On the stability of limit cycles for planar differential systems

We consider a planar differential system , , where P and Q are C¹ functions in some open set U⊆R², and . Let γ be a periodic orbit of the system in U. Let f(x,y):U⊆R²→R be a C¹ function such that

     

Claw Conditions for Heavy Cycles in Weighted Graphs

A graph is called a weighted graph when each edge e is assigned a nonnegative number w(e), called the weight of e. For a vertex v of a weighted graph, d^w(v) is the sum of the weights of the edges incident with v. For a subgraph H of a weighted graph G, the weight of H is the sum of the weights of the edges belonging to H. In this paper, we give a new sufficient condition for a weighted graph to have a heavy cycle. A 2-connected weighted graph G contains either a Hamilton cycle or a cycle of weight at least c, if G satisfies the following conditions: In every induced claw or induced modified claw F of G, (1) max{d^w(x),d^w(y)} c/2 for each non-adjacent pair of vertices x and y in F, and (2) all edges of F have the same weight.     

Improvement of duty-cycle heating compensation in NMR spin relaxation experiments

To reliably measure NMR relaxation properties of macromolecules is a prerequisite for precise experiments that identify subtle variations in relaxation rates, as required for the determination of rotational diffusion anisotropy, CSA tensor determination, advanced motional modeling or entropy difference estimations. An underlying problem with current NMR relaxation measurement protocols is maintaining constant sample temperature throughout the execution of the relaxation series especially when rapid data acquisition is required. Here, it is proposed to use a combination of a heating compensation and a proton saturation sequence at the beginning of the NMR relaxation pulse scheme. This simple extension allows reproducible, robust and rapid acquisition of NMR spin relaxation data sets. The method is verified with (15)N spin relaxation measurements for human ubiquitin.     

On Two-Dimensional Area-Preserving Diffeomorphisms with Infinitely Many Elliptic Islands

We consider two-parameter families of C ^r-smooth, r6, two-dimensional area-preserving diffeomorphisms that have structurally unstable simplest heteroclinic cycles. We find the conditions when diffeomorphisms under consideration possess infinitely many periodic generic elliptic points and elliptic islands.     

Signed domination numbers of directed graphs

The concept of signed domination number of an undirected graph (introduced by J. E. Dunbar, S. T. Hedetniemi, M. A. Henning and P. J. Slater) is transferred to directed graphs. Exact values are found for particular types of tournaments. It is proved that for digraphs with a directed Hamiltonian cycle the signed domination number may be arbitrarily small.     

Limit distributions for risk processes in case of claim amounts of finite expectation

We prove limit theorems for the distribution of $\text{Z}{}_{\mathrm{l}}\text{if {Z}{}_{\mathrm{l}}\text{:l ?}\text{R}{}_{\mathrm{+}}\text{}}$ is a risk process with claim amounts of finite mean. The results are illustrated by several examples and counterexamples.     

Transitive permutation groups and equipotent voting rules

Let F a two-alternative voting rule and G_F the subgroup of permutations of the voters under which F is invariant. Group theoretic properties of G_F provide information about the voting rule F. In particular, sets of imprimitivity of G_F describe the ‘committee decomposition’ structure of F and permutation group transitivity of G_F (equipotency) is shown to be closely connected with equal distribution of power among the voters. If equipotency replaces anonymity in the hypotheses of May's theorem, voting rules other than simple majority are possible. By combining equipotency with two additional social choice conditions a new characterization of simple majority rule is obtained. Equipotency is proposed as an important alternative to the more restrictive anonymity as a fairness criterion in social choice.     

Irreversible behavior of interacting systems. I. The approach to equilibrium

The Kac ring model is used to test the validity of some conjectures about irreversibility. If the whole system is regarded as the universe, then it is demonstrated that all clocks (subsystems) run in the same direction during those times when the universe is not in equilibrium. In addition, mathematical techniques are introduced by means of which the appropriate order parameter for large, finite Kac ring models can be evaluated asymptotically. It is shown that the relaxation of this order parameter to its equilibrium value of zero is not exponential.     

Systematics of the models of immune response and autoimmune disease

A dynamical model of normal immune response has been formulated in terms of cellular automata by Kaufmanet al. We generalize this model incorporating the antigens as a dynamical variable. This generalized model not only describes the kinetics of primary and secondary responses of humoral immunity, together with the appropriate memory cells, but also describes the vaccinated state as well as the states of low-dose and high-dose paralysis. Recently models of autoimmune response have also been developed in terms of discrete automata. But the models are underdetermined by the experimental facts, i.e., several models can account for the same set of observed biological facts. With an aim to find out how large this underdeterminacy is and how it can be reduced systematically, we have carried out an exhaustive computer-aided search of all those discrete three-cell and five-cell models of autoimmune response which at present cannot be ruled out by the existing biological informations. Out of the 3²⁵ possible five-cell models, only one fulfilled our criteria. We also carried out simulations of the dynamics of some of these models on a discrete lattice. We discuss the relevance of random interactions in the context of autoimmune disease.