Dual Space Research Articles

A semigroup approach to a reaction–diffusion system with cross-diffusion

In this paper, we prove existence, uniqueness and special properties of the solution to a parabolic–elliptic reaction–diffusion system with cross-diffusion in an open bounded domain in Rd, as well as in Rd,d=1, 3, for various classes of the system parameters. In the case of a bounded domain and of Neumann boundary conditions we develop an approach based on a semigroup technique in the dual space (H1(Ω))′. This gives the possibility of considering singular initial data, distributions or functionals that may represent relevant physical data. The influence of the system parameters upon the qualitative behavior of the solution is investigated and some sufficient conditions for the existence of global or local solutions, and for global solutions in the case of small initial data are provided. In the case of the whole space, the problem is studied for a singular diffusion expressed by a multivalued operator. Existence is proved by passing to the limit in an approximating problem for which existence follows via a semigroup approach in H−1(Rd), d=1, 3. An argument for the uniqueness is developed in Beppo Levi spaces. Finally, the results are illustrated by a model exhibiting a self-organized criticality.

A Generalized an Extended Fractional Mellin Transform and Parsevals Identity

Integral transform is one of the techniques in the function transformation methods. Integral transforms have been interesting tools for solving different problems arising in applied mathematics, mathematical physics and engineering science for at least two centuries. We have studied an extended fractional Mellin transform in the generalized sense. For this testing space E and its dual space E* are considered. We have investigated inversion formula by using inversion of classical Mellin transform and prove Parseval"s Identity for an extended fractional Mellin transform. Also discussed some results of this transform.

Robust representation learning for heterogeneous attributed networks

The aim of heterogeneous attributed network embedding is mapping network into low-dimensional representations while preserving topological structure and attributed content. However, when the content similarity of two closely related nodes is small in the heterogeneous attributed network, we find that the embedding vectors obtained by combining network structure information and content information are bad. To tackle this problem, we propose a new robust representation learning model for Heterogeneous Attributed Networks Embedding with Dual-Space (HANE-DS), whose idea is to embed heterogeneous attributed network into dual spaces. Firstly, a heterogeneous attributed network is converted to a heterogeneous structure network and a heterogeneous content network. Secondly, the network nodes of the heterogeneous structure network are embedded into the structure space by using our proposed the structure embedding model, while the network nodes of the heterogeneous content network are embedded into the content space by applying our proposed the content embedding model. Finally, by utilizing the learning embedding vectors for downstream tasks, our approach can capture more comprehensive, more prosperous, and more reasonable information. Up to present, this is the first paper that focuses on the heterogeneous attributed network embedding based on Dual-Space. Three real-world networks and five synthetic networks to evaluate the HANE-DS model, and experimental results show that the HANE-DS can outperform baselines. For classification and clustering, the HANE-DS is superior to the baseline methods. For link prediction, the HANE-DS is in the top three of the baseline methods. For recommendation, the HANE-DS is also ahead of the baseline methods in terms of HR@K. On the synthetic networks, it is verified that the baseline models are sensitive to the content similarity of two closely related nodes. In comparison, our model is insensitive to that situation. In summary, the HANE-DS is a robust model for the heterogeneous attributed network embedding.

Risk-neutral multiobjective optimal control of random Volterra integral equations

We analyze risk-neutral multiobjective optimal control problems, governed by Volterra integral equations with random inputs, and subject to expectation-type final/pure state constraints and mixed pointwise control-state constraints. Moreover, controls are unbounded, and inclusion-form pure state as well as control-state constraints are given by measurable set-valued mappings, whose images are nonempty closed subsets of infinite dimensional spaces. With such constraints, the multiplier corresponding to pure state constraints belongs to the dual space of the Banach space-valued continuous functions. We set up a decomposition result for this dual, and develop a Lagrange multiplier theorem for an abstract multi-criteria optimization model, in which a Robinson-type constraint qualification is applied. Then, making use of the obtained outcomes, exploiting higher integrability and regularity of the random field control/state variables, and employing Banach space-valued integration techniques, we derive Fritz-John necessary optimality conditions, with integrable functions and (countably additive) measures singular with respect to the Lebesgue measure as multipliers, for local weak Pareto solutions of studied problems.

Action-angle coordinates on coadjoint orbits and multiplicity free spaces from partial tropicalization

Coadjoint orbits and multiplicity free spaces of compact Lie groups are important examples of symplectic manifolds with Hamiltonian groups actions. Constructing action-angle variables on these spaces is a challenging task. A fundamental result in the field is the Guillemin-Sternberg construction of Gelfand-Zeitlin integrable systems for the groups K=Un,SOn. Extending these results to groups of other types is one of the goals of this paper.Partial tropicalizations are Poisson spaces with constant Poisson bracket. They provide a bridge between dual spaces of Lie algebras Lie(K)⁎ with linear Poisson brackets and polyhedral cones which parametrize the canonical bases of irreducible modules of G=KC.We generalize the construction of partial tropicalizations to allow for arbitrary cluster charts, and apply it to questions in symplectic geometry. For each regular coadjoint orbit of a compact group K, we construct an exhaustion by symplectic embeddings of toric domains. As a by product we are able to complete the proof of a long-standing conjecture due to Karshon and Tolman about the Gromov width of regular coadjoint orbits. We also construct an exhaustion by symplectic embeddings of toric domains for multiplicity free K-spaces.An essential tool in our study is the dual Poisson-Lie group K⁎ equipped with the Berenstein-Kazhdan potential Φ. Partial tropicalizations arise as tropical limits of K⁎, and the potential Φ defines the range of action variables of a Gelfand-Zeitlin type integrable system. Our results give rise to new questions and conjectures about the Poisson structure of K⁎ and Ginzburg-Weinstein Poisson isomorphisms Lie(K)⁎→K⁎.

Lusztig Factorization Dynamics of the Full Kostant–Toda Lattices

We study extensions of the classical Toda lattices at several different space-time scales. These extensions are from the classical tridiagonal phase spaces to the phase space of full Hessenberg matrices, referred to as the Full Kostant-Toda Lattice. Our formulation makes it natural to make further Lie-theoretic generalizations to dual spaces of Borel Lie algebras. Our study brings into play factorizations of Loewner-Whitney type in terms of canonical coordinatizations due to Lusztig. Using these coordinates we formulate precise conditions for the well-posedness of the dynamics at the different space-time scales. Along the way we derive a novel, minimal box-ball system for Full Toda that doesn't involve any capacities or colorings, as well as an extension of O'Connell's ODEs to Full Toda.

A Deep Double Ritz Method (D[formula omitted]RM) for solving Partial Differential Equations using Neural Networks

Residual minimization is a widely used technique for solving Partial Differential Equations in variational form. It minimizes the dual norm of the residual, which naturally yields a saddle-point (min–max) problem over the so-called trial and test spaces. In the context of neural networks, we can address this min–max approach by employing one network to seek the trial minimum, while another network seeks the test maximizers. However, the resulting method is numerically unstable as we approach the trial solution. To overcome this, we reformulate the residual minimization as an equivalent minimization of a Ritz functional fed by optimal test functions computed from another Ritz functional minimization. We call the resulting scheme the Deep Double Ritz Method (D2RM), which combines two neural networks for approximating trial functions and optimal test functions along a nested double Ritz minimization strategy. Numerical results on different diffusion and convection problems support the robustness of our method, up to the approximation properties of the networks and the training capacity of the optimizers.

Heat kernels and theory of Hardy spaces associated to Schrödinger operators on stratified groups

Let G be a stratified group and let ΔG be a sub-Laplacian on G. In this paper, we consider the Schrödinger operator L=ΔG+V, where the potential V is a nonnegative polynomial. We first prove the upper bound of the higher order derivatives of the heat kernel of L. We then establish a theory of Hardy spaces HLp(G) associated to L for the full range p∈(0,1]. Particularly, we prove that for any p∈(0,1], the Hardy space HLp(G) introduced in terms of nontangential or radial maximal functions associated to the semigroup e−tL admits a new atomic decomposition. Moreover, we provide the description of the dual spaces of these new Hardy spaces in terms of certain local Campanato spaces related to the potential V. As a byproduct, we obtain the maximal function characterizations for the local Hardy spaces associated to an arbitrary critical function.

A semantic and emotion‐based dual latent variable generation model for a dialogue system

AbstractWith the development of intelligent agents pursuing humanisation, artificial intelligence must consider emotion, the most basic spiritual need in human interaction. Traditional emotional dialogue systems usually use an external emotional dictionary to select appropriate emotional words to add to the response or concatenate emotional tags and semantic features in the decoding step to generate appropriate responses. However, selecting emotional words from a fixed emotional dictionary may result in loss of the diversity and consistency of the response. We propose a semantic and emotion‐based dual latent variable generation model (Dual‐LVG) for dialogue systems, which is able to generate appropriate emotional responses without an emotional dictionary. Different from previous work, the conditional variational autoencoder (CVAE) adopts the standard transformer structure. Then, Dual‐LVG regularises the CVAE latent space by introducing a dual latent space of semantics and emotion. The content diversity and emotional accuracy of the generated responses are improved by learning emotion and semantic features respectively. Moreover, the average attention mechanism is adopted to better extract semantic features at the sequence level, and the semi‐supervised attention mechanism is used in the decoding step to strengthen the fusion of emotional features of the model. Experimental results show that Dual‐LVG can successfully achieve the effect of generating different content by controlling emotional factors.

Driven one-dimensional noisy Kitaev chain

We study a one-dimensional Kitaev chain driven by the anisotropy parameter ${J}_{\ensuremath{-}}=t/{\ensuremath{\tau}}_{Q}$, in the presence of weak Gaussian noise in the parameter ${J}_{+}$. The system is prepared in the ground state of the Hamiltonian at the initial time $t\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$. The defect density and the residual energy at the end of the drive protocol reveals anti-Kibble Zurek (AKZ) behavior which deviates from the earlier reported linear in quench time dependence for the defect density. The entropy density at the end of the protocol is found to exhibit the signature of the AKZ behavior. In the context of the Kitaev chain, the two-point spin correlators are short ranged. The hidden topological order across the quantum phase transition point is probed via the nonlocal string order parameters. The two-point Majorana correlator and the hidden string correlator calculated in the final decohered state at the end of the noisy drive protocol is consistent with the AKZ picture. We have analyzed the kink statistics in the dual spin space, the first three cumulants at the end of the noisy drive also exhibit the AKZ scaling behavior for slower sweeps similar to the defect density. The kink distribution function is well approximated with the normal distribution with nonuniversal noise dependent mean and variance.

The common range of co-analytic Toeplitz operators on the Drury-Arveson space

We characterize the common range of the adjoints of cyclic multiplication operators on the Drury-Arveson space. We show that a function belongs to this common range if and only if its Taylor coefficients satisfy a simple decay condition. To achieve this, we introduce the uniform Smirnov class on the ball and determine its dual space. We show that the dual space of the uniform Smirnov class equals the dual space of the strictly smaller Smirnov class of the Drury-Arveson space, and that this in turn equals the common range of the adjoints of cyclic multiplication operators.

Topological properties of some multiplication operators on L(X)

A pair (u, ?) in X ? X?, where X is an infinite dimensional Banach space and X? its topological dual space, induces in a natural way two multiplication operators L?,u and R?,u on the Banach space L(X), defined by L?,u (T) (x) = ? (T(x)) u, and R?,u (T) (x) = ?(x)T(u), for all T in L(X) and x in X. In this paper, we present necessary and sufficient conditions for the compactness, demicompactness, stongly demicompactess, power compactness and Riesz property of this family of operators. We also establish sufficient conditions for the quasi-compactness and weak compactness of these operators. Finally, we show that the Dunford-Pettis property fails for the Banach space L(X) whenever either X or L(X) is reflexive.

The continuity of pseudo-differential operators on Local variable Hardy spaces and their dual spaces

The continuity of pseudo-differential operators on Local variable Hardy spaces and their dual spaces

THREE WEAK FORMULATIONS FOR AN OBSTACLE MODEL AND THEIR RELATIONSHIP

We consider an obstacle model mathematically described by means of a boundary value problem governed by PDE. Three possible vari­ational formulations are highlighted. The first one is a variational inequality of the first kind and the other two are mixed variational for­mulations with Lagrange multipliers in dual spaces. After we discuss the solvability of the three variational formulations under consider­ation we focus on the relationship between them. Subsequently, we address the recovery of the formulation in terms of PDE starting from the mixed variational formulations.

On the Cauchy dual operator and duality for Banach spaces of analytic functions

On the Cauchy dual operator and duality for Banach spaces of analytic functions

A Framework of Rhetorical Moves Designed to Scaffold the Research Proposal Development Process

Aim/Purpose: To provide a research proposal writing framework to help doctoral students argue and motivate their efforts at furthering the existing knowledge available to understand some phenomenon or theory. Background: I discuss how the cognitive process theory of writing and the science writing heuristic can lead to a set of rhetorical moves and question prompts that students can use to develop the content of their research proposals. Methodology: I searched the literature on research proposal writing and, more broadly, academic writing to locate teaching and learning concepts associated with my recent question prompt approach used to guide my doctoral students. I used search words such as “writing” and “question prompts.” My review led me to the cognitive process theory of writing and heuristic scaffolding. I searched further using keywords such as “rhetorical move” and “heuristic prompts.” I performed several iterations of literature searches and reviews. Contribution: Instead of guiding a linearly developed research proposal that begins with an Introduction and proceeds to a Literature Review and then a Research Design and Methods section, the framework reveals a research proposal’s underlying logical flow and content by describing five rhetorical moves: establishing a topic question from an interesting phenomenon, establishing research opportunities, selecting a research question, providing a tentative solution, and establishing a plan to investigate the solution. Thus, the framework contributes to scholarship about how educators can facilitate independent reflection and broader problem-solving at the doctoral research proposal development stage. Particularly for the social sciences, it reveals the promise of the cognitive process theory of writing, dual problem space model of reflection, and heuristic scaffolding as valuable theoretical perspectives for the supervision of the planning phase of doctoral research. Recommendations for Practitioners: Teachers and advisors may use the framework’s rhetorical moves and question prompts as cognitive scaffolds to help students navigate an ill-structured problem typical of doctoral research projects in the social sciences. The question type of scaffolding gives the research student more responsibility; rather than the thesis supervisor or advisor articulating a model or nominating the technique, they require the student to self-regulate and develop it independently. This helps deal with the oft-experienced circumstance in which the supervisor does not have time to interact directly and regularly with the student.

On δ- character in symmetric tω-Δ- Banach-Algebra

The aim of this paper is to provide information on new results from studies of symmetric tω-Δ- Banach Algebra. A new function d- character has been defined and apparent to be symmetric tω-Δ- Banaich Algebra. tω- approach dual cone space is introduced, with its symmetry properties indicated tω- ∆- Banaich algebra. Some new definitions, results and properties are given and discussed in this field.

Weak$^*$ closures and derived sets for convex sets in dual Banach spaces

The paper is devoted to the convex-set counterpart of the theory of weak$^*$ derived sets initiated by Banach and Mazurkiewicz for subspaces. The main result is the following: For every nonreflexive Banach space $\mathcal {X}$ and every countable successo

Solvability of a nonlinear integro-differential equation with fractional order using the Bernoulli matrix approach

<abstract> <p>Under some suitable conditions, we study the existence and uniqueness of a solution to a new modification of a nonlinear fractional integro-differential equation (<bold>NFIDEq</bold>) in dual Banach space C<sub>E</sub> (E, [0, T]), which simulates several phenomena in mathematical physics, quantum mechanics, and other domains. The desired conclusions are demonstrated with the use of fixed-point theorems after applying the theory of fractional calculus. The validation of the provided strategy has been done by utilizing the Bernoulli matrix approach (<bold>BMA</bold>) method as a numerical method. The major motivation for selecting the <bold>BMA</bold> approach is that it combines Bernoulli polynomial approximation with Caputo fractional derivatives and numerical integral transformation to reduce the <bold>NFIDEq</bold> to an algebraic system and then derive the numerical solution; additionally, the convergence analysis indicated that the proposed strategy has more precision than other numerical methods. Finally, as a verification of the theoretical work, we apply two examples with numerical results by using [Matlab R2022b], illustrating the comparisons between the exact solutions and numerical solutions, as well as the absolute error in each case is computed.</p> </abstract>

Non homogeneous dual wavelet frames and oblique extension principles in Hs(K)

In this paper, we introduce the notion of nonhomogeneous dual wavelet frames in Sobolev spaces over local fields. We provide the complete characterization of nonhomogeneous dual wavelet frames on local fields. Furthermore, we obtain a mixed oblique extension principle for such frames.