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This article suggests a collection of model-based and model-free output-feedback optimal solutions to a general ${H_{∞}}$ control design criterion of a continuous-time linear system. The goal is to obtain a static output-feedback controller while the design criterion is formulated with an exponential term, divergent or convergent, depending on the designer's choice. Two offline policy-iteration algorithms are presented first, which form the foundations for a family of online off-policy designs. These algorithms cover all different cases of partial or complete model knowledge and provide the designer with a collection of design alternatives. It is shown that such a design for partial model knowledge can reduce the number of unknown matrices to be solved online. In particular, if the disturbance input matrix of the model is given, off-policy learning can be done with no disturbance excitation. This alternative is useful in situations where a measurable disturbance is not available in the learning phase. The utility of these design procedures is demonstrated for the case of an optimal lane tracking controller of an automated car.

Due to many applications, the Choquet integral as a powerful tool for modeling non-deterministic problems needs to be further extended. Therefore the paper is devoted to a generalization of the Choquet integral. As a basis, the pseudo-integral for bounded integrand is extended to the case for nonnegative integrands at first, and then the generalized Choquet integral is defined. As special cases, pseudo-Choquet Stieltjes integrals, pseudo-fuzzy Stieltjes integrals, g-Choquet integrals, pseudo-(N)fuzzy integrals and pseudo-(S)fuzzy integrals are obtained, and various kinds of properties and convergence theorems are shown, meanwhile Markov, Jensen, Minkowski and Holder inequalities are proved. In the end, the generalized discrete Choquet integral is discussed. The obtained results for the generalized Choquet integral cover some previous results on different types of nonadditive integrals.

Knitting can efficiently fabricate stretchable and durable soft surfaces. These surfaces are often designed to be worn on solid objects as covers, garments, and accessories. Given a 3D model, we consider a knit for it wearable if the knit not only reproduces the shape of the 3D model but also can be put on and taken off from the model without deforming the model. This "wearability" places additional constraints on surface design and fabrication, which existing machine knitting approaches do not take into account. We introduce the first practical automatic pipeline to generate knit designs that are both wearable and machine knittable. Our pipeline handles knittability and wearability with two separate modules that run in parallel. Specifically, given a 3D object and its corresponding 3D garment surface, our approach first converts the garment surface into a topological disc by introducing a set of cuts. The resulting cut surface is then fed into a physically-based unclothing simulation module to ensure the garment's wearability over the object. The unclothing simulation determines which of the previously introduced cuts could be sewn permanently without impacting wearability. Concurrently, the cut surface is converted into an anisotropic stitch mesh. Then, our novel, stochastic, any-time flat-knitting scheduler generates fabrication instructions for an industrial knitting machine. Finally, we fabricate the garment and manually assemble it into one complete covering worn by the target object. We demonstrate our method's robustness and knitting efficiency by fabricating models with various topological and geometric complexities. Further, we show that our method can be incorporated into a knitting design tool for creating knitted garments with customized patterns.

Driven by real-life applications in geo-social networks, we study the problem of computing radius-bounded k-cores (RB-k-cores) that aims to find communities satisfying both social and spatial constraints. In particular, the model k-core (i.e., the subgraph where each vertex has at least k neighbors) is used to ensure the social cohesiveness, and a radius-bounded circle is used to restrict the locations of users in an RB-k-core. We explore several algorithmic paradigms to compute RB-k-cores, including a triple-vertex-based paradigm, a binary-vertex-based paradigm, and a paradigm utilizing the concept of rotating circles. The rotating-circle-based paradigm is further enhanced by several pruning techniques to achieve better efficiency. In addition, to find representative RB-k-cores, we study the diversified radius-bounded k-core search problem, which finds t RB-k-cores to cover the most number of vertices. We first propose a baseline algorithm that identifies the distinctive RB-k-cores after finding all the RB-k-cores. Beyond this, we design algorithms that can efficiently maintain the top-t candidate RB-k-cores and also achieve a guaranteed approximation ratio. Experimental studies on both real and synthetic datasets demonstrate that our proposed techniques can efficiently compute (diversified) RB-k-cores. Moreover, our techniques can be used to compute the minimum-circle-bounded k-core and significantly outperform the existing techniques.

International audience; This paper presents a computational method to test the stabilizability of discrete-time switched linear systems. The existence of a conic cover of the space on whose elements a convex condition holds is proved to be necessary and sufficient for stabilizability. An algorithm for computing a conic partition that satisfies the new necessary and sufficient condition is given. The algorithm, that allows also to determine bounds on the exponential convergence rate, is proved to overcome the conservatism of conditions equivalent to periodic stabilizability and is applied to a four dimensional system.

Attribute reduction, also called feature selection, is one of the most important issues of rough set theory, which is regarded as a vital preprocessing step in pattern recognition, machine learning, and data mining. Nowadays, high-dimensional mixed and incomplete data sets are very common in real-world applications. Certainly, the selection of a promising feature subset from such data sets is a very interesting, but challenging problem. Almost all of the existing methods generated a cover on the space of objects to determine important features. However, some tolerance classes in the cover are useless for the computational process. Thus, this article introduces a new concept of stripped neighborhood covers to reduce unnecessary tolerance classes from the original cover. Based on the proposed stripped neighborhood cover, we define a new reduct in mixed and incomplete decision tables, and then design an efficient heuristic algorithm to find this reduct. For each loop in the main loop of the proposed algorithm, we use an error measure to select an optimal feature and put it into the selected feature subset. Besides, to deal more efficiently with high-dimensional data sets, we also determine redundant features after each loop and remove them from the candidate feature subset. For the purpose of verifying the performance of the proposed algorithm, we carry out experiments on data sets downloaded from public data sources to compare with existing state-of-the-art algorithms. Experimental results showed that our algorithm outperforms compared algorithms, especially in classification accuracy.

EM enhancement is typically a lensing-type of effect by which a wide input beam couples to spatially localized resonant modes as for example Mie plasmons, void plasmons or Mie modes at high refractive index particles. Accordingly, EM enhancement is inherently a subwavelength phenomenon. Here, we explore a different paradigm for EM enhancement that is based on EM energy being collected over large time intervals by means of an ultra-slow-light waveguide. With ab-initio simulations, we study the dynamic evolution of such EM enhancement. After more than 7000 wave periods we observe that an intensity enhancement of more than 50 can cover an area of more than 15 $\lambda_{free^{2}}$ with $\lambda_{free}$ being the free-space wavelength.

In this paper, a new computationally efficient approach to solve fractional differential equations with Atangana–Baleanu operator is introduced. Controlled Picard’s method is employed for solving a class of fractional differential equations with order 0 α 1 . The proposed approach can cover wide range of integer and fractional orders differential equations due to the extra auxiliary parameter which enhances the convergence and is suitable for nonlinear differential equations. Two models of fractional Riccati equation are solved to validate and illustrate the accuracy of the new approach. Figures has been used to construct the results obtained from the presented approach. It is shown that the proposed method is efficient, credible, and easy to implement for various related problems in science and engineering.

The characterizing set-valued functions were introduced for uninorms with continuous underlying functions and these set-valued functions were useful for the complete characterization of such uninorms. In this work we study the characterizing functions of n-uninorms with continuous underlying t-norms and t-conorms. We will show that an n-uninorm with continuous underlying functions possesses n characterizing set-valued functions, where the graphs of these characterizing set-valued functions cover the set of all points of discontinuity of the respective n-uninorm. Moreover, for i=1,…,n, the i-th characterizing set-valued function divides the unit square into the two sets, where below the graph of the i-th characterizing set-valued function the n-uninorm attains values smaller than the local neutral element ei and above the graph of the i-th characterizing set-valued function the n-uninorm attains values greater than the local neutral element ei.

We present an efficient cutting-plane based approach to exactly solve a directed fixed charge network design (DFCND) problem, wherein the valid inequalities to the problem are generated using the polar duality approach. The biggest challenge in using this approach arises in constructing the polar dual of the problem. This would require enumerating all the extreme points of the convex hull of DFCND, which is computationally impractical for any instance of a reasonable size. Moreover, the resulting polar dual would be too large to solve efficiently, which is required at every iteration of the cutting-plane algorithm. The novelty of our solution approach lies in suggesting a way to circumvent this challenge by instead generating the violated facets, using polar duality, of the smaller substructures involving only a small subset of constraints and variables, obtained from 2-, 3-and 4-partitions of the underlying graph. For problem instances based on sparse graphs with zero flow costs, addition of these inequalities closes more than 20% of the optimality gap remaining after the addition of the knapsack cover inequalities used in the literature. This allows us to solve the problem instances in less than 400 s, on average, which otherwise take around 1000 s with the addition of only the knapsack cover inequalities, and around 4 hours for the Cplex MIP solver at its default setting.

Area coverage and network lifetime are two contradictory issues to the architecture development of a wireless sensor network (WSN). A satisfactory balance could be achieved by deploying abundant sensor nodes randomly and dividing them into k exclusive cover sets. Toward self-organized partition with higher efficiency, we address the problem from the perspective of networked potential games and propose a memorial mixed-response algorithm (MMRA), which is implemented in a distributed and synchronous manner. Being viewed as a game player, each sensor node first updates its memory using a temporary action, which is generated by following a mixed response rule. After this, the coordination evolves into the next iteration by each player randomly drawing an action from its memory with equal probabilities. We prove that our algorithm converges with probability 1 to a convention of Nash equilibria, with the worst approximation ratio strictly larger than 0.5. Moreover, it is also found that a tradeoff between solution efficiency and computation time could be achieved via the adjustment of the amount of randomness introduced via the memory length m as well as the probability pₘ, where better partition results are more likely to be generated using a larger m and smaller pₘ. Comparisons with existing distributed methods demonstrate the superiority of our method in terms of solution refinement as well as convergence speed.

This article investigates the state estimation issue of discrete-time Takagi-Sugeno fuzzy Markov jump affine systems that cover both traditional fuzzy Markov jump systems and fuzzy affine systems as two special cases. The original system is transformed into a Markov jump piecewise-affine system that varies with different operating regions. The fuzzy rules are dependent on the system mode, and accordingly, the partition of the state space is mode dependent. To analyze the stochastic stability of the estimation error system, a novel mode-dependent piecewise Lyapunov function is constructed, in which the region indices of both the plant state and the estimator state are mode dependent. Then, the existence for a simultaneously mode-dependent and region-dependent fuzzy affine estimator is investigated by virtue of the S-procedure and ellipsoidal outer approximation such that the estimation error system is stochastically stable with a prescribed H∞ performance. In the end, an illustrative example of a tunnel diode circuit is adopted to showcase the effectiveness and practicability of the developed state estimation strategy.

In this chapter we shall introduce what are referred to as the variational methods for data assimilation. Variational theory is the basis of many of the world’s operational numerical weather prediction and numerical ocean prediction schemes. We shall introduce the background error covariance matrix, along with the concept of model error. An important advancement that enabled four-dimensional variational (4D VAR) to be operationally viable was the introduction of incremental 4D VAR, which we shall introduce. We shall also cover the recent research into correlated observational error, as well as showing that 4D VAR is also the result of an optimal control problem.

In the presence of distributed energy resources (DERs), distribution networks (DNs) operate in multiple configurations. Maintaining complete observability is essential for the monitoring of the network during operation in any configuration. This article proposes a methodology for the optimal deployment of micro-phasor measurement units ( $\mu$ PMUs) in a DN operating in multiple configurations. A composite objective function has been formulated to minimize the number of $\mu$ PMUs while maximizing measurement redundancy. The optimal solution has been obtained considering the $\mu$ PMU with or without channel limits. The constraints have been formed simultaneously to cover all operational configuration of the network while eliminating repetition. The proposed methodology has been illustrated on an eight-node reconfigurable network. It has also been verified on several reconfigurable DNs, having 33, 34, 69, 118, 123, and 167 nodes. A comparison between the proposed methodology and recently published approaches has been made. The constraints for zero-injection nodes with various network configuration has also been considered to find optimal $\mu$ PMU deployment. The obtained optimal deployment of $\mu$ PMUs with or without channel limit satisfies the criteria of complete topological observability, topological in-dependency, and maximizes the sum of topology measurement redundancy.

Trapdoor stability has been widely studied by many researchers in the field of tunneling engineering. A general question being frequently asked is that why most sinkholes have a near-perfect circular shape on the ground surface. This could be possibly explained by the current numerical study using finite element limit analysis under axisymmetric condition, where upper and lower bound solutions of active circular trapdoors are determined. The failure study of sinkholes and the associated failure mechanisms in this paper are for non-homogeneous clay with a linear increase of strength with depth under various cover depth ratios and dimensionless strength gradients. A design equation for predicting the stability solutions is also developed based on the novel three dimensional solutions using axisymmetry.

It is remarked that the generalization of Jensen's inequality for pseudo-integrals (Pap and Strboja [14] ) is not a complete generalization of the classical Jensen's inequality, and a generalized Jensen's inequality for pseudo-integral with respect to ( ⊕ , ⊗ ) -pseudo-convex function is given in [28] . The present article is a continuation of the previous work. A new notion of ( ⊕ 1 , ⊗ 1 ) , ( ⊕ 2 , ⊗ 2 ) -pseudo-convex function is introduced, which generalize the notion of ( ⊕ , ⊗ ) -pseudo-convex function and many other previously generalizations. Motivated by the work of Kaluszka et al. [6] , related to Jensen's inequality with respect to different generalized fuzzy integrals, a new generalized Jensen's inequality between a pseudo-integral and general fuzzy integral, as well as between two different pseudo-integrals, with respect to ( ( ⊕ 1 , ⊗ 1 ) , ( ⊕ 2 , ⊗ 2 ) ) -pseudo-convex functions are proved. These results cover all previously obtained Jensen's inequalities for pseudo-integrals (Zhang and Pap [28] ) as well as the classical Jensen's inequality.

Graph clustering (also referred to as community detection) is an important topic in network analysis . Although a large amount of literature has been published on the problem, most of them are designed at the level of lower-order structure of networks, e.g., individual vertices and edges, and fail to capture higher-order information of networks. Recently, higher-order units (under the name of motifs) are introduced to graph clustering. These methods typically focus on constructing a motif-based hypergraph where higher-order information is preserved, and communities abstracted from the hypergraph usually achieve better accuracy. However, the hypergraph is often fragmented for a sparse network and contains a large number of isolated vertices that will be outliers of the identified community cover. To address the fragmentation problem, we propose an asymmetric triangle enhancement approach for graph clustering, in which a mixture of edges and asymmetric triangles is taken into consideration for cluster measures. We also design an approximation model to speed up the algorithm by estimating the measures. Extensive experiments on real and synthetic networks demonstrate the accuracy and efficiency of the proposed method.

Compared with natural images, aerial images collected by satellite sensors/aerial cameras can provide a much larger field of view and often contain multiple objects of interest (multiple labels). There are certain limitations of existing multilabel aerial image classification methods. First, label correlations were often ignored in previous MAIC work, and thus, multilabel classifiers failed to be self-adapted. Second, existing multilabeled data sets for aerial images only cover limited images with fixed labels. Therefore, the underlying semantic correlations of labels cannot be fully included, while such correlation information is implicitly used as common knowledge by human beings. To tackle these concerns, we propose a novel multilabel classification method for aerial images. Our contributions are twofold. First, as the first attempt, label correlations are inferred from both the specific data set and ConceptNet (a popular knowledge graph for common sense). Second, based on graph neural network (GNN), we propose a novel end-to-end aerial image classification model, named the multiple label concept graph (ML-CG). ML-CG builds a concept graph to describe the semantic correlations from both the label set and the ConceptNet. We also incorporate both semantic attention and label attention in the GNN to better extract meaningful information of image labels. Compared with state-of-the-art methods, the effectiveness of the proposed method is demonstrated on both the commonly used UCM data set and a recently proposed DFC15 data set with high image resolution.

We propose ε ↓ ( D → ) -logic as a formal foundation for the specification and development of event-based systems with data states. The framework is presented as an institution in the sense of Goguen and Burstall and the logic itself is parametrised by an underlying institution D → whose structures are used to model data states. ε ↓ ( D → ) -logic is intended to cover a broad range of abstraction levels from abstract requirements specifications up to constructive specifications. It uses modal diamond and box operators over complex actions adopted from dynamic logic. Atomic actions are pairs [inline-graphic not available: see fulltext] where e is an event and ψ a state transition predicate capturing the allowed reactions to the event. To write concrete specifications of recursive process structures we integrate (control) state variables and binders of hybrid logic. The semantic interpretation relies on event/data transition systems. For the presentation of constructive specifications we propose operational event/data specifications allowing for familiar, diagrammatic representations by state transition graphs. We show that ε ↓ ( D → ) -logic is powerful enough to characterise the semantics of an operational specification by a single ε ↓ ( D → ) -sentence. Thus the whole (formal) development process for event/data-based systems relies on ε ↓ ( D → ) -logic and its semantics as a common basis. It is supported by a variety of implementation constructors which can express, among others, event refinement and parallel composition. Due to the genericity of the approach, it is also possible to change a data state institution during system development when needed. All steps of our formal treatment are illustrated by a running example.

Numerosity, the set size of a group of items, helps guide behaviour and decisions. Non-symbolic numerosities are represented by the approximate number system. However, distinct behavioural performance suggests that small numerosities, i.e. subitizing range, are implemented differently in the brain than larger numerosities. Prior work has shown that neural populations selectively responding (i.e. hemodynamic responses) to small numerosities are organized into a network of topographical maps. Here, we investigate how neural populations respond to large numerosities, well into the ANS. Using 7 T fMRI and biologically-inspired analyses, we found a network of neural populations tuned to both small and large numerosities organized within the same topographic maps. These results demonstrate a continuum of numerosity preferences that progressively cover both the subitizing range and beyond within the same numerosity map, suggesting a single neural mechanism. We hypothesize that differences in map properties, such as cortical magnification and tuning width, underlie known differences in behaviour., Here, the authors show that the brain represents small and large numerosity ranges in a continuous topographic map, in line with the idea that differences in map properties underlie differences in perception.

Cross-domain scene classification identifies scene categories by learning knowledge from a labeled data set (source domain) to an unlabeled data set (target domain), where the source data and the target data are sampled from different distributions. A lot of domain adaptation methods are used to reduce the distribution shift across domains, and most existing methods assume that the source domain shares the same categories with the target domain. It is usually hard to find a source domain that covers all categories in the target domain. Some works exploit multiple incomplete source domains to cover the target domain. However, in such setting, the categories of each source domain are a subset of the target-domain categories, and the target domain contains ``unknown'' categories for each source domain. The existence of unknown categories results in the conventional domain adaptation unsuitable. Known and unknown categories should be treated separately. Therefore, a separation mechanism is proposed to separate the known and unknown categories in this article. First, multiple-source classifiers trained on the multiple source domains are used to coarsely separate the known/unknown categories in the target domain. The target images with high similarities to source images are selected as known categories, and the target images with low similarities are selected as unknown categories. Then, a binary classifier trained using the selected images is used to finely separate all target-domain images. Finally, only the known categories are implemented in the cross-domain alignment and classification. The target images get labels by integrating the hypotheses of multiple-source classifiers on the known categories. Experiments are conducted on three cross-domain data sets to demonstrate the effectiveness of the proposed method.

We study the Choquard equation with a local perturbation \begin{equation*} -\Delta u=\lambda u+(I_\alpha\ast|u|^p)|u|^{p-2}u+\mu|u|^{q-2}u,\ x\in \mathbb{R}^{N} \end{equation*} having prescribed mass \begin{equation*} \int_{\mathbb{R}^N}|u|^2dx=a^2. \end{equation*} For a $L^2$-critical or $L^2$-supercritical perturbation $\mu|u|^{q-2}u$, we prove nonexistence, existence and symmetry of normalized ground states, by using the mountain pass lemma, the Poho\v{z}aev constraint method, the Schwartz symmetrization rearrangements and some theories of polarizations. In particular, our results cover the Hardy-Littlewood-Sobolev upper critical exponent case $p=(N+\alpha)/(N-2)$. Our results are a nonlocal counterpart of the results in \cite{{Li 2021-4},{Soave JFA},{Wei-Wu 2021}}.

Reachability is a fundamental problem in graph analysis. In applications such as social networks and collaboration networks, edges are always associated with timestamps. Most existing works on reachability queries in temporal graphs assume that two vertices are related if they are connected by a path with non-decreasing timestamps (time-respecting) of edges. This assumption fails to capture the relationship between entities involved in the same group or activity with no time-respecting path connecting them. In this paper, we define a new reachability model, called span-reachability, designed to relax the time order dependency and identify the relationship between entities in a given time period. We adopt the idea of two-hop cover and propose an index-based method to answer span-reachability queries. Several optimizations are also given to improve the efficiency of index construction and query processing. We conduct extensive experiments on eighteen real-world datasets to show the efficiency of our proposed solution.

In this paper, we introduce the k-prize-collecting minimum power cover problem (k-PCPC). In this problem, we are given a point set V, a sensor set S on a plane and a parameter k with $$k\le |V|$$ . Each sensor can adjust its power and the covering range of sensor s with power p(D(s, r(s))) is a disk D(s, r(s)), where r(s) is the radius of disk D(s, r(s)) and $$p(D(s,r(s)))=c\cdot r(s)^{\alpha }$$ . The k-PCPC determines a disk set $$\mathcal {F}$$ such that at least k points are covered, where the center of any disk in $$\mathcal {F}$$ is a sensor. The objective is to minimize the total power of the disk set $$\mathcal {F}$$ plus the penalty of R, where R is the set of points that are not covered by $$\mathcal {F}$$ . This problem generalizes the well-known minimum power cover problem, minimum power partial cover problem and prize collecting minimum power cover problem. Our main result is to present a novel two-phase primal-dual algorithm for the k-PCPC with an approximation ratio of at most $$3^{\alpha }$$ .

Since their introduction by Erdős in 1950, covering systems (that is, finite collections of arithmetic progressions that cover the integers) have been extensively studied, and numerous questions and conjectures have been posed regarding the existence of covering systems with various properties. In particular, Erdős asked if the moduli can be distinct and all arbitrarily large, Erdős and Selfridge asked if the moduli can be distinct and all odd, and Schinzel conjectured that in any covering system there exists a pair of moduli, one of which divides the other. Another beautiful conjecture, proposed by Erdős and Graham in 1980, states that if the moduli are distinct elements of the interval [n, Cn], and n is sufficiently large, then the density of integers uncovered by the union is bounded below by a constant (depending only on C). This conjecture was confirmed (in a strong form) by Filaseta, Ford, Konyagin, Pomerance and Yu in 2007, who moreover asked whether the same conclusion holds if the moduli are distinct and sufficiently large, and $$\sum _{i=1}^k \frac{1}{d_i} < C$$ . Although, as we shall see, this condition is not sufficiently strong to imply the desired conclusion, as one of the main results of this paper we will give an essentially best possible condition which is sufficient. More precisely, we show that if all of the moduli are sufficiently large, then the union misses a set of density at least $$e^{-4C}/2$$ , where $$\begin{aligned} C = \sum _{i=1}^k \frac{\mu (d_i)}{d_i} \end{aligned}$$ and $$\mu $$ is a multiplicative function defined by $$\mu (p^i)=1+(\log p)^{3+\varepsilon }/p$$ for some $$\varepsilon > 0$$ . We also show that no such lower bound (i.e., depending only on C) on the density of the uncovered set holds when $$\mu (p^i)$$ is replaced by any function of the form $$1+O(1/p)$$ . Our method has a number of further applications. Most importantly, as our second main theorem, we prove the conjecture of Schinzel stated above, which was made in 1967. We moreover give an alternative (somewhat simpler) proof of a breakthrough result of Hough, who resolved Erdős’ minimum modulus problem, with an improved bound on the smallest difference. Finally, we make further progress on the problem of Erdős and Selfridge.

Fault diagnosis plays an important role in the measuring of the fault tolerance of an interconnection network, which is of great value in the design and maintenance of large-scale multiprocessor systems. As a classical variant of the hypercube, the Balanced Hypercube, denoted by BHn(n $$\ge$$ 1), has drawn a lot of research attention, and its $$g$$ -extra diagnosability has been studied to improve the network diagnostic ability. However, the current literatures on $$g$$ -extra diagnosability of BHn under the PMC model only cover the cases of $$g < 6$$ , and what’s more, seldom involve its $$g$$ -extra diagnosability under the MM* model, which is a great limitation on the research of BHn diagnosability. In this paper, the upper and lower bounds of the $$g$$ -extra diagnosability of the balanced hypercube are proved, respectively, based on the $$g$$ -extra connectivity by the contradiction method, and finally, the $$g$$ -extra diagnosability of BHn for $$2 \le g \le 2n - 1$$ under the PMC and MM* model is obtained, i.e., $$2\left[ {\left( {n - 2} \right)\lceil\frac{g - 1}{2}\rceil + n} \right] + g$$ . In addition, as a special case, the $$g$$ -extra diagnosability of the balanced hypercube for $$g = 2n$$ is proved to be $$2^{2n - 1} - 1$$ under the PMC and MM* model. In the end, simulation experiments are conducted to verify the effectiveness of our proposed theories. The conclusion of this paper has certain theory and application value for the research of BHn fault diagnosis.

The solution accuracy based on finite element interpolation strongly depends on mesh quality. Thus, the numerical manifold method advocates that if a finite element mesh is selected as the mathematical cover, the mesh should be uniform, where all elements are congruent squares or isosceles right triangles for two-dimensional problems. However, the elements close to the points of stress concentrations or singularities must be smaller than those further away from these points. This leads to an irregular mesh in which a few nodes of certain small elements are hung on the edges of adjacent larger elements. This study implements local refinement with arbitrary k-irregular meshes by selecting piecewise polynomials as the local approximations over physical patches and enforcing appropriate constraints on the piecewise polynomials. All the elements are ordinary isoparametric elements, and elements with variable numbers of nodes on edges are not required. No linear dependency issue exists. Furthermore, refinement with any number of mesh layers is developed so that solution error can be distributed as uniformly as possible. Typical examples with different singularities are designed to illustrate the effectiveness of the proposed procedure.

Rough set model accuracy varies when granules change. Refinement is one way to characterize granule evolutions, but it has not been formally defined in rough set theory so far. Topologically, a granule cover W is called a refinement of another granule cover F if there exists some F W ∈ F such that W ⊆ F W for each W ∈ W . In this article, we show that the concept refinement in topology is too abstract to describe the variability of the rough set model along with the change in granules. By abstracting from some important granule refinements, such as reductions and minimal descriptions, we propose two novel granule cover refinements named base-type refinements and base-preserving refinements, which combine both point-set topology and rough set theory. We analyze the accuracy relationships of some typical approximation operators between granule covers and their base-type or base-preserving refinements. We also show that higher accuracy implies finer granule covers for some approximation operators. These are promising results for rough set model accuracy comparisons. Furthermore, the results hint at how to avoid exponentially increased computation complexity on rough set model accuracy comparisons by checking refinement relationships of granule covers directly instead of computing on them using approximation operators.

The AlphaFold Protein Structure Database (AlphaFold DB, https://alphafold.ebi.ac.uk) is an openly accessible, extensive database of high-accuracy protein-structure predictions. Powered by AlphaFold v2.0 of DeepMind, it has enabled an unprecedented expansion of the structural coverage of the known protein-sequence space. AlphaFold DB provides programmatic access to and interactive visualization of predicted atomic coordinates, per-residue and pairwise model-confidence estimates and predicted aligned errors. The initial release of AlphaFold DB contains over 360,000 predicted structures across 21 model-organism proteomes, which will soon be expanded to cover most of the (over 100 million) representative sequences from the UniRef90 data set.

Data centers, the critical infrastructure underpinning Cloud computing, often employ Software-Defined Networks (SDN) to manage cluster, wide-area and enterprise networks. As the network forwarding in SDN is dynamically programmed by controllers, it is crucial to ensure that the controller intent is correctly translated into underlying forwarding rules. Therefore, detecting and locating forwarding anomalies in SDN is a fundamental problem in production networks. Existing research proposals, roughly categorized into probing-based, packet piggybacking-based, and flow statistics analysis-based, either impose significant overhead or do not provide sufficient coverage for certain forwarding anomalies. In this article, we propose ${\sf FADE}$ FADE , a controllable and passive measuring scheme to simultaneously deliver detection efficiency and accuracy. ${\sf FADE}$ FADE first analyzes the entire network topology and flow rules, and then computes a minimal set of flows that can cover all forwarding rules. For each selected network flow, ${\sf FADE}$ FADE decides the optimal number of monitoring positions on its path (much less than total number of hops), and installs dedicated rules to collect flow statistics. ${\sf FADE}$ FADE controls the installation and expiration of these rules, along with unique flow labels, to guarantee the accuracy of collected statistics, based on which ${\sf FADE}$ FADE algorithmically decides whether a forwarding anomaly is detected, and if so it further locates the anomaly. On top of ${\sf FADE}$ FADE , we propose ${\sf iFADE}$ iFADE (a more scalable version of ${\sf FADE}$ FADE ) to further optimize the usage and deployment of dedicated measurement rules. ${\sf iFADE}$ iFADE achieves over 40 percent rule reduction compared with ${\sf FADE}$ FADE . We implement a prototype of both ${\sf FADE}$ FADE and ${\sf iFADE}$ iFADE in about 12000 lines of code and evaluate the prototype extensively. The experiment results demonstrate ${\sf (i)}$ ( i ) ${\sf FADE}$ FADE and ${\sf iFADE}$ iFADE are accurate, e.g., they achieve over 95 percent true positive rate and 99 percent true negative rate in anomaly detection; ${\sf (ii)}$ ( ii ) ${\sf FADE}$ FADE and ${\sf iFADE}$ iFADE are lightweight, e.g., they reduce the overhead of control messages compared with state-of-the-art by about 50 and 90 percent, respectively.

We revisit the well-known work of K. Masuda in 1984 on the weak–strong uniqueness of L ∞ L 3 Leray–Hopf weak solutions of Navier–Stokes equation. We modify the argument, and extend the uniqueness result to the scaling critical anisotropic Lebesgue space with mixed-norms. As a consequence, our results cover the class of initial data and solutions which may be singular or decay with different rates along different spatial variables. The result relies on the establishment of several refined properties of solutions of the Stokes and Navier–Stokes equations in mixed-norm Lebesgue spaces which seem to be of independent interest.

In this paper we extend the similarity classifier to cover Dombi aggregation operators. The similarity classifier was earlier studied with the ordered weighted averaging (OWA) operator, the generalized mean, and other operators. We concentrate on the use of Dombi operators during aggregation of similarities within the classifier. Four Dombi aggregation operators applied here include: conjunctive, disjunctive, weighted conjunctive, and the product operator. From each of these operators, we form a variant of the similarity classifier. The proposed methods were tested on four real world medical datasets which include: fertility, lung cancer, Haberman’s survival, and liver disorder. The new classifiers achieved improved classification accuracies compared with earlier methods. Compared with the classifier using the generalized mean, the proposed method achieved an improvement of 17.80 % on fertility, 5.88 % on lung cancer, 0.65 % on Haberman’s survival, and 0.29 % on liver disorder datasets.

We extend classical nabla and delta Hardy–Copson type inequalities from $$\zeta >1$$ to $$0

We consider a special class of unipotent periods for automorphic forms on a finite cover of a reductive adelic group $$\mathbf {G}(\mathbb {A}_\mathbb {K})$$ G ( A K ) , which we refer to as Fourier coefficients associated to the data of a ‘Whittaker pair’. We describe a quasi-order on Fourier coefficients, and an algorithm that gives an explicit formula for any coefficient in terms of integrals and sums involving higher coefficients. The maximal elements for the quasi-order are ‘Levi-distinguished’ Fourier coefficients, which correspond to taking the constant term along the unipotent radical of a parabolic subgroup, and then further taking a Fourier coefficient with respect to a $${\mathbb K}$$ K -distinguished nilpotent orbit in the Levi quotient. Thus one can express any Fourier coefficient, including the form itself, in terms of higher Levi-distinguished coefficients. In companion papers we use this result to determine explicit Fourier expansions of minimal and next-to-minimal automorphic forms on split simply-laced reductive groups, and to obtain Euler product decompositions of certain Fourier coefficients.

In this paper, we consider symmetric $$\alpha $$ -stable processes on (unbounded) horn-shaped regions which are non-uniformly $$C^{1,1}$$ near infinity. By using probabilistic approaches extensively, we establish two-sided Dirichlet heat kernel estimates of such processes for all time. The estimates are very sensitive with respect to the reference function corresponding to each horn-shaped region. Our results also cover the case that the associated Dirichlet semigroup is not intrinsically ultracontractive. A striking observation from our estimates is that, even when the associated Dirichlet semigroup is intrinsically ultracontractive, the so-called Varopoulos-type estimates do not hold in general for symmetric stable processes on horn-shaped regions.

Decomposition-based multiobjective evolutionary algorithms (MOEAs) are a class of popular methods for solving the multiobjective optimization problems (MOPs), and have been widely studied in numerical experiments and successfully applied in practice. However, we know little about these algorithms from the theoretical aspect. In this paper, we present running time analysis of a simple MOEA with mutation and crossover based on the MOEA/D framework (MOEA/D-C) on five pseudo-Boolean functions. Our rigorous theoretical analysis shows that by properly setting the number of subproblems, the upper bounds of expected running time of MOEA/D-C obtaining a set of solutions to cover the Pareto fronts (PFs) of these problems are apparently lower than those of the one with mutation-only (MOEA/D-M). Moreover, to effectively obtain a set of solutions to cover the PFs of these problem, MOEA/D-C only needs to decompose these MOPs into several subproblems with a set of simple weight vectors while MOEA/D-M needs to find $\Omega (n)$ optimally decomposed weight vectors. This result suggests that the use of crossover in decomposition-based MOEA can simplify the setting of weight vectors for different problems and make the algorithm more efficient. This paper provides some insights into the working principles of MOEA/D and explains why some existing decomposition-based MOEAs work well in computational experiments.

We consider a class of elliptic problems in the half-space R + n with nonhomogeneous boundary conditions containing nonlinearities and critical singular potentials. We obtain existence and regularity results by means of a harmonic analysis approach based on a framework of weighted spaces in Fourier variables. This framework seems to be new in the context of elliptic boundary value problems and allows us to consider Hardy's potential λ 1 / | x | 2 in R + n and Kato's potential λ 2 / | x ′ | on the boundary ∂ R + n , as well as their versions with multiple poles, without using the so-called Kato and Hardy inequalities. Singular boundary forcing terms can also be addressed. Moreover, our results cover supercritical nonlinearities, such as ± u p in R + n and ± u q on ∂ R + n with integers p > 2 ⁎ − 1 and q > 2 ⁎ − 1 .

Cops and Robbers games have been studied for the last few decades in computer science and mathematics. As in general pursuit evasion games, pursuers (cops) seek to capture evaders (robbers); however, players move in turn and are constrained to move on a discrete structure, usually a graph, and know the exact location of their opponent. In 2017, Bonato and MacGillivray presented a general characterization of Cops and Robbers games in order for them to be globally studied. However, their model doesn't cover cases where stochastic events may occur, such as the robbers moving in a random fashion. In this paper we present a novel model with stochastic elements that we call a Generalized Probabilistic Cops and Robbers game (GPCR). A typical such game is one where the robber moves according to a probabilistic distribution, either because she is rather lost or drunk than evading, or because she is a robot. We present results to solve GPCR games, thus enabling one to study properties relating to the optimal strategies in large classes of Cops and Robbers games. Some classic Cops and Robbers games properties are also extended., 36 pages, submitted to the journal Theoretical Computer Science

In this study, some examples show that Hirsch's index (h-index) usually has some limitation in evaluations. In order to provide more choices and diversities for scientometrics evaluation index, this study discusses four types of extended indexes. We firstly use a type of variation of Cognitive Integral to serve as a new generalized Hirsch's index, called Cognitive Hirsch's index, and analyze some of its mathematical properties. Secondly, by introducing Reference Function generalized Integral, we further generalize and extend Cognitive Hirsch's index. Next, this study also proposes to transform any citation sequence into a unique 0-1 function. Subsequently, based on different shapes, i.e., systems of some special 0-1 functions, this paper then proposes a novel index which is measured by counting the extent to which a citation sequence's 0-1 function can cover some given function systems. Lastly, still based on the 0-1 functions, we present a new index by using some predetermined distributions.

Three main ambipolar compact models for Two-Dimensional (2D) materials based Field-Effect Transistors (2D-FETs) are reviewed: (1) Landauer model, (2) 2D Pao-Sah model, and (3) virtual Source Emission-Diffusion (VSED) model. For the Landauer model, the Gauss quadrature method is applied, and it summarizes all kinds of variants, exhibiting its state-of-art. For the 2D Pao-Sah model, the aspects of its theoretical fundamentals are rederived, and the electrostatic potentials of electrons and holes are clarified. A brief development history is compiled for the VSED model. In summary, the Landauer model is naturally appropriate for the ballistic transport of short channels, and the 2D Pao-Sah model is applicable to long-channel devices. By contrast, the VSED model offers a smooth transition between ultimate cases. These three models cover a fairly completed channel length range, which enables researchers to choose the appropriate compact model for their works.

A crucial operation of saturation theorem provers is deletion of subsumed formulas. Designers of proof calculi, however, usually discuss this only informally, and the rare formal expositions tend to be clumsy. This is because the equivalence of dynamic and static refutational completeness holds only for derivations where all deleted formulas are redundant, but the standard notion of redundancy is too weak: A clause C does not make an instance $$C\sigma $$ C σ redundant. We present a framework for formal refutational completeness proofs of abstract provers that implement saturation calculi, such as ordered resolution and superposition. The framework modularly extends redundancy criteria derived via a familiar ground-to-nonground lifting. It allows us to extend redundancy criteria so that they cover subsumption, and also to model entire prover architectures so that the static refutational completeness of a calculus immediately implies the dynamic refutational completeness of a prover implementing the calculus within, for instance, an Otter or DISCOUNT loop. Our framework is mechanized in Isabelle/HOL.

We prove the continuity of bounded solutions for a wide class of parabolic equations with $(p,q)$-growth $$ u_{t}-{\rm div}\left(g(x,t,|\nabla u|)\,\frac{\nabla u}{|\nabla u|}\right)=0, $$ under the generalized non-logarithmic Zhikov's condition $$ g(x,t,{\rm v}/r)\leqslant c(K)\,g(y,\tau,{\rm v}/r), \quad (x,t), (y,\tau)\in Q_{r,r}(x_{0},t_{0}), \quad 0, Comment: 32 pages

Given a graph $\Gamma$, its auxiliary \emph{square-graph} $\square(\Gamma)$ is the graph whose vertices are the non-edges of $\Gamma$ and whose edges are the pairs of non-edges which induce a square (i.e., a $4$-cycle) in $\Gamma$. We determine the threshold edge-probability $p=p_c(n)$ at which the Erd{\H o}s--Renyi random graph $\Gamma=\Gamma_{n,p}$ begins to asymptotically almost surely have a square-graph with a connected component whose squares together cover all the vertices of $\Gamma_{n,p}$. We show $p_c(n)=\sqrt{\sqrt{6}-2}/\sqrt{n}$, a polylogarithmic improvement on earlier bounds on $p_c(n)$ due to Hagen and the authors. As a corollary, we determine the threshold $p=p_c(n)$ at which the random right-angled Coxeter group $W_{\Gamma_{n,p}}$ asymptotically almost surely becomes strongly algebraically thick of order $1$ and has quadratic divergence.

An analogue of the Mukai map $$m_g: {\mathcal {P}}_g \rightarrow {\mathcal {M}}_g$$ m g : P g → M g is studied for the moduli $${\mathcal {R}}_{g, \ell }$$ R g , ℓ of genus g curves C with a level $$\ell $$ ℓ structure. Let $${\mathcal {P}}^{\perp }_{g, \ell }$$ P g , ℓ ⊥ be the moduli space of 4-tuples $$(S, {\mathcal {L}}, {\mathcal {E}}, C)$$ ( S , L , E , C ) so that $$(S, {\mathcal {L}})$$ ( S , L ) is a polarized K3 surface of genus g, $${\mathcal {E}}$$ E is orthogonal to $${\mathcal {L}}$$ L in $${{\,\mathrm{Pic}\,}}S$$ Pic S and defines a standard degree $$\ell $$ ℓ K3 cyclic cover of S, $$C \in \vert {\mathcal {L}} \vert $$ C ∈ | L | . We say that $$(S, {\mathcal {L}}, {\mathcal {E}})$$ ( S , L , E ) is a level $$\ell $$ ℓ K3 surface. These exist for $$\ell \le 8$$ ℓ ≤ 8 and their families are known. We define a level $$\ell $$ ℓ Mukai map $$r_{g, \ell }: {\mathcal {P}}^{\perp }_{g, \ell } \rightarrow {\mathcal {R}}_{g, \ell }$$ r g , ℓ : P g , ℓ ⊥ → R g , ℓ , induced by the assignment of $$(S, {\mathcal {L}}, {\mathcal {E}}, C)$$ ( S , L , E , C ) to $$ (C, {\mathcal {E}} \otimes {\mathcal {O}}_C)$$ ( C , E ⊗ O C ) . We investigate a curious possible analogy between $$m_g$$ m g and $$r_{g, \ell }$$ r g , ℓ , that is, the failure of the maximal rank of $$r_{g, \ell }$$ r g , ℓ for $$g = g_{\ell } \pm 1$$ g = g ℓ ± 1 , where $$g_{\ell }$$ g ℓ is the value of g such that $$\dim {\mathcal {P}}^{\perp }_{g, \ell } = \dim {\mathcal {R}}_{g,\ell }$$ dim P g , ℓ ⊥ = dim R g , ℓ . This is proven here for $$\ell = 3$$ ℓ = 3 . As a related open problem we discuss Fano threefolds whose hyperplane sections are level $$\ell $$ ℓ K3 surfaces and their classification.

Gravitational waves (GWs) may be magnified or de-magnified due to lensing. This phenomenon will bias the distance estimation based on the matched filtering technique. Via the multi-sphere ray-tracing technique, we study the GW magnification effect and selection effect with particular attention to the stellar-mass binary black holes (BBHs). We find that, for the observed luminosity distance $\lesssim 3~\mathrm{Gpc}$, which is the aLIGO/Virgo observational horizon limit, the average magnification keeps as unity, namely unbiased estimation, with the relative distance uncertainty $\sigma(\hat{d})/\hat{d}\simeq0.5\%\sim1\%$. Beyond this observational horizon, the estimation bias can not be ignored, and with the scatters $\sigma(\hat{d})/\hat{d} = 1\%\sim 15\%$. Furthermore, we forecast these numbers for Einstein Telescope. We find that the average magnification keeps closely as unity for the observed luminosity distance $\lesssim 90~\mathrm{Gpc}$. The luminosity distance estimation error due to lensing for Einstein Telescope is about $\sigma(\hat{d})/\hat{d} \simeq 10\%$ for the luminosity distance $\gtrsim 25~\mathrm{Gpc}$. Unlike the aLIGO/Virgo case, this sizable error is not due to the selection effect. It purely comes from the unavoidably accumulated lensing magnification. Moreover, we investigated the effects of the orientation angle and the BH mass distribution models. We found that the results are strongly dependent on these two components., Comment: 9 pages, 9 figures, published in MNRAS

In this paper, we discuss estimates on transition densities for subordinators, which are global in time. We establish the sharp two-sided estimates on the transition densities for subordinators whose L\'evy measures are absolutely continuous and decaying in mixed polynomial orders. Under a weaker assumption on L\'evy measures, we also obtain a precise asymptotic behaviors of the transition densities at infinity. Our results cover geometric stable subordinators, Gamma subordinators and much more., Comment: 47 pages. Example 3.4 is updated

We introduce the exceptional topological insulator (ETI), a non-Hermitian topological state of matter that features exotic non-Hermitian surface states which can only exist within the three-dimensional topological bulk embedding. We show how this phase can evolve from a Weyl semimetal or Hermitian three-dimensional topological insulator close to criticality when quasiparticles acquire a finite lifetime. The ETI does not require any symmetry to be stabilized. It is characterized by a bulk energy point gap, and exhibits robust surface states that cover the bulk gap as a single sheet of complex eigenvalues or with a single exceptional point. The ETI can be induced universally in gapless solid-state systems, thereby setting a paradigm for non-Hermitian topological matter., 6+ pages, 3 figures; 21 pages Supplemental Material

In this article, an omnidirectional dual-polarized antenna with low windage is presented within a wedge-profiled structure for supersonic onboard systems. Based on aerodynamic analysis, a wedge-profiled cavity is designed with colocated slots for omnidirectional and dual-polarized radiation. For horizontal polarization, two identical half-wavelength slots are etched onto the cavity sidewalls symmetrically. The vertical polarization is achieved by another horizontal slot, with the port isolation higher than 42 dB. A low windage coefficient of 0.11 is obtained within the wedge-profiled cavity with the dimensions of $41.6 \times 16 \times 30$ mm3 ( $0.34\lambda _{0} \times 0.13\lambda _{0} \times 0.24\lambda _{0}$ ). The proposed antenna is constructed and tested, with the measured results in agreement with the simulated ones. The operating bandwidths cover the band of 2.40–2.48 GHz, and the gain variations below 3.5 dB are realized in the azimuthal plane for both polarizations. The proposed wedgy antenna is with both electromagnetic and aerodynamic considerations, exhibiting the potentials in supersonic onboard communication systems.

Given a set R of m disjoint finite regions in the 2-dimensional plane, all regions having polygonal boundaries, and given a set D of n discs with fixed centers and radii, we consider the problem of finding a minimum cardinality subset D ∗ ⊆ D such that every point in R is covered by at least one disc in D ∗ . We show that this problem can be solved by using an iterative procedure that alternates between the solution of a traditional set-cover problem and the construction of the Laguerre–Voronoi diagram of a circle set. Computer experiments demonstrate the effectiveness of the proposed algorithm, particularly when the number | D ∗ | of discs necessary to cover R is low.

The article investigates the problem of state observation for discrete-time Markov jump piecewise-affine (PWA) systems, which cover both Markov jump systems and traditional PWA systems as two special classes. A new method for determining the admissible region-switching paths between adjacent time instants is proposed for both the plant state and the estimator state, so that both the computation cost and the conservatism of the theoretical results are reduced. To analyze the mean-square stability of the observation error system, a novel piecewise Lyapunov function is constructed, which depends on the current mode and the region indices of both the system and observer states. As the partitions of the state space are different for different modes, the region indices are dependent on the system mode. Then, mode-dependent PWA observers are designed by means of $\mathcal {S}$ -procedure and ellipsoidal outer approximation such that the observation error system is mean-square stable. The proposed observation strategy is demonstrated via an illustrative example.