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Absorbing sets are combinatorial structures in the Tanner graphs of low-density parity-check (LDPC) codes that have been shown to inhibit the high signal-to-noise ratio performance of iterative decoders over many communication channels. Absorbing sets of minimum size are the most likely to cause errors, and thus have been the focus of much research. In this paper, we determine the sizes of absorbing sets that can occur in general and left-regular LDPC code graphs, with emphasis on the range of \begin{document}$ b $\end{document} for a given \begin{document}$ a $\end{document} for which an \begin{document}$ (a,b) $\end{document} -absorbing set may exist. We identify certain cases of extremal absorbing sets that are elementary, a particularly harmful class of absorbing sets, and also introduce the notion of minimal absorbing sets which will help in designing absorbing set removal algorithms.

The generalized exponential rational function (GERF) method is used in this work to obtain analytic wave solutions to the Kudryashov-Sinelshchikov (KS) equation. The KS equation depicts the occurrence of pressure waves in mixtures of liquid-gas bubbles while accounting for thermal expansion and viscosity. By applying the GERF method to the KS equation, we obtain analytic solutions in terms of trigonometric, hyperbolic, and exponential functions, among others. These solutions include solitary wave solutions, dark-bright soliton solutions, singular soliton solutions, singular bell-shaped solutions, traveling wave solutions, rational form solutions, and periodic wave solutions. We discuss the two-dimensional and three-dimensional graphics of some obtained solutions under the accurate range space by selecting appropriate values for the involved arbitrary parameters to make this research more praiseworthy. The obtained analytic wave solutions specify the GERF method’s dependability, capability, trustworthiness, and efficiency.

Bitprobe complexity of the static membership problem has been widely investigated since Burhman et al. (2000) studied the worst-case bounds for a whole range of membership problems. Though tremendous progress has been made in recent years, as is evident from the remarkable papers due to Alon and Feige (2009) [1] , Viola (2012) [8] , and Lewenstein et al. (2014), among others, Nicholson et al. (2013) [5] in their survey of the state of the art of the area noted that even the first non-trivial scenario of two probe adaptive schemes storing two elements of the universe is not completely settled, in that the lower bound for this problem is still open. We propose a proof (Kesh and Sharma, 2019) [3] for the lower bound for the same problem, but for a restricted class of schemes. We believe that this proof makes progress over the ideas proposed by Radhakrishnan et al. (2001, 2010) towards the proof. It is our belief that the single restriction imposed on the general class of schemes towards our proof faithfully captures the dependencies among the elements that share the same bit in the datastructure, and we hope that the general lower bound proof can be developed using similar ideas.

We develop efficient algorithms for determining the number of global cubic fields of given discriminant. They are based on earlier work of Hasse for cubic number fields. Our new ingredients allow to enlarge the range of computations considerably. For explicit calculations we used Magma.

All current work on calculating approximate sets of a rough set inevitably requires the participation of all elements in the universe. However, it is cumbersome and causes a huge waste of resources when the universe is quite big and the computed set is small enough. By introducing the notion of invariant subspace in rough set theory, we skillfully reduce the range of elements involved in the computations of approximations of a rough set from the whole universe U to a suitable invariant subspace V of U, and then give the modular method within the reduced range. Moreover, we present the modular Boolean matrix method, such that the calculation of upper and lower approximations of rough sets can be converted into operations on modular matrices. Finally, the results in covering approximation space are generalized to fuzzy β-covering approximation space to calculate the upper and lower approximations of the crisp sets. In particular, an algorithm for calculating the value of β is proposed to make the involved information has the highest distinction degree. This β is more reliable and meaningful than the empirical one, and an example about COVID-19 is put forward to simply illustrate its application.

First, we present construction methods for interior operators on a meet semilattice. Second, under the assumption that the underlying meet semilattices constitute the range of an interior operator, we prove an ordinal sum theorem for countably many (finite or countably infinite) triangular norms on bounded meet semilattices, which unifies and generalizes two recent results: one by Dvořak and Holcapek and the other by some of the present authors. We also characterize triangular norms that are representable as the ordinal sum of countably many triangular norms on given bounded meet semilattices.

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.

By using the Banach fixed point theorem and Schauder fixed-point theorem for semilinear spaces, we study the existence of solutions to some class of boundary value problems for interval-valued differential equations on unbounded domains. Some sufficient conditions are provided in order to deduce the existence of solutions without switching points, and also for mixed solutions with a unique switching point. The influences of the range of the parameter in the boundary value condition has on the existence of solutions is also discussed. Finally, two examples are given to demonstrate the feasibility of the theorems.

In this paper we consider the fractional parts of a general sequence, for example the sequence $\alpha \sqrt{n}$ or $\alpha n^2$. We give a general method, which allows one to show that long-range correlations (correlations where the support of the test function grows as we consider more points) are Poissonian. We show that these statements about convergence can be reduced to bounds on associated Weyl sums. In particular we apply this methodology to the aforementioned examples. In so doing, we recover a recent result of Technau-Walker (2020) for the triple correlation of $\alpha n^2$ and generalize the result to higher moments. For both of the aforementioned sequences this is one of the only results which indicates the pseudo-random nature of the higher level ($m \ge 3$) correlations., Comment: 132 pages

The Lambert W function is defined as the multivalued inverse of the function w↦wew. It has a wide range of applications. We propose a new method to construct a high-precision analytical approximation of the two branches of W. The method is based on Pade approximation and Schroder's iteration. This method can also be extended to solve other transcendental equations in science and engineering.

In this paper, we introduce crisp utility functions with range in the class of fuzzy numbers. These utility functions can be employed to prove a generalization of the classical result of Cantor using partially ordered sets as domain of the utility functions. In particular, for every partially ordered countable set, we prove that it can be represented by an order preserving function (utility).

This paper proposes a method which generates a set of weights to synthesize sum or difference pattern with precisely controlled sidelobe level (SLL), null, and dynamic range ratio (DRR) for arbitrary arrays. Our pattern synthesis approach reduces mutual coupling between the neighboring elements and complexity of the feeding network design. However, the formulated optimization problem is nonconvex due to the nonconvex objective function and fractional DRR constraint. To tackle it, we firstly introduce two sets of auxiliary variables: one for DRR constraint and the other for sidelobe and null constraints. By doing so, we then decompose the original optimization problem into three sets of subproblems characterized by the auxiliary variables and weight variables. To facilitate the subproblem with weight variables reaching its optimum values, we derive an appropriate range of step size. Finally, we iteratively solve these subproblems to obtain the solution to the original problem. Extensive experiments employing non-equispaced linear and rectangular arrays, concentric ring array, and cylinder array, are implemented to demonstrate that the developed approach can accurately control SLLs, null and DRR for arbitrary arrays.

In a previous paper, we obtained several "compact versions" of Rubio de Francia's weighted extrapolation theorem, which allowed us to extrapolate the compactness of linear operators from just one space to the full range of weighted Lebesgue spaces, where these operators are bounded. In this paper, we study the extrapolation of compactness for bilinear operators in terms of bilinear Muckenhoupt weights. As applications, we easily recover and improve earlier results on the weighted compactness of commutators of bilinear Calder\'{o}n-Zygmund operators, bilinear fractional integrals and bilinear Fourier multipliers. More general versions of these results are recently due to Cao, Olivo and Yabuta (arXiv:2011.13191), whose approach depends on developing weighted versions of the Fr\'echet--Kolmogorov criterion of compactness, whereas we avoid this by relying on "softer" tools, which might have an independent interest in view of further extensions of the method., Comment: v3: final version, incorporated referee comments, to appear in Indagationes Mathematicae, 27 pages

We bound the variance and other moments of a random vector based on the range of its realizations, thus generalizing inequalities of Popoviciu and of Bhatia and Davis concerning measures on the line to several dimensions. This is done using convex duality and (infinite-dimensional) linear programming. The following consequence of our bounds exhibits symmetry breaking, provides a new proof of Jung’s theorem, and turns out to have applications to the aggregation dynamics modelling attractive–repulsive interactions: among probability measures on [Formula: see text] whose support has diameter at most [Formula: see text], we show that the variance around the mean is maximized precisely by those measures that assign mass [Formula: see text] to each vertex of a standard simplex. For [Formula: see text], the [Formula: see text] th moment—optimally centered—is maximized by the same measures among those satisfying the diameter constraint.

The non-linear conjugate gradient method is a very effective technique for addressing Large-Scale minimization problems, and it has a wide range of applications in Mathematics, Chemistry, Physics, Engineering, and Medicine, etc. In this paper, we present a new spectral conjugate gradient algorithm, a non-linear conjugate gradient algorithm, whose derivation is based on the Hisham–Khalil (KH) and Newton algorithms Based on Pure Conjugacy Condition, The importance of the research lies in finding an appropriate way to solve all kinds of linear and non-linear fuzzy equations because the Buckley and Qu's method is ineffective in solving all kinds of fuzzy equations and because the conjugate gradient method does not need a Hessian matrix (second partial derivatives of functions) in the solution. The descent property of the suggested method is shown provided that the step size meets the strong Wolfe conditions. In many circumstances, numerical results demonstrate that the novel technique is more efficient than the Fletcher–Reeves (FR) and Hisham–Khalil (KH) procedures in solving Fuzzy Nonlinear Equations.

This paper is devoted to studying the transcendental directions of entire solutions of $ f^{(n)}+A_{n-1}f^{(n-1)}+...+A_0f = 0 $, where $ n(\geq 2) $ is an integer and $ A_i(z)(i = 0, 1, ..., n-1) $ are entire functions of finite lower order. With some additional conditions, the set of common transcendental directions of non-trivial solutions, their derivatives and their primitives must have a definite range of measure. Moreover, we obtain the lower bound of the measure of the set defined by the common transcendental directions of Jackson difference operator of non-trivial solutions.

This paper deals with the global asymptotic stability of the zero solution of a certain class of incommensurate nonlinear fractional time-varying systems. By following the fractional comparison approach, two distinct new theorems are developed for the analysis of such class of systems where the state equations are associated with different fractional orders in the range 0–1. Based on these theorems, several new results are put forward for testing the asymptotic stability of such class of systems. Finally, three illustrative examples are provided where it is shown that some proposed asymptotic stability conditions are practically convenient for the analysis and control of such systems.

A century after the first attempts of Wentworth to characterise the shape of cobbles, our understanding of particle morphology is still expanding. A plethora of shape indices has been proposed in the literature to characterise the morphology of individual particles. This study aims to shed light on the merits and limitations of the indices currently used to characterise particle elongation, flatness and compactness, adopting a unified classification framework. Second, new indices are proposed to address the identified shortcomings. Third, a new particle classification system derived from the proposed indices is illustrated. It is shown the new system overcomes the misclassification of a range of particles that are incorrectly classified as bladed in the Zingg system.

Anomaly detection (AD) on hyperspectral images has been widely researched in recent decades due to its high practicability and wide range of application scenarios. Such AD methods derived from low-rank matrix decomposition (LRMD) have appeared rapidly and been applied effectively. However, most of them focused on the use of spectral information and neglected the abundant spatial characteristics. In this letter, a spectral-spatial total variation (TV) (SSTV) regularized low-rank matrix decomposition method with a Schatten 1/2 quasi-norm ( $S_{1/2}$ ) and denoising is proposed. First, to exploit the hyperspectral imagery (HSI) characteristics from the spectral perspective, we propose the low-rank matrix decomposition method with $S_{1/2}$ norm and image denoising modules. Second, we incorporate the SSTV regularization by employing a 2-D TV (TV) spatially and 1-D TV along the spectral dimension to realize the maximized utilization of spatial characteristics of HSI. Finally, the alternating direction multiplier method (ADMM) is brought in the calculating process to attain the consequent detection results. The superiority of the proposed method has been demonstrated by the excellent performance on three real datasets.

This paper is concerned with the stability of traveling waves for a Keller-Segel model with singular chemotactic term and zero chemo-attractant diffusion, where the model and the waves are established to explain the propagation of bacteria pulses along a capillary tube observed in Adler's experiment [1] . By applying the detailed spectral analysis, Evans function method and special transformations and combining with some numerical simulations, in some range of the parameters all the waves are shown to be spectrally stable in some exponentially weighted spaces, and in other range of parameters all the waves are shown to be unstable in any exponentially weighted spaces. The local well-posedness of the classical positive solution to the Cauchy problem of the model is also obtained by applying semigroup argument and some special transformations.

We study three numerical approximations of solutions of nonlocal diffusion evolution problems which are inspired in algorithms for computing the bilateral denoising filtering of an image, and which are based on functional rearrangements and on the Fourier transform. Apart from the usual time-space discretization, these algorithms also use the discretization of the range of the solution (quantization). We show that the discrete approximations converge to the continuous solution in suitable functional spaces, and provide error estimates. Finally, we present some numerical experiments illustrating the performance of the algorithms, specially focusing in the execution time.

Time-decaying harmonic oscillators yield dispersive estimates with weak decay, and change the threshold power of the nonlinearity between the short and the long range. In the non-critical case for the time-decaying harmonic oscillator, this threshold can be characterized by polynomial nonlinearities. However, in the critical case, it is difficult to characterize the threshold using only polynomial terms, and thus we use logarithmic nonlinear terms.

We investigate risk attitudes when the underlying domain of payoffs is finite and the payoffs are, in general, not numerical. In such cases, the traditional notions of absolute risk attitudes, that are designed for convex domains of numerical payoffs, are not applicable. We introduce comparative notions of weak and strong risk attitudes that remain applicable. We examine how they are characterized within the rank-dependent utility model, thus including expected utility as a special case. In particular, we characterize strong comparative risk aversion under rank-dependent utility. This is our main result. From this and other findings, we draw two novel conclusions. First, under expected utility, weak and strong comparative risk aversion are characterized by the same condition over finite domains. By contrast, such is not the case under non-expected utility. Second, under expected utility, weak (respectively: strong) comparative risk aversion is characterized by the same condition when the utility functions have finite range and when they have convex range (alternatively, when the payoffs are numerical and their domain is finite or convex, respectively). By contrast, such is not the case under non-expected utility. Thus, considering comparative risk aversion over finite domains leads to a better understanding of the divide between expected and non-expected utility, more generally, the structural properties of the main models of decision-making under risk.

In addition to the features of the two-parameter Chinese restaurant process (CRP), the restaurant under consideration has a cocktail bar and hence allows for a wider range of (bar and table) occupancy mechanisms. The model depends on three real parameters, $\alpha$ , $\theta_1$ , and $\theta_2$ , fulfilling certain conditions. Results known for the two-parameter CRP are carried over to this model. We study the number of customers at the cocktail bar, the number of customers at each table, and the number of occupied tables after n customers have entered the restaurant. For $\alpha>0$ the number of occupied tables, properly scaled, is asymptotically three-parameter Mittag–Leffler distributed as n tends to infinity. We provide representations for the two- and three-parameter Mittag–Leffler distribution leading to efficient random number generators for these distributions. The proofs draw heavily from methods known for exchangeable random partitions, martingale methods known for generalized Pólya urns, and results known for the two-parameter CRP.

A standard way to move particles in a SMC sampler is to apply several steps of a MCMC (Markov chain Monte Carlo) kernel. Unfortunately, it is not clear how many steps need to be performed for optimal performance. In addition, the output of the intermediate steps are discarded and thus wasted somehow. We propose a new, waste-free SMC algorithm which uses the outputs of all these intermediate MCMC steps as particles. We establish that its output is consistent and asymptotically normal. We use the expression of the asymptotic variance to develop various insights on how to implement the algorithm in practice. We develop in particular a method to estimate, from a single run of the algorithm, the asymptotic variance of any particle estimate. We show empirically, through a range of numerical examples, that waste-free SMC tends to outperform standard SMC samplers, and especially so in situations where the mixing of the considered MCMC kernels decreases across iterations (as in tempering or rare event problems)., revised version

Gα-transform, which is a comprehensive and essential form of Laplace-type integraltransforms, has both advantages and limitations. The purpose of this study is to consider the applicable range of Gα-transform in finding solutions of ordinary differential equations with variable coefficients. Finally, several examples are given to demonstrate the effectiveness of these results.

A multi-resolution quantizer is a sequence of quantizers where the output of a coarser quantizer can be deduced from the output of a finer quantizer. In this paper, we propose an asymptotically scale-invariant multi-resolution quantizer, which performs uniformly across any choice of average quantization step, when the length of the range of input numbers is large. Scale invariance is especially useful in worst case or adversarial settings, ensuring that the performance of the quantizer would not be affected greatly by small changes of storage or error requirements. We also show that the proposed quantizer achieves a tradeoff between rate and error that is arbitrarily close to the optimum., 12 pages, 2 figures. This paper is the extended version of a paper submitted to the IEEE International Symposium on Information Theory 2020

The subtrees and BC-subtrees (subtrees where any two leaves are at even distance apart) have been intensively studied in recent years. Such structures, under special constraints on degrees, have a wide range of applications in many fields. By way of an approach based on generating functions, we present novel recursive algorithms for enumerating various subtrees and BC-subtrees of maximum degree ≤k in trees. The algorithms are explained through detailed examples. We also briefly discuss, in trees, the densities of subtrees (resp. BC-subtrees) with maximum degree ≤k among all subtrees (resp. BC-subtrees). For a tree of order n, the novelly proposed algorithms have multiple advantages. (1) Novel ( k + 2 ) (resp. ( 2 k + 3 ) ) variable generating functions were introduced to construct the algorithms. (2) The proposed algorithms solved the fast enumerating problem of subtree (resp. BC-subtrees) with maximum degree constraint, and also make the subtree (resp. BC-subtrees) enumerating algorithms proposed by Yan and Yeh [1] (resp. Yang et al. [2] ) a special case of ours with k = n − 1 . (3) The time complexity of our algorithm for subtree (resp. BC-subtrees) is O ( k n ) (resp. O ( k n 2 ) ), which is much faster than the O ( n 2 ) (resp. O ( k n 3 ) ) time method based on algorithm proposed in [1] (resp. [2] ).

This paper studies an optimal reinsurance problem under the α-maximin mean-variance criterion proposed in Li et al. (2016) . We generalize ( Li et al., 2016 ) by considering a full range of ambiguity preferences and allowing for general form reinsurance contracts. For equilibrium reinsurance strategies, we find that the excess-of-loss form is unique for ambiguity-averse preferences but may not be optimal or unique for ambiguity-loving preferences. An insurer who is more ambiguous to the reference measure retains less risk if she is ambiguity-averse but does not necessarily retain more risk if she is ambiguity-loving and her ambiguity level is high. Our finding suggests that a highly ambiguity-loving preference may only manifest when the ambiguity level is very low, and hence, consistent with empirical studies, demonstrates that decision makers can be ambiguity-loving if they consider themselves more knowledgeable or competent than the other players.

This paper focuses on constructions of MDS self-dual codes from (extended) generalized Reed-Solomon (GRS) codes. Let $q = r^{2}$ be an odd prime power. We show that, there exists a $q$ -ary self-dual (extended) GRS code for each even length in the range $[{2r,3r-3}]$ , and for each singly even length in the range $[3r-1,4r]$ . This extends the only known consecutive range $[2,2r]$ to $[{2,3r-3}]$ for this case. Furthermore, our general constructions provide many MDS self-dual codes with new parameters which, to the best of our knowledge, were not reported before.

We prove that every term of the lower central series and Johnson filtrations of the Torelli subgroups of the mapping class group and the automorphism group of a free group is finitely generated in a linear stable range. This was originally proved for the second terms by Ershov and He., Comment: 37 pages, 3 figures. Major revision (especially to Section 5), to appear in JEMS

The continuous iteration of an exponential-type function allows to describe a whole range of growths between logarithmical and exponential. This principle has been applied to a class of lifetime di...

We present a unified approach to obtain sharp mean-squared and multiplicative inequalities of Hardy-Littlewood-Polya and Taikov types for multiple closed operators acting on Hilbert space. We apply our results to establish new sharp inequalities for the norms of powers of the Laplace-Beltrami operators on compact Riemannian manifolds and derive the well-known Taikov and Hardy-Littlewood-Polya inequalities for functions defined on the d-dimensional space in the limit case. Other applications include the best approximation of unbounded operators by linear bounded ones and the best approximation of one class by elements of another class. In addition, we establish sharp Solyar type inequalities for unbounded closed operators with closed range.

In this paper, we introduce a new two-dimensional fractional Stockwell transform using the kernel of the coupled fractional Fourier transform. We establish that the two-dimensional fractional Stockwell transform satisfies all the expected properties including Parseval identity and inversion formula. We also characterize the range of the fractional Stockwell transform on $$\mathscr {L}^2(\mathbb {R}^2)$$ and prove a convolution theorem of the transform. Finally, we prove the uncertainty principles for the coupled fractional Fourier transform as well as for the two-dimensional fractional Stockwell transform.

We propose a general formulation of nonconvex and nonsmooth sparse optimization problems with convex set constraint, which can take into account most existing types of nonconvex sparsity-inducing terms, bringing strong applicability to a wide range of applications. We design a general algorithmic framework of iteratively reweighted algorithms for solving the proposed nonconvex and nonsmooth sparse optimization problems, which solves a sequence of weighted convex regularization problems with adaptively updated weights. First-order optimality condition is derived and global convergence results are provided under loose assumptions, making our theoretical results a practical tool for analyzing a family of various reweighted algorithms. The effectiveness and efficiency of our proposed formulation and the algorithms are demonstrated in numerical experiments on various sparse optimization problems.

This paper proposes an advanced numerical-analytical approach for handling a class of fuzzy fractional differential equations involving Caputo-Fabrizio derivative with a non-singular kernel arsing in the medical sector. The solution methodology relies on the reproducing-kernel algorithm to generate analytical solutions in the form of a uniformly convergent series in the direct sum of the desired Hilbert spaces. The effectiveness of the method is analyzed by studying some theoretical, analytical, and stability results of the derived solutions based on the reproducing kernel theory. Numerical simulations are also provided in tables and graphs to demonstrate the reliability of this algorithm in solving fuzzy models using the new Caputo-Fabrizio fractional operator, especially for the drug pharmacokinetic model. The obtained results show the ability of the applied algorithm to solve a wide range of nonlinear fractional models emerging in pharmacology, medicine, and biochemistry.

Equation-free macroscale modelling is a systematic and rigorous computational methodology for efficiently predicting the dynamics of a microscale complex system at a desired macroscale system level. In this scheme, a given microscale model is computed in small patches spread across the space-time domain, with patch coupling conditions bridging the unsimulated space. For accurate predictions, care must be taken in designing the patch coupling conditions. Here we construct novel coupling conditions which preserve self-adjoint symmetry, thus guaranteeing that the macroscale model maintains some important conservation laws of the original microscale model. Consistency of the patch scheme’s macroscale dynamics with the original microscale model is proved for systems in 1D and 2D space, and these proofs immediately extend to higher dimensions. Expanding from a system with a single configuration to an ensemble of configurations establishes that the proven consistency also holds for cases where the microscale periodicity does not integrally fill the patches. This new self-adjoint patch scheme provides an efficient, flexible, and accurate computational homogenisation, as demonstrated here with canonical examples in 1D and 2D space based on heterogenous diffusion, and is applicable to a wide range of multiscale scenarios of interest to scientists and engineers.

Various authors have invoked discretized fractional Brownian motion (fBm) as a model for chain polymers with long-range interaction of monomers along the chain. We show that for these, in contrast to the Brownian case, linear forces are acting between all pairs of constituents, attractive for small Hurst index H and mostly repulsive when H is larger than 1/2. In the second part of this paper, we extend this study to periodic fBm and related models with a view to ring polymers with long range interactions.

In this paper a fairly complete mathematical model of CP-PLL, which reliable enough to serve as a tool for credible analysis of dynamical properties of these circuits, is studied. We refine relevant mathematical definitions of the hold-in and pull-in ranges related to the local and global stability. Stability analysis of the steady state for the charge-pump phase locked loop is non-trivial: straight-forward linearization of available CP-PLL models may lead to incorrect conclusions, because the system is not smooth near the steady state and may experience overload. In this work necessary details for local stability analysis are presented and the hold-in range is computed. An upper estimate of the pull-in range is obtained via the analysis of limit cycles. The study provided an answer to Gardner’s conjecture on the similarity of transient responses of CP-PLL and equivalent classical PLL and to conjectures on the infinite pull-in range of CP-PLL with proportionally-integrating filter.

Let ℋ {\mathcal{ {\mathcal H} }} be a complex Hilbert space and ℬ ( ℋ ) {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}) denotes the algebra of all bounded linear operators acting on ℋ {\mathcal{ {\mathcal H} }} . In this paper, we present some new pairs of generalized finite operators. More precisely, new pairs of operators ( A , B ) ∈ ℬ ( ℋ ) × ℬ ( ℋ ) \left(A,B)\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})\times {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}) satisfying: ∥ A X − X B − I ∥ ≥ 1 , for all X ∈ ℬ ( ℋ ) . \parallel AX-XB-I\parallel \ge 1,\hspace{1.0em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}X\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}). An example under which the class of such operators is not invariant under similarity orbit is given. Range kernel orthogonality of generalized derivation is also studied.

The product of two Gaussian Q -functions is actively used in computing the error probabilities of various digital modulation schemes. Evaluating symbol error probability (SEP) integral involving the product of Gaussian Q -functions over fading distributions is complicated. The closed-form solutions are not available always, therefore to derive these solutions approximations are used. In this paper, an approximation to the product of two Gaussian- Q functions as a sum of exponentials is presented. Further it is used to evaluate the error probabilities of modulation techniques over fading distributions in wide range of scenarios. The knowledge of moment generating function (MGF) of a fading model is sufficient enough to derive the closed-form solution to SEP integrals. Numerical results demonstrate superior accuracy of proposed approximation over other existing competing approximations. Furthermore the proposed solutions are fairly simple as the MGF of fading channels involve fundamental mathematical functions.

In this article, we do a study of common fixed point theorems for six self-maps in FM-Spaces using common limit in range property concerning two pairs of products of two different self-maps. We use the properties (CLRTH) and (CLRSR) along with contractive type implicit relations to prove our results. In support of our result, an example has been provided. Our findings are like those of Kumar and Chouhan [12]. Kumar and Chauhan demonstrated their primary result in [12] by improving and generalizing Aalam, Kumar, and Pants' [1] results. In past, many authors have done study of common fixed point using (E-A) property (like Aalam et. al. [1] proved results using this property), and then these results were improved and generalized by using common (E-A) property as this property is superior to (E-A) property, as the closeness of subspace is required to prove a required result on common fixed point by using these properties, which is a drawback. We improve and generalize all results on these properties using common limit in range property. The goal of this note is to refine and generalize Kumar and Chauhan's [12] results on a common fixed point, as well as some earlier comparable results.

Operational risk is one of the common risks in organizations, especially in banks, which has a wide range of errors in individual performance or system failure with process problems. In this paper, the mathematical model of operational risk to calculate the probability of the organization being convinced is presented, which is in the form of a Volterra integro-differenial equations and is solved by the Adomian decomposition method (ADM). Also, the ADM is applied in one dimension as semi-Adomian decomposition method (s-ADM). Comparison of results demonstrates proficiency of the ADM and s-ADMin this model.

We present a sparse Gauss-Newton solver for accelerated sensitivity analysis with applications to a wide range of equilibrium-constrained optimization problems. Dense Gauss-Newton solvers have shown promising convergence rates for inverse problems, but the cost of assembling and factorizing the associated matrices has so far been a major stumbling block. In this work, we show how the dense Gauss-Newton Hessian can be transformed into an equivalent sparse matrix that can be assembled and factorized much more efficiently. This leads to drastically reduced computation times for many inverse problems, which we demonstrate on a diverse set of examples. We furthermore show links between sensitivity analysis and nonlinear programming approaches based on Lagrange multipliers and prove equivalence under specific assumptions that apply for our problem setting., 10 pages, 7 figures

The purpose of the paper is to rectify a series of errors occurred in (Banerjee and Kundu in Lithuanian Math J 41: 379–392, 2020, Sahoo and Sarkar in Ann Alexandru Ioan Cuza Univ Math 66:81–92, 2020, Yuan et al. in Lithuanian Math J 58:249–262, 2018) for a particular situation. To get a fruitful solution and to overcome the issue, we introduce a new form of set sharing namely restricted set sharing, which is stronger than the usual one. We manipulate the newly introduced notion in this specific section of literature to resolve all the complications. Not only that we have subtly used the same sharing form to a well known unique range set (Frank and Reinders in Complex Var Theory Appl 37:185–193, 1998) to settle a long time unsolved problem.

We provide a general approach to Lipschitz regularity of solutions for a large class of vector-valued, nonautonomous variational problems exhibiting nonuniform ellipticity. The functionals considered here range amongst those with unbalanced polynomial growth conditions to those with fast, exponential type growth. The results obtained are sharp with respect to all the data considered and yield new, optimal regularity criteria even in the classical uniformly elliptic case. We give a classification of different types of nonuniform ellipticity, accordingly identifying suitable conditions to get regularity theorems., Comment: 83 pages

This paper reviews the literature on algorithms for solving bound-constrained mixed-integer derivative-free optimization problems and presents a systematic comparison of available implementations of these algorithms on a large collection of test problems. Thirteen derivative-free optimization solvers are compared using a test set of 267 problems. The testbed includes: (i) pure-integer and mixed-integer problems, and (ii) small, medium, and large problems covering a wide range of characteristics found in applications. We evaluate the solvers according to their ability to find a near-optimal solution, find the best solution among currently available solvers, and improve a given starting point. Computational results show that the ability of all these solvers to obtain good solutions diminishes with increasing problem size, but the solvers evaluated collectively found optimal solutions for 93% of the problems in our test set. The open-source solvers MISO and NOMAD were the best performers among all solvers tested. MISO outperformed all other solvers on large and binary problems, while NOMAD was the best performer on mixed-integer, non-binary discrete, small, and medium-sized problems.

We obtain generalizations of the uniform Sobolev inequalities of Kenig, Ruiz and the fourth author \cite{KRS} for Euclidean spaces and Dos Santos Ferreira, Kenig and Salo \cite{DKS} for compact Riemannian manifolds involving critically singular potentials $V\in L^{n/2}$. We also obtain the analogous improved quasimode estimates of the the first, third and fourth authors \cite{BSS} , Hassell and Tacy \cite{HassellTacy}, the first and fourth author \cite{SBLog}, and Hickman \cite{Hickman} as well as analogues of the improved uniform Sobolev estimates of \cite{BSSY} and \cite{Hickman} involving such potentials. Additionally, on $S^n$, we obtain sharp uniform Sobolev inequalities involving such potentials for the optimal range of exponents, which extend the results of S. Huang and the fourth author \cite{SHSo}. For general Riemannian manifolds we improve the earlier results in \cite{BSS} by obtaining quasimode estimates for a larger (and optimal) range of exponents under the weaker assumption that $V\in L^{n/2}$., Revised version to appear in Revista Matematica Iberoamericana

We provide a graphical treatment of SAT and #SAT on equal footing. Instances of #SAT can be represented as tensor networks in a standard way. These tensor networks are interpreted by diagrams of the ZH-calculus: a system to reason about tensors over C in terms of diagrams built from simple generators, in which computation may be carried out by transformations of diagrams alone. In general, nodes of ZH diagrams take parameters over C which determine the tensor coefficients; for the standard representation of #SAT instances, the coefficients take the value 0 or 1. Then, by choosing the coefficients of a diagram to range over B, we represent the corresponding instance of SAT. Thus, by interpreting a diagram either over the boolean semiring or the complex numbers, we instantiate either the decision or counting version of the problem. We find that for classes known to be in P, such as 2SAT and #XORSAT, the existence of appropriate rewrite rules allows for efficient simplification of the diagram, producing the solution in polynomial time. In contrast, for classes known to be NP-complete, such as 3SAT, or #P-complete, such as #2SAT, the corresponding rewrite rules introduce hyperedges to the diagrams, in numbers which are not easily bounded above by a polynomial. This diagrammatic approach unifies the diagnosis of the complexity of CSPs and #CSPs and shows promise in aiding tensor network contraction-based algorithms., Comment: In Proceedings QPL 2020, arXiv:2109.01534