Verification of system properties of polynomial systems using discrete-time approximations and set-based analysis

Magdeburg, Univ., Fak. für Elektrotechnik und Informationstechnik, Diss., 2015

Polynomial-time complexity for instances of the endomorphism problem in free groups

We say the endomorphism problem is solvable for an element W in a free group F if it can be decided effectively whether, given U in F, there is an endomorphism Φ of F sending W to U. This work analyzes an approach due to C. Edmunds and improved by C. Sims. Here we prove that the approach provides an efficient algorithm for solving the endomorphism problem when W is a two- generator word. We show that when W is a two-generator word this algorithm solves the problem in time polynomial in the length of U. This result gives a polynomial-time algorithm for solving, in free groups, two-variable equations in which all the variables occur on one side of the equality and all the constants on the other side.

Limit Cycles for a class of three dimensional polynomial differential systems

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Polynomial and linearized normal forms for almost periodic differential systems

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On the asymptotic expansion of the colored Jones polynomial for torus knots

In the asymptotic expansion of the hyperbolic specification of the colored Jones polynomial of torus knots, we identify different geometric contributions, in particular Chern-Simons invariant and Reidemeister torsion.

The Euler characteristic of a category

The Euler characteristic of a finite category is defined and shown to be compatible with Euler characteristics of other types of object, including orbifolds. A formula is proved for the cardinality of a colimit of sets, generalizing the classical inclusion-exclusion formula. Both rest on a generalization of Rota's Möbius inversion from posets to categories.

Chebyshev polynomial approximation to approximate partial differential equations

This paper suggests a simple method based on Chebyshev approximation at Chebyshev nodes to approximate partial differential equations. The methodology simply consists in determining the value function by using a set of nodes and basis functions. We provide two examples. Pricing an European option and determining the best policy for chatting down a machinery. The suggested method is flexible, easy to program and efficient. It is also applicable in other fields, providing efficient solutions to complex systems of partial differential equations.

Algebraic properties of isochronous centers of polynomial quadratic-like Hamiltonian systems

Projecte de recerca elaborat a partir d’una estada a la School of Mathematics and Statistics de la University of Plymouth, United Kingdom, entre abril juliol del 2007.Aquesta investigació és encara oberta i la memòria que presento constitueix un informe de la recerca que estem duent a terme actualment. En aquesta nota estudiem els centres isòcrons dels sistemes Hamiltonians analítics, parant especial atenció en el cas polinomial. Ens centrem en els anomenats quadratic-like Hamiltonian systems. Diverses propietats dels centres isòcrons d'aquest tipus de sistemes van ser donades a [A. Cima, F. Mañosas and J. Villadelprat, Isochronicity for several classes of Hamiltonian systems, J. Di®erential Equations 157 (1999) 373{413]. Aquell article estava centrat principalment en el cas en que A; B i C fossin funcions analítiques. El nostre objectiu amb l'estudi que estem duent a terme és investigar el cas en el que aquestes funcions són polinomis. En aquesta nota formulem una conjectura concreta sobre les propietats algebraiques que venen forçades per la isocronia del centre i provem alguns resultats parcials.

Polynomial functors and trees

We explore the relationship between polynomial functors and trees. In the first part we characterise trees as certain polynomial functors and obtain a completely formal but at the same time conceptual and explicit construction of two categories of rooted trees, whose main properties we describe in terms of some factorisation systems. The second category is the category Ω of Moerdijk and Weiss. Although the constructions are motivated and explained in terms of polynomial functors, they all amount to elementary manipulations with finite sets. Included in Part 1 is also an explicit construction of the free monad on a polynomial endofunctor, given in terms of trees. In the second part we describe polynomial endofunctors and monads as structures built from trees, characterising the images of several nerve functors from polynomial endofunctors and monads into presheaves on categories of trees. Polynomial endofunctors and monads over a base are characterised by a sheaf condition on categories of decorated trees. In the absolute case, one further condition is needed, a projectivity condition, which serves also to characterise polynomial endofunctors and monads among (coloured) collections and operads.

A new look at the Heston characteristic function

A new expression for the characteristic function of log-spot in Heston model is presented. This expression more clearly exhibits its properties as an analytic characteristic function and allows us to compute the exact domain of the moment generating function. This result is then applied to the volatility smile at extreme strikes and to the control of the moments of spot. We also give a factorization of the moment generating function as product of Bessel type factors, and an approximating sequence to the law of log-spot is deduced.

Polynomial functors and polynomial monads

We study polynomial functors over locally cartesian closed categories. After setting up the basic theory, we show how polynomial functors assemble into a double category, in fact a framed bicategory. We show that the free monad on a polynomial endofunctor is polynomial. The relationship with operads and other related notions is explored.

Characteristic reflection patterns in the southeast Canadian cordillera, northern Appalachians and Swiss Alps

There are some striking similarities and some differences between the seismic reflection sections recorded across the fold and thrust belts of the southeast Canadian Cordillera, Quebec-Maine Appalachians and Swiss Alps. In the fold and thrust belts of all three mountain ranges, seismic reflection surveys have yielded high-quality images of. (1) nappes (thin thrust sheets) stacked on top of ancient continental margins; (2) ramp anticlines in the hanging walls of faults that have ramp-flat or listric geometries; (3) back thrusts and back folds that developed during the terminal phases of orogeny; and (4) tectonic wedges and regional decollements. A principal result of the Cordilleran and Appalachian deep crustal studies has been the recognition of master decollements along which continental margin strata have been transported long distances, whereas a principal result of the Swiss Alpine deep crustal program has been the identification of the Adriatic indenter, a crustal-scale wedge that caused delamination of the European lithosphere. Significant crustal roots are observed beneath the fold and thrust belts of the Alps, southeast Canadian Cordillera and parts of the southern Appalachians, but such structures beneath the northern Appalachians have probably been removed by post-orogenic collapse and/or crustal attenuation associated with the Mesozoic opening of the Atlantic Ocean.

Weak-type estimates for Haar shift operators: Sharp power on the A(p) characteristic

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Inversion of analytic characteristic functions and infinite convolutions of exponential and Laplace densities

This paper shows that certain quotients of entire functions are characteristic functions. Under some conditions, we provide expressions for the densities of such characteristic functions which turn out to be generalized Dirichlet series which in turn can be expressed as an infinite linear combination of exponential or Laplace densities. We apply these results to several examples.

Characteristic bimodal profiles of RNA polymerase II at thousands of active mammalian promoters.

BACKGROUND: In mammals, ChIP-seq studies of RNA polymerase II (PolII) occupancy have been performed to reveal how recruitment, initiation and pausing of PolII may control transcription rates, but the focus is rarely on obtaining finely resolved profiles that can portray the progression of PolII through sequential promoter states. RESULTS: Here, we analyze PolII binding profiles from high-coverage ChIP-seq on promoters of actively transcribed genes in mouse and humans. We show that the enrichment of PolII near transcription start sites exhibits a stereotypical bimodal structure, with one peak near active transcription start sites and a second peak 110 base pairs downstream from the first. Using an empirical model that reliably quantifies the spatial PolII signal, gene by gene, we show that the first PolII peak allows for refined positioning of transcription start sites, which is corroborated by mRNA sequencing. This bimodal signature is found both in mouse and humans. Analysis of the pausing-related factors NELF and DSIF suggests that the downstream peak reflects widespread pausing at the +1 nucleosome barrier. Several features of the bimodal pattern are correlated with sequence features such as CpG content and TATA boxes, as well as the histone mark H3K4me3. CONCLUSIONS: We thus show how high coverage DNA sequencing experiments can reveal as-yet unnoticed bimodal spatial features of PolII accumulation that are frequent at individual mammalian genes and reminiscent of transcription initiation and pausing. The initiation-pausing hypothesis is corroborated by evidence from run-on sequencing and immunoprecipitation in other cell types and species.