Non-generic Cases Research Articles

Symmetric tensor decomposition

We present an algorithm for decomposing a symmetric tensor, of dimension n and order d, as a sum of rank-1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for binary forms. We recall the correspondence between the decomposition of a homogeneous polynomial in n variables of total degree d as a sum of powers of linear forms (Waring’s problem), incidence properties on secant varieties of the Veronese variety and the representation of linear forms as a linear combination of evaluations at distinct points. Then we reformulate Sylvester’s approach from the dual point of view. Exploiting this duality, we propose necessary and sufficient conditions for the existence of such a decomposition of a given rank, using the properties of Hankel (and quasi-Hankel) matrices, derived from multivariate polynomials and normal form computations. This leads to the resolution of systems of polynomial equations of small degree in non-generic cases. We propose a new algorithm for symmetric tensor decomposition, based on this characterization and on linear algebra computations with Hankel matrices. The impact of this contribution is two-fold. First it permits an efficient computation of the decomposition of any tensor of sub-generic rank, as opposed to widely used iterative algorithms with unproved global convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding uniqueness conditions and for detecting the rank.

Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom. Nongeneric cases

In this paper the problem of classification of integrable natural Hamiltonian systems with $n$ degrees of freedom given by a Hamilton function which is the sum of the standard kinetic energy and a homogeneous polynomial potential $V$ of degree $k>2$ is investigated. It is assumed that the potential is not generic. Except for some particular cases a potential $V$ is not generic, if it admits a nonzero solution of equation $V'(\vd)=0$. The existence of such solution gives very strong integrability obstructions obtained in the frame of the Morales-Ramis theory. This theory gives also additional integrability obstructions which have the form of restrictions imposed on the eigenvalues $(\lambda_1,...,\lambda_n)$ of the Hessian matrix $V''(\vd)$ calculated at a non-zero $\vd\in\C^n$ satisfying $V'(\vd)=\vd$. Furthermore, we show that similarly to the generic case also for nongeneric potentials some universal relations between $(\lambda_1,...,\lambda_{n})$ calculated at various solutions of $V'(\vd)=\vd$ exist. We derive them for case $n=k=3$ applying the multivariable residue calculus. We demonstrate the strength of the obtained results analysing in details the nongeneric cases for $n=k=3$. Our analysis cover all the possibilities and we distinguish those cases where known methods are too weak to decide if the potential is integrable or not. Moreover, for $n=k=3$ thanks to this analysis a three-parameter family of potentials integrable or super-integrable with additional polynomial first integrals which seemingly can be of an arbitrarily high degree with respect to the momenta was distinguished.

Scaling of saddle-node bifurcations: degeneracies and rapid quantitative changes

The scaling of the time delay near a ‘bottleneck’ of a generic saddle-node bifurcation is well known to be given by an inverse square-root law. We extend the analysis to several non-generic cases for smooth vector fields. We proceed to investigate C0 vector fields. Our main result is a new phenomenon in two-parameter families having a saddle-node bifurcation upon changing the first parameter. We find distinct scalings for different values of the second parameter ranging from power laws with exponents in (0, 1) to scalings given by O(1). We illustrate this rapid quantitative change of the scaling law by an overdamped pendulum with varying length.

Cyclicity of several planar graphics and ensembles through three singular points without generic conditions

This paper investigates the number and distribution of the limit cycles bifurcated from several graphics and ensembles through a saddle-node P 0 and two hyperbolic saddles P 1 and P 2 for the non-generic cases of r 1 ( 0 ) = 1 , r 2 ( 0 ) ≠ 1 and r 1 ( 0 ) ≠ 1 , r 2 ( 0 ) = 1 , where r 1 ( 0 ) and r 2 ( 0 ) are the hyperbolicity ratio of the saddles P 1 and P 2 , respectively. For the case of r 1 ( 0 ) = 1 , r 2 ( 0 ) ≠ 1 , we suppose that the connection from P 0 to P 2 and the connection from P 0 to P 1 keep unbroken. We prove that these graphics and ensembles are of finite cyclicity respectively. Moreover, the cyclicity is linearly dependent on the order of the neutral saddle P 1 if P 2 is contractive and r 2 ( 0 ) ∈ Q . We also show that the nearer r 2 ( 0 ) is close to 1, the more the limit cycles are bifurcated. For the case of r 1 ( 0 ) ≠ 1 , r 2 ( 0 ) = 1 , we obtain that these graphics and ensembles are of finite cyclicity respectively if P 1 is of finite order and the hp-connection from P 0 to P 2 keeps unbroken.

BPCONT: An Auto Driver for the Continuation of Branch Points of Algebraic and Boundary-Value Problems

BPcont, a driver for the software package Auto for the numerical continuation of simple branch points of algebraic and boundary-value problems, is described in detail. Simple branch points are points in the continuation space where two solution branches intersect transversally. Detection and accurate location of branch points along the continuation of a solution branch and switching to the continuation of the other branch are implemented in Auto and other software packages, but branch point continuation has not been fully supported. BPcont fills this gap. It handles both generic problems, where branch point continuation is performed in two extra parameters, and nongeneric cases, where one extra parameter is typically involved due to problem-specific symmetries. This paper presents the extended algebraic and boundary-value problems which define branch point continuation, discusses their initialization at branch point detection and the symmetry-breaking technique used to automatically handle nongeneric cases, and describes the BPcont implementation. Several examples, including generic and nongeneric algebraic problems, generic and nongeneric periodic boundary-value problems, and a nongeneric nonperiodic boundary-value problem, are also presented. (A corrected version of this paper has been appended to the originally posted pdf.)

The behaviour of resonances in Hecke triangular billiards under deformation

The right-hand boundary of Artin's billiard on the Poincaré half-plane is continuously deformed to generate a class of chaotic billiards which includes fundamental domains of the Hecke groups Γ(2, n) at certain values of the deformation parameter. The quantum scattering problem in these open chaotic billiards is described and the distributions of both real and imaginary parts of the resonant eigenvalues are investigated. The transitions to arithmetic chaos in the cases n ∊ {4, 6} are closely examined and the explicit analytic form for the scattering matrix is given together with the Fourier coefficients for the scattered wavefunction. The n = 4 and 6 cases have an additional set of regular equally spaced resonances compared to Artin's billiard (n = 3). For a general deformation, a numerical procedure is presented which generates the resonance eigenvalues and the evolution of the eigenvalues is followed as the boundary is varied continuously which leads to dramatic changes in their distribution. For deformations away from the non-generic arithmetic cases, including that of the tiling Hecke triangular billiard n = 5, the distributions of the positions and widths of the resonances are consistent with the predictions of a random matrix theory.

Transversals to Line Segments in Three-Dimensional Space

We completely describe the structure of the connected components of transversals to a collection of n line segments in ℝ3. Generically, the set of transversals to four segments consists of zero or two lines. We catalog the non-generic cases and show that n≥ 3 arbitrary line segments in ℝ3 admit at most n connected components of line transversals, and that this bound can be achieved in certain configurations when the segments are coplanar, or they all lie on a hyperboloid of one sheet. This implies a tight upper bound of n on the number of geometric permutations of line segments in ℝ3.

Darboux points and integrability of Hamiltonian systems with homogeneous polynomial potential

In this paper we study the integrability of natural Hamiltonian systems with a homogeneous polynomial potential. The strongest necessary conditions for their integrability in the Liouville sense have been obtained by a study of the differential Galois group of variational equations along straight line solutions. These particular solutions can be viewed as points of a projective space of dimension smaller by one than the number of degrees of freedom. We call them Darboux points. We analyze in detail the case of two degrees of freedom. We show that, except for a radial potential, the number of Darboux points is finite and it is not greater than the degree of the potential. Moreover, we analyze cases when the number of Darboux points is smaller than maximal. For two degrees of freedom the above-mentioned necessary condition for integrability can be expressed in terms of one nontrivial eigenvalue of the Hessian of potential calculated at a Darboux point. We prove that for a given potential these nontrivial eigenvalues calculated for all Darboux points cannot be arbitrary because they satisfy a certain relation which we give in an explicit form. We use this fact to strengthen maximally the necessary conditions for integrability and we show that in a generic case, for a given degree of the potential, there is only a finite number of potentials which satisfy these conditions. We also describe the nongeneric cases. As an example we give a full list of potentials of degree four satisfying these conditions. Then, investigating the differential Galois group of higher order variational equations, we prove that, except for one discrete family, among these potentials only those which are already known to be integrable are integrable. We check that a finite number of potentials from the exceptional discrete family are not integrable, and we conjecture that all of them are not integrable.

Singularities of the envelope of curves tangent to a semi-cubic cusp

We study the envelope of one-parameter families of smooth curves tangent to a curve having a semi-cubic cusp such that the radius of curvature at the tangency point vanishes when this point approaches the cusp. We show that, in general, the closure of the envelope has two semi-cubic cusps at the same point, one of which is the given cusp tangent to the same straight line. We study also nongeneric and dual cases and normal forms of the equation of the envelope.

On covariant embeddings of a linear functional equation with respect to an analytic iteration group in some non-generic cases

In the paper On covariant embeddings of a linear functional equation with respect to an analytic iteration group [3] the authors described the problem of covariant embeddings of a linear functional equation with respect to analytic iteration groups and solved it in the generic cases. However, some cases remained unsolved. In this paper we present the solution for these open cases.

The bubble algebra: structure of a two-colour Temperley–Lieb Algebra

We define new diagram algebras providing a sequence of multiparameter generalizations of the Temperley–Lieb algebra, suitable for the modelling of dilute lattice systems of two-dimensional statistical mechanics. These algebras give a rigorous foundation to the various 'multi-colour algebras' of Grimm, Pearce and others. We determine the generic representation theory of the simplest of these algebras, and locate the nongeneric cases (at roots of unity of the corresponding parameters). We show by this example how the method used (Martin's general procedure for diagram algebras) may be applied to a wide variety of such algebras occurring in statistical mechanics. We demonstrate how these algebras may be used to solve the Yang–Baxter equations.

General soliton matrices in the Riemann–Hilbert problem for integrable nonlinear equations

We derive the soliton matrices corresponding to an arbitrary number of higher-order normal zeros for the matrix Riemann–Hilbert problem of arbitrary matrix dimension, thus giving the complete solution to the problem of higher-order solitons. Our soliton matrices explicitly give all higher-order multisoliton solutions to the nonlinear partial differential equations integrable through the matrix Riemann–Hilbert problem. We have applied these general results to the three-wave interaction system, and derived new classes of higher-order soliton and two-soliton solutions, in complement to those from our previous publication [Stud. Appl. Math. 110, 297 (2003)], where only the elementary higher-order zeros were considered. The higher-order solitons corresponding to nonelementary zeros generically describe the simultaneous breakup of a pumping wave (u3) into the other two components (u1 and u2) and merger of u1 and u2 waves into the pumping u3 wave. The two-soliton solutions corresponding to two simple zeros generically describe the breakup of the pumping u3 wave into the u1 and u2 components, and the reverse process. In the nongeneric cases, these two-soliton solutions could describe the elastic interaction of the u1 and u2 waves, thus reproducing previous results obtained by Zakharov and Manakov [Zh. Éksp. Teor. Fiz. 69, 1654 (1975)] and Kaup [Stud. Appl. Math. 55, 9 (1976)].

Optimal realizations of controlled unitary gates

The controlled-not gate and the single qubit gates are considered elementary gates in quantum computing. It is natural to ask how many such elementary gates are needed to implement more elaborate gates or circuits. Recall that a controlled-U gate can be realized with two controlled-not gates and four single qubit gates. We prove that this implementation is optimal if and only if the matrix U satisfies the conditions trU\neq 0, tr(UX)\neq 0, and detU\neq 1. We also derive optimal implementations in the remaining non-generic cases. realizations of controlled unitary gates (pp139-155) G. Song and A. Klappenecker The controlled-not gate and the single qubit gates are considered elementary gates in quantum computing. It is natural to ask how many such elementary gates are needed to implement more elaborate gates or circuits. Recall that a controlled-U gate can be realized with two controlled-not gates and four single qubit gates. We prove that this implementation is optimal if and only if the matrix U satisfies the conditions trU\neq 0, tr(UX)\neq 0, and detU\neq 1. We also derive optimal implementations in the remaining non-generic cases.

Efficient Design with Interdependent Valuations

We study efficient, Bayes-Nash incentive compatible mechanisms in a social choice setting that allows for informational and allocative externalities. We show that such mechanisms exist only if a congruence condition relating private and social rates of information substitution is satisfied. If signals are multi-dimensional, the congruence condition is determined by an integrability constraint, and it can hold only in nongeneric cases where values are private or a certain symmetry assumption holds. If signals are one-dimensional, the congruence condition reduces to a monotonicity constraint and it can be generically satisfied. We apply the results to the study of multi-object auctions, and we discuss why such auctions cannot be reduced to one-dimensional models without loss of generality.

COMPUTATION OF SIMPLEST NORMAL FORMS OF DIFFERENTIAL EQUATIONS ASSOCIATED WITH A DOUBLE-ZERO EIGENVALUE

In this paper a method is presented for computing the simplest normal form of differential equations associated with the singularity of a double zero eigenvalue. Based on a conventional normal form of the system, explicit formulae for both generic and nongeneric cases are derived, which can be used to compute the coefficients of the simplest normal form and the associated nonlinear transformation. The recursive algebraic formulae have been implemented on computer systems using Maple. The user-friendly programs can be executed without any interaction. Examples are given to demonstrate the computational efficiency of the method and computer programs.

Solutions of Nonlinear Planar Elliptic Problems with Triangle Symmetry

In this paper we continue the study of the nodal domain structure of doubly periodic solutions of certain nonlinear elliptic problems initiated in Fifeet al.(Physica D100(1997), 257–278). More precisely, we consider small amplitude solutions ofΔu+λf(u)=0 in R2whose nodal domains consist of equilateral triangles tiling the plane. If this equation is suitably perturbed, then for genericfwe prove the existence of unique nearby solutions with triangle symmetry and show how their nodal domain geometry breaks up. Furthermore, we treat the non-generic rectangular cases which had to be excluded in Fifeet al.(Physica D100(1997), 257–278) as well as other nodal domain structures.

Onset of cooperative entrainment in limit-cycle oscillators with uniform all-to-all interactions: bifurcation of the order function

The concept of order function was previously proposed as the key to a general theory of macroscopic mutual (or cooperative) entrainment in large populations of limit-cycle oscillators with weak interactions such that every element is linked to all the other, as well as with weakly dispersed intrinsic frequencies, that is, limit-cycle oscillators that can be modeled by globally coupled phase oscillators with distributed frequencies. Following previous work, a bifurcation theory of the order function is developed on the basis of its self-consistent functional equation to elucidate, in particular, generic scaling behavior of such systems at the onset of cooperative entrainment. Among other results, when the onset is not abrupt, the critical exponent of fundamental order parameters turns out to generically differ from the conventional value 1 2 taken by the well-studied sinusoidal coupling model as well as by typical mean-fields models of thermodynamic phase transitions to which coupled-oscillator models investigated here are analogous. The theory also reveals what happens in nongeneric cases. Moreover, a criterion is found of whether the bifurcation is normal or inverted. All these analytical results and predictions are verified not only by numerically solving the equation of the order function, but also by numerical simulations. Although this paper is mainly concerned with the critical behaviors, noncritical regimes are also explored to demonstrate overall power of the order function theory by reproducing simulation results such as average-frequency spectra. The theory, however, keeps some room to be further generalized. A finding which suggests this is put forth.

The Set of 2-by-3 Matrix Pencils — Kronecker Structures and Their Transitions under Perturbations

The set (or family) of 2-by-3 matrix pencils $A - \lambda B$ comprises 18 structurally different Kronecker structures (canonical forms). The algebraic and geometric characteristics of the generic and the 17 nongeneric cases are examined in full detail. The complete closure hierarchy of the orbits of all different Kronecker structures is derived and presented in a closure graph that shows how the structures relate to each other in the 12-dimensional space spanned by the set of 2-by-3 pencils. Necessary conditions on perturbations for transiting from the orbit of one Kronecker structure to another in the closure hierarchy are presented in a labeled closure graph. The node and arc labels shows geometric characteristics of an orbit’s Kronecker structure and the change of geometric characteristics when transiting to an adjacent node, respectively. Computable normwise bounds for the smallest perturbations $(\delta A,\delta B)$ of a generic 2-by-3 pencil $A - \lambda B$ such that $( A + \delta A ) - \lambda ( B + \delta B )$ has a specific nongeneric Kronecker structure are presented. First, explicit expressions for the perturbations that transfer $A - \lambda B$ to a specified nongeneric form are derived. In this context tractable and intractable perturbations are defined. Second, a modified GUPTRI that computes a specified Kronecker structure of a generic pencil is used. Perturbations devised to impose a certain nongeneric structure are computed in a way that guarantees one will find a Kronecker canonical form (KCF) on the closure of the orbit of the intended KCF. Both approaches are illustrated by computational experiments. Moreover, a study of the behaviour of the nongeneric structures under random perturbations in finite precision arithmetic (using the GUPTRI software) show for which sizes of perturbations the structures are invariant and also that structure transitions occur in accordance with the closure hierarchy. Finally, some of the results are extended to the general m-by-$( m + 1 )$ case.

On a nonstandard two-parametric quantum algebra and its connections withU p, q (gl(2)) andU p, q(gl(1/1))

A quantum algebraU p, q (ζ,H,X ±) associated with a nonstandardR-matrix with two deformation parameters (p, q) is studied and, in particular, its universal ℛ-matrix is derived using Reshetikhin's method. Explicit construction of the (p, q)-dependent nonstandardR-matrix is obtained through a coloured generalized boson realization of the universal ℛ-matrix of the standardU p, q(gl(2)) corresponding to a nongeneric case. General finite dimensional coloured representation of the universal ℛ-matrix ofU p, q(gl(2)) is also derived. This representation, in nongeneric cases, becomes a source for various (p, q)-dependent nonstandardR-matrices. Superization ofU p, q(ζ,H,X ±) leads to the super-Hopf algebraU p, q(gl(1/1)). A contraction procedure then yields a (p, q)-deformed super-Heisenberg algebraU p, q(sh(1)) and its universal ℛ-matrix.

On Randomized Semi-algebraic Test Complexity

We investigate the impact of randomization on the complexity of deciding membership in a (semi-)algebraic subset X ⊂ R m . Examples are exhibited where allowing for a certain error probability ϵ in the answer of the algorithms the complexity of decision problems decreases. A randomized (Ω k , {=, ≤})-decision tree ( k ⊆ R a subfield) over m will be defined as a pair ( T, μ) where μ a probability measure on some R n and T is a (Ω k , {=, ≤})-decision tree over m + n. We prove a general lower bound on the randomized decision complexity for testing membership in an irreducible algebraic subset X ⊂ R m and apply it to k-generic complete intersection of polynomials of the same degree, extending results in P. Bürgisser, T. Lickteig, and M.. Shub ((1992), J. Complexity 8, 203-215) and P. Bürgisser ((1992), Research Report No. 8578-CS, Univ. Bonn). We also give applications to nongeneric cases, such as graphs of elementary symmetric functions, SL( m, R ), and determinant varieties, extending results in (Lickteig, T. (1990), Tech. Rep. TR-90-90-052 Int. Comp. Science Inst., Berkeley, and Univ. Tübingen, Habilitationsschrift). The results are based on the methods from that paper and P. Bürgisser and T. Lickteig (1992), J. Pure Appl. Algebra 81, 247-267).