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The importance of Locally Repairable Codes (LRCs) lies on their applications in distributed storage systems (DSSs). There are two approaches to repair multiple failed nodes: the parallel approach, in which a set of failed nodes are simultaneously repaired; and the sequential method, wherein the failed nodes are repaired successively by making use of known nodes, including those already repaired. LRCs in the joint sequential-parallel mode were investigated with the aim of reducing the repair time in the sequential mode, and in this study, we continue the investigation by providing LRCs with higher repair tolerance. We propose a construction of generator matrices for binary LRCs based on back-circulant Latin squares and $t$ -dimensional permutation cubes. The codes based on back-circulant Latin squares have locality $r=5$ and availability $t \geq 4 $ in the parallel mode and, in the case where $t+1$ is neither even nor a multiple of 3, they possess a short block length compared to their counterparts [Tamo and Barg (2014) and Wang and Zhang (2015)]. We present LRCs based on $t$ -dimensional permutation $m$ -cubes with block length $m^{t}$ , locality $r=2m-3$ , and availability $t \geq 3$ in parallel mode. These codes can repair any set of failed nodes of size up to $2^{t}-1$ in the sequential mode in at most $t-1$ steps. It is shown that these codes are overall local with repair tolerance $2t$ in the parallel mode. Finally, we introduce the LRC-AL class of codes, a new class of LRCs which satisfies the property that for any pair of nodes $i,j$ , there is a repair set for the failed node $i$ that contains the live node $j$ .

A hybrid multi-state orbital angular momentum-multi pulse-position modulation ( $N$ OAM-MPPM) scheme over gamma-gamma free-space optical ( ${\Gamma \Gamma }$ -FSO) channel is studied in this paper. In our study, all atmospheric and pointing error impacts are taken into account. Expressions for the parameters of ${\Gamma \Gamma }$ -FSO-pointing error channel are derived. In addition, approximate-tight upper bounds on the bit-error rates (BERs) of $N$ OAM and $N$ OAM-MPPM techniques are developed over ${\Gamma \Gamma }$ -FSO-pointing error channels, considering the influences of beam divergence and pointing error (PE). The ${\Gamma \Gamma }$ -FSO-PE channel parameters and the BER expressions are evaluated numerically and verified by simulation. It turned out that the analytical results are nearly the same as those obtained from simulation under different turbulence scenarios and OAM modes. The results demonstrate that under variable turbulence conditions, the $N$ OAM-MPPM technique outperforms both ordinary $N$ OAM and MPPM systems. Furthermore, different deep learning (DL) techniques, namely random forest (RF), convolution neural network (CNN), and auto-encoder (AE), are employed to get the optimum classification accuracy using different datasets of $N$ OAM-MPPM over ${\Gamma \Gamma }$ -PE channel model. Finally, the results indicate that AE has the best performance metrics compared to other models using different datasets.

Large-scale complex network data poses significant challenges for the analytic ideas and tools to monitor and analyze complex networks. As classical structure descriptors, average path length ( $L$ ), global and local network efficiency ( $E_{glob}$ and $E_{loc}$ ), general and loop clustering coefficient ( $C_{3}$ and $D$ ), play significant roles in complex network analysis. In this paper, we make use of resistive distance instead of the shortest path distance to suggest resistive distance descriptors, i.e., $L_{r}$ , $\Omega _{glob}$ and $\Omega _{loc}$ , $\Omega _{3}$ and $D_{r}$ . We investigate all the resistive descriptors on classical WS, BA and ER models and find some interesting phenomenons. On one hand, $L_{r}$ and $\Omega _{3}$ (resp., $\Omega _{glob}$ and $\Omega _{loc}$ ) can be used to characterize the features of small-word networks. On the other hand, $L_{r}$ , $\Omega _{3}$ , and $\Omega _{glob}$ can be utilized to measure network invulnerability. To access the effectiveness of the resistive distance descriptors, we conduct extensive numerical simulations on synthetic and real networks. In comparison with the baselines, the proposed metrics show competitive performance on classical network models and are more efficient for networks with small and medium size.

A $(m,n)$ sonar sequence is an $m\times n$ array with exactly one dot in each column and where all lines connecting two dots in the array are distinct as vectors. These arrays are known to have many applications such as sonar and radar detection and these are studied as a particular case of Golomb rectangles or two-dimensional Sidon sets. The main open problem for sonar sequences is: for fixed $m$ , find the largest $n$ for which there is an $(m,n)$ sonar sequence, these sequences are called the best sonar sequences. The extended sonar sequences are generalizations of sonar sequences where each column has at most one dot, the motivation to study these arrays are the best results obtained when applied to radar and sonar detection. In this paper, we give the best sonar sequences with $m\leq 100$ obtained from an exhaustive computational search based on the Caicedo, Ruiz and Trujillo constructions and we present new constructions of extended sonar sequences that use Sidon sets.

In this paper, we consider the performance of exclusive-OR (XOR) rule in detecting the presence or absence of a deterministic signal in bivariate Gaussian noise. Signals, when present at the two sensors, are assumed unequal, whereas the noise components have identical marginal distribution but are correlated. The sensors send their one-bit quantized data to a fusion center, which then employs the XOR rule to arrive at the final decision. Here we prove that, in the limit as the correlation coefficient $r$ approaches 1, the optimum fusion rule for both parallel and tandem topologies is XOR with identical, alternating partitions (XORAP) of the observations at the sensors. We further quantify the asymptotic decrease of the Bayes error of XORAP towards zero as a constant multiplied by $\sqrt {1-r}$ , as $r$ approaches 1. When compared to the asymptotic Bayes error of CLRT, which decreases to zero exponentially fast, as a function of $1/(1-r)$ , the Bayes error of XORAP decreases to zero much slower.

In this paper, we present a Model-Free Stochastic Inverse Optimal Control (IOC) algorithm for the discrete-time infinite-horizon stochastic linear quadratic regulator (LQR). Our proposed algorithm exploits the richness of the available system trajectories to recover the control gain $K$ and cost function parameters $(Q,R)$ in a low (space, sample, and computational) complexity manner. By leveraging insights on the stochastic LQR, we guarantee well-posedness of the Model-Free Stochastic IOC LQR via satisfaction of the Certainty Equivalence optimality conditions. The exact solution of the control gain $K$ is recovered via a deterministic, low complexity Least Squares approach. Using $K$ , we solve a completely model-free non-iterative SemiDefinite Programming (SDP) problem to obtain a unique (up to a scalar ambiguity) $(Q,R)$ , in which optimality and feasibility are jointly ensured. Via derivation of the sample complexity bounds, we show that the non-asymptotic performance of the Model-Free Stochastic IOC LQR can be characterized by the signal-to-noise (SNR) ratio of the finite set of system state and input signals. We present a model-based version of the algorithm for the special case where $(A,B)$ is available, and we, further, provide the extension to the Stochastic Model-Free IOC linear quadratic tracking (LQT) case.

Let $G$ be an additive group of order $v$ . A $k$ -element subset $D$ of $G$ is called a $(v, k, \lambda, t)$ -almost difference set if the expressions $g-h$ , for $g$ and $h$ in $D$ , represent $t$ of the non-identity elements in $G$ exactly $\lambda $ times and every other non-identity element $\lambda + 1$ times. Almost difference sets are highly sought after as they can be used to produce functions with optimal nonlinearity, cyclic codes, and sequences with three-level autocorrelation. A set of positive integers $A$ is called a Golomb ruler if the difference between two distinct elements of $A$ are different. In this paper, we use Singer type Golomb rulers to construct new families of almost difference sets. Additionally, we constructed 2-adesigns from these almost difference sets.

$(\alpha,\beta)$ -Pythagorean fuzzy set is a very efficient way of dealing with uncertainty. In this article, we have introduced the notions of $(\alpha,\beta)$ - Pythagorean fuzzy subring and $(\alpha,\beta)$ - Pythagorean fuzzy ideal of a ring. Further, we have briefly described various results related to it. Also, we have discussed the level subring of an $(\alpha,\beta)$ - Pythagorean fuzzy subring. Moreover, we have studied the direct product and ring homomorphism of $(\alpha,\beta)$ - Pythagorean fuzzy subrings.

In this paper, a class of Clifford-valued neutral-type recurrent neural networks with $D$ operator is explored. By using non-decomposition method and the Banach fixed point theorem, we obtain several sufficient conditions for the existence of anti-periodic solutions for Clifford-valued neutral-type recurrent neural networks with $D$ operator. By using the proof by contradiction and inequality techniques, we obtain the global exponential synchronization of anti-periodic solutions for Clifford-valued neutral-type recurrent neural networks with $D$ operator. Finally, we give one example to illustrate the feasibility and effectiveness of main results.

A category theory constitutes a convenient conceptual apparatus to organize the worlds of mathematical entities. The concept of fuzzy natural transformation as an abstract mapping on functors is one of the essential concepts of this theory. If we admit a piece of non-commutativity in its definition diagrams, then the ‘upward’ and the ‘downward’ diagram parts generate different result sets. In this way, we can introduce the concept of fuzzy natural transformation. We deal with the multi-fuzzy natural transformation if such a transformation is based on a multi-diagram. This paper aims to describe the multi-fuzzy natural transformation situation when the symmetric difference of the result is finite. It allows us to organize the whole spectrum of the result sets in a unique quotient algebra $\mathcal {P}^{comp}(\omega)/\mathrm {fin}$ . Different algebraic properties of this structure will be explored, and a piece of classical encoding theory will be reconstructed in the environment determined by this algebra. In particular, the concepts of the $k$ -multi similarity, the abstract $k$ -multi similarity, the $k$ -similarity balls, and the abstract $k$ -similarity balls are introduced as some generalization of the idea of Hamming’s distances and Hamming’s balls. Finally, it is shown how to automate some verification processes in this context using an R-based programming environment.

In this paper, a new two-steps design strategy is introduced for the optimal rational approximation of the fractional-order Butterworth filter. At first, the weighting factors of the summation between the ${n}^{\text {th}}$ -order and the $(n + 1)^{\text {th}}$ -order Butterworth filters are optimally determined. Subsequently, this model is employed as an initial point for another optimization routine, which minimizes the magnitude-frequency error relative to the $(n+\alpha)^{\text {th}}$ -order, where $\alpha \in (0, 1)$ , Butterworth filter. The proposed approximant demonstrates improved performance about the magnitude mean squared error compared to the state-of-the-art design for six decades of bandwidth, but the introduced approach does not require a fractional-order transfer function model and the approximant of the $s^\alpha $ operator. The proposed strategy also avoids the use of the cascading technique to yield higher-order fractional-order Butterworth filter models. The performance of the proposed $1.5^{\text {th}}$ -order Butterworth filter in follow-the-leader feedback topology is verified through SPICE simulations and its hardware implementation based on Analog Devices AD844AN-type current feedback operational amplifier.

Given a specific propagation speed $h$ in a social network $G(V, E)$ , an influence circle(IC) of a node $s$ in time $t$ is a node set of its influenced nodes, where the distance between $s$ and its expected influenced node $w$ is less than radius $r=ht$ . Different from the Influence Maximization(IM) problem which finds a set of $k$ initial seed nodes in a network so that the expected size of cascade is maximized, the aim of the proposed influence circle covering (ICCovering) problem in this work is to find a minimum number of seeds or ICs to cover the whole network. The general approach for this covering problem is greedy strategy, which iteratively selects a seed with the largest influence circle. However, the upper bound of greedy algorithms does not perform very well, and the value will increase further as the network scale expands. In this paper, we propose an $\alpha $ -approximation partitioning algorithm for large-scale social networks, where $\alpha $ is the maximum number of outer edges of Voronoi cells appeared in the partition. The algorithm divides the input graph into smaller cells so that each cell can be solved separately, and a feasible solution to the input object can be constructed by combining the solutions of the smaller cells. When solving the smaller cells, we adopt the linear programming method. In order to improve its effectiveness and efficiency, we also propose two optimization algorithms. Extensive experiments on real social networks confirm the superiorities and effectiveness of our solution.

It is well known that the stochastic optimization problem can be regarded as one of the most hard problems since, in most of the cases, the values of $f$ and its gradient are often not easily to be solved, or the $F(\cdot, \xi)$ is normally not given clearly and (or) the distribution function $P$ is equivocal. Then an effective optimization algorithm is successfully designed and used to solve this problem that is an interesting work. This paper designs stochastic bigger subspace algorithms for solving nonconvex stochastic optimization problems. A general framework for such algorithm is presented for convergence analysis, where the so-called the sufficient descent property, the trust region feature, and the global convergence of the stationary points are proved under the suitable conditions. In the worst-case, we will turn out that the complexity is competitive under a given accuracy parameter. We will proved that the $SFO$ -calls complexity of the presented algorithm with diminishing steplength is $O\left({\epsilon ^{-{\frac {1}{1-\beta }}}}\right)$ and the $SFO$ -calls complexity of the given algorithm with random constant steplength is $O(\epsilon ^{-2})$ respectively, where $\beta \in (0.5,1)$ and $\epsilon $ is accuracy and the needed conditions are weaker than the quasi-Newton methods and the normal conjugate gradient algorithms. The detail algorithm framework with variance reduction is also proposed for experiments and the nonconvex binary classification problem is done to demonstrate the performance of the given algorithm.

This paper proposes a simple $1^{\mathrm {st}}$ -order disturbance observer (DO)-based optimal control (DOBOC) strategy for surface-mounted permanent magnet synchronous motor (SPMSM) drives. A detailed comparative analysis of three reduced-order (i.e., $1^{\mathrm {st}}$ , $2^{\mathrm {nd}}$ , and $3^{\mathrm {rd}}$ ) DOs has been presented in terms of structural complexity along with transient and steady-state performances. A simple $1^{\mathrm {st}}$ -order DOBOC is proposed with a considerably simple and easy gain tuning procedure without any complex equation and time-consuming weighting matrices selection while ensuring a competitive control performance. Also, the stability of three DOBOCs is proven using the Lyapunov function, which has not been carried out in the previous studies. Furthermore, this paper provides a detailed procedure to systematically select the controller and observer gains of three reduced-order (i.e., $1^{\mathrm {st}}$ , $2^{\mathrm {nd}}$ , and $3^{\mathrm {rd}}$ ) DOBOC by experiment. Comparative experiments are performed to confirm the feasibility of the proposed $1^{\mathrm {st}}$ -order DOBOC using a prototype SPMSM test-bed with a TI TMS320F28335 DSP. Finally, verification results and comparative table are presented to compare the speed tracking performance (e.g., transient response, undershoot (US), steady-state error (SSE), and robustness to load torque disturbances) of three reduced-order DOBOC methods under speed and load torque step-changes.

The issues of mean-square admissibility and synthesis of It $\hat {o}$ -type stochastic singular systems (SSSs) under Brownian parameter perturbations are introduced in this article. For ease of computation, a novel sufficient condition is given to guarantee autonomous systems are mean-square admissible via strict linear matrix inequalities(LMIs). Furthermore, own to the measurable of the system states, both state feedback controller and observer-based controller (OBC) for It $\hat {o}$ -type SSSs are investigated. However, in It $\hat {o}$ -type SSSs, because the state of the system and the observer can be affected by Brownian fluctuation, it is not feasible that the observer and control gains design are completely separate. To this end, an innovative design approach is also proposed to solve the controller and observer parameters simultaneously in form of strict LMIs. Finally, three examples are introduced to demonstrate the effectiveness of the proposed method.

High-contrast transmitarray (HCTA) metasurfaces, named as metalines in this article, a category of lithographically defined on-chip diffractive elements with minimized scattering loss, enables parallel on-chip optical signal processing. While high-performance devices with different functionalities are reported, most of them are static, and their optical functions are fixed after fabrication. However, dynamically tunable metasurfaces are of essential need in many modern optical systems. Here, we employ an inverse design method, combining FDTD parameter sweep with Fourier-optics simulations, to design thermally-controlled varifocal metalenses. Thermal tunability is accomplished through the thermo-optical effect of silicon. Maximum focal length tunability of 12% and 24% are demonstrated for $400~\mu \text{m}$ and $200~\mu \text{m}$ aperture varifocal metalens doublets at the wavelength of $1.55~\mu \text{m}$ , respectively. The focusing efficiency is more than 74% for the whole temperature range. The $400~\mu \text{m}$ and $200~\mu \text{m}$ aperture metalenses are able to concentrate the input beam to spot sizes less than one wavelength and two wavelengths, respectively. This paves the way for dynamically controlled silicon photonics computational chips.

Convolutional neural networks (CNNs) have developed to become powerful models for various computer vision tasks ranging from object detection to semantic segmentation. However, most of the state-of-the-art CNNs cannot be deployed directly on edge devices such as smartphones and drones, which need low latency under limited power and memory bandwidth. One popular, straightforward approach to compressing CNNs is network slimming, which imposes $\ell_1$ regularization on the channel-associated scaling factors via the batch normalization layers during training. Network slimming thereby identifies insignificant channels that can be pruned for inference. In this paper, we propose replacing the $\ell_1$ penalty with an alternative nonconvex, sparsity-inducing penalty in order to yield a more compressed and/or accurate CNN architecture. We investigate $\ell_p (0 < p < 1)$, transformed $\ell_1$ (T$\ell_1$), minimax concave penalty (MCP), and smoothly clipped absolute deviation (SCAD) due to their recent successes and popularity in solving sparse optimization problems, such as compressed sensing and variable selection. We demonstrate the effectiveness of network slimming with nonconvex penalties on three neural network architectures -- VGG-19, DenseNet-40, and ResNet-164 -- on standard image classification datasets. Based on the numerical experiments, T$\ell_1$ preserves model accuracy against channel pruning, $\ell_{1/2, 3/4}$ yield better compressed models with similar accuracies after retraining as $\ell_1$, and MCP and SCAD provide more accurate models after retraining with similar compression as $\ell_1$. Network slimming with T$\ell_1$ regularization also outperforms the latest Bayesian modification of network slimming in compressing a CNN architecture in terms of memory storage while preserving its model accuracy after channel pruning., version1 published in ISVC'20; version 2: fixed typo; version3 is the extended version and submitted to a journal; version 4: more typos fixed, official version will be on IEEE Access

For a connected graph $\mathcal {G}$ , the distance $d(u, v)$ between any two vertices $u$ , $v~\in ~\mathcal {V(G)}$ is defined as; $d(u,v)=\min _{\mathcal {P}_{u,v}}\{\text {length of}~\mathcal {P}_{uv}\}$ i.e the minimum length of path connecting any two vertices. Two vertices $u$ , $v$ of a graph $G$ are said to be resolved by a vertices $\mathcal {W}$ of a graph $\mathcal {G}$ if $d(u,w) \neq d(v,w)$ . An ordered set $\mathcal {W} = \{w_{1},w_{2},w_{3},\ldots,w_{k}\}\,\,\in \,\,V(G)$ is resolving set for a connected graph $G$ , if for any two vertices $u$ , $v$ there exists $w_{i} \in W$ such that $d(u,w_{i}) \neq d(v, w_{i})$ . The representation of vertices $v$ w.r.t $\mathcal {W}$ is represented by $r(v|\mathcal {W})$ which is $k$ -vector( $k$ -tuple) $(d(v,w_{1}),d(v,w_{2}),d(v,w_{3}),\ldots,d(v,w_{k}))$ . If the representation of all the vertices of graph $\mathcal {G}$ are different w.r.t $\mathcal {W}$ then $\mathcal {W}$ is the resolving set for $G$ [1]. A resolving set that contains minimum numbers of vertices is metric basis for $\mathcal {G}$ . The cardinality of smallest resolving set is the metric dimension of $\mathcal {G}$ , represented by $dim(\mathcal {G})$ [2], [3]. A family of connected graph $\mathcal {G}$ has unbounded metric dimension if $dim(\mathcal {G})$ depends on order of a graph. In this paper, we show that total graph of path power three and four have unbounded metric dimension. We also proved some results on edges of power of path graph.

Let $R=\mathbb {F}_{2}+u\mathbb {F}_{2}+v\mathbb {F}_{2}+uv\mathbb {F}_{2}$ be a finite non-chain ring, where $u^{2}=u$ , $v^{2}=v$ , $uv=vu$ . We give the lower and upper bounds on the covering radius of different types of repetition codes for Chinese Euclidean distance over $R$ . Furthermore, we determine the upper bound on the covering radius of block repetition codes, simplex codes of types $\alpha $ and $\beta $ , MacDonald codes of types $\alpha $ and $\beta $ for Chinese Euclidean distance over $R$ .

Recently, deep neural network (DNN) schemes based on big data-driven methods have been successfully applied to image classification, communication, translation of language, speech recognition, etc. However, more efforts are still needed to apply them to complex robust nonlinear filter design in signal processing, especially for the robust nonlinear $H_{\infty }$ filter design for robust state estimation of nonlinear stochastic signal system under uncertain external disturbance and output measurement noise. In general, the design problem of robust nonlinear $H_{\infty }$ filter needs to solve a complex Hamilton-Jacobi-Isaacs equation (HJIE), which is not easily solved analytically or numerically. Further, the robust nonlinear $H_{\infty }$ filter is not easily designed by training DNN directly via conventional big data schemes. In this paper, a novel robust $H_{\infty }$ HJIE-embedded DNN-based filter design is proposed as a co-design of $H_{\infty }$ filtering algorithm and DNN learning algorithm for the robust state estimation of nonlinear stochastic signal systems with external disturbance and output measurement noise. In the proposed robust $H_{\infty }$ DNN-based filter design, we have proven that when the approximation error of HJIE by the trained DNN through Adam learning algorithm approaches to 0, the HJIE-embedded DNN-based filter will approach the robust nonlinear $H_{\infty }$ filter of nonlinear stochastic signal system with uncertain external disturbance and output measurement noise. Finally, a trajectory estimation problem of 3-D geometry incoming nonlinear stochastic missile system by the proposed robust $H_{\infty }$ HJIE-embedded DNN-based filter scheme through the measurement by the sensor of radar system with external disturbance and measurement noise is given to illustrate the design procedure and validate its robust $H_{\infty }$ filtering performance when compared with the extended Kalman filter and particle filter.

In this paper, we use the CSS and Steane’s constructions to establish quantum error-correcting codes (briefly, QEC codes) from cyclic codes of length $6p^{s}$ over $\mathbb F_{p^{m}}$ . We obtain several new classes of QEC codes in the sense that their parameters are different from all the previous constructions. Among them, we identify all quantum MDS (briefly, qMDS) codes, i.e., optimal quantum codes with respect to the quantum Singleton bound. In addition, we construct quantum synchronizable codes (briefly, QSCs) from cyclic codes of length $6p^{s}$ over $\mathbb F_{p^{m}}$ . Furthermore, we give many new QSCs to enrich the variety of available QSCs. A lot of them are QSCs codes with shorter lengths and much larger minimum distances than known non-primitive narrow-sense BCH codes.

The segmentation of piecewise polynomial signals arises in a variety of scientific and engineering fields. When a signal is modeled as a piecewise polynomial, the key then becomes the detection of breakpoints followed by curve fitting and parameter estimation. This paper proposes HOPS, a fast High-Order Polynomial Segmenter, which is based on $\ell _{0}$ -penalized least-square regression. While the least-squares regression ensures fitting fidelity, the $\ell _{0}$ penalty takes the number of breakpoints into account. We show that dynamic programming can be applied to find the optimal solution to this problem and that a pruning strategy and matrix factorization can be utilized to accelerate the execution speed. Finally, we provide some illustrative examples, and compare the proposed method with state-of-the-art alternatives.

One of the teleportation versions is applied to convey an entangled two-qubit system via a class of a two-qubit XYZ Heisenberg chain model. At thermal equilibrium, the exact solution of the XYZ model with an external magnetic field and the Kaplan–Shekhtman–Entin-Wohlman–Aharony interaction (KSEWA-interaction) is proposed in $X$ , and $Z$ direction. The teleported quantum Fisher information is studied, where we estimate the weight parameter $\theta $ , temperature and the strength parameter of KSEWA-interaction. It is found that, the strength of the KSEWA-interaction and the magnetic field be of benefit in the case of estimating the weight function. Meanwhile, the temperature decreases the teleported information.

In this paper, we optimize code parameters of finite-length spatially coupled low-density parity-check (SC-LDPC) codes, represented by a set of code parameters $(l,r,w,L,M)$ . Although the finite-length scaling behavior of SC-LDPC codes was studied in the existing literature, the previous works impose a constraint such that the coupling width $w$ is equal to the variable node degree $l$ and they do not focus on optimizing the code parameters for given code and decoder specifications such as the code rate, frame size, and decoding complexity. In order to optimize the code parameters with the target specifications, we first extend the scaling law of SC-LDPC codes without the constraint $w=l$ . Using the scaling law formulated with a new variable $w$ , we show that the coupling width $w$ directly affects the slope of the performance curve and performance comparisons are given to investigate trade-offs inherent in the code parameters. It is shown that there are trade-offs for the code parameters in the perspective of the asymptotic performance limit, code rate, and scaling behaviors. In addition, the scaling law allows us to find the optimal code parameter set showing the best finite-length performance. Interestingly, the optimal code parameter set $(l,r,w)$ varies depending on the coupling length $L$ and uncoupled code length $M$ that determine the code and decoder specifications, which means there is no specific code parameter set prevailing over different kinds of applications. Finally, we illustrate this result using the investigated trade-offs on the code parameters, which gives us useful insight on how to choose the code parameters.

A metaloop antenna (1FD-MetaLPA), made of $C$ -type metaatoms and having a single feed point, $F_{\mathrm {L}}$ ’, is created to radiate a left-hand circularly polarized (LHCP) axial beam at a frequency of $f_{\mathrm {LH-1FD}} =2.55$ GHz and a right-hand circularly polarized (RHCP) axial beam at a frequency of $f_{\mathrm {RH-1FD}} =3.45$ GHz. It is revealed that the maximum gains at $f_{\mathrm {LH-1FD}}$ and $f_{\mathrm {RH-1FD}}$ are unbalanced. Subsequently, a number of $C$ -type metaatoms are replaced by $N$ -type metaatoms. The replacement generates a balanced gain of 7.1 dBi with an LHCP gain bandwidth of 7.7% and an RHCP gain bandwidth of 8.3%. The radiation efficiency under the balanced gain is 75% at $f_{\mathrm {LH-1FD}}$ and 29% at $f_{\mathrm {RH-1FD}}$ . Next, a large metaloop antenna that radiates a conical beam and has a feed point, $F_{\mathrm {O}}$ , is concentrically added to the outside of the gain-balanced 1FD-MetaLPA. This antenna system is designated as the MetaLPA-plus. It is found that the MetaLPA-plus radiates an LHCP tilted beam at $f_{\mathrm {LH-1FD}}$ and an RHCP tilted beam at $f_{\mathrm {RH-1FD}}$ . The angle of depression (tilt angle) using equal amplitude excitation is $\theta _{\mathrm {max}} = 20^{\circ }$ at $f_{\mathrm {LH-1FD}}$ and $\theta _{\mathrm {max}} = 15^{\circ }$ at $f_{\mathrm {RH-1FD}}$ . These tilted CP radiation beams are rotated around the antenna axis with change in the excitation phases at points $F_{\mathrm {L}}$ ’ and $F_{\mathrm {O}}$ . During the beam-steering, the gain for an LHCP wave is nearly constant, with a value between 5 dBi and 6.5 dBi. This also holds true for an RHCP wave.

It is well-known now that orbital angular momentum (OAM) modes can be a potential technology to take full advantages of the space resources in the fiber-based optical communication system because of its infinite states. However, the corresponding fiber design for OAM modes to meet various requirements in the fiber-based communication system is still not enough. For the dispersion property, there are various attributes given in the standard of ITU-T series, which has not yet been proposed for the OAM modes. In this paper, we propose and design a non-zero dispersion-shifted air-core ring fiber for OAM mode, which can offer more options on dispersion values. A negative dispersion of −1.246 ps/(nm $\cdot $ km) or a positive dispersion of 1.541 ps/(nm $\cdot $ km) at 1550 nm with < 1.5 ps/(nm $\cdot $ km) dispersion variation across the C band or < 4 ps/(nm $\cdot $ km) dispersion variation across the C and L bands for OAM $_{1,1}$ mode is achieved in the numerical simulation.

This paper deals with decomposition of complete graphs on $n$ vertices into circulant graphs with reduced degree $r< n-1$ . They are denoted as $C_{n}(a_{1}, a_{2}, {\dots }, a_{m})$ , where $a_{1}$ to $a_{m}$ are generators. Mathematical labeling for such bigger (higher order and huge size) and complex (strictly regular with so many triangles) graphs is very difficult. That is why after decomposition, an edge irregular $k$ -labeling for these subgraphs is computed with the help of algorithmic approach. Results of $k$ are computed by implementing this iterative algorithm in computer. Using the values of $k$ , an upper bound for edge irregularity strength is suggested for $C_{n}(a_{1}, a_{2}, {\dots }, a_{m})$ that is ${\vert E\vert }/{2}\log _{2} \vert V\vert $ .

We demonstrate the power of the simulation-based invariant verification technique through two case studies in which it is formally verified that two mutual exclusion protocols, MCS protocol and Anderson protocol, enjoy the mutual exclusion property by the simulation-based invariant verification technique. We initially attempted formally verifying that the two protocols enjoy the property by the induction-based invariant verification technique. We successfully completed the formal proof for MCS protocol by the simulation-based invariant verification technique earlier than the one by the induction-based invariant verification technique even though we started the latter earlier than the former. We did not successfully complete the formal proof for Anderson protocol by the induction-based invariant verification technique as of the time of this paper submission. We define a variant of simulation, “observably equivalent simulations,” and prove some theorems on them to conduct the MCS case study. Given a state machine $M$ , another state machine $M'$ constructed by introducing auxiliary variables into $M$ observably equivalently simulates $M$ . In this sense, $M'$ can be regarded as an abstract version of $M$ .

Herein, we investigate the system spectral efficiency (SSE) and system energy efficiency (SEE) for massive two-way multiple-input-multiple-output (MIMO) relaying networks while considering imperfect channel state information (CSI) with $K$ -pair users in the presence of the hybrid relaying processing. First, we obtain the general expression of the signal-to-interference-and-noise-ratio (SINR) for the considered system when the antenna number of the relay station (RS) $M$ tends to be significantly large. Next, we consider the power scaling scheme (PSS) and achieve asymptotic SSE and SEE. We set the pilot sequence transmit power $P_{P}$ to be fixed $P_{P}$ and variable $P_{P}$ . For a nonvanishing SINR, we achieve different power scaling cases under fixed and variable $P_{P}$ . For a fixed $P_{P}$ value, the optimal PSS of the SSE can be obtained when the transmit power of each user can be scaled by $M$ , whereas that of the SEE can be achieved when the transmit power of both the RS and each user terminal can be reduced by $M$ . Based on the variable value of $P_{P}$ , the optimal PSS’ of both the SSE and the SEE can be used when the transmit power of both RS and each user can be scaled by $M^{1/3}$ . Thus, we conclude that the required SSE and SEE performances can be attained in multi-pair massive MIMO two-way relaying networks with a considerably low transmit power at RS and/or the transmit power of each user. The analytical results for the same are validated via simulation.

To protect image data privacy, a secure and fast image encryption algorithm using hyper-chaos based key generator and vector operation is proposed. Firstly, we introduce a novel post-process method to create a key matrix for significantly reducing the iterations of the hyperchaotic system from $W\,\,\times \,\,H$ /4 to $2W$ for an image size of $W\,\,\times \,\,H$ . And within the post-process, a random selector driven by the piecewise linear chaotic map ensures the good randomness and unpredictability of the generated key matrix. Secondly, the vector operation is employed to accelerate the cipher block chaining based diffusion process, so the plain image can be parallel encrypted row by row and column by column with high efficiency. Thirdly, we utilize a Logistic map to quickly produce an initial vector for the vectorized diffusion process. Finally, the proposed algorithm is evaluated by some common security and performance tests. Experimental results show that the cipher image can pass all tests of NIST SP 800–22 with $P$ -values $\gg ~0.01$ , its correlation coefficient between pixels is close to 0, and the entropy is greater than 7.999. Combined with other results of security tests, we can safely conclude that the proposed image encryption algorithm provides adequate protection against statistical, brute-force, chosen-plaintext, and other common types of attacks. In addition, the time complexity is in the order of ${O}$ ( $W+H$ ) and the average encryption time of 512 $\times $ 512 images is only 0.023s. The results indicate that our algorithm with high security and fast speed can meet the requirements of real-time confidential transmission of massive image data.

We introduce a consistent and efficient method to construct self-dual codes over $GF(q)$ using symmetric matrices and eigenvectors from a self-dual code over $GF(q)$ of smaller length where $q \equiv 1 \pmod 4$ . Using this method, which is called a ‘symmetric building-up’ construction, we improve the bounds of the best-known minimum weights of self-dual codes with lengths up to 40, which have not significantly improved for almost two decades. We focus on a class of self-dual codes, which includes double circulant codes. We obtain 2967 new self-dual codes over $GF(13)$ and $GF(17)$ up to equivalence. We also compute the minimum weights of quadratic residue(QR) codes that were previously unknown. These are a [20, 10, 10] QR self-dual code over $GF(23)$ , [24, 12, 12] QR self-dual codes over $GF(29)$ and $GF(41)$ , and a [32, 16, 14] QR self-dual code over $GF(19)$ . They have the highest minimum weights so far.

The proposed work focuses on improving the performance of the traditional nonlinear saturation controller (NSC) algorithm. In this work, a tuning mechanism can be implemented by providing the NSC control unit with the measured speed $\Omega $ of the rotor from a shaft encoder device in order to introduce a tuned NSC (TNSC). In other words, we are going to tune the NSC’s natural frequency ${\omega _{c}}$ at one half of $\Omega $ such that $\Omega =2{\omega _{c}}$ . This TNSC is adopted to reduce the vibratory amplitudes of a 16-pole constant-stiffness rotor-Active magnetic bearings AMBs) system for a wide range of speeds and eccentricities. The whole controlled system is studied mathematically to seek its approximate solutions via the multiple scales technique. Different relations between the rotor’s amplitudes and its parameters are plotted so as to verify the proposed tuning mechanism and its crucial role in improving the NSC’s performance.

The 5G and beyond networks will intrinsically accommodate a wide range of use-case scenarios and expand the limit of legacy mobile systems. The 5G network architecture can handle the seamless operation of various wireless channels in a heterogeneous environment. The $\eta $ - $\mu $ fading model is well-suited for versatile channels as it adapts to different fading behaviors in a broad-range propagation for non-line-of-sight (NLOS) circumstances. This paper evaluates the performance of heterogeneous wireless networks using $\eta $ - $\mu $ fading channel under mobility conditions. We incorporated the random waypoint (RWP) model with $\eta $ - $\mu $ distribution to model the dynamic behavior of non-homogeneous fading. The derivation of expressions for the probability density function (PDF) and cumulative distribution function (CDF) of the received signal power for a mobile network in all three-dimensional topologies is extracted. Consequently, the outage probability (OP) and average bit error rate (ABER) are analyzed to quantify the performance of the mobile system. The effect of co-channel interference (CCI) is investigated based on a desired and interfering signal transmitted in mobile networks. The proposed novel-form can characterize the performance of a mobile user, and the derivation is useful for measuring the effect of noise and interference on the signal. Finally, the novel-form applicability analyzes the impact of mobility incorporated in different fading channels such as Nakagami-m, Nakagami-q (Hoyt), Rayleigh, and one-sided Gaussian distributions.

Graph theory is widely used to analyze the structure models in chemistry, biology, computer science, operations research and sociology. Molecular bonds, species movement between regions, development of computer algorithms, shortest spanning trees in weighted graphs, aircraft scheduling and exploration of diffusion mechanisms are some of these structure models. Let $G = (V_{G}, E_{G})$ be a connected graph, where $V_{G}$ and $E_{G}$ represent the set of vertices and the set of edges respectively. The idea of the edge version of metric dimension is based on the distance of edges in a graph. Let $R_{E_{G}}$ be the smallest set of edges in a connected graph $G$ that forms a basis such that for every pair of edges $e_{1},e_{2}~\in ~E_{G}$ , there exists an edge $e~\in ~R_{E_{G}}$ for which $d_{E_{G}}(e_{1}, e)~\neq ~d_{E_{G}}(e_{2}, e)$ holds. In this paper, we show that the family of circulant graphs $C_{n}(1,2)$ is the family of graphs with constant edge version of metric dimension.

Previous studies on magnetization AC loss of high temperature superconducting (HTS) coated conductors (CCs) taken into account the ideally sinusoidal external magnetic field, or non-sinusoidal by considering only the harmonics in phase with the fundamental component. However, realistically magnetic field often contains harmonics with different phase angles and amplitudes. In this paper, magnetization AC loss in HTS CC was studied, considering the phase angle $\varphi $ of each harmonic; assuming superconductor is subjected to external magnetic field which is distorted by the $3^{\mathrm {rd}}$ and the $5^{\mathrm {th}}$ harmonics. Phase angle of each harmonic was considered in the range of 0 to $\pi $ and amplitude of fundamental magnetic field, was assumed as 10, 20, 50, and 100 mT. Different distortion level of magnetic field was considered too. It is concluded that phase angle of field harmonic has a significant effect on magnetization loss of HTS CCs.

The current analysis is carried out for the Al2O3 nanoparticles transportation through a permeable cubic geometry under the influence of the magnetic force through a hot cubic object. The physical phenomenon described in the basic equations together with the Maxwell’s equations. Then D3Q19 model is sketched for the discretization of the velocity vectors for using the $3D$ Lattice Boltzmann Method (LBM). The impact of Brownian motion in the Al2O3-H2O nanofluids is considered by taking the Koo–Kleinstreuer model into consideration. The nanoparticles transportation under the impacts of the buoyancy and Lorentz forces, and permeability is studied with LBM. Numerical simulations are performed for different values of magnetic parameter, Darcy’s and Rayleigh numbers. The validation of the applied technique is presented in Fig. 3 in the form of isotherms by comparing the obtained results with Calcagni et al.. For the enhancement of the heat transfer analysis a quantitative comparison of the current study is presented in Fig. 4. All the results for various parameters are presented in the form of isotherms. The obtained results show the efficient conduction for higher values of $Ha$ . The larger values of $Da$ show reduction in the boundary layer thickness. The average Nusselt number $Nu_{ave} $ enhances with higher values of the permeability parameter. The efficiency of the implemented technique is demonstrated in Table 4, where the Nusselt number is tabulated for distinct numbers of $Gr$ and $Ha$ , and are compared with the available literature.

A particularly original negative group delay (NGD) theory of $\Delta $ -topology is developed in the present paper. The NGD three-port topology is a passive circuit purely constituted by capacitors network. The model of the proposed $\Delta $ -topology impedance 3-D matrix is analytically established in function of the capacitor elements. Then, the S-matrix model is derived by using the Y-to-S transform. By considering the S-matrix frequency responses, a bandpass (BP) NGD analysis of the capacitive $\Delta $ -topology is originally elaborated. It is theoretically demonstrated that the passive $\Delta $ -topology is susceptible to behave as a BP NGD circuit under an analytical condition between the constituting capacitor values. The design feasibility of the BP NGD function is experimentally verified with lumped capacitor components-based $\Delta $ -circuit proof of concept. An electronic circuit board constituted by purely capacitive-network $\Delta $ -circuit is fabricated as an original $\Delta $ -circuit prototype. The tested board is constituted by arbitrary chosen capacitors, 100 nF, 10 nF and 0.1 nF. As expected, the calculation, simulation and measurement results, which are in very good agreement, confirm the BP NGD behavior. It is observed from measurement that it generates NGD of about −18.1 ns at a frequency of about 0.55 MHz and, lower and upper cut-off frequencies of about 0.33 MHz and 1.71 MHz. It is noteworthy that the transmission and reflection coefficients at very low frequency are independent of the capacitor values and analytically equal to 2/3 and 1/3, respectively.

Pattern matching is a fundamental problem in theoretical computer science. The algorithms for pattern matching and the study on the statistics of pattern matching have found enormous applications in practical fields. In this paper, we revisit the Markov embedding approach for studying pattern matching in repeated experiments. For any pattern of length $m$ over alphabet $\Sigma $ , we show that the mean and variance of the waiting time of the pattern in iid experiments can be computed in $O(m)$ time based on Markov embedding technique, improving over the $O(|\Sigma | \cdot m)$ and $O(m^{2})$ naïve bounds. Our method extends to computing the $k$ -th moment of the waiting time, and it extends to computing other related statistics about pattern matching in repeated experiments, and it also extends to the case of Markov dependent experiments.

Let $\mathbb {Z}_{4}$ be the integer ring of residue classes modulo 4. In this paper, we construct four infinite families of $\mathbb {Z}_{4}$ -codes with one nonzero Lee weight by their generator matrices. Furthermore, we study the linearity of their Gray images and obtain a family of optimal binary codes.

Based on the common existence of the mixed road, a switched controller with consideration of the difference between estimation optimal and current flux (EOCFD) is presented in the lattice hydrodynamic model of traffic flow. Based on the Hurwitz criteria and the $H_{\infty }$ -norm, stability conditions for the curved road and straight road scenarios are obtained with the transfer functions $G_{0}$ and $G_{1}$ respectively. By analyzing the Bode-plot of transfer functions $G_{0}$ and $G_{1}$ , stability analysis is performed with the feedback gain $k$ , the radian $\theta _{j}$ and the curvature radius $R$ . These theoretical results indicate that the switched control scheme prompts the traffic flow to be more stable in both curved and straight roads. Compared with Cheng’s model, numerical simulations with multiple perturbations confirm that the switched control scheme can further suppress the traffic congestion with a lower feedback gain on mixed road.

The main difficulty in the traditional nonlinear $H_{\infty }$ control design lies in how to solve the nonlinear partial differential Hamilton-Jacobi-Isaacs equation (HJIE), especially for nonlinear time-varying systems. In this study, a novel HJIE-embedded DNN $H_{\infty }$ control scheme is proposed to be efficiently trained for nonlinear $H_{\infty }$ stabilization and tracking control designs of nonlinear dynamic systems with the external disturbance. The proposed DNN-based $H_{\infty }$ control approach not only capitalizes on the availability of theoretical partial differential HJIE but also reduces the amount of empirical data and the complexity to train HJIE-embedded DNN. We have shown that the proposed DNN-based $H_{\infty }$ control scheme can approach the theoretical result of $H_{\infty }$ robust control when the training error approaches zero and the asymptotic stability is also guaranteed if the nonlinear time-varying system is free of external disturbance. The proposed method could be easily extended to DNN-based $H_{\infty }$ reference tracking control of nonlinear systems for more practical applications. Finally, two examples, including $({i})$ an $H_{\infty }$ stabilization of nonlinear time-varying system and $(ii)$ an $H_{\infty }$ unmanned aerial vehicle (UAV) reference tracking control system, are proposed to illustrate the design procedure and to demonstrate the effectiveness of our DNN-based $H_{\infty }$ method.

Multitype quasi-cyclic (QC) low-density parity-check (LDPC) codes are a class of protograph LDPC codes lifted cyclically from protographs with multiple edges, represented by two weight and slope matrices. For a given weight-matrix, an approach is proposed to find the maximum-achievable girth $g_{\max }$ of the corresponding multitype QC-LDPC codes by some inevitable chains having less complexity than the existing methods. This advantage leads to some new patterns of the weight matrices such that the corresponding codes have some improvements in terms of the maximum-achievable girths or the minimum-distance upper-bounds. In continue, for a given weight-matrix with maximum-achievable girth $g_{\max }$ , some slope-matrices are constructed by a depth-first search algorithm for which the corresponding multitype QC-LDPC codes with even girth $g$ , $g\le g_{\max }$ , have smaller lengths, higher rates, or larger minimum-distances than the state-of-the-art achievements. Simulation results show that the constructed codes have some error-rate advantages than PEG, random-like, CCSDS, and 802.11n/ac IEEE standard LDPC codes.

This paper initially proposes an optimization problem and after presents its optimal solution. Then, this result is applied to obtain relaxed conditions to design controllers for nonlinear plants described by Takagi-Sugeno (TS) models, based on fuzzy Lyapunov function (FLF) and Linear Matrix Inequalities (LMI). The FLF is given by $V(x(t)) = x(t)^{T}P(\alpha (x(t)))x(t)$ , where $x(t)$ is the plant state vector, $P(\alpha (x(t))) = \alpha _{1}(x(t))P_{1} + \alpha _{2}(x(t))P_{2} + \cdots + \alpha _{r}(x(t))P_{r}$ , $P_{i}=P_{i}^{T} > 0$ and $\alpha _{i}(x(t))$ is the weight related to the local model $i$ in the representation of the plant by TS fuzzy models, for $i=1,2,\cdots,r$ . When one calculates the time derivative of this $V(x(t))$ , it appears the term $x(t)^{T}\dot {P}(\alpha (x(t)))x(t)$ , that is usually handled using conservative upper bounds, supposing that the bounds of the time derivative of $\alpha _{i}(x(t))$ , $i=1,2,\cdots,r$ , are available. The main result of this paper is a procedure to obtain optimal upper bounds for the term $x(t)^{T}\dot {P}(\alpha (x(t)))x(t)$ , such that they contemplate the maximum value and are always smaller than or equal to the maximum value. It is a relevant result on this subject, because these optimal upper bounds do not add any constraint. With these optimal upper bounds, a relaxed design method for stabilization of TS fuzzy models is proposed. Two numerical examples illustrate the effectiveness of this procedure.

The stabilization of periodic switched $k$ -valued logical networks is investigated in this paper, and some new results are presented. The system considered consists of several $k$ valued logical networks and these networks run in a periodic switching law. First, by using the Cheng product of matrices, a periodic switched $k$ -valued logical (control) network is transformed into a discrete dynamic system which is written as an algebraic form. Second, the switching-state space and the switching-input-state space are defined. Then combining with the algebraic form, some necessary and sufficient conditions for the stability and the stabilization are obtained. An algorithm to find the input sequence that stabilizes the system is also provided. Finally, illustrative examples are given to support the proposed new results.

Aydogdu et al. studied the standard forms of generator and parity-check matrices of $\mathbb {Z}_{2}\mathbb {Z}_{2}[u^{3}]$ -additive codes, and presented generators of $\mathbb {Z}_{2}\mathbb {Z}_{2}[u^{3}]$ -additive cyclic codes (Finite Fields Appl. 48: 241-260, 2017). In this paper, we investigate some other useful properties of $\mathbb {Z}_{2}\mathbb {Z}_{2}[u^{3}]$ -additive codes, including asymptotically good $\mathbb {Z}_{2}\mathbb {Z}_{2}[u^{3}]$ -additive cyclic codes and $\mathbb {Z}_{2}\mathbb {Z}_{2}[u^{3}]$ -additive complementary dual codes. The present paper can be viewed as a necessary complementary part of Aydogdu’s work.

In this paper, we compare conventional saddle type FinFETs to partial isolation type saddle FinFETs (Pi-FinFETs) using 3D TCAD simulations to examine the effect of single charge traps for proper prediction of leakage current. We simulated single charge traps at various locations in the drain region, and analyzed how the traps affect leakage current. Our results show that Pi-FinFETs enhanced the leakage current characteristics given the presence of a single charge trap. Also, it was found that Pi-FinFETs exhibit half the $\text{F}_{\mathrm {TAT}}$ of S-FinFETs. Based on the results from our analysis method, where we use $\text{I}_{\mathrm {off}}$ fluctuation, the $\text{F}_{\mathrm {TAT}}$ , the $\sigma _{\mathrm {F}}$ and the $\text{P}_{\mathrm {F}}$ parameters to accurately compare performance, and present device design guidelines aimed at improving DRAM refresh characteristics.

Beyond fifth-generation (5G) heterogeneous networks (HetNets) are facing the challenge of accommodating enormous mobile users and data traffic with scarce spectrum resources in the microWave ( $\mu $ Wave) band. In this work, we consider both challenges in HetNets comprising large size, high power base station (LHB) and relay operating in $\mu $ Wave band and small size, low power base station (SLB) and device-to-device (D2D) operating in millimeter wave (mmWave) band. We formulate user association optimization problems to pitch user association schemes based on downlink uplink decoupled (DU-DPL) access against traditional downlink uplink coupled (DU-CPL) access in HetNets to gauge the performance of these access schemes in terms of accommodating users and spectrum efficiency in $\mu $ Wave and mmWave bands. Formulated problems are non-deterministic polynomial-time hard (NP-hard) and solved using $\epsilon $ -optimal algorithm. Simulation results show the edge of DU-DPL access over DU-CPL access in terms of users association and spectrum efficiency.

This paper addresses the $\mathcal {H}_{2}$ , $\mathcal {H}_\infty $ and mixed $\mathcal {H}_{2}/\mathcal {H}_\infty $ formation static output control of continuous-time linear multi-agent systems with Markovian switching network topologies. It is assumed that the operation mode of the network topology cannot be directly measured but, instead, can be estimated by an imperfect detector. To model this problem, we consider a continuous-time hidden Markov model, in which the hidden component represents the real operation mode of the network topology while the observed component represents the information emitted from the detector and available for the controller. It is also assumed that only a partial information from the state variables of the multi-agent systems is available. By using an LMI (linear matrix inequality) formulation, a distributed static output controller which switches according to the detector information is designed to guarantee the stability in the mean square sense of the closed loop system, as well as an upper-bound for an index performance. Three situations are considered for the performance criteria: the $\mathcal {H}_{2}$ norm, the $\mathcal {H}_\infty $ norm, and the mixed $\mathcal {H}_{2}/\mathcal {H}_\infty $ norms. The paper is concluded with a numerical example to illustrate the effectiveness of the theoretical results.

The resistance distance is an intrinsic metric on graphs that have been extensively studied by many physicists and mathematicians. The resistance distance between two vertices of a simple connected graph $G$ is equal to the resistance between two equivalent points on an electrical network, constructed to correspond to $G$ , with each edge being replaced by a unit resistor. Hypercube $Q_{n}$ is one of the most efficient and versatile topological structures of the interconnection networks, which received much attention over the past few years. The folded $n$ -cube graph is obtained from hypercube $Q_{n}$ by merging vertices of the hypercube $Q_{n}$ that are antipodal, i.e., lie at a distance $n$ . Folded $n$ -cube graphs have been studied in parallel computing as a potential network topology. The folded $n$ -cube has the same number of vertices but half the diameter as compared to hypercubes which play an important role in analyzing the efficiency of interconnection networks. We intend is to minimize the diameter. In this study, we will compute the resistance distance between any two vertices of the folded $n$ -cube by using the symmetry method and classic Kirchhoff’s equations. This method is beneficial for distance-transitive graphs. As an application, we will also give an example and compute the resistance distance in the Biggs-Smith graph, which shows the competency of the proposed method.

This paper considers the problem of expanding a language class that can be proven by a non-interactive zero-knowledge proof system (NIZK) in a black-box manner in the common reference string model. Namely, given NIZKs for two languages, ${\mathcal L}_{0}$ and ${\mathcal L}_{1}$ , can we construct an NIZK for ${\mathcal L}_{0} \vee {\mathcal L} _{1}$ in a black-box manner? NIZKs for disjunctive languages have a large number of applications, such as electronic voting. Therefore, such a black-box construction may enable the efficient constructions of such applications. However, Abe et al. (PKC 2020) showed that this is impossible if the two given NIZKs are simulation-sound. In this paper, we prove that it is also impossible if the two given NIZKs are constructed by the commit-and-prove methodology that is typically used in many cryptographic protocols, including NIZKs. This result suggests that if we want to augment the capability of NIZKs in terms of the languages they can prove, we should rely on certain properties or structures of the underlying NIZKs, such as algebraic structures.