Random walks in two-dimensional complexes

A 2-dimensional complex is a union of a finite number of quarter planes ₊ ² having some boundaries in common. The most interesting example is the union of all 2-dimensional faces of ₊ ^N . We consider maximally homogeneous random walks on such complexes and obtain necessary and sufficient conditions for ergodicity, null recurrence and transience up to some non-zero assumptions which are of measure 1 in the parameter space.The problem we address in this paper is of theoretical range. However, the results can be applied to performance evaluation of some telecommunication systems (e.g. local area networks) viewed as interacting queues. To enforce this assertion, a detailed example of coupled queues in differentregimes is presented.     

A periodicity theorem for the octahedron recurrence

The octahedron recurrence lives on a 3-dimensional lattice and is given by . In this paper, we investigate a variant of this recurrence which lives in a lattice contained in . Following Speyer, we give an explicit non-recursive formula for the values of this recurrence and use it to prove that it is periodic of period n+m. We then proceed to show various other hidden symmetries satisfied by this bounded octahedron recurrence. An earlier version of this work has circulated under the name “A coboundary category defined using the octahedron recurrence.”     

Numerically satisfactory solutions of Kummer recurrence relations

Pairs of numerically satisfactory solutions as for the three-term recurrence relations satisfied by the families of functions , , are given. It is proved that minimal solutions always exist, except when and z is in the positive or negative real axis, and that is minimal as whenever . The minimal solution is identified for any recurrence direction, that is, for any integer values of and . When the confluent limit , with fixed, is the main tool for identifying minimal solutions together with a connection formula; for , is the main tool to be considered.     

Economic implications of poor access to antenatal care in rural and remote Western Australian Aboriginal communities: An individual sampling model of pregnancy

Background

Australian Aboriginal women attend antenatal care less frequently and experience poorer pregnancy outcomes than non-Aboriginal women. Improving access to antenatal care is recognised as a means to improve pregnancy outcomes for mother and baby.

Objective

To estimate the costs of inadequate antenatal care and provide baseline measurements and implications for policy that targets improving access to care in rural and remote Western Australian (WA) Aboriginal communities.

Methods

An individual sampling model of pregnancy was developed that simulated hypothetical women with pregnancy events and outcomes observed in the WA Aboriginal population. Weekly pregnancy events were modelled via logistic regression according to maternal characteristics, events during pregnancy and current gestation, with adequate and inadequate care (?4 and <4 antenatal visits) to reflect differences in outcomes reported in the literature. The pregnancy model simulated clinical management including antenatal visits, hospitalisations, and transfers to tertiary care.

Results

The mean cost of pregnancy was AUD$8985, with a large difference depending on access to antenatal care (AUD$7635 and AUD$10,216 for adequate and inadequate care respectively). The main difference in costs resulted from neonatal care (AUD$1021 vs AUD$3205 for adequate and inadequate care respectively). In a rural community with 150 births per year, up to AUD$123,082 may be spent to improve access to care at no extra cost to the total current expenditure (AUD$1,347,733).

Conclusions

The large difference in pregnancy costs between those receiving adequate and inadequate care demonstrates that additional expenditure on improving access to antenatal care may be cost-effective and should be further investigated.     

A thermodynamic classification of real numbers

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A new classification scheme for real numbers is given, motivated by ideas from statistical mechanics in general and work of Knauf (1993) [16] and Fiala and Kleban (2005) [8] in particular. Critical for this classification of a real number will be the Diophantine properties of its continued fraction expansion.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=qnPF2QS4cRg.     

A Favard theorem for orthogonal rational functions on the unit circle

We consider forn=0, 1,... the nested spaces _n of rational functions of degreen at most with given poles . Given a finite measure supported on the unit circle, we associate with it a nested orthogonal basis of rational functions ₀,..., _n for _n ,n=0, 1,.... These _n satisfy a recurrence relation that generalizes the recurrence for Szeg polynomials.In this paper we shall prove a Favard type theorem which says that if one has a sequence of rational functions _{n n} which are generated by such a recurrence, then there will be a measure supported on the unit circle to which they are orthogonal. We shall give a sufficient condition for the uniqueness of this measure.     

Recurrence Criteria for Generalized Dirichlet Forms

We develop sufficient analytic conditions for recurrence and transience of non-sectorial perturbations of possibly non-symmetric Dirichlet forms on a general state space. These form an important subclass of generalized Dirichlet forms which were introduced in Stannat (Ann Scuola Norm Sup Pisa Cl Sci (4) 28(1):99–140, 1999). In case there exists an associated process, we show how the analytic conditions imply recurrence and transience in the classical probabilistic sense. As an application, we consider a generalized Dirichlet form given on a closed or open subset of \(\mathbb {R}^d\) which is given as a divergence free first-order perturbation of a non-symmetric energy form. Then, using volume growth conditions of the sectorial and non-sectorial first-order part, we derive an explicit criterion for recurrence. Moreover, we present concrete examples with applications to Muckenhoupt weights and counterexamples. The counterexamples show that the non-sectorial case differs qualitatively from the symmetric or non-symmetric sectorial case. Namely, we make the observation that one of the main criteria for recurrence in these cases fails to be true for generalized Dirichlet forms.     

Is the recurrence relation for orthogonal polynomials always stable?

Attention is drawn to a phenomenon of pseudostability in connection with the three-term recurrence relation for discrete orthogonal polynomials. The computational implications of this phenomenon are illustrated in the case of discrete Legendre and Krawtchouk polynomials. The phenomenon also helps to explain a form of instability in Stieltjes's procedure for generating recursion coefficients of discrete orthogonal polynomials.Work supported in part by the National Science Foundation under grant DMS-9023403.     

On projective N-curvature inheritance

The concept of a Lie recurrence was introduced by the first author?[6]. It is an infinitesimal transformation $\overline{x}^{i}={x}^{i}+\varepsilon {v}^{i}({x}^{j})$ with respect to which the Lie derivative of a curvature tensor is proportional to itself. Apart from other results related to a Lie recurrence, it was established that the Weyl projective curvature tensor is Lie recurrent with respect to a Lie recurrence but its converse is not necessarily true. However, an infinitesimal transformation with respect to which the Weyl projective curvature tensor and the Ricci tensor are Lie recurrent, is necessarily a Lie recurrence. Singh?[12] studied an infinitesimal transformation with respect to which the Lie derivative of the curvature tensor is proportional to itself and called such transformation as curvature inheritance. Obviously, a curvature inheritance is nothing but a Lie recurrence. Singh?[13] also considered a curvature inheritance which is a projective motion and called it a projective curvature inheritance. Gatoto and Singh [1,2] studied $\widetilde{K}$ -curvature inheritance and projective $\widetilde{K}$ -curvature inheritance. Pandey and Pandey?[9] studied $\widetilde{K}$ projective Lie recurrence. Mishra and Yadav?[3] studied projective curvature inheritance in an NP-F _n . In the present paper we have established that an infinitesimal transformation in a Finsler space is Lie recurrence if and only if the normal projective curvature tensor is Lie recurrent. A part from this result we have generalized almost all theorems of Mishra and Yadav?[3].     

On the mean value of the index of composition of an integral ideal

Text

In this paper, we study the mean value of the index of composition of an integral ideal, and get an asymptotic formula for it.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=oAdWsFz3gKM.     

Regularity of solutions and interfaces of a generalized porous medium equation inR N

Summary We consider the Cauchy problem for the generalized porous medium equation u_t=(u) where u=u(x, t), xRⁿ and t>0, and the initial datum u(x, 0) is assumed to be nonnegative, integrable mid to nave compact support. The nonlinearity (u) is a C¹ function defined for uO which grows like a power of u. Our assumptions generalize the porous medium case, (u)=u^m, m>1, and also include the equation of the Marshak waves. This problem has finite speed of propagation. We estimate the rate of growth of the support of the solution with precise estimates for t 0 and t. Our main result deals with the regularity of the solutions. We show that after a certain time t₀ the pressure, defined by v=(u), with (u)=(u)/u and (0)=0, is a Lipschitz-continuous function of x and t and the interface is a Lipschitz-continuous surface in R^N+1; the solution u is Hölder continuous for all times t> 0.Both authors partially supported by CAICYT, Project 2805-83. The second author also supported by USA-Spain Joint Research Grant CCB-8402023.     

Link maps and the geometry of their invariants

The geometry of two types of link homotopy invariants of a link map f:S^pS^qS^m is discussed. The first one is the -invariant which greatly generalizes the classical notion of linking number. The second one, the -invariant, is closely related to the linking behaviour of f|s^p with only the double point set of f|S^q, and therefore measures (to some extend) the obstruction to embedding S^q. These invariants are related by a Hopf invariant homomorphism. In many cases link maps are classified up to link homotopy here, and a setting is provided e.g. for future injectivity results for . Also the image of is studied, yielding an interesting double filtration of stable homotopy groups of spheres.     

On the Structure of Cimpact Simply Connected Manifolds of Positive Sectional Curvature

We present a first structure theorem for compact simply connected positively curved manifolds with arbitrarily small pinching constants: For each nN and 0<1, there exists a positive number V = V(n,) such that if (M,g) is a compact simply connected n-dimensional Riemannian manifold with sectional curvature 0相似文献   

Class Numbers of Indefinite Binary Quadratic Forms

Let be the class number of properly equivalent primitive binary quadratic forms of discriminant . The case of indefinite forms is considered. Assuming that the extended Riemann hypothesis for some fields of algebraic numbers holds, the following results are proved. 1. Let be an arbitrarily slow monotonically increasing function such that . Then

Feller Processes Generated by Pseudo-Differential Operators: On the Hausdorff Dimension of Their Sample Paths

Let {X _t} _t0 be a Feller process generated by a pseudo-differential operator whose symbol satisfiesÇⁿ|q(Ç,)|c(1=)()) for some fixed continuous negative definite function (). The Hausdorff dimension of the set {X _t:tE}, E [0, 1] is any analytic set, is a.s. bounded above by dim E. is the Blumenthal–Getoor upper index of the Levy Process associated with ().     

FE simulation of human trachea swallowing movement before and after the implantation of an endoprothesis

Background and aims

Nowadays interventions associated to the implantation of tracheal stents in patients with airway pathologies, are a very common surgery that in the post-operating period can bear many problems such as migration of the stent, development of granulation tissue at the edges of the stent with overgrowth of the tracheal lumen, or accumulation of secretions inside the prosthesis. Among the movements that trachea carries out, swallowing seems to drive harmful consequences for the tracheal tissues surrounding the prosthesis. In this work a finite element model of a human trachea has been developed and used to analyze its behavior during swallowing.

Material and methods

In the present work, a complete human trachea finite element model based on experimental study was developed. The real swallowing movement of two patients before and after the implantation of Dumon prosthesis was used to simulate and then analyze the effect that the tracheal implant has on the stress response of the trachea and on the physiological capacity to swallow.

Results

In both studied cases with an implanted Dumon prosthesis, patients showed a decrease of their ability to swallow; one lost 26.4% and the other one 18.9% of their tracheal ascending movements. Besides, the prosthesis implantation caused an increase of the stresses located in the superior contact border between the tracheal wall and the prosthesis. It could be seen that the resulting force equivalent to the elevating tracheal muscle forces for degluting, was around F = 13.5 N for the two patients both before and after the stent implantation.

Conclusion

The implantation of a Dumon prosthesis modifies the mechanical response of the trachea altering its stress distribution and its ascending movement.     

The Peterson recurrence formula for the chromatic discriminant of a graph

The absolute value of the coefficient of q in the chromatic polynomial of a graph G is known as the chromatic discriminant of G and is denoted α(G). There is a well known recurrence formula for α(G) that comes from the deletion-contraction rule for the chromatic polynomial. In this paper we prove another recurrence formula for α(G) that comes from the theory of Kac- Moody Lie algebras. We start with a brief survey on many interesting algebraic and combinatorial interpretations of α(G). We use two of these interpretations (in terms of acyclic orientations and spanning trees) to give two bijective proofs for our recurrence formula of α(G).     

A Variant of the ORTHOMIN(2) Method for Singular Linear Systems

For singular linear systems A x=b, ORTHOMIN(2) is known theoretically to attain the minimum residual min _xR nb–A x₂ under a certain condition. However, in the actual computation with finite precision arithmetic, the residual is often observed to be reduced further than the theoretically expected level. Therefore, we propose a variant of ORTHOMIN(2), which is mathematically equivalent to the original ORTHOMIN(2) method, but uses recurrence formulas that are different from those of ORTHOMIN(2); they contain alternative expressions for the auxiliary vector and the recurrence coefficients. Although our implementation has the same computational costs as ORTHOMIN(2), numerical experiments on singular systems show that our implementation is more accurate and less affected by rounding errors than ORTHOMIN(2).     

The diffusive phase of a model of self-interacting walks

Summary We consider simple random walk onZ ^d perturbed by a factor exp[T ^–P J _T], whereT is the length of the walk and . Forp=1 and dimensionsd2, we prove that this walk behaves diffusively for all – < <₀, with ₀ > 0. Ford>2 the diffusion constant is equal to 1, but ford=2 it is renormalized. Ford=1 andp=3/2, we prove diffusion for all real (positive or negative). Ford>2 the scaling limit is Brownian motion, but ford2 it is the Edwards model (with the wrong sign of the coupling when >0) which governs the limiting behaviour; the latter arises since for ,T ^–p J _T is the discrete self-intersection local time. This establishes existence of a diffusive phase for this model. Existence of a collapsed phase for a very closely related model has been proven in work of Bolthausen and Schmock.     

A note on Poincaré recurrence for semigroup actions via Furstenberg family

This paper presents an explanation of recurrence for semigroup actions from two points of view. The first one considers the concept of recurrence in the Poincaré sense as stated in Souza (Semigroup Forum 83:351–370, 2011) while the second one considers the notion of recurrence via a Fursternberg family as defined in Akin (Recurrence in Topological Dynamics. Furstenberg and Ellis Action. Plenum Press, New York, 1997). A relationship between these notions of recurrence is presented.