Superposition Operators Research Articles

An existence theory for nonlinear superposition operators of mixed fractional order

An existence theory for nonlinear superposition operators of mixed fractional order

The nonlinear superposition operator on harmonic fock spaces

We describe the conditions under which the superposition operator maps one harmonic Fock space into another. We further proved that all superposition operators on the spaces are both bounded and globally Lipschitz continuous. The condition for invertibility is also characterized. In the end, we identify all the fixed points of the operator on the spaces.

Order Bounded and 2-Complex Symmetric Weighted Superposition Operators on Fock Spaces

One aim of the paper is to characterize some complex symmetric and 2-complex symmetric bounded weighted superposition operators on Fock spaces respect to the conjugations J and Jr,s,t defined by Jf(z)=f(z¯)¯ and Jr,s,tf(z)=teszf(rz+s¯)¯. Another aim is to characterize the order bounded weighted superposition operators from one Fock space into another Fock space.

S-asymptotically ω periodic solutions of heat equation

We mean by an S-asymptotically ω-periodic function any continuous and bounded function from the real axis to a Banach space that converges to a periodic function as t tends to infinity. We study in this paper the existence, uniqueness and differentiable dependence on the S-asymptotically ω-periodic mild solution of a heat equation on the space of continuous functions from a non-empty n-dimensional bounded domain with a Lipschitz boundary to the real axis. The dependence of the solution concerns the initial conditions, more precisely when the initial conditions is a S-asymptotically ω-periodic function we study the differentiable dependence of the S-asymptotically ω-periodic solution of heat equation. To show our main result in this work we introduce the properties of the superposition operator, or also called Nemytskii operator, in the space of S-asymptotically ω-periodic functions. The notion of derivation for the last operator will also be highlighted. We also use in this paper semi-groups which have become important tools for differential equations. In this study, our focus shifted from seeking an s-asymptotically w-periodic solution for our heat equation, which is a problem in dynamic systems, to one in functional analysis. More precisely, our strategy consists of applying the implicit function theorem on a certain operator that we constructed on our workspaces in order to achieve the objective described in our main theorem. In fact our theorem gives conditions to ensure that around a mild S-asymptotically ω-periodic solution of our heat equation with an initial value, there exists a regular (in the usual sens) mild S-asymptotically ω-periodic solution which depends on a neighboring initial condition.

S-asymptotically ω-Periodic Solutions of generalized Liénard

The S-asymptotically ω-periodic functions are a continuous and bounded functions from the real axis to a Banach space that converges to a periodic functions as tends to infinity. Starting from the zero solution, we prove in this work the existence and uniqueness of S-asymptotically ω-periodic solution of generalized Liénard's differential Equation. We stady after that the regular dependence of this solution with a certain parameter in Banach space, present in our equation, and with the forcing term hwo possesses a similar nature as the later. For this, our approach will be to use a perturbation method around an equilibrium. More precisely when the forcing is a Sasymptotically ω-periodic function we study the differentiable dependence of the S-asymptotically ω-periodic solution of Liénard equation. In this study, we changed our initial objective, which was to search for na S asymptotically ω-periodic solution for our Lienard equation, a problem relating to dynamical systems, towards an approach based on functional analysis. Concretely, we adopted a strategy consisting of using the implicit function theorem on a specific operator that we defined in our workspaces. This approach allowed us to achieve the objective stated in our main theorem. To realize our aim, we use the Nemytskii operators (also called superposition operators) and state some properties on these operators. We have also extended the well-established result on the almost periodic function of the derivative of an almost periodic function to the context of S-asymptotically ω-periodic cases. Finally, and to close our work, we give a corollary which presents a particular case of our main theorem.

Multi‐Wavelength Metasurface Empowered Cryptography for Heightened Security and Improved Fidelity

AbstractInformation security holds paramount importance in the contemporary information society. Here, a novel cryptography method is proposed based on the multi‐wavelength complex‐amplitude metasurface, which can achieve a high security level by generating a frequency‐selective key as an additional layer of protection. Moreover, the quality of the decrypted holographic images can be significantly improved by employing the complex‐amplitude encoding method and the modified visual secret sharing scheme, which adopts a new superposition operator to prevent fidelity loss. As a proof‐of‐concept demonstration, a prototype metasurface sample is designed, fabricated, and characterized, where the measured results agree very well with the numerical ones and the design goals. The proposed method can reinforce security level and improve fidelity of the decrypted message, rendering it appealing for applications in communication and anti‐counterfeiting.

An existence theory for superposition operators of mixed order subject to jumping nonlinearities

We consider a superposition operator of the form ∫[0,1](−Δ)sudμ(s), for a signed measure µ on the interval of fractional exponents [0,1] , joined to a nonlinearity whose term of homogeneity equal to one is ‘jumping’, i.e. it may present different coefficients in front of the negative and positive parts. The signed measure is supposed to possess a positive contribution coming from the higher exponents that overcomes its negative contribution (if any). The problem taken into account is also of ‘critical’ type, though in this case the critical exponent needs to be carefully selected in terms of the signed measure µ. Not only the operator and the nonlinearity considered here are very general, but our results are new even in special cases of interest and include known results as particular subcases. The possibility of considering operators ‘with the wrong sign’ is also a complete novelty in this setting.

Narrow operators on [formula omitted]-complete lattice-normed spaces

We extend some of main results of Abasov et al., 2016, Fotiy et al., (2020), Mykhaylyuk et al., (2015), Pliev and Fang, (2017), Pliev and Popov, (2014) to the setting of orthogonally additive operators on lattice-normed spaces. We introduce a new class of C-complete lattice-normed spaces which strictly includes a class of Banach–Kantorovich spaces. The first main result of the paper asserts that every laterally-to-norm continuous C-compact orthogonally additive operator T:X→Y from an atomless C-complete lattice-normed space X to a Banach space Y is narrow. As a non expecting consequence of the first main result we obtain necessary and sufficient conditions for a nonlinear superposition operator TN:E(X)→E(X) to be C-compact. We also show that the sum of two orthogonally additive operators G and T, where G is narrow and T is laterally-to-norm continuous and C-compact, is a narrow operator. In the last part of the article we investigate dominated orthogonally additive operators. In particular we show the narrowness of dominated operators taking values in a Banach sequence space Y or in a separable symmetric operator space CE.

A fixed point result for DND-convex-power condensing mappings

In this paper, based on the so-called degree of nondensifiability (DND), we introduce the concept of DND-convex-power condensing mapping which generalizes that of the DND-condensing one. A fixed point result for this new class of mappings is proved, and with some examples we evidence the differences between it and other fixed point theorems of the same type. As application of our results, we prove under suitable conditions the existence of fixed point for the superposition operator, defined in the Banach spaces of the continuous functions from [Formula: see text] and with values in a Banach space.

Spectral of the nonlinear weighted superposition operator on Fock spaces

We follow several approaches in nonlinear spectral theory and determine the various spectral forms for the nonlinear weighted superposition operator on Fock spaces. The results show that most of the forms introduced so far coincide and contain singeltons. The classical, asymptotic, and connected eigenvalues, and some numerical ranges of the operator are also identified. We further prove that the operator is both linear and odd asymptotically with respect to the pointwise multiplication operator on the spaces.

On the identification and optimization of nonsmooth superposition operators in semilinear elliptic PDEs

We study an infinite-dimensional optimization problem that aims to identify the Nemytskii operator in the nonlinear part of a prototypical semilinear elliptic partial differential equation (PDE) which minimizes the distance between the PDE-solution and a given desired state. In contrast to previous works, we consider this identification problem in a low-regularity regime in which the function inducing the Nemytskii operator is a-priori only known to be an element of H1loc(ℝ). This makes the studied problem class a suitable point of departure for the rigorous analysis of training problems for learning-informed PDEs in which an unknown superposition operator is approximated by means of a neural network with nonsmooth activation functions (ReLU, leaky-ReLU, etc.). We establish that, despite the low regularity of the controls, it is possible to derive a classical stationarity system for local minimizers and to solve the considered problem by means of a gradient projection method. The convergence of the resulting algorithm is proven in the function space setting. It is also shown that the established first-order necessary optimality conditions imply that locally optimal superposition operators share various characteristic properties with commonly used activation functions: They are always sigmoidal, continuously differentiable away from the origin, and typically possess a distinct kink at zero. The paper concludes with numerical experiments which confirm the theoretical findings.

Superposition operators between normal weight bloch spaces

Superposition operators between normal weight bloch spaces

The nonlinear superposition operators between some analytic function spaces

In this paper we characterize the nonlinear superposition operators mapping the space Q s \mathcal {Q}_s into the Dirichlet-type space D α p \mathcal {D}^p_\alpha and answer basically the remaining open cases provided by Galanopoulos, Girela, and Auxiliadora Márquez [J. Math. Anal. Appl. 463 (2018), pp. 659–680]. Meanwhile, we give some sufficient-necessary conditions for the superposition operators between Q s \mathcal {Q}_s spaces in all cases.

Iterated function systems based on the degree of nondensifiability

In the present paper we introduce the concept of iterated function systems (IFS) having at least one \phi -condensing mapping which belongs to the finite set of self-mappings that define the IFS. It is shown the existence of an invariant for those IFS. Whenever all the self-mappings are \phi -condensing we prove that the invariant set is compact. We propose some applications of those IFS having \phi -condensing self-mappings to the superposition operator defined on the Banach space \mathcal{C}([0,1]) .

Upper semicontinuous convex-valued selectors of a Nemytskii operator with nonconvex values and evolution inclusions with maximal monotone operators

On the space of continuous functions, we consider a multivalued superposition operator, a Nemytskii operator valued in the space of p-integrable functions with 1≤p<∞. The Nemytskii operator is generated by a multivalued mapping defined on the direct product of a segment of the real line and a separable reflexive Banach space and having closed values in this space. At each point of the phase variable, the mapping either has a closed graph and is convex-valued or it is lower semicontinuous in some neighborhood of this point. In recent years, such mappings are called mappings with mixed semicontinuity properties. We prove that the restriction of the Nemytskii operator onto any compact subset of the space of continuous functions with the sup-norm has a multivalued selector with convex closed values that is upper semicontinuous in the weak topology of the space of p-integrable functions. This result generalizes some well-known results of the author valid in the case when the multivalued mapping generating the Nemytskii operator has compact values. A multivalued selector theorem is used to study an evolution inclusion in a separable Hilbert space with a time-dependent maximal monotone operator and a composite perturbation that is the sum of two multivalued mappings. The values of the first multivalued mapping are closed, bounded, not necessarily convex sets. The second multivalued mapping has closed values and possesses mixed semicontinuity properties. The existence of an absolutely continuous solution is proved. The proof is based on the theorem on an upper semicontinuous convex-valued selector and the author's theorem on selectors continuous in parameter and passing through fixed points of a parameter-dependent multivalued mapping with closed, nonconvex, decomposable values in the space of p-integrable functions as well as on the classical Ky Fan's fixed point theorem. Our approach is fairly simple, concise and clear as compared to other known approaches based on discretization.

On the superposition operator in the space of functions of Hωq ([0,1

In this paper, we obtained a necessary and sufficient condition for the embedding H?q([0,1])? IBVqp([0,1]), where IBVq p denotes the set of functions of bounded q-integral p-variation. Additionally, the conditions for the composition and superposition operators were provided to map the space H?q([0,1]) into itself, by which these operators were bounded. Finally, we applied these results to examine the existence and uniqueness of solutions to Hammerstein integral equations in the space of H?q([0,1]).

Some results on the space of bounded second &lt;i&gt;κ&lt;/i&gt;-variation functions

&lt;abstract&gt;&lt;p&gt;In this paper, we prove that if a globally Lipschitz non-autonomous superposition operator maps the space of functions of bounded second $ \kappa $-variation into itself then its generator function satisfies a Matkowski condition. We also provide conditions for the existence and uniqueness of solutions of the Hammerstein and Volterra equations in this space.&lt;/p&gt;&lt;/abstract&gt;

A two-parameter family of non-parametric, deformed exponential manifolds

We construct a new family of non-parametric statistical manifolds by means of a two-parameter class of deformed exponential functions, that includes functions with power-law, linear and sublinear rates of growth. The manifolds are modelled on weighted, mixed-norm Sobolev spaces that are especially suited to this purpose, in the sense that an important class of nonlinear superposition operators (those used in the construction of divergences and tensors) act continuously on them. We analyse variants of these operators, that map into “subordinate” Sobolev spaces, and evaluate the associated gain in regularity. With appropriate choice of parameter values, the manifolds support a large variety of the statistical divergences and entropies appearing in the literature, as well as their associated tensors, eg. the Fisher-Rao metric. Manifolds of finite measures and probability measures are constructed; the latter are shown to be smoothly embedded submanifolds of the former.

General solution and Green’s function for fluid-saturated infinite and semi-infinite orthotropic poro-thermoelastic materials

In this study, two-dimensional (2D) steady-state general solution and Green’s function for orthotropic materials under the influence of poro-thermoelasticity (PTE) are investigated. Using the superposition principle and operator theory, the complete general solution of corresponding governing equations is calculated and stated in terms of harmonic functions. Based on these compact general solutions, the 2D Green’s function for a steady line fluid source and line heat source in the interior of an infinite orthotropic PTE plane and on the surface of a semi-infinite orthotropic PTE plane is presented. Consequently, eight newly implemented harmonic functions with undetermined constants are obtained. The equilibrium, boundary, and continuous conditions are used to obtain undetermined constants, and the corresponding coupled field is obtained by inserting these harmonic functions into the general solution. All coupled field’s components are described in terms of elementary functions, which can be used as the benchmarks for various numerical codes and approximate solutions.

Viscoelastic transcranial wave propagation modeling for neuronavigated focused ultrasound procedures in humans

Transcranial focused ultrasound (FUS) has gained significant attention as a non-invasive therapeutic technique that can reach both cortical and subcortical targets in the human brain. However, the skull barrier distorts the ultrasound beam and introduces significant losses that need to be accounted for. In this talk, I will present our efforts to develop high precise transcranial ultrasound modeling based on the open-source viscoelastic wave propagation solver BabelViscoFDTD and its application in neuronavigated procedures for FUS-based neuromodulation. The viscoelastic wave equation is expressed in isotropic stress tensors and displacement vectors, and their nodes are placed in a staggered grid arrangement. The solver uses a Cartesian O(4)-space O(2)-time FDTD scheme. A narrow-band quality factor models attenuation. A reduction of staircase artifacts can be enabled using a superposition operator. BabelViscoFDTD is optimized for multiple computing GPU and CPU backends (CUDA, OpenCL, Metal, and X86-64) and includes GPU-accelerated solvers of the Rayleigh-Sommerfeld equation and Bio-Heat thermal equations. We adapted the viscoelastic and thermal solvers to neuronavigate two FUS devices: A 128-element concaved array operable at 250 kHz and 700 kHz and a 4-ring concaved array operable at 500 kHz. The high degree of optimization of the solvers allow that numerical procedures (neuronavigation and modeling) can be performed in an Apple M1 Max (64 GB RAM) laptop computer.