Computing the permanental polynomials of bipartite graphs by Pfaffian orientation

AbstractThe permanental polynomial of a graph G is π(G,x)≜per(xI−A(G)). From the result that a bipartite graph G admits an orientation Ge such that every cycle is oddly oriented if and only if it contains no even subdivision of K2,3, Yan and Zhang showed that the permanental polynomial of such a bipartite graph G can be expressed as the characteristic polynomial of the skew adjacency matrix A(Ge). In this note we first prove that this equality holds only if the bipartite graph G contains no even subdivision of K2,3. Then we prove that such bipartite graphs are planar. Unexpectedly, we mainly show that a 2-connected bipartite graph contains no even subdivision of K2,3 if and only if it is planar 1-cycle resonant. This implies that each cycle is oddly oriented in any Pfaffian orientation of a 2-connected bipartite graph containing no even subdivision of K2,3. Accordingly, we give a way to compute the permanental polynomials of such graphs by Pfaffian orientation

Biological Electron Transfer Mediators in Model Membrane Systems

The overall objective of this research was to create chemically well-defined liquid redox membranes, each of which would be capable of mediating a process of continuous electron transfer between an aqueous reductant on one side and an aqueous oxidant on the other. The transport of electrons between the two otherwise separate aqueous phases would be effected by membrane-bound reversible redox-active carrier molecules. It has been proposed that such molecules function in certain electron transfer processes associated with biological membranes. In particular, the isoprenoid benzoquinone coenzyme Q is thought to carry electrons (and protons) across the inner mitochondrial membrane during respiration, whilst the related naphthoquinone vitamin K acts as a carrier in photosynthetic and bacterial electron transfer. Since these two molecules were also found to possess all the attributes required of 'ideal' carriers, they were chosen for incorporation in all the redox membranes investigated during this research, in the hope that their behaviour in these simple model systems could be used to elucidate their biological function. Preliminary investigations were carried out to find suitable aqueous reductants and reducible substrates. A wide variety of common redox reagents, and several biological molecules, were tested for their ability to reduce coenzyme Q10 and vitamin K1, or to reoxidise their quinol forms, in biphasic reactions in which the quinone/quinol in hexane solution was equilibrated anaerobically over the aqueous reagent. A number of criteria were specified defining 'ideal' reductants, substrates and carriers, and control experiments were performed to assess the ideality of the reagents used. In general terms, coenzyme Q was found to be reduced by a wider range of reductants than vitamin K, while dihydrovitamin K was much more easily reoxidised than reduced coenzyme Q. Both quinones reacted readily at pH 7 with reduced methyl viologen and flavin mononucleotide free radicals, which behaved as ideal reductants. Electron paramagnetic resonance spectroscopy revealed the presence of semiquinone free radical species of coenzyme Q and vitamin K at the hexane/aqueous interface during these reactions. To establish continuous membrane-mediated electron transfer processes, experiments were conducted in an H-shaped reaction vessel in which the quinone solution (in hexane) formed a bridge between the aqueous reductant and aqueous oxidant in the lower limbs. Reduced methyl viologen, in large molar excess over the quinone and substrate, was routinely used as a reductant, and the kinetics of reduction of various substrates were followed spectrophotometrically. Both coenzyme Q and vitamin K were shown to function as reversible electron carriers in these bulk membrane systems, but the rate of substrate reduction was always very much slower when coenzyme Q was used. For this reason, more detailed studies were restricted to vitamin K as electron carrier. The kinetics of reduction of methylene blue by dihydrovitamin K were first order with respect to the oxidised dye, and the measured rate constant was consistent with diffusion control on the substrate side of the interface. The reaction was inhibited to varying degrees by the addition of different amphipathic phospholipids to the membrane. With cytocnrome c as substrate, the biphasic reaction with dihydrovitamin K was no longer diffusion-controlled, but was determined by mechanistic factors. The very slow, apparently zero-order reaction was greatly stimulated by the addition of the mitochondrial phospholipid cardio-lipin to the membrane, and the variation of the reaction rate with ionic strength of the aqueous phase could be explained in terms of binding between the acidic phospholipid and the basic protein. Such interactions are proposed to be of importance in the functional organisation of the mitochondrial membrane. Having established the abilities of coenzyme Q and vitamin K to act as electron carriers across a bulk hydrocarbon phase, attempts were made to improve the biological model by reducing the thickness of the membrane to the dimensions of a lipid bilayer. The two model systems studied were planar bimolecular lipid membranes (BLM) and closed unilamellar lipid vesicles (liposomes). The stabilities and thicknesses of a large number of BLM formed from a selection of amphipathic lipids and lipid mixtures were examined, and 'recipes' were found for membrane-forming solutions which yielded stable bilayers containing either vitamin K or coenzyme Q. A membrane cell was developed, allowing electron transfer across ultrathin membranes to be followed spectrophotometrically. Unfortunately, the instability of lipid bilayers within the apparatus did not allow kinetics experiments to be performed using BLM. However, electron transfer across thicker lens membranes, mediated by vitamin K, was demonstrated. (Abstract shortened by ProQuest.)

Periodic time dependent Hamiltonian systems and applications

[eng] A dynamical system is one that evolves with time. This definition is so diffuse that seems to be completely useless, however, gives a good insight of the vast range of applicability of this field of Mathematics has. It is hard to track back in the history of science to find the origins of this discipline. The works by Fibonacci, in the twelfth century, concerning the population growth rate of rabbits can be already considered to belong to the above mentioned field. Newton's legacy changed the prism through the humankind watched the universe and established the starting shot of several areas of knowledge including the study of difierential equations. Newton's second law relates the acceleration, the second derivative of the position of a body with the net force acting upon it. The formulation of the law of universal gravitation settled the many body problem, the fundamental question around the field of celestial mechanics has grown. Newton itself solved the two body problem, providing an analytical proof of Kepler's laws. In the subsequent years a number of authors, among of them Euler and Lagrange, exhausted Newton's powerful ideas but none of them was able to find a closed solution of the many body problem. By the end of the nineteenth century, Poincaré changed again the point of view: The French mathematician realized that the many body problem could not be solved in the sense his predecessors expected, however, many other fundamental questions could be addressed by studying the solutions of not quantitatively but by means of their geometrical and topological properties. The ideas that bloomed in Poincaré's mind are nowadays a source of inspiration for modern scientist facing problems located along all the spectrum of human knowledge. Poincaré understood that invariant structures organize the long term behaviour of the solutions of the system. Invariant objects are, therefore, the skeleton of the dynamics. These invariant structures and their linear normal behaviour are to be analyzed carefully and this shall lead to a good insight on global aspects of the phase space. For nonintegrable systems the task of studying invariant objects and their stability is, in general, a problem which is hard to be handled rigorously. Usually, the hypotheses needed to prove specific statements on the solutions of the systems reduce the applicability of the results. This is especially relevant in physical problems: Indeed, we cannot, for instance, choose the mass of Sun to be suficiently small. The advent of the computers changed the way to undertake studies of dynamical systems. The task of writing programs for solving, numerically, problems related to specific examples is, at the present time, as important as theoretical studies. This has two main consequences: On the first hand, more involved models can be chosen to study real problems and this allow us to understand better the relation between abstract concepts and physical phenomena. Secondly, even when facing fundamental questions on dynamics, the numerical studies give us data from which build our theoretical developments. Nowadays, a large number of commercial (or public) software packages helps scientist to study simple problems avoiding the tedious work to master numerical algorithms and programming languages. These programs are coded to work in the largest possible number of different situations, therefore, they do not have the eficiency that programs written specifically for a certain purpose have. Some of the computations presented in this dissertation cannot be performed by using commercial software or, at least, not in a reasonable amount of time. For this reason, a large part of the work presented here has to do with coding and debugging programs to perform numerical computations. These programs are written to be highly eficient and adapted to each problem. At the same time, the design is done so that specific blocks of the code can be used for other computations, that is, there exist a commitment between eficiency and reusability which is hard to achieve without having full control on the code. Under these guiding principles we undertake the study of applied dynamical systems according to the following stages: From a particular problem we get a simple model, then perform a number of numerical experiments that permits us to understand the invariant objects of the system, with that information, we can isolate the relevant phenomena and identify the key elements playing a role on it. Next, we try to find an even simpler model in which we can develop theoretical arguments and produce theorems that, with more effort, can be generalized or related to other problems which, in principle, seem to be difierent to the original one. Paraphrasing Carles Simó, from a physical problem we can take the lift to the abstract world, use theoretical arguments, come out with conclusions and, finally, lift down to the real world and apply these conclusions to specific problems (maybe not only the original one). This methodology has been developed in the last decades over the world when it turned out to outstand among the most powerful approaches to cope with problems in applied mathematics. The group of Dynamical Systems from Barcelona has been one of the bulwarks of this development from the late seventies to the present days. Following the guidelines presented in the previous section, we concern with several problems, mostly from the field of celestial mechanics but we also deal with a phenomenon coming from high energy physics. All these situations can be modeled by means of periodically time dependent Hamiltonian systems. To cope with those investigations, we develop software which can be used to perform computations in any periodically perturbed Hamiltonian system. We split the contents of this dissertation in two parts. The first one is devoted to general tolos to handle periodically time dependent Hamiltonians, even though we fill this first part with a number of illustrating examples, the goal is to keep the exposition in the abstract setting. Most of the contents of Part I deal with the development of software used to be applied in the second part. Some of the software has not been applied to the specific contents of Part II, this is left for future work. We also devote a whole chapter to some theoretical issues that, while are motivated by physical problems, they fall out of the category of periodic time dependent Hamiltonians. This splitting of contents has the intention of reecting, somehow, the basic methodological principles presented in the previous paragraph, keeping separated the abstract and the physical world but keeping in mind the lift

Stochastic Analysis

The meeting took place on May 30-June 3, 2011, with over 55 people in attendance. Each day had 6 to 7 talks of varying length (some talks were 30 minutes long), except for Thursday: the traditional hike was moved to Thursday due to the weather (and weather on thursday was indeed fine). The talks reviewed directions in which progress in the general field of stochastic analysis occurred since the last meeting of this theme in Oberwolfach three years ago. Several themes were covered in some depth, in addition to a broad overview of recent developments. Among these themes a prominent role was played by random matrices, random surfaces/planar maps and their scaling limits, the KPZ universality class, and the interplay between SLE (Schramm-Loewner equation) and the GFF (Gaussian free field)

Degeneracies in the Eigenvalue Spectrum of Quantum Graphs

In this dissertation, we analyze the spectrum of the Laplace operator on graphs. In particular, we are interested in generic eigenpairs. We consider a wide range of vertex conditions on vertices of a quantum graph. Furthermore, we also investigate the eigenfunctions, showing that generically they do not vanish on vertices, unless this is unavoidable due to presence of looping edges. In the proof, the simplicity of eigenvalues and non-vanishing of eigenvalues are tightly interconnected; each property is assisting in the proof of the other (the proof is done by induction). The proof is geometric in nature and uses local modifications of the graph to reduce it to previously considered cases. We also consider an application of the result to the study of the secular manifold of a graph, showing that for large classes of graphs, the set of smooth points of the manifold has exactly two connected components. The spectrum of a symmetric quantum graph is also considered. We aim to give explicit and computation-oriented formulas for extracting the part of a Schrödinger operator on a graph which corresponds to a particular irreducible representation of the graph's symmetry. Starting with a representation of the symmetry by its action on the space of directed bonds of the graph, we find a basis which block-diagonalizes both the representation and the bond scattering matrix of the graph. The latter leads to a factorization of the secular determinant into factors that correspond to irreducible representation of the symmetry group

The role of hyperbolic invariant objects: From Arnold diffusion to biological clocks

El marc d'aquesta tesi són els objectes invariants hiperbòlics (tors amb bigotis, cicles límit, NHIM,. . .), que constitueixen, per aquesta tesi, els objectes essencials per a l'estudi de diversos problemes des de la difusió d'Arnold fins als rellotges biològics. Treballem en tres temes diferents des d'un enfocament tant teòric com numèric, amb una especial atenció per a les aplicacions, especialment en neurobiologia:· Existència de difusió d'Arnold per a sistemes Hamiltonians a priori inestables· Algorismes numèrics ràpids per al càlcul de tors invariants i els "bigotis" associats, per a sistemes Hamiltonians utilitzant el mètode de la parametrització.· Càlcul d'isòcrones i corbes de resposta de fase (PRC) en sistemes neurobiològics usant el mètode de la parametrització.En la primera part de la tesi, hem considerat el cas d'un sistema Hamiltonià a priori inestable amb 2+1/2 graus de llibertat sotmès a una pertorbació de tipus general. "A priori inestable" significa que el sistema no pertorbat presenta un punt d'equilibri hiperbòlic amb una òrbita homoclínica associada. El resultat principal d'aquesta part de la tesi és que per a un conjunt genèric de pertorbacions prou regulars, el sistema presenta el fenòmen de la difusió d'Arnold, és a dir, existeixen trajectòries la variable acció de les quals experimenta un canvi d'ordre 1. La demostració es basa en un estudi detallat de les zones ressonants i els objectes invariants generats en elles, i ofereix una descripció completa de la geografia de les ressonàncies generades per una pertorbació genèrica.En la segona part d'aquest memòria, desenvolupem mètodes numèrics eficients que requereixen poca memòria i operacions per al càlcul de tors invariants i els "bigotis" associats en sistemes Hamiltonians (aplicacions simplèctiques i camps vectorials Hamiltonians).En particular, això inclou els objectes invariants involucrats en el mecanisme de la difusió d'Arnold, estudiat en el capítol anterior. Els algorismes es basen en el mètode de la parametrització i segueixen de prop demostracions recents del teorema KAM que no usen variables acció-angle. Donem detalls de la implementació numèrica que hem dut a terme i mostrem alguns exemples.En la darrera part de la tesi relacionem problemes de temps en sistemes biològics amb algunes eines conegudes de sistemes dinàmics. En particular, usem el mètode de la parametrització i les simetries de Lie per a calcular numèricament les isòcrones i les corbes de resposta de fase (PRC) associades a oscil·ladors i ho apliquem a diversos models biològics ben coneguts. A més a més, aconseguim estendre el càlcul de PRCs en un entorn de l'oscil·lador. Les PRCs són útils per a l'estudi de la sincronització d'oscil·ladors acoblats i una eina bàsica en biologia experimental (ritmes circadians, acoblament sinàptic i elèctric de neurones,. . . )

Computation and Physics in Algebraic Geometry

Physics provides new, tantalizing problems that we solve by developing and implementing innovative and effective geometric tools in nonlinear algebra. The techniques we employ also rely on numerical and symbolic computations performed with computer algebra. First, we study solutions to the Kadomtsev-Petviashvili equation that arise from singular curves. The Kadomtsev-Petviashvili equation is a partial differential equation describing nonlinear wave motion whose solutions can be built from an algebraic curve. Such a surprising connection established by Krichever and Shiota also led to an entirely new point of view on a classical problem in algebraic geometry known as the Schottky problem. To explore the connection with curves with at worst nodal singularities, we define the Hirota variety, which parameterizes KP solutions arising from such curves. Studying the geometry of the Hirota variety provides a new approach to the Schottky problem. We investigate it for irreducible rational nodal curves, giving a partial solution to the weak Schottky problem in this case. Second, we formulate questions from scattering amplitudes in a broader context using very affine varieties and D-module theory. The interplay between geometry and combinatorics in particle physics indeed suggests an underlying, coherent mathematical structure behind the study of particle interactions. In this thesis, we gain a better understanding of mathematical objects, such as moduli spaces of point configurations and generalized Euler integrals, for which particle physics provides concrete, non-trivial examples, and we prove some conjectures stated in the physics literature. Finally, we study linear spaces of symmetric matrices, addressing questions motivated by algebraic statistics, optimization, and enumerative geometry. This includes giving explicit formulas for the maximum likelihood degree and studying tangency problems for quadric surfaces in projective space from the point of view of real algebraic geometry