Betti numbers of toric varieties and eulerian polynomials

It is well-known that the Eulerian polynomials, which count permutations in $S_n$ by their number of descents, give the $h$-polynomial/$h$-vector of the simple polytopes known as permutohedra, the convex hull of the $S_n$-orbit for a generic weight in the weight lattice of $S_n$. Therefore the Eulerian polynomials give the Betti numbers for certain smooth toric varieties associated with the permutohedra. In this paper we derive recurrences for the $h$-vectors of a family of polytopes generalizing this. The simple polytopes we consider arise as the orbit of a non-generic weight, namely a weight fixed by only the simple reflections $J=\{s_{n},s_{n-1},s_{n-2} \cdots,s_{n-k+2},s_{n-k+1}\}$ for some $k$ with respect to the $A_n$ root lattice. Furthermore, they give rise to certain rationally smooth toric varieties $X(J)$ that come naturally from the theory of algebraic monoids. Using effectively the theory of reductive algebraic monoids and the combinatorics of simple polytopes, we obtain a recurrence formula for the Poincar\'e polynomial of $X(J)$ in terms of the Eulerian polynomials

The Morse Lemma in Infinite Dimensions via Singularity Theory

An infinite dimensional Morse lemma is proved using the deformation lemma from singularity theory. It is shown that the versions of the Morse lemmas due to Palais and Tromba are special cases. An infinite dimensional splitting lemma is proved. The relationship of the work here to other approaches in the literature in discussed

A Study of Optimal 4-bit Reversible Toffoli Circuits and Their Synthesis

Optimal synthesis of reversible functions is a non-trivial problem. One of the major limiting factors in computing such circuits is the sheer number of reversible functions. Even restricting synthesis to 4-bit reversible functions results in a huge search space (16! {\approx} 2^{44} functions). The output of such a search alone, counting only the space required to list Toffoli gates for every function, would require over 100 terabytes of storage. In this paper, we present two algorithms: one, that synthesizes an optimal circuit for any 4-bit reversible specification, and another that synthesizes all optimal implementations. We employ several techniques to make the problem tractable. We report results from several experiments, including synthesis of all optimal 4-bit permutations, synthesis of random 4-bit permutations, optimal synthesis of all 4-bit linear reversible circuits, synthesis of existing benchmark functions; we compose a list of the hardest permutations to synthesize, and show distribution of optimal circuits. We further illustrate that our proposed approach may be extended to accommodate physical constraints via reporting LNN-optimal reversible circuits. Our results have important implications in the design and optimization of reversible and quantum circuits, testing circuit synthesis heuristics, and performing experiments in the area of quantum information processing.Comment: arXiv admin note: substantial text overlap with arXiv:1003.191

Observation and inverse problems in coupled cell networks

A coupled cell network is a model for many situations such as food webs in ecosystems, cellular metabolism, economical networks... It consists in a directed graph $G$, each node (or cell) representing an agent of the network and each directed arrow representing which agent acts on which one. It yields a system of differential equations $\dot x(t)=f(x(t))$, where the component $i$ of $f$ depends only on the cells $x_j(t)$ for which the arrow $j\rightarrow i$ exists in $G$. In this paper, we investigate the observation problems in coupled cell networks: can one deduce the behaviour of the whole network (oscillations, stabilisation etc.) by observing only one of the cells? We show that the natural observation properties holds for almost all the interactions $f$

Forced patterns near a Turing-Hopf bifurcation

We study time-periodic forcing of spatially-extended patterns near a Turing-Hopf bifurcation point. A symmetry-based normal form analysis yields several predictions, including that (i) weak forcing near the intrinsic Hopf frequency enhances or suppresses the Turing amplitude by an amount that scales quadratically with the forcing strength, and (ii) the strongest effect is seen for forcing that is detuned from the Hopf frequency. To apply our results to specific models, we perform a perturbation analysis on general two-component reaction-diffusion systems, which reveals whether the forcing suppresses or enhances the spatial pattern. For the suppressing case, our results explain features of previous experiments on the CDIMA chemical reaction. However, we also find examples of the enhancing case, which has not yet been observed in experiment. Numerical simulations verify the predicted dependence on the forcing parameters.Comment: 4 pages, 4 figure

Canonical Characteristic Sets of Characterizable Differential Ideals

We study the concept of canonical characteristic set of a characterizable differential ideal. We propose an efficient algorithm that transforms any characteristic set into the canonical one. We prove the basic properties of canonical characteristic sets. In particular, we show that in the ordinary case for any ranking the order of each element of the canonical characteristic set of a characterizable differential ideal is bounded by the order of the ideal. Finally, we propose a factorization-free algorithm for computing the canonical characteristic set of a characterizable differential ideal represented as a radical ideal by a set of generators. The algorithm is not restricted to the ordinary case and is applicable for an arbitrary ranking.Comment: 26 page

Bifurcations in nonlinear models of fluid-conveying pipes supported at both ends

Stationary bifurcations in several nonlinear models of fluid conveying pipes fixed at both ends are analyzed with the use of Lyapunov-Schmidt reduction and singularity theory. Influence of gravitational force, curvature and vertical elastic support on various properties of bifurcating solutions are investigated. In particular the conditions for occurrence of supercritical and subcritical bifurcations are presented for the models of Holmes, Thurman and Mote, and Paidoussis.Comment: to appear in Journal of Fluids and Structures; 6 figure

A torus bifurcation theorem with symmetry

Hopf bifurcation in the presence of symmetry, in situations where the normal form equations decouple into phase/amplitude equations is described. A theorem showing that in general such degeneracies are expected to lead to secondary torus bifurcations is proved. By applying this theorem to the case of degenerate Hopf bifurcation with triangular symmetry it is proved that in codimension two there exist regions of parameter space where two branches of asymptotically stable two-tori coexist but where no stable periodic solutions are present. Although a theory was not derived for degenerate Hopf bifurcations in the presence of symmetry, examples are presented that would have to be accounted for by any such general theory

Equivariant singularity theory with distinguished parameters: Two case studies of resonant Hamiltonian systems

We consider Hamiltonian systems near equilibrium that can be (formally) reduced to one degree of freedom. Spatio-temporal symmetries play a key role. The planar reduction is studied by equivariant singularity theory with distinguished parameters. The method is illustrated on the conservative spring-pendulum system near resonance, where it leads to integrable approximations of the iso-energetic Poincaré map. The novelty of our approach is that we obtain information on the whole dynamics, regarding the (quasi-) periodic solutions, the global configuration of their invariant manifolds, and bifurcations of these.