Dual Space Research Articles

Preserving tradition: the role of community customs and sustainable practices in traditional house preservation on Nias Island, Indonesia

ABSTRACT Indonesia is home to over 300 ethnicities, many of which have traditional communities living in rainforest settlements. Unfortunately, the number of traditional houses is decreasing because 20th century introduction to Christianity, post-independence agenda in 1970, 21st century tourism and globalisation. Bawömataluo Village, Nias Island, has the largest number of traditional wooden houses (52%) that are still preserved (Atmanti and Uekita 2023). This research focuses on Bawömataluo Village, taking 30 samples of omo hada, observing the renovation activities, and the connection between the rainforest and the settlement. The investigation finds out that there is a continuity of the duality of space, the use of similar Nias native wooden materials inside the house even after modifications, the rainforest holds a strong connection with the living heritage, enough availability of the local materials, building technique and mutual custom in the living heritage will help the community to preserve their traditional house for future generation.This research finally provides calculations of materials required for traditional houses and assess the number of trees should be conserved for the preservation of traditional houses for a span of 100 years. This research finding contributes to heritage preservation in both academic and practical applications.

Distributed online constrained nonconvex optimization in dynamic environments over directed graphs

This paper is concerned with distributed online constrained nonconvex optimization problems of minimizing a global cost function decomposed by a sum of local smooth (possibly nonconvex) cost functions. This type of problems occupy a significant component of online learning in dynamic environments which are commonly involved in time-varying (TV) digraphs. Moreover, the network topology of TV digraphs is considered to be more general where related weight matrices are permitted to be row stochastic. Aiming at tackling these intricate challenges effectively, we adopt a valid primal–dual framework decomposing the multiple coupled constraints into individual node-related constraints. Additionally, by integrating a compensation error scheme, a novel primal dual mirror descent (PDMD) algorithm is proposed which employs two sequence of variables respectively serving as compensation error terms for bidirectional mirror mapping processes between primal space and dual space. Under some wild conditions, we theoretically prove that the proposed method can sublinearly reach the stationary point. In numerical simulations, four numerical examples are used to illustrate the validity and superiority of the proposed algorithm with contrast algorithms.

Weakly compact sets and Riesz representation theorem in variable exponent Lebesgue-Bochner sequence spaces

In this paper, we establish three intrinsic criteria for weakly compact sets in variable exponent Lebesgue-Bochner sequence spaces ℓ(Φ)(X), where each criterion's conditions are both necessary and sufficient. We also present a Riesz-type representation theorem for the dual spaces of these sequence spaces. Additionally, we apply these criteria to Lebesgue-Bochner sequence spaces. We provide a necessary and sufficient condition for the boundedness criterion of weak compactness, which answers an open question in sequence spaces posed by Hernandez et al. Our findings demonstrate that weak compactness cannot be directly extrapolated from a Banach space X to ℓ(Φ)(X), such as ℓ∞(X).

Orthoisomorphisms between lattices of geometric tripotents in dual spaces of facially symmetric spaces

In this paper we consider properties of bijection between lattices of geometric tripotents in dual spaces of two facially symmetric spaces. More precisely, we will prove that if a bijection is an order isomorphism and preserves orthogonality in one direction, then it is an orthoisomorphism, and vice versa. We will also define order isomorphisms and orthoisomorphisms between some classes of lattices of geometric tripotents in the dual space of a facially symmetric space

Local dual spaces and primary decomposition

Generalizing the concept of the Macaulay inverse system, we introduce a way to describe localizations of an ideal in a polynomial ring. This leads to an approach to the differential primary decomposition as a description of the affine scheme defined by the ideal.

The accelerating universe in a noncommutative analytically continued foliated quantum gravity

Abstract Based on an analytically continued Riemannian foliated quantum gravity super-Hamiltonian, known as branch cut quantum gravity (BCQG) we propose a novel approach to investigating the effects of noncommutative geometry on a minisuperspace of variables, influencing the acceleration behavior of the Universe’s wave function and the cosmic scale factor. Noncommutativity is introduced through a deformation of the conventional Poisson algebra, enhanced with a symplectic metric. The resulting symplectic manifold provides a natural setting that enables an isomorphism between canonically conjugate dual vector spaces, spanning the BCQG cosmic scale factor and its complementary quantum counterpart. Using this formulation, we describe the dynamic evolution of the Universe’s wave function, the cosmic scale factor, and its complementary quantum image. Our results strongly suggest that the noncommutative algebra induces late-time accelerated growth of the wave function, the Universe’s scale factor, and its complementary quantum counterpart, offering a new perspective on explaining the accelerating cosmic expansion rate and the inflationary period. In contrast to the inflationary model, where inflation requires a remarkably fine-tuned set of initial conditions in a patch of the Universe, analytically continued non-commutative foliated quantum gravity captures short and long scales, driving the evolutionary dynamics of the Universe through a reconfiguration of the primordial cosmic content of matter and energy. This reconfiguration is encapsulated into a quantum field potential, which leads to the generation of relic gravitational waves, a topic for future investigation. Graphical representations and contour plots indicate a characteristic torsion (or twist) deformation of spacetime geometry. This result introduces new speculative elements regarding the reconfiguration of matter and energy as a driver of spacetime torsion deformation, generating relic gravitational waves and serving as an alternative topological mechanism for the Universe’s acceleration. However, these assumptions require further investigation.

Enhancing Image Dehazing Efficiency with Dual Colour Space Attentional Deep Networks

Aims and Background: Image dehazing is an essential task in computer-vision, aimed at enhancing the clarity and visibility of images degraded by fog and haze. Traditional methods often struggle with handling the complex scattering effects of haze and maintaining the natural colours of the scene. Objectives and Methodology: To address these challenges, we propose a novel approach termed Dual Colour Space Attentional Deep Network (DCSADN) for efficient image dehazing. Our method leverages the advantages of two-colour spaces: RGB and YCbCr, to boost the network's skill to capture both the luminance and chrominance information, thereby improving the dehazing performance. We employ a multi-stage training strategy to optimize the performance of the DCSADN. This multi-stage approach ensures that the network generalizes well across different types of haze conditions. Extensive experiments demonstrate the superiority of our method over state-of-the-art dehazing techniques. Results: The results indicate that our approach not only achieves higher quantitative scores in terms of Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index (SSIM) but also produces dehazed images with more natural colours and details. Conclusion: The findings reveal that the dual colour space processing and attentional modules are crucial for the enhanced performance of our method and providing a valuable tool for improving image quality in hazy conditions.

On Some Topological Properties of Ch(X) and of the Dual Operator

The hypo-topology on the algebra C(X) of real-valued continuous functions defined on a Tychonoff space 2X f ∈ C(X) with its hypograph, hypof = {(x,t) ∈ X x R:f(x) ≥ t}. This topology is very useful in the calculus of variations and in optimization theory (e.g. Maximization problems). We denote C(X) with the hypo-topology by Ch(X). Our study deals with fundamental properties of these function spaces, and with the linear operators on them and as well as the characterization of the topological properties of Ch(X) in terms of topological properties of the base space X. We are studying the linear operator between the functional algebras Ch(X) and the Ch(Y). We are primarily concerned with the continuity of the evaluation functional, the general evaluation and the continuity of the characters of Ch(X) before the investigation of the properties of the dual operator f* of a continuous function f:X→Y. This operator is defined by f*:Ch (Y)→Ch (X), where f*(g)=gof for all g ∈ C(Y). The continuity of f* enables us to characterize the continuity of an algebra homomorphism of the type φ:Ch (Y)→Ch (X) for a realcompact space Y. For such a space, we present a type of Riesz theorm with states that an algebra homomorphism φ: Ch(Y)→Ch(X) is continuous if and only if there exists a unique hypo-function f: X→Y such that φ = f*. There after, we give the equivalence between the properties of f and those of f*. The study of the continuous linear functional on Ch(X) helps us to compute the topological dual space of this algebra. We show here that this dual space is useful only when the set of isolated points of X is dense.

Some Lie Algebra Structures on Symmetric Powers

Let k be a field of any characteristic, V a finite-dimensional vector space over k, and S d ( V * ) be the d-th symmetric power of the dual space V * . Given a linear map φ on V and an eigenvector w of φ , we prove that the pair ( φ , w ) can be used to construct a new Lie algebra structure on S d ( V * ) . We prove that this Lie algebra structure is solvable, and in particular, it is nilpotent if φ is a nilpotent map. We also classify the Lie algebras for all possible pairs ( φ , w ) , when k = C and V is two-dimensional.

Eccentric p-Summing Lipschitz Operators and Integral Inequalities on Metric Spaces and Graphs

The extension of the concept of p-summability for linear operators to the context of Lipschitz operators on metric spaces has been extensively studied in recent years. This research primarily uses the linearization of the metric space M afforded by the associated Arens–Eells space, along with the duality between M and the metric dual space M# defined by the real-valued Lipschitz functions on M. However, alternative approaches to measuring distances between sequences of elements of metric spaces (essentially involved in the definition of p-summability) exist. One approach involves considering specific subsets of the unit ball of M# for computing the distances between sequences, such as the real Lipschitz functions derived from evaluating the difference in the values of the metric from two points to a fixed point. We introduce new notions of summability for Lipschitz operators involving such functions, which are characterized by integral dominations for those operators. To show the applicability of our results, in the last part of this paper, we use the theoretical tools obtained in the first part to analyze metric graphs. In particular, we show new results on the behavior of numerical indices defined on these graphs satisfying certain conditions of summability and symmetry.

EBC-Net: 3D semi-supervised segmentation of pancreas based on edge-biased consistency regularization in dual perturbation space.

Deep learning technology has made remarkable progress in pancreatic image segmentation tasks. However, annotating 3D medical images is time-consuming and requires expertise, and existing semi-supervised segmentation methods perform poorly in the segmentation task of organs with blurred edges in enhanced CT such as the pancreas. To address the challenges of limited labeled data and indistinct boundaries of regions of interest (ROI). We propose Edge-Biased Consistency Regularization (EBC-Net). 3D edge detection is employed to construct edge perturbations and integrate edge prior information into limited data, aiding the network in learning from unlabeled data. Additionally, due to the one-sidedness of a single perturbation space, we expand the dual-level perturbation space of both images and features to more efficiently focus the model's attention on the edges of the ROI. Finally, inspired by the clinical habits of doctors, we propose a 3D Anatomical Invariance Extraction Module and Anatomical Attention to capture anatomy-invariant features. Extensive experiments have demonstrated that our method outperforms state-of-the-art methods in semi-supervised pancreas image segmentation. Moreover, it can better preserve the morphology of pancreatic organs and excel at edges region accuracy. Incorporated with edge prior knowledge, our method mixes disturbances in dual-perturbation space, which shifts the network's attention to the fuzzy edge region using a few labeled samples. These ideas have been verified on the pancreas segmentation dataset.

Deep Fourier Residual method for solving time-harmonic Maxwell's equations

Solving PDEs with machine learning techniques has become a popular alternative to conventional methods. In this context, Neural networks (NNs) are among the most commonly used machine learning tools, and in those models, the choice of an appropriate loss function is critical. In general, the main goal is to guarantee that minimizing the loss during training translates to minimizing the error in the solution at the same rate. In this work, we focus on the time-harmonic Maxwell's equations, whose weak formulation takes H0(curl,Ω) as the space of test functions. We propose a NN in which the loss function is a computable approximation of the dual norm of the weak-form PDE residual. To that end, we employ the Helmholtz decomposition of the space H0(curl,Ω) and construct an orthonormal basis for this space in two and three spatial dimensions. Here, we use the Discrete Sine/Cosine Transform to accurately and efficiently compute the discrete version of our proposed loss function. Moreover, in the numerical examples we show a high correlation between the proposed loss function and the H(curl)-norm of the error, even in problems with low-regularity solutions.

Duality, BMO and Hankel operators on Bernstein spaces

In this paper we deal with the problem of describing the dual space (Bκ1)⁎ of the Bernstein space Bκ1, that is the space of entire functions of exponential type (at most) κ>0 whose restriction to the real line is Lebesgue integrable. We provide several characterizations, showing that such dual space can be described as a quotient of the space of entire functions of exponential type κ whose restriction to the real line are in a suitable BMO-type space, or as the space of symbols b for which the Hankel operator Hb is bounded on the Paley–Wiener space Bκ/22. We also provide a characterization of (Bκ1)⁎ as the BMO space w.r.t. the Clark measures of the inner function ei2κz on the upper half-plane, in analogy with the known description of the dual of backward-shift invariant 1-spaces on the torus. Furthermore, we show that the orthogonal projection Pκ:L2(R)→Bκ2 induces a bounded operator from L∞(R) onto (Bκ1)⁎.Finally, we show that Bκ1 is the dual space of a suitable VMO-type space or as the space of symbols b for which the Hankel operator Hb on the Paley–Wiener space Bκ/22 is compact.

Dual-space distribution metric-based evolutionary algorithm for multimodal multi-objective optimization

Multimodal multi-objective optimization problems (MMOPs) are characterized by having multiple global or local Pareto solution sets. In most of multimodal multi-objective evolutionary algorithms, the distribution of population in objective space and decision space cannot be taken into account, simultaneously. In this paper, an innovative evolutionary algorithm based on dual-space distribution metric for multimodal multi-objective optimization (DsDMEA) is designed to solve this problem. Specifically, the two enhanced performance metrics are coupled together as a single dual-space distribution metric (DsDM) serving as the criterion for environmental selection. DsDMEA explores solutions on Pareto front and Pareto solutions sets that are not associated with the reference point, which show limited contributions to the distribution in dual spaces. Then DsDMEA removes any bi-noncontributing solutions using DsDM during environmental selection until the population size is met. This achieves a balance between the distribution of solutions and convergence in decision space. At the same time, this paper introduces a novel fitness computation method, enabling the precise localization of local Pareto fronts within the population. Finally, eight state-of-the-art multimodal multi-objective evolutionary algorithms are adopted to make a comparison on two types of benchmark to verify the performance of the DsDMEA. The experimental results demonstrated the DsDMEA effectiveness in solving MMOPs with local problems and traditional MMOPs.

From kinetic flocking model of Cucker–Smale type to self-organized hydrodynamic model

We investigate the hydrodynamic limit problem for a kinetic flocking model. We develop a GCI-based Hilbert expansion method, and establish rigorously the asymptotic regime from the kinetic Cucker–Smale model with a confining potential in a mesoscopic scale to the macroscopic limit system for self-propelled individuals, which is derived formally by Aceves-Sánchez et al. in 2019. In the traditional kinetic equation with collisions, for example, Boltzmann-type equations, the key properties that connect the kinetic and fluid regimes are: the linearized collision operator (linearized collision operator around the equilibrium), denoted by [Formula: see text], is symmetric, and has a nontrivial null space (its elements are called collision invariants) which include all the fluid information, i.e. the dimension of Ker([Formula: see text]) is equal to the number of fluid variables. Furthermore, the moments of the collision invariants with the kinetic equations give the macroscopic equations. The new feature and difficulty of the corresponding problem considered in this paper is: the linearized operator [Formula: see text] is not symmetric, i.e. [Formula: see text], where [Formula: see text] is the dual of [Formula: see text]. Moreover, the collision invariants lies in Ker([Formula: see text]), which is called generalized collision invariants (GCI). This is fundamentally different with classical Boltzmann-type equations. This is a common feature of many collective motions of self-propelled particles with alignment in living systems, or many active particle system. Another difficulty (also common for active system) is involved by the normalization of the direction vector, which is highly nonlinear. In this paper, using Cucker–Smale model as an example, we develop systematically a GCI-based expansion method, and micro–macro decomposition on the dual space, to justify the limits to the macroscopic system, a non-Euler-type hyperbolic system. We believe our method has wide applications in the collective motions and active particle systems.

Duality of spaces and the origin of integral reflection conditions

The dualism between direct and reciprocal space is at the origin of well known relations between basis vectors in the two spaces. It is shown that when a coordinate system corresponding to a non-primitive unit cell is adopted, this dualism has to be handled with care. In particular, the reciprocal of a non-primitive unit cell is not a unit cell but a region in reciprocal space that does not represent a unit of repetition by translation. The basis vectors do not correspond to reciprocal-space cell lengths, contrary to what is stated even in the core CIF dictionary. The corresponding unit cell is a multiple of this region. The broken correspondence between basis vectors and unit cell is at the origin of the integral reflection conditions.

Semantic Reconstruction of Multimodal Process Data With Dual Latent Space Constraints

Semantic Reconstruction of Multimodal Process Data With Dual Latent Space Constraints

On dual Kadec norms

Let (X,‖⋅‖) be a Banach space such that all w⁎-convergent sequences in the dual unit sphere SX⁎ are also norm convergent. Then the weak⁎ and norm topologies agree on SX⁎. By known results it implies that X has a renorming whose dual is locally uniformly rotund, hence also C1-Fréchet smooth. In particular, X is an Asplund space. Our results also lend an alternative proof of the celebrated Josefson-Nissenzweig theorem.

Enhancing Underwater Images through Multi-Frequency Detail Optimization and Adaptive Color Correction

This paper presents a novel underwater image enhancement method addressing the challenges of low contrast, color distortion, and detail loss prevalent in underwater photography. Unlike existing methods that may introduce color bias or blur during enhancement, our approach leverages a two-pronged strategy. First, an Efficient Fusion Edge Detection (EFED) module preserves crucial edge information, ensuring detail clarity even in challenging turbidity and illumination conditions. Second, a Multi-scale Color Parallel Frequency-division Attention (MCPFA) module integrates multi-color space data with edge information. This module dynamically weights features based on their frequency domain positions, prioritizing high-frequency details and areas affected by light attenuation. Our method further incorporates a dual multi-color space structural loss function, optimizing the performance of the network across RGB, Lab, and HSV color spaces. This approach enhances structural alignment and minimizes color distortion, edge artifacts, and detail loss often observed in existing techniques. Comprehensive quantitative and qualitative evaluations using both full-reference and no-reference image quality metrics demonstrate that our proposed method effectively suppresses scattering noise, corrects color deviations, and significantly enhances image details. In terms of objective evaluation metrics, our method achieves the best performance in the test dataset of EUVP with a PSNR of 23.45, SSIM of 0.821, and UIQM of 3.211, indicating that it outperforms state-of-the-art methods in improving image quality.

Stabilized distributed online mirror descent for multi-agent optimization

In the domain of multi-agent networks, distributed online mirror descent (DOMD) and distributed online dual averaging (DODA) play pivotal roles as fundamental algorithms for distributed online convex optimization. However, in contrast to DODA, DOMD fails when employed with a dynamic learning rate sequence. To bridge this gap, we introduce two novel variants of DOMD by incorporating a distributed stabilization step in primal space and dual space, respectively. We demonstrate that our stabilized DOMD algorithms achieve a sublinear bound with a sequence of dynamic learning rates. We further evolve our dual-stabilized DOMD by integrating a lazy communicated subgradient descent step, resulting in a re-indexed DODA. This establishes a connection between the two types of distributed algorithms, which enhances our understandings of distributed optimization. Moreover, we extend our proposed algorithms to handle the case of exponentiated gradient, where the iterate is constrained within a simplex probability. Finally, we conduct extensive numerical simulations to validate our theoretical analysis.