Alessandro Arsie

University of Toledo ·  Department of Mathematics and Statistics

We show that a bi-flat F-structure $(\nabla,\circ,e,\nabla^*,*,E)$ on a manifold $M$ defines a differential bicomplex $(d_{\nabla},d_{E\circ\nabla^*})$ on forms with value on the tangent sheaf of the manifold. Moreover, the sequence of vector fields defined recursively by $d_{\nabla}X_{(\alpha+1)}=d_{L\nabla^*}X_{(\alpha)}$ coincide with the coeffi...

In this paper, we generalize the Givental theory for Frobenius manifolds and cohomological field theories to flat F-manifolds and F-cohomological field theories. In particular, we define a notion of Givental cone for flat F-manifolds, and we provide a generalization of the Givental group as a matrix loop group acting on them. We show that this acti...

In this paper, we investigate collision orbits of two identical bodies placed on the surface of a two-dimensional sphere and interacting via an attractive potential of the form $V(q)=-\cot(q)$, where $q$ is the angle formed by the position vectors of the two bodies. We describe the $\omega$-limit set of the variables in the symplectically reduced s...

Generalizing a construction presented in Arsie and Lorenzoni (Lett Math Phys 107:1919–1961, 2017), we show that the orbit space of B2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begi...

This paper is devoted to the high codimension bifurcations of a classical predator–prey system with Allee effects and generalized Holling type III functional response \(p(x)=\frac{mx^2}{ax^2+bx+1}\) where \(b>-2\sqrt{a}\). We show that the maximal orders of nilpotent saddle, cusp singularity and weak focus are all three. The unfoldings of a cusp si...

The transition between strong and weak Allee effects in prey provides a simple regime shift in ecology. In this paper, we study the interplay between the functional response of Holling type IV and both strong and weak Allee effects. The model investigated here presents complex dynamics and high codimension bifurcations. In particular, nilpotent cus...

Generalizing a construction presented in [3], we show that the orbit space of $B_2$ less the image of coordinate lines under the quotient map is equipped with two Dubrovin-Frobenius manifold structures which are related respectively to the defocusing and the focusing nonlinear Schr\"odinger (NLS) equations. Motivated by this example, we study the c...

We define the double ramification hierarchy associated to an F-cohomological field theory and use this construction to prove that the principal hierarchy of any semisimple (homogeneous) flat F-manifold possesses a (homogeneous) integrable dispersive deformation at all orders in the dispersion parameter. The proof is based on the reconstruction of a...

In this paper, we study relations between various natural structures on F-manifolds. In particular, given an arbitrary Riemannian F-manifold, we present a construction of a canonical flat F-manifold associated to it. We also describe a construction of a canonical homogeneous Riemannian F-manifold associated to an arbitrary exact homogeneous flat pe...

In this paper we study relations between various natural structures on F-manifolds. In particular, given an arbitrary Riemannian F-manifold we present a construction of a canonical flat F-manifold associated to it. We also describe a construction of a canonical homogeneous Riemannian F-manifold associated to an arbitrary exact homogeneous flat penc...

We define the double ramification hierarchy associated to an F-cohomological field theory and use this construction to prove that the principal hierarchy of any semisimple (homogeneous) flat F-manifold possesses a (homogeneous) integrable dispersive deformation at all orders in the dispersion parameter. The proof is based on the reconstruction of a...

The transition between strong and weak Allee effects in prey provides a simple regime shift in ecology. A deteriorating environment changes weak Allee effects into strong ones. In this paper, we study the interplay between the functional response of Holling type IV and both strong and weak Allee effects. The model investigated here presents complex...

In this paper we generalize the Givental theory for Frobenius manifolds and cohomological field theories to flat F-manifolds and F-cohomological field theories. In particular, we define the notion of Givental cone for flat F-manifolds, and we provide a generalization of the Givental group as a matrix loop group acting on them. We show that this act...

This paper presents a mathematical analysis of a modified tubular bell. Tubular bells are often modeled as a vibrating beam with free boundary conditions. However, most orchestral tubular bells contain a single cap attached to one end of the tube. This cap affects both the strike note as well as the relative frequencies of the overtones. However, t...

We address the problem of the asymptotic reducibility of a system of analytic quasi-periodic linear functional differential equations extending the results of M.S. Bortei and V.I. Fodchuk [3] to the case of (possibly) complex eigenvalues. At the same time, we provide a simpler proof of the asymptotic reducibility that bypasses the method of acceler...

Designing a robot swarm requires a swarm designer to understand the trade-offs unique to a swarm. The most basic design decisions are how many robots there should be in the swarm and the individual robot size. These choices in turn impact swarm cost and robot interference, and therefore swarm performance. The underlying physical reasons for why the...

Leveraging on the work of De Vore and Zuazua, we further explore their methodology and deal with two open questions presented in their paper. We show that for a class of linear evolutionary PDEs the admissible choice of relevant parameters used to construct the near-optimal sampling sequence is not influenced by the spectrum of of the operator cont...

We present a procedure to construct a bi-flat $F$-manifold structure on the orbit space of well-generated complex reflection groups. We apply this procedure to the exceptional well-generated complex reflection groups of rank $2$ and $3$, we show that the dual connection coincides with a Dunkl-Kohno-type connection associated with such groups and ar...

In this paper, we study the extension of the minimizing equal mass parallelogram solutions which was derived by Chen in 2001 [2]. Chen's solution was minimizing for one quarter of the period [0; T], where numerical integration had been used in his proof. In this paper we extend Chen’s solution in the reduced space to [0; 4T] and we show that this e...

We extend some of the results proved for scalar equations in Arsie et al. (2015, Nonlinearity, 28) and Arsie et al. (2015, Proc. R. Soc. A, 471, doi:10.1098/rspa.2014.0124), to the case of systems of integrable conservation laws. In particular, for such systems we prove that the eigenvalues of a matrix obtained from the quasilinear part of the syst...

We study an isospectral flow (Lax flow) that provides an explicit deformation from upper Hessenberg complex matrices to normal matrices, extending to the complex case and to the case of normal matrices the results of [2]. The Lax flow is given bydAdt=[[A†,A]du,A], where brackets indicate the usual matrix commutator, [A,B]:=AB−BA, A† is the conjugat...

We propose an extension of the Dubrovin–Zhang perturbative approach to the study of normal forms for non-Hamiltonian integrable scalar conservation laws. The explicit computation of the first few corrections leads to the conjecture that such normal forms are parameterized by one single functional parameter, named the viscous central invariant. A co...

In this paper we study $F$-manifolds equipped with multiple flat connections (and multiple $F$-products), that are required to be compatible in a suitable sense. Multi-flat $F$-manifolds are the analogue for $F$-manifolds of Frobenius manifolds with multi-Hamiltonian structures. We show that a necessary condition for the existence of such multiple...

We study normal forms of scalar integrable dispersive (non necessarily Hamiltonian) conservation laws via the Dubrovin-Zhang perturbative scheme. Our computations support the conjecture that such normal forms are parametrised by infinitely many arbitrary functions that can be identified with the coefficients of the quasilinear part of the equation....

In this paper, we extend purely non-local Hamiltonian formalism to a class of Riemannian F-manifolds, without assumptions on the semisimplicity of the product $\circ$ or on the flatness of the connection $\nabla$. In the flat case we show that the recurrence relations for the principal hierarchy can be re-interpreted using a local and purely non-lo...

In this paper we introduce an isospectral flow (Lax flow) that deforms real Hessenberg matrices to Jacobi matrices isospectrally. The Lax flow is given by(Formula presented.) where brackets indicate the usual matrix commutator, [A, B] : = AB−BA, AT is the transpose of A and the matrix [AT, A]du is the matrix equal to [AT, A] along diagonal and uppe...

Motivated by the theory of integrable PDEs of hydrodynamic type and by the generalization of Dubrovin's duality in the framework of F-manifolds due to Manin (2005) [7], we consider a special class of F-manifolds, called bi-flat F-manifolds.A bi-flat F-manifold is given by the following data (M,∇1,∇2,∘,∗,e,E), where (M,∘) is an F-manifold, e is the...

We consider the action of a special class of reciprocal transformation on the principal hierarchy associated to a semisimple $F$-manifold with compatible flat structure $(M,\circ,\nabla,e)$. Under some additional assumptions, the hierarchy obtained applying these reciprocal transformations is also associated to an $F$-manifold with compatible flat...

We introduce a bracket on 1-forms defined on ${\cal J}^{\infty}(S^1, \mathbb{R}^n)$, the infinite jet extension of the space of loops and prove that it satisfies the standard properties of a Poisson bracket. Using this bracket, we show that certain hierarchies appearing in the framework of $F$-manifolds with compatible flat connection $(M, \nabla,...

Motivated by the theory of integrable PDEs of hydrodynamic type and by the generalization of Dubrovin's duality in the framework of $F$-manifolds due to Manin [22], we consider a special class of $F$-manifolds, called bi-flat $F$-manifolds. A bi-flat $F$-manifold is given by the following data $(M, \nabla_1,\nabla_2,\circ,*,e,E)$, where $(M, \circ)...

F-manifolds with eventual identities were introduced by Manin in [17]. In this work, we investigate what these structures entail from the point of view of integrable partial differential equations of hydrodynamic type, especially in terms of the related recurrence relations. After having defined new equivalence relations for connections compatible...

A widely applied strategy for workload sharing is to equalize the workload assigned to each resource. In mobile multiagent systems, this principle directly leads to equitable partitioning policies whereby: 1) the environment is equitably divided into subregions of equal measure; 2) one agent is assigned to each subregion; and 3) each agent is respo...

In this paper we study some properties of bi-Hamiltonian deformations of Poisson pencils of hydrodynamic type. More specifically, we are interested in determining those structures of the fully deformed pencils that are inherited through the interaction between structural properties of the dispersionless pencils (in particular exactness or homogenei...

In this paper we are interested in non trivial bi-Hamiltonian deformations of the Poisson pencil $\omega_{\lambda}=\omega_2+\lambda \omega_1=u\delta'(x-y)+\f{1}{2}u_x\delta(x-y)+\lambda\delta'(x-y)$. Deformations are generated by a sequence of vector fields $\{X_2, X_4,...\}$, where each $X_{2k}$ is homogenous of degree $2k$ with respect to a gradi...

This work deals with the problem of locating the omega-limit set Omega of a bounded solution of a given autonomous vector field f on a Riemannian manifold. Assuming to know that the omega-limit set 52 is contained in an embedded submanifold S and using an auxiliary function that we call height function W for f and S. we show how to obtain a better...

The present work deals with a dynamical system of the form A¿ = [[N, A^T+A], A]+¿[[A^T, A], A], where A¿ is an n×n real matrix, N is constant n×n real matrix, [A, B] = AB-BA, and ¿ is a positive constant. In particular, the purpose of this work is to establish structure preserving properties of this dynamical system for tr...

In this paper, we consider a class of dynamic vehicle routing problems, in which a number of mobile agents in the plane must visit target points generated over time by a stochastic process. It is desired to design motion coordination strategies in order to minimize the expected time between the appearance of a target point and the time it is visite...

This work deals with the problem of locating the omega-limit set of a bounded solution of a given autonomous vector field on a Riemannian manifold. The derived results extend LaSalle's invariance principle in such a way that the newly obtained conditions provide in certain situations a more refined statement about the location of the omega-limit se...

The most widely applied resource allocation strategy is to balance, or equalize, the total workload assigned to each resource. In mobile multi-agent systems, this principle directly leads to equitable partitioning policies in which (i) the workspace is divided into subregions of equal measure, (ii) there is a bijective correspondence between agents...

The most widely applied resource allocation strategy is to balance, or equalize, the total workload assigned to each resource. In mobile multi-agent systems, this principle directly leads to equitable partitioning policies in which (i) the workspace is divided into subregions of equal measure, (ii) there is a bijective correspondence between agents...

In this paper, we consider a class of dynamic vehicle routing problems, in which a number of mobile agents in the plane must visit target points generated over time by a stochastic process. It is desired to design motion coordination strategies in order to minimize the expected time between the appearance of a target point and the time it is visite...

We consider motion planning of human self- rotations; that is, human-body rotations without external torques, which are common in microgravity, diving, and gymnastics. In these cases it may be difficult to formulate an objective function that leads to maneuvers that are appropriate for humans to perform in high-stress situations. For example, cogni...

We present an analysis of the motion planning problem for a driftless quantized control system on SO(3). The system under investigation depends on two control quanta, beta₁ and beta₂. We are able to provide efficient finite complexity motion plans which steer the system arbitrarily close to any given target configuration as so...

This work is dedicated to Professor Roger W. Brockett on the occasion of his 70th birthday Abstract. This paper deals with a dynamical system of the form ˙ A =[[N, A T + A],A]+ν[[A T,A],A], where A is an n × n real matrix, N is a constant n × n real matrix, ν is a positive constant and [A, B] =AB − BA. In particular, the purpose of this paper is to...

IntroductionProblem formulationControl policy descriptionPerformance analysis in light loadA performance analysis for sTP, mTP/FG and mTP policiesSome numerical resultsConclusions References

The aim of this paper is twofold. On one hand we present an approach to the general problem of nonlinear control in the framework of (differentiable) groupoids, which, in our opinion deserves further in- vestigation. On the other hand, using recently-developed algebraic tools, we show that for a control system whose state space is a semisimple Lie...

Quite recently, Alexeev and Nakamura proved that if Y is a stable semi-Abelic variety (SSAV) of dimension g equipped with the ample line bundle OY (1), which deforms to a principally polarized Abelian variety, then OY (n) is very ample as soon as n 2g + 1, that is n 5 in the case of surfaces. Here it is proved, via elementary methods of projective...

The notion of geometric k-normality for curves is introduced in complete generality and is investigated in the case of nodal and cuspidal curves living on several types of surfaces. We discuss and suggest some applications of this notion to the study of Severi varieties of nodal curves on surfaces of general type and on P^2 .

We define stacks of uniform cyclic covers of Brauer-Severi schemes, proving that they can be realized as quotient stacks of open subsets of representations, and compute the Picard group for the open substacks parametrizing smooth uniform cyclic covers. Moreover, we give an analogous description for stacks parametrizing triple cyclic covers of Braue...

In this note we are interested in the graded modulesM k=I(k)/Ik and Nk : = [`(Ik )] /IkN_k : = \overline {I^k } /I^k , whereI is a saturated ideal in the homogeneous coordinate ringS=K[x0,…,xn] of ℙn,I (k) is the symbolic power and [`(Ik )]\overline {I^k } is the saturation of the ordinary power. Very little is known about these modules, and we p...

In this note, we consider the natural functorial rational map φ from the (restricted) Hilbert scheme Hilb(d, g, r) to the moduli space Mg, associating the nondegenerate projective model p(C) of a smooth curve C to its isomorphism class [C]. We prove that φ is non constant in a neighbourhood of p(C), for any [C] ∈ U ⊂ Mg (where g ≥ 1 and U is a dens...

Generalizing the construction of the Maslov class [μΛ] for a Lagrangian embedding in a symplectic vector space, we prove that it is possible to give a consistent definition of the class [μΛ] for any Lagrangian submanifold of a Calabi–Yau manifold. Moreover, extending a result of Morvan in symplectic vector spaces, we prove that [μΛ] can be represen...

Developing a previous idea of Faltings, we characterize the complete intersections of codimension 2 in P^n, n>=3, over an algebraically closed field of any characteristic, among l.c.i. X, as those that are subcanonical and scheme-theoretically defined by p<=n-1 equations. Moreover, we give some other results assuming that the normal bundle of X ext...

In this paper we wish to show how to compute the support of caustics related to geometrical solutions (Lagrangian submanifolds) of the geometrical Cauchy problem for the eikonal equation, a special case of Hamilton-Jacobi equation. Although the computation is carried out for the simple Hamiltonian H (q.p) = (1/2)p2 on T→ℝ2, we can treat cases with...