Examples Of Non-uniqueness Research Articles

Uniqueness and nonuniqueness for the [formula omitted] minimization source localization problem with three measurements

The source localization problem has been studied by many researchers and applied to a wide range of areas including Global Positioning System (GPS) localization. Although there have been studies examining closed form solutions for cases with not so large noise, only the existence of local solution in L2 minimization settings is known. In this paper, L1 minimization with three measurements is studied, and nonuniqueness examples for the minimization are provided. Futher, the uniqueness of the solution in certain conditions is stated and proven. The proof is based on the analysis for the extreme points of the objective function. The singular points are the three source points and the three measurement circles. The critical points are the Fermat point and some points located inside of one measurement circle and outside two other circles. The numbers of the latter critical points are infinite for the equilateral measurement triangle and one for nonequilateral measurement triangle. The analysis for these critical points is also provided. Finally, a few numerical implementations are provided to clarify the uniqueness and nonuniqueness of the problem.

Infinitely many non-uniqueness examples for Cauchy problems of the two-dimensional wave and Schrödinger equations

In 1963, Kumano-go presented one non-uniqueness example for the two-dimensional wave equation with a time-dependent potential. We construct infinitely many non-uniqueness examples with different wave numbers at infinity for Cauchy problems of the two-dimensional wave equation and Schrödinger equation as a generalization of the construction by Kumano-go.

Measure solutions to a system of continuity equations driven by Newtonian nonlocal interactions

We prove global-in-time existence and uniqueness of measure solutions of a nonlocal interaction system of two species in one spatial dimension. For initial data including atomic parts we provide a notion of gradient-flow solutions in terms of the pseudo-inverses of the corresponding cumulative distribution functions, for which the system can be stated as a gradient flow on the Hilbert space \begin{document}$ L^2(0,1)^2 $\end{document} according to the classical theory by Brezis. For absolutely continuous initial data we construct solutions using a minimising movement scheme in the set of probability measures. In addition we show that the scheme preserves finiteness of the \begin{document}$ L^m $\end{document} -norms for all \begin{document}$ m\in [1,+\infty] $\end{document} and of the second moments. We then provide a characterisation of equilibria and prove that they are achieved (up to time subsequences) in the large time asymptotics. We conclude the paper constructing two examples of non-uniqueness of measure solutions emanating from the same (atomic) initial datum, showing that the notion of gradient flow solution is necessary to single out a unique measure solution.

Degenerate elliptic equations and nonuniqueness of solutions to the Kolmogorov equation

In this paper we propose a new method of constructing examples of nonuniqueness of probability solutions by reducing the stationary Fokker-Planck-Kolmogorov equation to a degenerate elliptic equation on a bounded domain.

On the Feasibility of Detecting Thin Conductive Layers from Field Measurements on the Surface of a Medium

A two-dimensional environment in which the fields are described by the Helmholtz equation is considered. A linearized formulation of the problem of the recovery of medium parameters (an inverse problem for the Helmholtz equation) is studied. The conditions for the unique detection of thin conductive layers are established. Examples of nonuniqueness of the solution of the inverse problem from information that initially seemed redundant even for a unique solution are given.

On the reconstruction of inducing dipole directions and susceptibilities from knowledge of the magnetic field on a sphere

ABSTRACTReconstructing magnetizations from measurements of the generated magnetic potential is generally non-unique. The non-uniqueness still remains if one restricts the magnetization to those induced by an ambient magnetic dipole field (i.e. the magnetization is described by a scalar susceptibility and the dipole direction). Here, we investigate the situation under the additional constraint that the susceptibility is either spatially localized in a subregion of the sphere or that it is band-limited. If the dipole direction is known, then the susceptibility is uniquely determined under the spatial localization constraint while it is only determined up to a constant under the assumption of band-limitedness. If the dipole direction is not known, uniqueness is lost again. However, we show that all dipole directions that could possibly generate the measured magnetic potential need to be zeros of a certain polynomial which can be computed from the given potential. We provide examples of non-uniqueness of the dipole direction and examples on how to find admissible candidates for the dipole direction under the spatial localization constraint.

Global Convergence of a Sticky Particle Method for the Modified Camassa--Holm Equation

In this paper, we prove convergence of a sticky particle method for the modified Camassa--Holm equation (mCH) with cubic nonlinearity in one dimension. As a byproduct, we prove global existence of weak solutions $u$ with regularity: $u$ and $u_x$ are space-time BV functions. The total variation of $m(\cdot ,t)=u(\cdot, t)-u_{xx}(\cdot,t)$ is bounded by the total variation of the initial data $m_0$. We also obtain $W^{1,1}(\mathbb{R})$-stability of weak solutions when solutions are in $ L^\infty(0,\infty;W^{2,1}(\mathbb{R}))$. (Notice that peakon weak solutions are not in $W^{2,1}(\mathbb{R})$.) Finally, we provide some examples of nonuniqueness of peakon weak solutions to the mCH equation.

Examples of Non-Uniqueness of the Equilibrium States for a Floating Ball

We provide a numerical algorithm for numerically approximating a centrally located floating ball. We give examples of equilibria, and we present non-unique cases for the same physical parameters when the density of the ball is either greater than the supporting liquid (heavy) or lighter than the density of the vapor above (light). We classify the non-uniqueness by analyzing a function related to the force balance. We derive the potential energy of these states, and make comparisons of the non-unique cases. In the cases of both the light and heavy floating balls, the evidence presented supports the conjecture that when there are two equilibria, the one with lower energy corresponds to the location of triple junction (between the ball, the vapor and the liquid) that is closer to the equator of the ball.

Uniqueness of rarefaction waves in multidimensional compressible Euler system

We show that 1D rarefaction wave solutions are unique in the class of bounded entropy solutions to the multidimensional compressible Euler system. Such a result may be viewed as a counterpart of the recent examples of non-uniqueness of the shock wave solutions to the Riemann problem, where infinitely many solutions are constructed by the method of convex integration.

Algebraic independence of elements in immediate extensions of valued fields

Refining a constructive combinatorial method due to MacLane and Schilling, we give several criteria for a valued field that guarantee that all of its maximal immediate extensions have infinite transcendence degree. If the value group of the field has countable cofinality, then these criteria give the same information for the completions of the field. The criteria have applications to the classification of valuations on rational function fields. We also apply the criteria to the question of which extensions of a maximal valued field, algebraic or of finite transcendence degree, are again maximal. In the case of valued fields of infinite p-degree, we obtain the worst possible examples of nonuniqueness of maximal immediate extensions: fields which admit an algebraic maximal immediate extension as well as one of infinite transcendence degree.

Time reversal in photoacoustic tomography and levitation in a cavity

A class of photoacoustic acquisition geometries in is considered such that the spherical mean transform admits an exact filtered back projection reconstruction formula. The reconstruction is interpreted as a time reversion mirror that reproduces exactly an arbitrary source distribution in the cavity. A series of examples of non-uniqueness of the inverse potential problem is constructed based on the same geometrical technique.

Nonuniqueness for a parabolic SPDE with $\frac{3}{4}-\varepsilon $-Hölder diffusion coefficients

Motivated by Girsanov's nonuniqueness examples for SDEs, we prove nonuniqueness for the parabolic stochastic partial differential equation (SPDE) \[\frac{\partial u}{\partial t}=\frac{\Delta}{2}u(t,x) +\bigl|u(t,x)\bigr|^{\gamma}\dot{W}(t,x),\qquad u(0,x)=0.\] Here $\dot{W}$ is a space-time white noise on ${\mathbb {R}}_+\times {\mathbb {R}}$. More precisely, we show the above stochastic PDE has a nonzero solution for $0<\gamma<3/4$. Since $u(t,x)=0$ solves the equation, it follows that solutions are neither unique in law nor pathwise unique. An analogue of Yamada-Watanabe's famous theorem for SDEs was recently shown in Mytnik and Perkins [Probab. Theory Related Fields 149 (2011) 1-96] for SPDE's by establishing pathwise uniqueness of solutions to \[\frac{\partial u}{\partial t}=\frac{\Delta}{2}u(t,x)+\sigma \bigl(u(t,x)\bigr)\dot{W}(t,x)\] if $\sigma$ is H\"{o}lder continuous of index $\gamma>3/4$. Hence our examples show this result is essentially sharp. The situation for the above class of parabolic SPDE's is therefore similar to their finite dimensional counterparts, but with the index $3/4$ in place of $1/2$. The case $\gamma=1/2$ of the first equation above is particularly interesting as it arises as the scaling limit of the signed mass for a system of annihilating critical branching random walks.

Scattering of time-harmonic electromagnetic plane waves by perfectly conducting diffraction gratings

Consider the scattering of time-harmonic electromagnetic plane waves by a doubly periodic surface in ℝ3. The medium above the surface is supposed to be homogeneous and isotropic with a constant dielectric coefficient, while the material below is perfectly conducting. This paper is concerned with the existence of quasiperiodic solutions for any frequency. Based on an equivalent variational formulation established by the mortar technique of Nitsche, we verify the existence of solutions for a broad class of incident waves including plane waves. The only assumption is that the grating profile is a Lipschitz biperiodic surface. Note that the solvability result of the present paper covers the resonance case where Rayleigh frequencies are allowed. Finally, non-uniqueness examples are presented in the resonance case and in the case of TE or TM polarization for classical gratings.

An example of non-uniqueness of a simple partial fraction of the best uniform approximation

For arbitrary natural n ≥ 2 we construct an example of a real continuous function, for which there exists more than one simple partial fraction of order ≤ n of the best uniform approximation on a segment of the real axis. We prove that even the Chebyshev alternance consisting of n+1 points does not guarantee the uniqueness of the best approximation fraction. The obtained results are generalizations of known non-uniqueness examples constructed for n = 2, 3 in the case of simple partial fractions of an arbitrary order n.

On domain perturbation for super-linear Neumann problems and a question of Y. Lou, W-M Ni and L. Su

We prove a new domain variation result for Neumann problems and apply it to give new examples of non-uniqueness of positive solutions.

Uniqueness and Nonuniqueness for Ricci Flow on Surfaces: Reverse Cusp Singularities

We extend the notion of what it means for a complete Ricci flow to have a given initial metric, and consider the resulting well-posedness issues that arise in the two-dimensional case. On one hand, we construct examples of nonuniqueness by showing that surfaces with cusps can evolve either by keeping the cusps or by contracting them. On the other hand, by adding a noncollapsedness assumption for the initial metric, we establish a uniqueness result.

Global uniqueness in determining polygonal periodic structures with a minimal number of incident plane waves

In this paper, we investigate the inverse problem of recovering a two-dimensional perfectly reflecting diffraction grating from the scattered waves measured above the structure. Inspired by a novel idea developed by Bao et al (Trans. Am. Math. Soc. to appear), we present a complete characterization of the global uniqueness in determining polygonal periodic structures using a minimal number of incident plane waves. The idea in this paper combines the reflection principle for the Helmholtz equation and the dihedral group theory. We characterize all periodic polygonal structures that cannot be identified by one incident plane wave, including the resonance case where a Rayleigh frequency is allowed. Furthermore, we show that those unidentifiable gratings provide non-uniqueness examples for appropriately chosen wave number and incident angles. We also indicate and fix a gap in the proof of the main theorem of Elschner and Yamamoto (2007 Z. Anal. Anwend. 26 165–77) and generalize the uniqueness results of that paper.

On Uniqueness and Solvability in The Linear Problem of Ship Waves

This work is devoted to the question of uniqueness and solvability for the linear problem of ship waves, describing the forward motion of rigid totally submerged bodies in an unbounded fluid with a free surface. The statement of the problem is discussed with some attention paid to the conditions at infinity. For the problem, Green's identity and boundary integral equations on the wetted surface of the bodies are derived; equivalence of the equations and the boundary-value problem is proved. New criteria and sufficient conditions of unique solvability are suggested. A uniqueness theorem is proved in the form of simple bounds of possible nonuniqueness parameters. Algorithms for verification of uniqueness and finding nonuniqueness examples are developed and numerical results illustrating the theoretical considerations are given.

An identity for a quasilinear ODE and its applications to the uniqueness of solutions of BVPs

The following boundary value problem (1.1) ( φ p ( u ′ ) ) ′ + a ( x ) f ( u ) = 0 , x 0 < x < x 1 , (1.2) u ( x 0 ) = u ( x 1 ) = 0 , is considered, where φ p ( s ) = | s | p − 2 s , p > 1 , a ∈ C 1 [ x 0 , x 1 ] , a ( x ) > 0 for x ∈ [ x 0 , x 1 ] , and f ∈ C 1 ( R ) . An identity for solutions of (1.1) and its linearized equation is derived. Some applications of the identity to uniqueness of solutions of problem (1.1)–(1.2) are presented. Non-uniqueness examples for problem (1.1)–(1.2) are also established. Moreover the results obtained here are applied to the study of radially symmetric solutions of the Dirichlet problem for elliptic equations in annular domains.

On nonuniqueness of solutions to elliptic equations for probability measures

We consider the set of all probability measures μ on R d satisfying an elliptic equation L ∗ μ = 0 in the weak worm. We give sufficient conditions in order that this set contains at least two different elements. We also construct new examples of nonuniqueness for such equations.