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Let X be a compact and connected Riemann surface of genus g ≥ 2. In this paper, moduli spaces of Higgs bundles over X with structure group SL(n, ℂ), for n ≥ 3, and Spin(2n, ℂ), for n ≥ 4, are considered. In the case of structure group SL(n, ℂ), two involutions of the Higgs bundle moduli space are defined and an alternative proof is given to show, using specific properties of the structure group, that the stable fixed points can be described as certain Higgs pairs with structure group SO(n, ℂ) or Sp(n, ℂ). Moreover, the notions of stability, semistability, and polystability of the obtained Higgs pairs are established. Also, two involutions of the moduli space of Higgs bundles with structure group Spin(2n, ℂ) are defined for n ≥ 4 analogously and it is shown, also using specific properties of the group, that their stable fixed points can be described as certain Higgs pairs whose structure group is of the form Spin(2r + 1, ℂ) × Spin(2n − 2r − 1, ℂ) for some 0 ≤ r ≤ n − 1. [ABSTRACT FROM AUTHOR]

The construction industry has long been plagued by low productivity and high injury and fatality rates. Robots have been envisioned to automate the construction process, thereby substantially improving construction productivity and safety. Despite the enormous potential, teaching robots to perform complex construction tasks is challenging. We present a generalizable framework to harness human teleoperation data to train construction robots to perform repetitive construction tasks. First, we develop a teleoperation method and interface to control robots on construction sites, serving as an intermediate solution toward full automation. Teleoperation data from human operators, along with context information from the job site, can be collected for robot learning. Second, we propose a new method for extracting keyframes from human operation data to reduce noise and redundancy in the training data, thereby improving robot learning efficacy. We propose a hierarchical imitation learning method that incorporates the keyframes to train the robot to generate appropriate trajectories for construction tasks. Third, we model the robot's visual observations of the working space in a compact latent space to improve learning performance and reduce computational load. To validate the proposed framework, we conduct experiments teaching a robot to generate appropriate trajectories for excavation tasks from human operators' teleoperations. The results suggest that the proposed method outperforms state-of-the-art approaches, demonstrating its significant potential for application. [ABSTRACT FROM AUTHOR]

For a free filter F on ω , endow the space N F = ω ∪ { p F } , where p F ∉ ω , with the topology in which every element of ω is isolated whereas all open neighborhoods of p F are of the form A ∪ { p F } for A ∈ F . Spaces of the form N F constitute the class of the simplest non-discrete Tychonoff spaces. The aim of this paper is to study them in the context of the celebrated Josefson–Nissenzweig theorem from Banach space theory. We prove, e.g., that, for a filter F, the space N F carries a sequence ⟨ μ n : n ∈ ω ⟩ of normalized finitely supported signed measures such that μ n (f) → 0 for every bounded continuous real-valued function f on N F if and only if F ∗ ≤ K Z , that is, the dual ideal F ∗ is Katětov below the asymptotic density ideal Z . Consequently, we get that if F ∗ ≤ K Z , then: (1) if X is a Tychonoff space and N F is homeomorphic to a subspace of X, then the space C p ∗ (X) of bounded continuous real-valued functions on X contains a complemented copy of the space c 0 endowed with the pointwise topology, (2) if K is a compact Hausdorff space and N F is homeomorphic to a subspace of K, then the Banach space C(K) of continuous real-valued functions on K is not a Grothendieck space. The latter result generalizes the well-known fact stating that if a compact Hausdorff space K contains a non-trivial convergent sequence, then the space C(K) is not Grothendieck. [ABSTRACT FROM AUTHOR]

AbstractLet K be a compact subgroup of a locally compact group G. In this work, we initiate an investigation of the Lebesgue-Fourier algebra S1 A(G/K) on the coset space G/K with pointwise multiplication; after some general results on the Banach algebra S1 A(G/K), as our main result, we give some necessary conditions for that S1 A(G/K) be approximately amenable in terms of algebraic and topological properties of G and K . [ABSTRACT FROM AUTHOR]

Let X be a compact metric space, G be a countable discrete infinite amenable group continuously acting on X and $ \mathcal {F} $ F be a Følner sequence of G. In this paper, we acquire some ergodic properties of the discrete amenable group action $ (X, G) $ (X,G). By virtue of these properties, we prove the existence of syndetic sensitivity of the minimal $ \mathcal {F} $ F-centre of attraction of a point $ x\in X $ x∈X provided that the minimal $ \mathcal {F} $ F-centre of attraction of x is not S-generic and admits a dense set of quasi-weakly almost periodic points relative to $ \mathcal {F} $ F of $ (X, G) $ (X,G). The presented results enhance some main results of [Z. Chen and X. Dai, Chaotic dynamics of minimal centre of attraction of discrete amenable group actions, J. Math. Anal. Appl. 456 (2017), pp. 1397–1414.]. [ABSTRACT FROM AUTHOR]

In this paper, the entropy of discrete Heisenberg group actions is considered. Let α be a discrete Heisenberg group action on a compact metric space X. Two types of entropies, h ~ (α) and h(α) are introduced, in which h ~ (α) is defined in Ruelle's way and h(α) is defined via the natural extension of α. It is shown that when X is the torus and α is induced by integer matrices then h ~ (α) is zero and h(α) can be expressed via the eigenvalues of the matrices. [ABSTRACT FROM AUTHOR]

A model space is a subspace of the Hardy space which is invariant under the backward shift, and a truncated Toeplitz operator is the compression of a Toeplitz operator on some model space. In this paper we prove a necessary and sufficient condition for the commutator of two truncated Toeplitz operators on a model space to be compact. [ABSTRACT FROM AUTHOR]

We study the existence of conformal metrics on noncompact Riemannian manifolds with noncompact boundary, which are complete as metric spaces and have negative constant scalar curvature in the interior and negative constant mean curvature on the boundary. These metrics are constructed on smooth manifolds obtained by removing d-dimensional submanifolds from certain n-dimensional compact spaces locally modelled on generalized solid cones. We prove the existence of such metrics if and only if d > n - 2 2 . Our main theorem is inspired by the classical results by Aviles–McOwen and Loewner–Nirenberg, known in the literature as the "singular Yamabe problem". [ABSTRACT FROM AUTHOR]

We consider periodic difference equations in infinite-dimensional Banach spaces possessing compact asymptotically stable set $ {\mathcal A} $ A . It is established that such $ {\mathcal A} $ A persists under spatial discretizations by means of projection methods as nearby closed and bounded uniformly asymptotically stable sets, which moreover converge to $ {\mathcal A} $ A in the Hausdorff metric for increasingly more accurate schemes. The proof is based on a Lyapunov function guaranteed by an ambient converse theorem and a pullback construction. As application serves integrodifference equations on spaces of continuous and p-integrable functions over a compact habitat. [ABSTRACT FROM AUTHOR]

In this paper, we construct odd finitely summable spectral triples based on length functions of bounded doubling on noncommutative solenoids. Our spectral triples induce a Leibniz Lip-norm on the state spaces of the noncommutative solenoids, giving them the structure of Leibniz quantum compact metric spaces. By applying methods of R. Floricel and A. Ghorbanpour, we also show that our odd spectral triples on noncommutative solenoids can be considered as inductive limits of spectral triples on rotation algebras. In the final section, we prove a noncommutative version of Wiener's lemma and show that our odd spectral triples can be defined to have an associated smooth dense subalgebra which is stable under the holomorphic functional calculus, thus answering a question of B. Long and W. Wu. The construction of the smooth subalgebra also extends to the case of nilpotent discrete groups. [ABSTRACT FROM AUTHOR]

Image steganalysis is used to detect the presence of hidden data. Recent studies have shown that deep convolutional neural networks (CNNs) applied to steganalysis exhibit excellent performance. However, current network architectures have deepened layers to pursue an ultimate local receptive field, overlooking the boundary and overall information of the image. As a result, the network fails to effectively extract steganographic feature information. In this paper, we propose a method that effectively captures both boundary and global information. We process the images through segmentation and padding, followed by treatment with four symmetric sub-networks with shared parameters and structures to acquire more comprehensive steganographic features. By integrating two loss functions into the traditional cross-entropy loss, we can train a more compact feature space, thereby enhancing network performance. Experiments were conducted on the BOSSbase1.01 dataset, using two widely employed steganography methods, namely WOW (wavelet obtained weights) and SUNIWARD (spatial universal wavelet relative distortion), for comparison. Results show the proposed model demonstrates superior performance on various payloads. [ABSTRACT FROM AUTHOR]

Recently, there has been a lot of research in Quantum Cellular Automata (QCA) technology because it promises low power consumption, low complexity, low latency, and compact space. Simultaneously, approximate arithmetic, a new paradigm in computing, streamlines the computational process and emerges as a low-power, high-performance design approach for arithmetic circuits. Furthermore, the XOR gate has been widely used in digital design and is a basic building block that can be used in many upcoming technologies. The full adder (FA) circuit is a key component of QCA technology and is utilized in arithmetic logic operations including subtraction, multiplication, and division. A great deal of research has been done on the design of approximate FA, full subtractor (FS), full adder/subtractor (FA/S), and 4-bit ripple carry adder (RCA) based on XOR logic, establishing them as essential components in the creation of QCA-based arithmetic circuits. This study presents three new and effective QCA-based circuits, based on XOR logic: an approximate FA, an approximate FS, an approximate FA/S, and an approximate 4-bit ripple carry adder (RCA). Interestingly, some designs have inputs on one side and outputs on the other, making it easier to reach the components without being encircled by other cells and leading to a more effective circuit design. In particular, a delay of 0.5 clock phases, an area of 0.01 μm2, and implementation utilizing just 11 cells was accomplished in the approximate FA and subtractor designs. In a similar vein, the estimated FA/S designs showed 0.5 clock phase delay, 0.01 μm2 area, and 12 cells used for implementation. An approximate 4-bit RCA is proposed using 64 QCA cells. The effectiveness of these designs is evaluated through functional verification with the QCADesigner program. According to simulation results, these proposed solutions not only function well but significantly outperform previous ideas in terms of speed and space. The proposed FA, FS, and RCA designs surpassed the previous best designs by 21%, 21%, and 43%, respectively, in terms of cell count. [ABSTRACT FROM AUTHOR]

In this study, a nonlinear degenerate parabolic equation is used to describe a nonlinear-Laplacian equation process that arises in many areas of science and engineering in mechanics, quantum physics, and chemical design. This work has the objective of proving the existence of the local weak solution of a nonlinear p(x)- Laplacian equation by the compactness theorem. The uniformly local characteristics of the solutions for the gradients by estimating the regularization problem and using the Moser iterative techniques. Moreover, some properties of the local solutions depend on uniformly bounded situations and the Lp(x)-norm to the gradient is considered. [ABSTRACT FROM AUTHOR]

Let $ (G, X) $ (G , X) be a G-system, which means that X is a compact Hausdorff space and G is an infinite topological group continuously acting on X, and let μ be a G-invariant measure of $ (G, X) $ (G , X). In this paper, we introduce the concepts of rigidity, uniform rigidity and μ-Ω-equicontinuity of $ (G,X) $ (G , X) with respect to an infinite sequence Ω of G and the notions of μ-Ω-equicontinuity and μ-Ω-mean-equicontinuity of a function $ f\in L^2(\mu) $ f ∈ L 2 (μ) with respect to an infinite sequence Ω of G. Then we give some equivalent conditions for $ f\in L^2(\mu) $ f ∈ L 2 (μ) and $ (G,X) $ (G , X) to be rigid, respectively. In addition, if G is commutative and X satisfies the first axiom of countability, we present some equivalent conditions for $ (G,X) $ (G , X) to be uniformly rigid. [ABSTRACT FROM AUTHOR]

Using elementary probabilistic methods, in particular a variant of the Weak Law of Large Numbers related to the Bernoulli distribution, we prove that for every infinite compact spaces K and L the product K\times L admits a sequence \langle \mu _n\colon n\in \mathbb {N}\rangle of normalized signed measures with finite supports which converges to 0 with respect to the weak* topology of the dual Banach space C(K\times L)^*. Our approach is completely constructive—the measures \mu _n are defined by an explicit simple formula. We also show that this result generalizes the classical theorem of Cembranos [Proc. Amer. Math. Soc. 91 (1984), pp. 556–558] and Freniche [Math. Ann. 267 (1984), pp. 479–486] which states that for every infinite compact spaces K and L the Banach space C(K\times L) contains a complemented copy of the space c_0. [ABSTRACT FROM AUTHOR]

The weak mean equicontinuity for a countable discrete amenable group G acting continuously on a compact metrizable space X is studied. It is shown that weak mean equicontinuity, continuously pointwise ergodicity and uniformity are coincided. Moreover, we prove that (X, G) is mean equicontinuous if and only if the product system (X × X , G) is weak mean equicontinuous. [ABSTRACT FROM AUTHOR]

The main purpose of this work is to create a type of topological spaces namely "almost star α-Hurewicz spaces" and study its properties, and besides, the concepts of α-compact space, α-Hurewicz space, star α- Hurewicz space and strongly star α-Hurewicz space. Many properties of α-Hurewicz spaces and almost α-Hurewicz are investigated. This allows us to provide new examples of explicit descriptions of spaces as well as some types of α-covering for spaces such as α-compact and α-Lindelof space and using as a tool to prove important results in topological spaces. In addition, a certain connection of α-Hurewicz space with the Hurewicz space and almost α-Hurewicz space was considered. There is a relationship between the version of the strongly star α-Hurewicz property and star α-Hurewicz property with star α-compact property and almost star α-Hurewicz. Some of the examples that make a distinction between the properties are mentioned and reviewed, showing that the concepts are not equivalent. Results on the preservation of the properties of α-Hurewicz and almost-α-Hurewicz spaces are included such as behavior under subspaces, products, α-irresolute function, and mappings with other forms of topological spaces. [ABSTRACT FROM AUTHOR]

We obtain a complete description of divergent Palais–Smale sequences for the prescribed Ricci curvature functional on compact homogeneous spaces. As an application, we prove the existence of saddle points on generalized Wallach spaces and several types of generalized flag manifolds. We also describe the image of the Ricci map in some of our examples. [ABSTRACT FROM AUTHOR]

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In this research article, we present a new class of soft compact spaces and soft Lindelöf spaces, we identify the idea of soft strongly b*--compact and soft strongly b*--Lindelöf spaces and we supply multiple interesting examples. As well as we mention that the inaugurated spaces are conserved under soft strongly b*--irresolute mappings and we look into definite of results which connect an extensive soft topology with the showing soft spaces. As well as we inquiry the features and attributive of soft strongly b*--connected spaces and discuss and identify its relationship with soft connectedness. [ABSTRACT FROM AUTHOR]

We investigate a widely used application of compactness of bounded linear operators T : X(B) →Y, where X(B) is a Banach space of holomorphic functions on the open unit ball B ⊂ CN and Y is a Banach space. In particular, we show that compactness of the operator when X(B) is not reflexive, is not a sufficient condition for the property that every bounded sequence (fn)n∈N in X(B) such that fn→0 with respect to the compact open topology as n→∞, implies that T(fn)→0 with respect to the norm of Y as n→∞. [ABSTRACT FROM AUTHOR]

In this paper, we can prove that: (1) Any non-degenerate strongly harmonic map from a compact Finsler manifold with non-negative flag curvature to a Finsler manifold with non-positive flag curvature must be totally geodesic. (2) If (M, F) is a compact Landsberg space with sectional flag curvature, then any non-degenerate strongly harmonic map from (M, F) with non-negative Ricci curvature to a Finsler manifold with non-positive flag curvature must be totally geodesic, which generalizes the result of Eells and Sampson (Amer. J. Math. 86: 106-160, 1964). [ABSTRACT FROM AUTHOR]

Let D = {{ C : ||| < 1} be the open unit disk in the complex plane C and let H(D) be the space of all holomorphic functions on D. For a non-negative integer n and a function f H(D), the nthh order differentiation operator is defined as Dnf = f(n). The weighted composition operator together with nthh order differentiation operator give rise to a new operator generally termed as generalized weighted composition operator denoted by Wn, and is defined by Wn, f(() = (()f(n)(((()), f H(D); D, where H(D) and is a holomorphic self-map of D. This operator is basically the combination of multiplication operator MM, composition operator CC and nthh order differentiation operator Dn. We study the boundedness and compactness of this operator between Dirichlet-type spaces and Bloch-type spaces. [ABSTRACT FROM AUTHOR]

We continue the study of truncated matrix-valued moment problems begun in [12]. Let q ∈ N. Suppose that (X , X) is a measurable space and E is a finite-dimensional vector space of measurable mappings of X into H q , the Hermitian q × q matrices. A linear functional Λ on E is called a moment functional if there exists a positive H q -valued measure μ on (X , X) such that Λ (F) = ∫ X 〈 F , d μ 〉 for F ∈ E. In this paper a number of special topics on the truncated matricial moment problem are treated. We restate a result from [11] to obtain a matricial version of the flat extension theorem. Assuming that X is a compact space and all elements of E are continuous on X we characterize moment functionals in terms of positivity and obtain an ordered maximal mass representing measure for each moment functional. The set of masses of representing measures at a fixed point and some related sets are studied. The class of commutative matrix moment functionals is investigated. We generalize the apolar scalar product for homogeneous polynomials to the matrix case and apply this to the matricial truncated moment problem. [ABSTRACT FROM AUTHOR]

We consider a Markov control model with Borel state space, metric compact action space, and transitions assumed to have a density function with respect to some probability measure satisfying some continuity conditions. We study the optimization problem of maximizing the probability of visiting some subset of the state space infinitely often, and we show that there exists an optimal stationary Markov policy for this problem. We endow the set of stationary Markov policies and the family of strategic probability measures with adequate topologies (namely, the narrow topology for Young measures and the $ws^\infty$ -topology, respectively) to obtain compactness and continuity properties, which allow us to obtain our main results. [ABSTRACT FROM AUTHOR]

We characterize subhomogeneity for twisted étale groupoid |$\text{C}^{*}$| -algebras and obtain an upper bound on their nuclear dimension. As an application, we remove the principality assumption in recent results on upper bounds on the nuclear dimension of a twisted étale groupoid |$\text{C}^{*}$| -algebra in terms of the dynamic asymptotic dimension of the groupoid and the covering dimension of its unit space. As a non-principal example, we show that the dynamic asymptotic dimension of any minimal (not necessarily free) action of the infinite dihedral group |$D_{\infty }$| on an infinite compact Hausdorff space |$X$| is always one. So if we further assume that |$X$| is second-countable and has finite covering dimension, then |$C(X)\rtimes _{r} D_{\infty }$| has finite nuclear dimension and is classifiable by its Elliott invariant. [ABSTRACT FROM AUTHOR]

For the set I (X) of probability measures on a compact Hausdorff space X, we propose a new way to introduce the topology by using the open subsets of the space X. Then, among other things, we give a new proof that for a compact Hausdorff space X, the space I (X) is also a compact Hausdorff space. For a Tychonoff space X, we consider the topological space I τ (X) of τ -smooth idempotent probability measures on X and show that the space I τ (X) is Čech-complete if and only if the given space X is Čech-complete. [ABSTRACT FROM AUTHOR]

In order to treat the degree of belief rationally, Baoding Liu created uncertainty theory. An uncertain variable, as a measurable function from an uncertainty space to the set of real numbers, is a basic concept in uncertainty theory. It is very meaningful to study its properties. Lusin's theorem is one of the most classical theorems in measure theory that reveals the close relationship between measurable and continuous functions, and has important significance. In this paper, we give three pairs of continuity conditions for uncertain measures, and present that every pair reveals duality, which is a kind of symmetry between objects. Furthermore, it is demonstrated that these continuity conditions are equivalent. And, we also prove that these three pairs of continuity conditions and the condition: if { Λ n } is a sequence of open sets and Λ n ↘ ∅ , then lim n → ∞ M { Λ n } = 0 are equivalent in compact metric spaces. It is shown that Lusin's theorem for uncertain variables holds if and only if the uncertain measure satisfies any of the above continuity conditions in a compact metric space. And, Lusin's theorem can be applied to uncertain variables with symmetric or asymmetric distributions. Finally, we provide several examples to illustrate applications of Lusin's theorem for uncertain variables. As far as we know, our results are new in uncertainty theory. [ABSTRACT FROM AUTHOR]

A non-autonomous Caputo fractional differential equation (FDE) of order α ∈ (0 , 1) in R d with a driving system on a compact base space P is shown to generate a skew-product semi-flow on C α × P , where C α is the space of continuous functions f : R + → R d with a weighted norm giving uniform convergence on compact time subsets. This skew-product semi-flow is then shown to have a bounded and closed attractor when the vector field of the Caputo FDE satisfies a uniform dissipativity condition. It attracts bounded sets of constant initial functions f in here C α. The properties and structure of this attractor in C α × P are also discussed. [ABSTRACT FROM AUTHOR]

Let \Gamma be a finitely generated group which admits an action by homeomorphisms on a metrizable space X. We show that there is a metric on X defining the original topology such that for this metric, the action is by bi-Lipschitz transformations. [ABSTRACT FROM AUTHOR]

We prove that order convergence on a Boolean algebra turns it into a compact convergence space if and only if this Boolean algebra is complete and atomic. We also show that on an Archimedean vector lattice, order intervals are compact with respect to order convergence if and only the vector lattice is complete and atomic. Additionally we provide a direct proof of the fact that uo convergence on an Archimedean vector lattice is induced by a topology if and only if the vector lattice is atomic. [ABSTRACT FROM AUTHOR]

Let (K,d) be a connected compact metric space and p\in (1, \infty). Under the assumption of Kigami [ Conductive homogeneity of compact metric spaces and construction of p-energy , Memoirs of the European Mathematical Society, vol. 5, Europea Mathematical Society (EMS), Berline, 2023, Assumption 2.15] and the conductive p-homogeneity, we show that \mathcal {W}^p(K)\subset C(K) holds if and only if p>\operatorname {dim}_{AR}(K,d), where \mathcal {W}^p(K) is Kigami's (1,p)-Sobolev space and \operatorname {dim}_{AR}(K,d) is the Ahlfors regular dimension. [ABSTRACT FROM AUTHOR]

AbstractIt is known that both the Menger and Hurewicz property of a Tychonoff space X can be described by the way X is placed in its Čech-Stone compactification βX. We provide analogous characterizations for the projective versions of the properties of Menger and Hurewicz. [ABSTRACT FROM AUTHOR]

The N-star compactifications of frames are the frame-theoretic counterpart of the N-point compactifications of locally compact Hausdorff spaces. A π -compactification of a frame L is a compactification constructed using a special type of a basis called a π -compact basis; the Freudenthal compactification is the largest π -compactification of a rim-compact frame. As one of the main results, we show that the Freudenthal compactification of a regular continuous frame is the least upper bound for the set of all N-star compactifications. A compactification whose right adjoint preserves disjoint binary joins is called perfect. We establish a class of frames for which N-star compactifications are always perfect. For the class of zero-dimensional frames, we construct a compactification which is isomorphic to the Banaschewski compactification and the Freudenthal compactification; in some special case, this compactification is isomorphic to the Stone–Čech compactification. [ABSTRACT FROM AUTHOR]

An optimal portable microfluidic device should ensure least number of accessories for versatile field application. Typically, in such a device fabricated with polydimethylsiloxane (PDMS), the transport of fluid is enabled through a nonconventional pumping mechanism. This pumping system has been demonstrated to utilize the relatively high air permeability of polydimethyl siloxane (PDMS) to transport small volume fluid. In the recent past, microporous PDMS has replaced PDMS in this capacity. Microporous PDMS is typically fabricated through a series of steps where a sacrificial template is used to infiltrate the polymer. This template is removed after the polymer undergoes curing. This method has consistently produced a spongy structure that is nonrigid, sticky, and moist rendering it unwieldy. In this work, we present a novel concept of using camphor (Cinnamomum camphora) as a template to fabricate a dry polymeric sponge. The proposed sponge is molded on a sublimable material, camphor to avoid the additional step of template dissolution. The sponge is demonstrated to be stiff yet flexible rendering it convenient to be compacted into a confined space. Additionally, the sponge is dry and nonsticky as compared to structures that have been generated through sugar leaching. [ABSTRACT FROM AUTHOR]

The Latent Space Perspicacity and Interpretation Enhancement (LS-PIE) framework enhances dimensionality reduction methods for linear latent variable models (LVMs). This paper extends LS-PIE by introducing an optimal latent discovery strategy to automate identifying optimal latent dimensions and projections based on user-defined metrics. The latent condensing (LCON) method clusters and condenses an extensive latent space into a compact form. A new approach, latent expansion (LEXP), incrementally increases latent dimensions using a linear LVM to find an optimal compact space. This study compares these methods across multiple datasets, including a simple toy problem, mixed signals, ECG data, and simulated vibrational data. LEXP can accelerate the discovery of optimal latent spaces and may yield different compact spaces from LCON, depending on the LVM. This paper highlights the LS-PIE algorithm's applications and compares LCON and LEXP in organising, ranking, and scoring latent components akin to principal component analysis or singular value decomposition. This paper shows clear improvements in the interpretability of the resulting latent representations allowing for clearer and more focused analysis. [ABSTRACT FROM AUTHOR]

This study focused on establishing the existence and multiplicity of solutions for gauged nonlinear Schrödinger equations set on the plane with sign-changing potentials. Our findings contribute to the extension of recent advancements in this area of research. Initially, we examined scenarios where the potential function V is lower-bounded and the function space has a compact embedding into Lebesgue spaces. Subsequently, we addressed more complex cases characterized by a sign-changing potential V and a function space that fails to compactly embed into Lebesgue spaces. The proofs of our results are based on the Trudinger-Moser inequality, the application of variational methods, and the utilization of Morse theory. [ABSTRACT FROM AUTHOR]

This paper deals with the existence and multiplicity results for a bi‐nonlocal problem without the Ambrosetti–Rabinwitz condition, in the context of variable exponents of Sobolev spaces on compact Riemannian manifolds. Using the mountain pass theorem, we obtain that our problem admits at least nontrivial weak solutions. Also, using the fountain and dual fountain theorems, we obtain the existence of infinitely many nontrivial high or small energy solutions. [ABSTRACT FROM AUTHOR]

In this article, we establish that any uniformly local Mizoguchi-Takahashi contraction is actually a set-valued contraction due to Feng and Liu on a metrically convex complete metric space. Through an example, we demonstrate that this result need not hold on any arbitrary metric space. Furthermore, when the metric space is compact, we derive that any Mizoguchi-Takahashi local contraction and Nadler local contraction are equivalent. Moreover, a result related to invariant best approximation is established. [ABSTRACT FROM AUTHOR]

We study a family of invariants of compact metric spaces that combines the Curvature Sets defined by Gromov in the 1980 s with Vietoris–Rips Persistent Homology. For given integers k ≥ 0 and n ≥ 1 we consider the dimension k Vietoris–Rips persistence diagrams of all subsets of a given metric space with cardinality at most n. We call these invariants persistence sets and denote them as D n , k VR . We first point out that this family encompasses the usual Vietoris–Rips diagrams. We then establish that (1) for certain range of values of the parameters n and k, computing these invariants is significantly more efficient than computing the usual Vietoris–Rips persistence diagrams, (2) these invariants have very good discriminating power and, in many cases, capture information that is imperceptible through standard Vietoris–Rips persistence diagrams, and (3) they enjoy stability properties analogous to those of the usual Vietoris–Rips persistence diagrams. We precisely characterize some of them in the case of spheres and surfaces with constant curvature using a generalization of Ptolemy's inequality. We also identify a rich family of metric graphs for which D 4 , 1 VR fully recovers their homotopy type by studying split-metric decompositions. Along the way we prove some useful properties of Vietoris–Rips persistence diagrams using Mayer–Vietoris sequences. These yield a geometric algorithm for computing the Vietoris–Rips persistence diagram of a space X with cardinality 2 k + 2 with quadratic time complexity as opposed to the much higher cost incurred by the usual algebraic algorithms relying on matrix reduction. [ABSTRACT FROM AUTHOR]

This paper demonstrates several of Ostrowski-type inequalities for fuzzy number functions and investigates their connections with other inequalities. Specifically, employing the Aumann integral and the Kulisch–Miranker order, as well as the inclusion order on the space of real and compact intervals, we establish various Ostrowski-type inequalities for fuzzy-valued mappings ( F · V · M s). Furthermore, by employing diverse orders, we establish connections with the classical versions of Ostrowski-type inequalities. Additionally, we explore new ideas and results rooted in submodular measures, accompanied by examples and applications to illustrate our findings. Moreover, by using special functions, we have provided some applications of Ostrowski-type inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Besov spaces on compact Lie groups in a subelliptic setting, that is, associated with a family of vector fields, satisfying the Hörmander condition, and their corresponding sub-Laplacian. Embedding properties between subelliptic Besov spaces and Besov spaces associated to the Laplacian on the group are proved. We link the description of subelliptic Sobolev spaces with the matrix-valued quantisation procedure of pseudo-differential operators to provide sharp subelliptic Sobolev and Besov estimates for operators in the $ (\rho,\delta) $ (ρ , δ) -Hörmander classes. In contrast with the available results in the literature in the setting of compact Lie groups, we allow Fefferman-type estimates in the critical case $ \rho =\delta. $ ρ = δ. Interpolation properties between Besov spaces and Triebel–Lizorkin spaces are also investigated. [ABSTRACT FROM AUTHOR]

An embedded series resonator for compact co‐polarized circular patch antenna pairs is presented in this paper. With a compact center spacing (CS) of 0.32λ0 and edge spacing (ES) of 0.03λ0 (λ0 is the free‐space wavelength at the center frequency), strong mutual coupling inevitably arises. However, by introducing a cross‐shaped fence and two quarter‐wavelength open‐ended slots onto a circular patch, port isolation of the first proposed antenna pair is significantly enhanced by 25.2 dB within a bandwidth of 6.2%. Moreover, the effectiveness of the embedded series resonator is demonstrated by achieving a wider 20‐dB decoupling bandwidth of 10.4% for the second proposed compact bandwidth‐enhanced antenna pair. For validation, both proposed prototypes are fabricated and measured, showing good agreement between measurements and simulations. To demonstrate the universality, a 1 × 8 patch array with an embedded series resonator can achieve a beam scanning angle of ±70◦ with a gain attenuation of 2.6 dB. Benefiting from compact size, high isolation, and enhanced decoupling bandwidth, the proposed designs hold promising potential for application in multiple‐input multiple‐output system and antenna array. [ABSTRACT FROM AUTHOR]

2D caricature editing has shown superior performance. However, 3D exaggerated caricature face (ECF) modeling with flexible shape and texture editing capabilities is far from achieving satisfactory high-quality results. This paper aims to model shape and texture variations of 3D caricatures in a learnable parameter space. To achieve this goal, we propose a novel framework for highly controllable editing of 3D caricatures. Our model mainly consists of the texture and shape hyper-networks, texture and shape Sirens, and a projection module. Specifically, two hyper-networks take the texture and shape latent codes as inputs to learn the compact parameter spaces of the two Siren modules. The texture and shape Sirens are leveraged to model the deformation variations of textural styles and geometric shapes. We further incorporate precise control of the camera parameters in the projection module to enhance the quality of generated ECF results. Our method allows flexible editing online and swapping textural features between 3D caricatures. For this purpose, we contribute a 3D caricature face dataset with textures for training and testing. Experiments and user evaluations demonstrate that our method is capable of generating diverse high-fidelity caricatures and achieves better editing capabilities than state-of-the-art methods. [ABSTRACT FROM AUTHOR]

Let M = Sn /Γ and h be a nontrivial element of finite order p in π1(Μ), where the integers n, p ≥ 2, Γ is a finite abelian group which acts freely and isometrically on the n-sphere and therefore M is diffeomorphic to a compact space form. In this paper, we prove that there are infinitely many non-contractible closed geodesics of class [h] on the compact space form with Cr-generic Finsler metrics, where 4 ≤ r ≤ ∞. The conclusion also holds for Cr-generic Riemannian metrics for 2 ≤ r ≤ ∞. The proof is based on the resonance identity of non-contractible closed geodesics on compact space forms. [ABSTRACT FROM AUTHOR]

Graph anomaly detection aims to identify the nodes that display significantly different behavior from the majority. However, existing methods neglect the combined interaction between the network structure and node attributes, resulting in suboptimal latent representations of nodes due to network noise. In this paper, we introduce a novel approach called adversarial regularized attributed network embedding (ARANE) for graph anomaly detection. ARANE addresses this issue by forcing normal nodes to inhabit a compact manifold in the latent space, taking into account both the network structure and node attributes. It ensures that data points from the normal class, originating from different distributions, are distributed within a single compact latent space, while excluding anomalies from this region. ARANE employs a dual-encoder architecture consisting of an attribute encoder and a structure encoder. The attribute encoder learns node attribute embeddings, while the structure encoder focuses on learning structure embeddings. To obtain high-quality node embeddings for effective anomaly detection, we apply adversarial learning to regularize the learned embeddings separately in both the structure and attribute spaces. Furthermore, we introduce a fusion module that combines the final node embeddings derived from the structure and attribute spaces. These joint embeddings serve as inputs to a dual-decoder for graph reconstruction, where the resulting reconstruction errors are utilized as anomaly scores for anomaly detection. Extensive experiments conducted on real-world attributed networks demonstrate the superior effectiveness of our proposed method compared to state-of-the-art approaches. • We introduce ARANE for one-class graph classification. It uses network structure & node attributes to tightly cluster normal nodes, excluding anomalies. • Our method uses a compact fusion module, capturing interactions between structural & attribute data in networks, for seamless integration. • Extensive experiments on real-world networks prove ARANE's effectiveness as a top solution for one-class graph classification. [ABSTRACT FROM AUTHOR]

In this paper, we study almost finiteness and almost finiteness in measure of nonfree actions. Let α:G↷X$\alpha:G\curvearrowright X$ be a minimal action of a locally finite‐by‐virtually Z${\mathbb {Z}}$ group G$G$ on the Cantor set X$X$. We prove that under certain assumptions, the action α$\alpha$ is almost finite in measure if and only if α$\alpha$ is essentially free. As an application, we obtain that any minimal topologically free action of a virtually Z${\mathbb {Z}}$ group on an infinite compact metrizable space with the small boundary property is almost finite. This is the first general result, assuming only topological freeness, in this direction, and these lead to new results on uniform property Γ$\Gamma$ and Z$\mathcal {Z}$‐stability for their crossed product C∗$C^*$‐algebras. Some concrete examples of minimal topological free (but nonfree) subshifts are provided. [ABSTRACT FROM AUTHOR]

In this paper, I introduce the properties EPCℵ1$\mathrm{EPC}_{\aleph _1}$ and ProjCes(E)$\mathrm{ProjCes}(E)$ for forcing notions and show that it is consistent that the forcing axiom for EPCℵ1+ProjCes(E)$\mathrm{EPC}_{\aleph _1}+ \mathrm{ProjCes}(E)$ forcing notions holds, the continuum hypothesis fails, and an ultrapower of the field of reals has the property β1$\beta _1$. This provides a partial solution to H. Woodin's question concerning the existence of discontinuous homomorphisms on the Banach algebra of all complex‐valued continuous functions on a compact space. Furthermore, we prove that the uniformization of a coloring of a ladder system on a stationary–costationary set E$E$ is an example of an EPCℵ1+ProjCes(ω1∖E)$\mathrm{EPC}_{\aleph _1}+ \mathrm{ProjCes}(\omega _1 \setminus E)$ forcing notion. As a corollary, it is consistent that a nonfree Whitehead group exists, the continuum hypothesis fails, and an ultrapower of the field of reals has the property β1$\beta _1$. [ABSTRACT FROM AUTHOR]