Non-generic Cases Research Articles

Stationary broken parity states in active matter models.

We demonstrate that several nonvariational continuum models commonly used to describe active matter as well as other active systems exhibit nongeneric behavior: each model supports asymmetric but stationary localized states even in the absence of pinning at heterogeneities. Moreover, such states only begin to drift following a drift-transcritical bifurcation as the activity increases. Asymmetric stationary states should only exist in variational systems, i.e., in models with gradient structure. In other words, such states are expected in passive systems, but not in active systems where the gradient structure of the model is broken by activity. We identify a "spurious" gradient dynamics structure of these models that is responsible for this nongeneric behavior, and determine the types of additional terms that render the models generic, i.e., with asymmetric states that appear via drift-pitchfork bifurcations and are generically moving. We provide detailed illustrations of our results using numerical continuation of resting and steadily drifting states in both generic and nongeneric cases.

Topological edge states of 1D chains and index theory

We provide an elementary proof and refinement of a well-known idea from physics: a chiral-symmetric local Hamiltonian on a half-space has the same signed number of edge-localized states with energies in the bulk bandgap as its bulk winding number. The requirement of non-elementary methods to relate generic and non-generic cases is emphasized. Our hands-on approach complements a quick abstract proof based on the classical index theory of Toeplitz operators.

On the persistency of excitation

The result of J.C. Willems et al. “A note on persistency of excitation”, System & Control Letters, 2005 gives identifiability conditions for system identification as well as data-driven representations for data-driven control. The existing proofs however are proofs by contradiction and do not give insight into the assumptions of controllability and persistency of excitation of the input. Moreover, the existing proofs do not clarify how conservative the assumptions are. We provide an alternative constructive proof for the single-input case. It is shown that persistency of excitation of order more than the time horizon is needed in nongeneric cases, corresponding to special initial conditions. The special initial conditions are explicitly characterized in terms of the solution of a Sylvester equation. Another contribution of the paper is a representation of a scalar persistently exciting input of a finite order as an output of an autonomous linear time-invariant system.

Closed-form meromorphic solutions of some third order boundary layer ordinary differential equations

We introduce a general third order non-linear autonomous ODE which covers many ODEs coming from boundary layer problems, like the Falkner-Skan equation and the Cheng-Minkowycz equation. Using Wiman-Valiron theory and complex analytic methods recently developed, for the generic cases, it is shown that all their meromorphic solutions must be rational, or rational in one exponential, and then we find all of them explicitly. For a few non-generic cases, some solutions, which are meromorphic or singlevalued, are also obtained. Our results also explain why it is so difficult to obtain new closed-form solutions of the Falkner-Skan equation.

An analysis of two degenerate double-Hopf bifurcations

<abstract><p>The generic double-Hopf bifurcation is presented in detail in literature in textbooks like references. In this paper we complete the study of the double-Hopf bifurcation with two degenerate (or nongeneric) cases. In each case one of the generic conditions is not satisfied. The normal form and the corresponding bifurcation diagrams in each case are obtained. New possibilities of behavior which do not appear in the generic case were found.</p></abstract>

Orthogonal projections on hyperplanes intertwined with unitaries

Fix a point in a finite-dimensional complex vector space and consider the sequence of iterates of this point under the composition of a unitary map with the orthogonal projection on the hyperplane orthogonal to the starting point. We prove that, generically, the series of the squared norms of these iterates sums to the dimension of the underlying space. This leads us to construct a (device-dependent) dimension witness for quantum systems which involves the probabilities of obtaining certain strings of outcomes in a sequential yes–no measurement. The exact formula for this series in non-generic cases is provided as well as its analogue in the real case.

Shape programming of a magnetic elastica

We consider a cantilever beam which possesses a possibly non-uniform permanent magnetization, and whose shape is controlled by an applied magnetic field. We model the beam as a plane elastic curve and we suppose that the magnetic field acts upon the beam by means of a distributed couple that pulls the magnetization towards its direction. Given a list of target shapes, we look for a design of the magnetization profile and for a list of controls such that the shapes assumed by the beam when acted upon by the controls are as close as possible to the targets, in an averaged sense. To this effect, we formulate and solve an optimal design and control problem leading to the minimization of a functional which we study by both direct and indirect methods. In particular, we prove that minimizers exist, solve the associated Lagrange-multiplier formulation (besides non-generic cases), and are unique at least for sufficiently low intensities of the controlling magnetic fields. To achieve the latter result, we use two nested fixed-point arguments relying on the Lagrange-multiplier formulation of the problem, a method which also suggests a numerical scheme. Various relevant open question are also discussed.

Lower bounds for the local cyclicity for families of centers

In this paper, we are interested in how the local cyclicity of a family of centers depends on the parameters. This fact was pointed out in [21], to prove that there exists a family of cubic centers, labeled by CD3112 in [25], with more local cyclicity than expected. In this family, there is a special center such that at least twelve limit cycles of small amplitude bifurcate from the origin when we perturb it in the cubic polynomial general class. The original proof has some crucial missing points in the arguments that we correct here. We take advantage of a better understanding of the bifurcation phenomenon in nongeneric cases to show two new cubic systems exhibiting 11 limit cycles and another exhibiting 12. Finally, using the same techniques, we study the local cyclicity of holomorphic quartic centers, proving that 21 limit cycles of small amplitude bifurcate from the origin, when we perturb in the class of quartic polynomial vector fields.

The equivalence problem for generic four-dimensional metrics with two commuting Killing vectors

We consider the equivalence problem of four-dimensional semi-Riemannian metrics with the $2$-dimensional Abelian Killing algebra. In the generic case we determine a semi-invariant frame and a fundamental set of first-order scalar differential invariants suitable for solution of the equivalence problem. Genericity means that the Killing leaves are not null, the metric is not orthogonally transitive (i.e., the distribution orthogonal to the Killing leaves is non-integrable), and two explicitly constructed scalar invariants $C_\rho$ and $\ell_{\mathcal C}$ are nonzero. All the invariants are designed to have tractable coordinate expressions. Assuming the existence of two functionally independent invariants, we solve the equivalence problem in two ways. As an example, we invariantly characterise the Van den Bergh metric. To understand the non-generic cases, we also find all $\Lambda$-vacuum metrics that are generic in the above sense, except that either $C_\rho$ or $\ell_{\mathcal C}$ is zero. In this way we extend the Kundu class to $\Lambda$-vacuum metrics. The results of the paper can be exploited for invariant characterisation of classes of metrics and for extension of the set of known solutions of the Einstein equations.

The Root Extraction Problem for Generic Braids

We show that, generically, finding the k-th root of a braid is very fast. More precisely, we provide an algorithm which, given a braid x on n strands and canonical length l, and an integer k > 1 , computes a k-th root of x, if it exists, or guarantees that such a root does not exist. The generic-case complexity of this algorithm is O ( l ( l + n ) n 3 log n ) . The non-generic cases are treated using a previously known algorithm by Sang-Jin Lee. This algorithm uses the fact that the ultra summit set of a braid is, generically, very small and symmetric (through conjugation by the Garside element Δ ), consisting of either a single orbit conjugated to itself by Δ or two orbits conjugated to each other by Δ .

Nikulin Involutions and the CHL String

We study certain even-eight curve configurations on a specific class of Jacobian elliptic K3 surfaces with lattice polarizations of rank ten. These configurations are associated with K3 double covers, some of which are elliptic but not Jacobian elliptic. Several non-generic cases corresponding to K3 surfaces of higher Picard rank are also discussed. Finally, the results and the construction in question are interpreted in the context of the string dualities linked with the eight-dimensional CHL string.

Separability and entanglement of two qubits density matrices using Lorentz transformations

Explicit separability of general two qubits density matrices is related to Lorentz transformations. We use the 4-dimensional form [Formula: see text] of the Hilbert–Schmidt (HS) decomposition of the density matrix. For the generic case, in which Lorentz transformations diagonalize [Formula: see text] into [Formula: see text], we give relations between the [Formula: see text] and the [Formula: see text]. In particular, we consider two cases: (a) Two qubits density matrices with one pair of linear terms in the HS decomposition. (b) Two qubits density matrices with two or three symmetric pairs of linear terms. Some of the theoretical results are demonstrated by numerical calculations. The four non-generic cases (which may be reduced to case (a) are analyzed and the non-generic property is related explicitly to Lorentz velocity [Formula: see text] which is not reachable physically.

Lagrangian Transport and Chaotic Advection in Two-Dimensional Anisotropic Systems

Enhanced geothermal systems (EGSs) are a promising concept to make geothermal power generation available in many regions worldwide. However, in current EGSs generally only a fraction of the geothermal reservoir is effectively accessed by the production fluid, resulting in suboptimal performance. Recent studies in the literature on a two-dimensional Darcy flow in a circular reservoir driven by reoriented injector–producer wells demonstrated that well configurations and pumping schemes designed on the basis of chaos theory enable efficient distribution of the production fluids throughout the entire reservoir. Key to this is accomplishment of chaotic advection, i.e. the rapid dispersion and stretching of material fluid regions, by a “proper” flow forcing. However, these studies concern isotropic reservoirs, while geothermal reservoirs typically are highly anisotropic. Our theoretical/computational study expands on said studies by investigating the impact of anisotropy on fundamental aspects of the Lagrangian transport of production fluids. This reveals that anisotropy generically eliminates key organizing mechanisms in the Lagrangian transport, viz. symmetries, and thus tends to promote disorder and, inherently, chaotic advection. However, symmetries are partially preserved—and thus order and coherence partially restored—in non-generic cases such as pumping schemes employing an even number of injector–producer wells and well configurations aligned with the anisotropy. Symmetry associated with well alignment in fact appears crucial to an intriguing “order within chaos” observed only in such cases: prolonged confinement of fluid to subregions of chaotic areas.

On the Sato-Tate conjecture for non-generic abelian surfaces

We prove many non-generic cases of the Sato-Tate conjecture for abelian surfaces as formulated by Fité, Kedlaya, Rotger and Sutherland, using the potential automorphy theorems of Barnet-Lamb, Gee, Geraghty and Taylor.

Compléments sur les extensions entre séries principales p-adiques et modulo p de GL(F)

We complete the results of a previous article. Let $G$ be a split connected reductive group over a finite extension $F$ of $\mathbb{Q}_p$. When $F=\mathbb{Q}_p$, we determine the extensions between unitary continuous $p$-adic and smooth mod $p$ principal series of $G(\mathbb{Q}_p)$ without assuming the centre of $G$ connected nor the derived group of $G$ simply connected. This shows a new phenomenon: there may exist several non-isomorphic non-split extensions between two distinct principal series. We also complete the computations of self-extensions of a principal series in the non-generic cases when the centre of $G$ is connected. Finally, we determine the extensions of a principal series of $G(F)$ by an "ordinary" representation of $G(F)$ (i.e. parabolically induced from a special representation twisted by a character). In order to do so, we compute Emerton's $\delta$-functor $\mathrm{H^\bullet Ord}_{B(F)}$ of derived ordinary parts with respect to a Borel subgroup on an ordinary representation of $G(F)$.

Classification of critical sets and their images for quadratic maps of the plane

We provide a complete classification of the critical sets and their images for quadratic maps of the real plane. Critical sets are always conic sections, which provides a starting point for the classification. The generic cases, maps whose critical sets are either ellipses or hyperbolas, was published by Delgado et al. in 2013. This work completes the classification by including all the nongeneric cases: the empty set, a single point, a single line, a parabola, two parallel lines, two intersecting lines, or the whole plane. We describe all possible images for each critical set case and illustrate the geometry of representative maps for each case.

An Inductive Construction of Rigid Panel-Hinge Graphs and Their Applications to Form Design

A panel-hinge framework is a structure composed of rigid panels connected by hinges. It was recently proved that at a generic position, the rigidity of panel-hinge frameworks can be predicted by examining a certain combinatorial property of the underlying graph. In this study, we apply such combinatorial characteristics to create design forms. When considering the application of design forms, we must also take into account non-generic cases. In this paper we develop two new approaches; the first one that the method inductively generates non-generic rigid panel-hinge frameworks consisting of orthogonal panels and the second one that inductively generates non-generic rigid panel-hinge frameworks based on fractal geometry coupled with space-filling convex polyhedron as a construction unit. We will give examples of forms created by the proposed method in order to demonstrate the applicability of the proposed methods to design forms.

Principal Poincaré Pontryagin function associated to some families of Morse real polynomials

It is known that the principal Poincaré Pontryagin function is generically an Abelian integral. We give a sufficient condition on monodromy to ensure that it is also an Abelian integral in non-generic cases.In non-generic cases it is an iterated integral. Uribe (2006 J. Dyn. Control. Syst. 12 109–34, 2009 J. Diff. Eqns 246 1313–41) gives in a special case a precise description of the principal Poincaré Pontryagin function, an iterated integral of length at most 2, involving logarithmic functions with only 1 ramification at a point at infinity. We extend this result to some non-isomonodromic families of real Morse polynomials.

TANGENT-POINT SELF-AVOIDANCE ENERGIES FOR CURVES

We study a two-point self-avoidance energy [Formula: see text] which is defined for all rectifiable curves in ℝn as the double integral along the curve of 1/rq. Here r stands for the radius of the (smallest) circle that the is tangent to the curve at one point and passes through another point on the curve, with obvious natural modifications of this definition in the exceptional, non-generic cases. It turns out that finiteness of [Formula: see text] for q ≥ 2 guarantees that γ has no self-intersections or triple junctions and therefore must be homeomorphic to the unit circle 𝕊1 or to a closed interval I. For q > 2 the energy [Formula: see text] evaluated on curves in ℝ3 turns out to be a knot energy separating different knot types by infinite energy barriers and bounding the number of knot types below a given energy value. We also establish an explicit upper bound on the Hausdorff-distance of two curves in ℝ3 with finite [Formula: see text]-energy that guarantees that these curves are ambient isotopic. This bound depends only on q and the energy values of the curves. Moreover, for all q that are larger than the critical exponent q crit = 2, the arclength parametrization of γ is of class C1, 1-2/q, with Hölder norm of the unit tangent depending only on q, the length of γ, and the local energy. The exponent 1 - 2/q is optimal.

Characterization and construction of the nearest defective matrix via coalescence of pseudospectral components

Let A be a matrix with distinct eigenvalues and let w ( A ) be the distance from A to the set of defective matrices (using either the 2-norm or the Frobenius norm). Define Λ ϵ , the ϵ -pseudospectrum of A, to be the set of points in the complex plane which are eigenvalues of matrices A + E with ‖ E ‖ < ϵ , and let c ( A ) be the supremum of all ϵ with the property that Λ ϵ has n distinct components. Demmel and Wilkinson independently observed in the 1980s that w ( A ) ⩾ c ( A ) , and equality was established for the 2-norm by Alam and Bora (2005). We give new results on the geometry of the pseudospectrum near points where first coalescence of the components occurs, characterizing such points as the lowest generalized saddle point of the smallest singular value of A - zI over z ∈ C . One consequence is that w ( A ) = c ( A ) for the Frobenius norm too, and another is the perhaps surprising result that the minimal distance is attained by a defective matrix in all cases. Our results suggest a new computational approach to approximating the nearest defective matrix by a variant of Newton’s method that is applicable to both generic and nongeneric cases. Construction of the nearest defective matrix involves some subtle numerical issues which we explain, and we present a simple backward error analysis showing that a certain singular vector residual measures how close the computed matrix is to a truly defective matrix. Finally, we present a result giving lower bounds on the angles of wedges contained in the pseudospectrum and emanating from generic coalescence points. Several conjectures and questions remain open.