

Non-Born–Oppenheimer quantum chemistry on the fly with continuous path branching due to nonadiabatic and intense optical interactions

We extend our formerly proposed theory for non-Born-Oppenheimer electronic and nuclear wavepacket dynamics within on-the-fly scheme [T. Yonehara, S. Takahashi, and K. Takatsuka, J. Chem. Phys. 130, 214113 (2009)] to a case of nonadiabatic dynamics under an intense laser field: electron wavepacket in a molecule is propagated in attosecond time-scale along non-Born-Oppenheimer nuclear paths that smoothly branch due to nonadiabatic coupling and/or optical interactions. Such branching paths are determined consistently with the motion of the electron wavepackets. Furthermore, these nuclear paths are quantized in terms of Gaussian wavepackets (action decomposed function), which can be applied to nonclassical paths. Both electronic wavepacket dynamics and quantization of non-Born-Oppenheimer paths are generalized so as to include the direct effects of the classical vector potential of electromagnetic fields. In the second half of this paper, we perform numerical studies to explore nonadiabatic dynamics in a laser field by examining two cases: one is a two-state model system having an avoided crossing, and the other is two-state dynamics in HF molecule on the two low lying ab initio potential curves. Both are placed in laser fields. With the former system, we survey some basic properties of the coupling of nonadiabatic dynamics and laser interaction varying the relevant coupling parameters such as the laser timing with respect to the incident of nonadiabatic transition. This investigation will set a foundation for the future studies of control of electronic states in realistic multidimensional molecular systems. Application to the latter system shows that non-Born-Oppenheimer quantum chemistry in laser fields is indeed useful in the study of dynamics in ab initio level. Through the comparison with full quantum data, we verify that the formalism and methodology developed here work accurately. Furthermore, we attain some basic insight about the characteristics of molecules in laser fields.

Branching with Uncertain Semantics

Branching with Uncertain Semantics

A limit theorem for trees of alleles in branching processes with rare neutral mutations

We are interested in the genealogical structure of alleles for a Bienaymé–Galton–Watson branching process with neutral mutations (infinite alleles model), in the situation where the initial population is large and the mutation rate small. We shall establish that for an appropriate regime, the process of the sizes of the allelic sub-families converges in distribution to a certain continuous state branching process (i.e. a Jiřina process) in discrete time. Itô’s excursion theory and the Lévy–Itô decomposition of subordinators provide fundamental insights for the results.

Dynamics of the time to the most recent common ancestor in a large branching population

If we follow an asexually reproducing population through time, then the amount of time that has passed since the most recent common ancestor (MRCA) of all current individuals lived will change as time progresses. The resulting “MRCA age” process has been studied previously when the population has a constant large size and evolves via the diffusion limit of standard Wright–Fisher dynamics. For any population model, the sample paths of the MRCA age process are made up of periods of linear upward drift with slope +1 punctuated by downward jumps. We build other Markov processes that have such paths from Poisson point processes on ℝ++×ℝ++ with intensity measures of the form λ⊗μ where λ is Lebesgue measure, and μ (the “family lifetime measure”) is an arbitrary, absolutely continuous measure satisfying μ((0, ∞))=∞ and μ((x, ∞))<∞ for all x>0. Special cases of this construction describe the time evolution of the MRCA age in (1+β)-stable continuous state branching processes conditioned on nonextinction—a particular case of which, β=1, is Feller’s continuous state branching process conditioned on nonextinction. As well as the continuous time process, we also consider the discrete time Markov chain that records the value of the continuous process just before and after its successive jumps. We find transition probabilities for both the continuous and discrete time processes, determine when these processes are transient and recurrent and compute stationary distributions when they exist. Moreover, we introduce a new family of Markov processes that stands in a relation with respect to the general (1+β)-stable continuous state branching process and its conditioned version that is similar to the one between the family of Bessel-squared diffusions and the unconditioned and conditioned Feller continuous state branching process.