Non-trivial Kernel Research Articles

Analysis of zero modes for Dirac operators with magnetic links

In this paper we provide a means to approximate Dirac operators with magnetic fields supported on links in S3 (and R3) by Dirac operators with smooth magnetic fields. Then we proceed to prove that under certain assumptions, the spectral flow of paths along these operators is the same in both the smooth and the singular case. We recently characterized the spectral flow of such paths in the singular case. This allows us to show the existence of new smooth, compactly supported magnetic fields in R3 for which the associated Dirac operator has a non-trivial kernel. Using Clifford analysis, we also obtain criteria on the magnetic link for the non-existence of zero modes.

A remark about weak fillings

Let L be a closed manifold of dimension n≥2 which admits a totally real embedding into Cn. Let ST*L be the space of rays of the cotangent bundle T*L of L, and let DT*L be the unit disk bundle of T*L defined by any Riemannian metric on L. We observe that ST*L endowed with its standard contact structure admits weak symplectic fillings W which are diffeomorphic to DT*L and for which any closed Lagrangian submanifold N⊂W has the property that the map H1(N,R)→H1(W,R) has a nontrivial kernel. This relies on a variation on a theorem by Laudenbach and Sikorav.

On the Persistence of the Eigenvalues of a Perturbed Fredholm Operator of Index Zero under Nonsmooth Perturbations

Let $H$ be a real Hilbert space and denote by $S$ its unit sphere. Consider the nonlinear eigenvalue problem $Lx + \epsilon N(x) = \lambda x$, where $\epsilon, \lambda \in \mathbb R$, $L : H \to H$ is a bounded self-adjoint (linear) operator with nontrivial kernel and closed image, and $N : H \to H$ is a (possibly) nonlinear perturbation term. A unit eigenvector $\bar x \in S \cap \mathrm {Ker} L$ of $L$ (corresponding to the eigenvalue $\lambda = 0$) is said to be persistent if it is close to solutions $x \in S$ of the above equation for small values of the parameters $\epsilon \neq 0$ and $\lambda$. We give an affirmative answer to a conjecture formulated by R. Chiappinelli and the last two authors in an article published in 2008. Namely, we prove that, if $N$ is Lipschitz continuous and the eigenvalue $\lambda = 0$ has odd multiplicity, then the sphere $S\cap \mathrm {Ker} L$ contains at least one persistent eigenvector. We provide examples in which our results apply, as well as examples showing that, if the dimension of $\mathrm {Ker} L$ is even, then the persistence phenomenon may not occur.

A Note on Constraint Preconditioning

We extend the results derived in Keller, Gould, and Wathen [SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1300--1317] for constraint preconditioning. In particular, we consider the case where the leading block of the system matrix as well as that of the preconditioner are nonsymmetric and have nontrivial kernels. We also analyze the form of the preconditioner with negated constraints, which ensures that under reasonable assumptions the preconditioned system is diagonalizable, while preserving the nonunit eigenvalues and negating some unit eigenvalues.

Existence results for a class of high order differential equation associated with integral boundary conditions at resonance

By using Mawhin’s continuation theorem, we provide some sufficient conditions for the existence of solution for a class of high order differential equations of the form $$x^{(n)} =f(t,x,x^{\prime },\dots ,x^{(n-1)})\,, \quad t \in [0, 1]\,,$$ associated with the integral boundary conditions at resonance. The interesting point is that we shall deal with the case of nontrivial kernel of arbitrary dimension corresponding to high order differential operator which will cause some difficulties in constructing the generalized inverse operator.

A Non-trivial Ghost Kernel for Complex Projective Spaces with Symmetries

It is shown that the ghost kernel for certain equivariant stable cohomotopy groups of projective spaces is non-trivial. The proof is based on the Borel cohomology Adams spectral sequence and the calculations with the Steenrod algebra afforded by it.

A concrete realization of the slow-fast alternative for a semilinear heat equation with homogeneous Neumann boundary conditions

Abstract We investigate the asymptotic behavior of solutions to a semilinear heat equation with homogeneous Neumann boundary conditions. It was recently shown that the nontrivial kernel of the linear part leads to the coexistence of fast solutions decaying to 0 exponentially (as time goes to infinity), and slow solutions decaying to 0 as negative powers of t. Here we provide a characterization of slow/fast solutions in terms of their sign, and we show that the set of initial data giving rise to fast solutions is a graph of codimension one in the phase space.

Optimal decay estimates for the general solution to a class of semi-linear dissipative hyperbolic equations

We consider a class of semi-linear dissipative hyperbolic equations in which the operator associated to the linear part has a nontrivial kernel. Under appropriate assumptions on the nonlinear term, we prove that all solutions decay to 0, as t \to \infty , at least as fast as a suitable negative power of t . Moreover, we prove that this decay rate is optimal in the sense that there exists a nonempty open set of initial data for which the corresponding solutions decay exactly as that negative power of t . Our results are stated and proved in an abstract Hilbert space setting, and then applied to partial differential equations.

End-to-end verification of information-flow security for C and assembly programs

Protecting the confidentiality of information manipulated by a computing system is one of the most important challenges facing today's cybersecurity community. A promising step toward conquering this challenge is to formally verify that the end-to-end behavior of the computing system really satisfies various information-flow policies. Unfortunately, because today's system software still consists of both C and assembly programs, the end-to-end verification necessarily requires that we not only prove the security properties of individual components, but also carefully preserve these properties through compilation and cross-language linking. In this paper, we present a novel methodology for formally verifying end-to-end security of a software system that consists of both C and assembly programs. We introduce a general definition of observation function that unifies the concepts of policy specification, state indistinguishability, and whole-execution behaviors. We show how to use different observation functions for different levels of abstraction, and how to link different security proofs across abstraction levels using a special kind of simulation that is guaranteed to preserve state indistinguishability. To demonstrate the effectiveness of our new methodology, we have successfully constructed an end-to-end security proof, fully formalized in the Coq proof assistant, of a nontrivial operating system kernel (running on an extended CompCert x86 assembly machine model). Some parts of the kernel are written in C and some are written in assembly; we verify all of the code, regardless of language.

Inverse problem for Oskolkov's system of equations

The inverse problem with unknown source function is studied for the nonlinear Oskolkov's system of partial differential equations that describes the dynamics of viscoelastic Kelvin–Voigt fluid. It is reduced to the problem for a first order semilinear ordinary differential equation with operator coefficients in Banach spaces. The main difficulty of the problem is a presence of the operator with a nontrivial kernel at the derivative. By the results on the unique solvability of the abstract problem, obtained by the authors before, the existence of a unique classical solution for the Oskolkov's system inverse problem is proved. Copyright © 2015 John Wiley & Sons, Ltd.

Parameterized complexity of control and bribery for d-approval elections

A d-Approval election consists of a set C of candidates and a set V of votes, where each vote v can be presented as a set of d candidates. For a vote v∈V, the d-Approval voting protocol assigns one point to each candidate in v. The candidate getting the most points from all votes wins the election. An important aspect of studying election systems is the strategic behavior such as control and bribery problems. The control by deleting votes problem decides whether for a given election (C,V), a specific candidate c, and an integer k, it is possible to delete at most k votes such that c wins the resulting election. In the control by adding votes setting, one has two sets V and U of votes and asks for a subset U′⊆U such that |U′|≤k and c becomes the winner in V∪U′. The bribery problem has the same input as the vote deleting control problem and asks for changing at most k votes to make c win. All three problems have been shown NP-hard. We initialize the study of the parameterized complexity of these problems and present a collection of tractability and intractability results. In particular, we derive a polynomial-size problem kernel for the standard parameterization of the control by deleting votes problem, the seemingly first non-trivial problem kernel for the control problem of elections.

Polynomial completion of symplectic jets and surfaces containing involutive lines

Motivated by work of Dragt and Abell on accelerator physics, we study the completion of symplectic jets by polynomial maps of low degrees. We use Andersén-Lempert Theory to prove that symplectic completions always exist, and we prove the degree bound conjectured by Dragt and Abell in the physically relevant cases. However, we disprove the degree bound for $$3$$ -jets in dimension 4. This follows from the fact that if $$\Sigma $$ is the disjoint union of $$r=7$$ involutive lines in $$\mathbb P^3$$ , then $$\Sigma $$ is contained in a degree $$d=4$$ hypersurface, i.e., the restriction morphism $$\iota :H^0(\mathbb P^3,\mathcal O_{\mathbb P^3}(4))\rightarrow H^0(\Sigma ,\mathcal O_{\Sigma }(4))$$ has a nontrivial kernel (Todd). We give two new proofs of this fact, and finally we show that if $$(r,d)\ne (7,4)$$ then the map $$\iota $$ has maximal rank.

Non-scattering wavenumbers and far field invisibility for a finite set of incident/scattering directions

We investigate a time harmonic acoustic scattering problem by a penetrable inclusion with compact support embedded in the free space. We consider cases where an observer can produce incident plane waves and measure the far field pattern of the resulting scattered field only in a finite set of directions. In this context, we say that a wavenumber is a non-scattering wavenumber if the associated relative scattering matrix has a non-trivial kernel. Under certain assumptions on the physical coefficients of the inclusion, we show that the non-scattering wavenumbers form a (possibly empty) discrete set. Then, in a second step, for a given real wavenumber and a given domain , we present a constructive technique to prove that there exist inclusions supported in for which the corresponding relative scattering matrix is null. These inclusions have the important property to be impossible to detect from far field measurements. The approach leads to a numerical algorithm, which is described at the end of the paper and which allows us to provide examples of (approximated) invisible inclusions.

Conformally covariant operators and conformal invariants on weighted graphs

Let \(G\) be a finite connected simple graph. We define the moduli space of conformal structures on \(G\). We propose a definition of conformally covariant operators on graphs, motivated by Graham et al. (J Lond Math Soc 46:557–565, 1992). We provide examples of conformally covariant operators, which include the edge Laplacian and the adjacency matrix on graphs. In the case where such an operator has a nontrivial kernel, we construct conformal invariants, providing discrete counterparts of several results in Canzani et al. (2014; Electron Res Announc Math Sci 20:43–50, 2013) established for Riemannian manifolds. In particular, we show that the nodal sets and nodal domains of null eigenvectors are conformal invariants.

A novel approach to construct the adjoint problem for a first-order functional integro-differential equation with general nonlocal condition

In this work, with the aim of determining Green’s solution or generalized Green’s solution, we propose a novel constructive approach by which a linear or specific nonlinear problem involving general linear nonlocal condition for a first-order functional ordinary integro-differential equation with general nonsmooth coefficients satisfying some general properties such as p-integrability and boundedness is reduced to an integral equation. A system of two integro-algebraic equations, called “the adjoint system,” is constructed for this problem. Green’s functional for the problem with trivial kernel and generalized Green’s functional for the problem with nontrivial kernel are the unique solutions to the specific cases of this adjoint system. Green’s functional and generalized Green’s functional have two components. Their first components correspond to Green’s function and generalized Green’s function for the problem, respectively. Some illustrative applications are provided with known and unknown results.

On independence of some pseudocharacters on braid groups

It is proved that the pseudocharacter defined on the braid group by the signature of braid closures is linearly independent of all pseudocharacters obtained from the twist number via the Malyutin operators, provided that the number of strands is greater than 4. This pseudocharacter is shown to have a nontrivial kernel part. It is observed that the operators $I$ and $R$ defined by Malyutin on the space of pseudocharacters satisfy the Heisenberg relation, and that some of Malyutin’s results are standard consequences of this fact.

Faddeev eigenfunctions for two-dimensional Schrödinger operators via the Moutard transformation

We demonstrate how the Moutard transformation of two-dimensional Schrodinger operators acts on the Faddeev eigenfunctions on the zero energy level and present some explicitly computed examples of such eigenfunctions for smooth fast decaying potentials of operators with non-trivial kernel and for deformed potentials which correspond to blowing up solutions of the Novikov-Veselov equation.

Normal families of functions for subelliptic operators and the theorems of Montel and Koebe

A classical theorem of Montel states that a family of holomorphic functions on a domain Ω⊆C, uniformly bounded on the compact subsets of Ω, is a normal family. The aim of this paper is to obtain a generalization of this result in the subelliptic setting of families of solutions u to Lu=0, where L belongs to a wide class of real divergence-form PDOs, comprising sub-Laplacians on Carnot groups, subelliptic Laplacians on arbitrary Lie groups, as well as the Laplace–Beltrami operator on Riemannian manifolds. To this end, we extend another remarkable result, due to Koebe: we characterize the solutions to Lu=0 as fixed points of suitable mean-value operators with non-trivial kernels. A suitable substitute for the Cauchy integral formula is also provided. Finally, the local-boundedness assumption is relaxed, by replacing it with Lloc1-boundedness.

Regularization of an autoconvolution problem in ultrashort laser pulse characterization

An ill-posed inverse problem of autoconvolution type is investigated. This inverse problem occurs in non-linear optics in the context of ultrashort laser pulse characterization. The novelty of the mathematical model consists in a physically required extension of the deautoconvolution problem beyond the classical case usually discussed in literature: (i) For measurements of ultrashort laser pulses with the self-diffraction SPIDER method, a stable approximate solution of an autocovolution equation with a complex-valued kernel function is needed. (ii) The considered scenario requires complex functions both, in the solution as well as in the right-hand side of the integral equation. Since, however, noisy data are available not only for amplitude and phase functions of the right-hand side, but also for the amplitude of the solution, the stable approximate reconstruction of the associated smooth phase function represents the main goal of the paper. An iterative regularization approach will be described that is specifically adapted to the physical situation in pulse characterization, using a non-standard stopping rule for the iteration process of computing regularized solutions. The opportunities and limitations of regularized solutions obtained by our approach are illustrated by means of several case studies for synthetic noisy data and physically realistic complex-valued kernel functions. Based on an example with focus on amplitude perturbations, we show that the autoconvolution equation is locally ill-posed everywhere. To date, the analytical treatment of the impact of noisy data on phase perturbations remains an open question. However, we show its influence with the help of numerical experiments. Moreover, we formulate assertions on the non-uniqueness of the complex-valued autoconvolution problem, at least for the simplified case of a constant kernel. The presented results and figures associated with case studies illustrate the ill-posedness phenomena also for the case of non-trivial complex kernel functions.

PERSISTENCE OF THE NORMALIZED EIGENVECTORS OF A PERTURBED OPERATOR IN THE VARIATIONAL CASE

AbstractLet H be a real Hilbert space and denote by S its unit sphere. Consider the nonlinear eigenvalue problem Ax + ε B(x) =δ x, where A: H → H is a bounded self-adjoint (linear) operator with nontrivial kernel Ker A, and B: H → H is a (possibly) nonlinear perturbation term. A unit eigenvector x0 ∈ S∩ Ker A of A (thus corresponding to the eigenvalue δ=0, which we assume to be isolated) is said to be persistent, or a bifurcation point (from the sphere S∩ Ker A), if it is close to solutions x ∈ S of the above equation for small values of the parameters δ ∈ ℝ and ε ≠ 0. In this paper, we prove that if B is a C1 gradient mapping and the eigenvalue δ=0 has finite multiplicity, then the sphere S∩ Ker A contains at least one bifurcation point, and at least two provided that a supplementary condition on the potential of B is satisfied. These results add to those already proved in the non-variational case, where the multiplicity of the eigenvalue is required to be odd.