Duality Mapping Research Articles

Majorana zero mode-soliton duality and in-gap and BIC bound states in modified Toda model coupled to fermion

A two-dimensional field theory of a fermion chirally coupled to Toda field plus a scalar self-coupling potential is considered. Using techniques of integrable systems we obtain analytical zero modes, in-gap states and bound states in the continuum (BIC) for topological configurations of the scalar field. Fermion-soliton duality mappings are uncovered for the bound state spectrum, which interpolates the weak and strong coupling sectors of the model and give rise to novel Thirring-like and multi-frequency sine-Gordon models, respectively. The non-perturbative effects of the back-reaction of the fermion bound states on the kink are studied and it is shown that the zero mode would catalyze the emergence of a new kink with lower topological charge and greater slope at the center, in the strong coupling limit of the model. For special topological charges and certain relative phases of the fermion components the kinks can host Majorana zero modes. The Noether, topological and a novel nonlocal charge densities satisfy a formula of the Atiyah-Patodi-Singer-type. Our results may find applications in several branches of non-linear physics, such as confinement in QCD2, braneworld models, high Tc superconductivity and topological quantum computation. We back up our results with numerical simulations for continuous families of topological sectors.

Unequal time commutators in Friedmann universes: deterministic evolution of massless fields

We analyze minimally coupled massless scalar field in a Friedmann (FRW) universe in conformal co-ordinates to model the evolution of tensor perturbations and study the structure of the Wightman function therein. Using a duality map from a power law FRW universe to the de Sitter universe for such fields we obtain unequal time commutation relations between quantum field variables. We demonstrate that the commutation relations are invariant under state change and/or vacuum state selection. Using such commutators it is then possible to construct out of time ordered commutators (OTOC) in the FRW universes. The OTOCs are supposed to suggest the onset of chaotic behavior during the quantum evolution, we see that in case of Friedmann universes, unlike the scalar perturbations, the causal structure arrests the growth of quantum tensor perturbations for all the relevant epochs of the universe e.g. the de Sitter phase, the radiation dominated and the matter dominated era. Therefore the semi classical results of having a large backreaction and omnipresent noise in certain branches of the Friedmann universes with massless fields such as the tensor perturbations remain robust and stable.

On the Generalization of a Multiplicity Result

In this work, we shifted a recent multiplicity result by B. Ricceri from a Hilbert space to a Banach space setting by making use of a duality mapping relative to some increasing function. Using the min–max arguments, we provide conditions for an action functional to have at least two global minima.

Tikhonov regularization for a class of generalized hemivariational inequality in Banach spaces

This paper aims to study in a reflexive Banach space a class of generalized hemivariational inequality problem involving a set-valued mapping and a nonlinear perturbation. We study the solvability of the considered generalized hemivariational inequality problem and give some properties of its solutions by introducing some new coercivity conditions and hemivariational inequality property. Moreover, by means of the normalized duality mapping, we introduce the Tikhonov regularization for the problem in order to get an approximating sequence of its solutions, whose weak convergence is proved under mild conditions.

Coupled Variational Inequalities: Existence, Stability and Optimal Control

In this paper, we introduce and investigate a new kind of coupled systems, called coupled variational inequalities, which consist of two elliptic mixed variational inequalities on Banach spaces. Under general assumptions, by employing Kakutani-Ky Fan fixed point theorem combined with Minty technique, we prove that the set of solutions for the coupled variational inequality (CVI, for short) under consideration is nonempty and weak compact. Then, two uniqueness theorems are delivered via using the monotonicity arguments, and a stability result for the solutions of CVI is proposed, through the perturbations of duality mappings. Furthermore, an optimal control problem governed by CVI is introduced, and a solvability result for the optimal control problem is established. Finally, to illustrate the applicability of the theoretical results, we study a coupled elliptic mixed boundary value system with nonlocal effect and multivalued boundary conditions, and a feedback control problem involving a least energy condition with respect to the control variable, respectively.

"A new Krasnoselskii’s type algorithm for zeros of strongly monotone and Lipschitz mappings"

"For q > 1, let E be a q-uniformly smooth real Banach space with dual space E∗. Let A : E → E∗ be a Lipschitz and strongly monotone mapping such that A^{−1}(0) ̸= ∅. For given x_1 ∈ E, let {x_n} be generated iteratively by the algorithm : x_{n+1} = x_n − λJ^{−1}(Ax_n), n ≥ 1, where J is the normalized duality mapping from E into E∗ and λ is a positive real number choosen in a suitable interval. Then it is proved that the sequence {xn} converges strongly to x∗, the unique point of A^{−1}(0). Our theorems are applied to the convex minimization problem. Futhermore, our technique of proof is of independent interest."

Convex optimization in sums of Banach spaces

We characterize the solution of a broad class of convex optimization problems that address the reconstruction of a function from a finite number of linear measurements. The underlying hypothesis is that the solution is decomposable as a finite sum of components, where each component belongs to its own prescribed Banach space; moreover, the problem is regularized by penalizing some composite norm of the solution. We establish general conditions for existence and derive the generic parametric representation of the solution components. These representations fall into three categories depending on the underlying regularization norm: (i) a linear expansion in terms of predefined “kernels” when the component space is a reproducing kernel Hilbert space (RKHS), (ii) a non-linear (duality) mapping of a linear combination of measurement functionals when the component Banach space is strictly convex, and, (iii) an adaptive expansion in terms of a small number of atoms within a larger dictionary when the component Banach space is not strictly convex. Our approach generalizes and unifies a number of multi-kernel (RKHS) and sparse-dictionary learning techniques for compressed sensing available in the literature. It also yields the natural extension of the classical spline-fitting techniques in (semi-)RKHS to the abstract Banach-space setting.

Some Results on Normalized Duality Mappings and Approximating Fixed Points in Convex Real Modular Spaces

In this paper, the concept of normalized duality mapping has introduced in real convex modular spaces. Then, some of its properties have shown which allow dealing with results related to the concept of uniformly smooth convex real modular spaces. For multivalued mappings defined on these spaces, the convergence of a two-step type iterative sequence to a fixed point is proved

Existence and approximate controllability of non-autonomous functional impulsive evolution inclusions in Banach spaces

In this paper, we are concerned with the approximate controllability results for a class of impulsive functional differential control systems involving time dependent operators in Banach spaces. First, we show the existence of a mild solution for non-autonomous functional impulsive evolution inclusions in separable reflexive Banach spaces with the help of the evolution family and a generalization of the Leray-Schauder fixed point theorem for multi-valued maps. In order to establish sufficient conditions for the approximate controllability of our problem, we first consider a linear-quadratic regulator problem and obtain the optimal control in the feedback form, which contains the resolvent operator consisting of duality mapping. With the help of this optimal control, we prove the approximate controllability of the linear system and hence derive sufficient conditions for the approximate controllability of our problem. Moreover, in this paper, we rectify several shortcomings of the related works available in the literature, namely, proper identification of resolvent operator in Banach spaces, characterization of phase space in the presence of impulsive effects, and lack of compactness of the operator h(⋅)↦∫0⋅U(⋅,s)h(s)ds:L1([0,T];Y)→C([0,T];Y), where Y is a Banach space and U(⋅,⋅) is the evolution family, etc. Finally, we provide a concrete example to illustrate the efficiency of our results.

Strong and weak-star stabilisation of bilinear systems on nonreflexive Banach spaces

This paper deals with bilinear systems evolving in a Banach state space (which is not necessarily reflexive). To the best of our knowledge, strong and weak-star stabilisation of control systems in nonreflexive state spaces has not been studied yet. This work is a first attempt towards providing sufficient conditions that guarantee the weak-star stabilisation of the system at hand. The strong stabilisation is stated as well. The approach uses results relying on convex analysis and some properties of the duality mapping and linear semigroup theory. Applications to reaction–diffusion and transport equations are further presented.

Approximation-solvability of population biology systems based on p-Laplacian elliptic inequalities with demicontinuous strongly pseudo-contractive operators

The aim of this paper is to investigate existence, uniqueness and convergence of approximants of nonzero positive weak solutions for a class of population biology systems, which are models of one species based on p-Laplacian elliptic inequalities with demicontinuous strongly pseudo-contractive operators and involved logistic growth and harvesting rates. Toward this end, we foremost develop a kind of new general variational inequality principles with generalized duality mappings in reflexive Banach spaces. Then, we employ the new principle to obtain our main results for a general p-Laplacian elliptic inequality and the population biology systems. We note that the results are different from those relevant work of p-Laplacian elliptic inequalities in the literature, it is because a pseudo-contractive operator may not be an S-contractive operator and vice versa.

QED driven QAOA for network-flow optimization

We present a general framework for modifying quantum approximate optimization algorithms (QAOA) to solve constrained network flow problems. By exploiting an analogy between flow-constraints and Gauss' law for electromagnetism, we design lattice quantum electrodynamics (QED)- inspired mixing Hamiltonians that preserve flow constraints throughout the QAOA process. This results in an exponential reduction in the size of the configuration space that needs to be explored, which we show through numerical simulations, yields higher quality approximate solutions compared to the original QAOA routine. We outline a specific implementation for edge-disjoint path (EDP) problems related to traffic congestion minimization, numerically analyze the effect of initial state choice, and explore trade-offs between circuit complexity and qubit resources via a particle-vortex duality mapping. Comparing the effect of initial states reveals that starting with an ergodic (unbiased) superposition of solutions yields better performance than beginning with the mixer ground-state, suggesting a departure from the ``short-cut to adiabaticity" mechanism often used to motivate QAOA.

Simplified Levenberg–Marquardt method in Banach spaces for nonlinear ill-posed operator equations

In 2011, Jin Qinian has proposed a Levenberg–Marquardt method, by making use of duality mapping and the Bregman distance, to get an approximate solution of a nonlinear ill-posed operator equation in Banach space using an a posteriori parameter choice strategy and Morozov-type stopping rule. The method considered by Jin Qinian was an extension of the method proposed by M. Hanke in 1997 for the Hilbert space case. In this paper, we suggest a modified variant of the method, namely, the simplified Levenberg–Marquardt scheme in Banach spaces. The advantage of the method considered in the paper is that, it is also applicable for the operator equation with non-smooth operator. We establish convergence of the method under a modified a posteriori parameter choice strategy which is more feasible than the one considered by Jin Qinian (2011). Numerical example to demonstrate the validity of the considered method is discussed.

Width-Based Backward Search

It has been shown recently that duality mapping is a viable strategy to turn progression (forward search) into regression (backward search), but the experimental results suggest that the dual versions of standard IPC benchmarks are quite difficult to solve for heuristic search planners. We aim to study the performance of width-based planners over regression. Our experiments show that width-based search can solve dual problems efficiently when the goal state is restricted to single fluent, but it becomes challenging when the goal state contains conjunctive fluents. We then show that the backward version of best-first width-search (BFWS) with the evaluation function f5, BFWS(f5), and its polynomial variant, k-BFWS(f5), are not competitive with their forward versions, but can be orthogonal over the IPC benchmarks. Hence, we propose a front-to-end bidirectional search k-BDWS-e and its front-to-front variant by integrating forward and backward k-BFWS(f5) with the additional intersection check between expanded states whose novelty is 1 in the opposite Close list. Practical findings on the challenges of regression in classical planning are briefly discussed.

Duality Mapping for Schatten Matrix Norms

In this paper, we fully characterize the duality mapping over the space of matrices that are equipped with Schatten norms. Our approach is based on the analysis of the saturation of the Hölder inequality for Schatten norms. We prove in our main result that, for the duality mapping over the space of real-valued matrices with Schatten-p norm is a continuous and single-valued function and provide an explicit form for its computation. For the special case p = 1, the mapping is set-valued; by adding a rank constraint, we show that it can be reduced to a Borel-measurable single-valued function for which we also provide a closed-form expression.

Transport signature of the magnetic Berezinskii-Kosterlitz-Thouless transition

Motivated by recent experimental progress in 2D magnetism, we theoretically study spin transport in 2D easy-plane magnets at finite temperatures across the Berezinskii-Kosterlitz-Thouless (BKT) phase transition, by developing a duality mapping to the 2+1D electromagnetism with the full account of spin’s finite lifetime. In particular, we find that the non-conservation of spin gives rise to a distinct signature across the BKT transition, with the spin current decaying with distance power-law (exponentially) below (above) the transition; this is detectable in the proposed experiment with NiPS_33 and CrCl_33.

Orthogonality in smooth countably normed spaces

We generalize the concepts of normalized duality mapping, J-orthogonality and Birkhoff orthogonality from normed spaces to smooth countably normed spaces. We give some basic properties of J-orthogonality in smooth countably normed spaces and show a relation between J-orthogonality and metric projection on smooth uniformly convex complete countably normed spaces. Moreover, we define the J-dual cone and J-orthogonal complement on a nonempty subset S of a smooth countably normed space and prove some basic results about the J-dual cone and the J-orthogonal complement of S.

Distortional weak-coupling instability of Bogoliubov Fermi surfaces

Centrosymmetric multiband superconductors which break time-reversal symmetry generically have two-dimensional nodes, i.e., Fermi surfaces of Bogoliubov quasiparticles. We show that the coupling of the electrons to the lattice always leads to a weak-coupling instability of such a state towards spontaneous breaking of inversion symmetry at low temperatures. This instability is driven by a Cooper logarithm in the internal energy but the order parameter is not superconducting but distortional. We present a comprehensive symmetry analysis and introduce a measure that allows to compare the strengths of competing distortional instabilities. Moreover, we discuss the instability using an effective single-band model. This framework reveals a duality mapping of the effective model which maps the distortional order parameter onto a superconducting one, providing a natural explanation for the Cooper logarithm and the weak-coupling nature of the instability. Finally, we consider the possibility of a pair-density wave state when inversion symmetry is broken. We find that it can indeed exist but does not affect the instability itself.

General p-curl systems and duality mappings on Sobolev spaces for Maxwell equations

We study a general p-curl system arising from a model of type-II superconductors. We show several trace theorems that hold on either a Lipschitz domain with small Lipschitz constant or on a C^{1,1} domain. Certain duality mappings on related Sobolev spaces are computed and used to establish surjectivity results for the p-curl system. We also solve a nonlinear boundary value problem for a general p-curl system on a C^{1,1} domain and provide a variational characterization of the first eigenvalue of the p-curl operator.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/116/abstr.html

Variational and Topological Methods for a Class of Nonlinear Equations which Involves a Duality Mapping

The purpose of this paper is to show the existence results for the following abstract equation Jpu = Nfu,where Jp is the duality application on a real reflexive and smooth X Banach space, that corresponds to the gauge function φ(t) = tp-1, 1 < p < ∞. We assume that X is compactly imbedded in Lq(Ω), where Ω is a bounded domain in RN, N ≥ 2, 1 < q < p∗, p∗ is the Sobolev conjugate exponent.Nf : Lq(Ω) → Lq′(Ω), 1/q + 1/q′ = 1, is the Nemytskii operator that Caratheodory function generated by a f : Ω × R → R which satisfies some growth conditions. We use topological methods (via Leray-Schauder degree), critical points methods (the Mountain Pass theorem) and a direct variational method to prove the existence of the solutions for the equation Jpu = Nfu.