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Many studies to find solutions for the optimum synthesis problems of linkage mechanisms for path, motion or function generation have appeared in the literature. However, their main focus has been on the development of optimization algorithms or synthesis methods without the handling of the defect problems or only with consideration of the same assembly mode. Hwang and Chen's pioneering work proposed a defect-free optimum synthesis method with constraint equations to eliminate order, circuit and branch defects for Stephenson III six-bar motion generators. However, their proposed constraint equations for the three types of defects are incomplete or not clear enough. In this discussion, we not only examine these faults but also offer the correct and complete constraints to eliminate the three types of defects.

Bu çalışmanınamacı, matematik öğretimine yönelik bir teori olan Semiyotik ArabuluculukTeorisi’nin Türkiye’de tanıtılmasını sağlamak ve uygulanabilirliğine yönelikbir örnek sunmaktır. Teorinin temel kavramlarının ve özelliklerinintanıtılmasının ardından, uygulanabilirliğini tartışmak adına; bir nesne olarakseçilen kâğıdın; kâğıt katlama etkinliklerinde kullanımının semiyotikpotansiyeli analiz edilmiş ve bir simetri eksenine sahip dörtgenlerin(ikizkenar yamuk ve deltoid) inşasında ve kavramsal olarak oluşturulmasındaortaya çıkan işaretlerin analizi yapılmıştır. Çalışma 10. sınıfta okumakta olan35 öğrenci ile gerçekleştirilen bir durum çalışmasıdır. Öğrenciler dörderli vebeşerli gruplara ayrılmıştır. Bu makalede sadece bir grubun çalışması raporlanmıştır.Kâğıt katlamanın semiyotik potansiyelinin analizi sonucunda öğrencilerinçakışma, iz vasıtasıyla simetri, eşlik gibi kavramları zihinlerindecanlandırmasına ve eşkenar dörtgen, kare, yamuk gibi bildikleri dörtgenlerinözellikleri üzerine derin düşünmelerine yardımcı olan bir nesne olduğugörülmüştür. Kâğıt, Semiyotik Arabuluculuk Teorisi’nin bir nesnesi olarakkullanılabilecek özellikleri taşımaktadır. Söz konusu teorinin sınıflardakullanılması için ise öğretmenin yönlendirici rolünün ne denli önemli olduğubelirlenmiştir., The aim of this study is to introduce Theory ofSemiotic Mediation, which is a theory regarding mathematics teaching, in Turkeyand present an example regarding its practice. After describing the maincharacteristics and concepts of Theory of Semiotic Mediation, in order todiscuss the practice of the theory, semiotic potential of a sheet of paper,which was chosen as an artifact of Theory of Semiotic Mediation, was analyzedin paper folding activities and also the signs were analyzed that occurred inthe building and conceptual constructing of quadrilaterals with only onesymmetry axe (isosceles trapezoid and rhombus). The study was a case studywhich was carried out with 35 10th grade students. Students were separated fouror five each. Only one groups’ activity was reported in this paper. As a resultof the analysis of the semiotic potential of it, it has been seen that paperfolding is an artifact that helps students to reconstruct concepts suchsymmetry and equality by means of coincide and crease and helps to think deeplyon the quadrilaterals as rectangle, square, trapezoid and etc. which studentshave already known. A sheet of paper could be used as an artifact of Theory ofSemiotic Mediation. In order to be used the mentioned theory in classes, it wasfounded that teachers’ guidance role was quite essential.

The theoretical models recently used by Chandra [Computational Condensed Matter 27 (2021) e00556] for computing the Gruneisen parameter are not valid at extreme compression but are valid at small compressions. Values of critical pressure for the validity of the Chandra models are about 24 GPa for e -Fe and 3.5 GPa for NaCl. It is emphasized that the third order Gruneisen parameter assumed by Chandra to remain constant for materials under the effect of pressure is not consistent with the recent investigations. We have developed a method for determining the volume dependence of Gruneisen parameter by taking into account the pressure dependence of third order Gruneisen parameter. The results have been obtained for e -Fe and NaCl which differ appreciably at high pressures from those computed by Chandra taking a constant value of third order Gruneisen parameter.

In several recent papers, attempts have been made to apply Wardowski's method of $F$-contractions in order to obtain fixed point results for single and multivalued mappings in $b$-metric spaces. In this article, it is shown that in most cases the conditions imposed on respective mappings are too strong and that the results can be obtained directly, i.e., without using most of the properties of auxiliary function $F$.

We highlight some problems in this paper. They lead to question the model itself.

In the paper the second order half-linear delay differential equation ( r ( t ) | u ' ( t ) | α - 1 u ' ( t ) ) ' + p ( t ) | u ( ? ( t ) ) | α - 1 u ( ? ( t ) ) = 0 , α 1 is studied under the condition ? ∞ r - 1 / α ( t ) d t < ∞ . The oscillation criterion proved by Sun and Meng (2006, Theorem 2.2) is carefully examined and improved by simplifying and sharpening the estimates used in the original proof and removing unnecessary assumptions. Among others, using elementary arguments we show that an arbitrary positive nondecreasing function ? from this criterion can be safely and with no loss of generality replaced by a constant function. An extension to other related equations, such as neutral equations is also provided.

The main result in Y. O. Hamidoune's paper ‘Adding distinct congruence classes' (Combin. Probab. Comput.7 (1998) 81–87) is as follows. If S is a generating subset of a cyclic group G such that 0 ∉ S and |S| ⩾ 5, then the number of sums of the subsets of S is at least min(|G|, 2|S|). Unfortunately, the argument of the author, who, sadly, passed away in 2011, relies on a lemma whose proof is incorrect; in fact, the lemma is false for all cyclic groups of even order. In this short note we point out this mistake, correct the proof, and discuss why the main result is actually true for all finite abelian groups.

On a recent paper Benchohra and Mostefai [2] presented some existence results for an integral equation of fractional order with multiple time delay in Banach spaces. In contrast to the classical case, when assumptions are expressed in terms of the strong topology, they considered another case, namely with the weak topology. It has some consequences for the proof. We present here some comments and corrections.

We revisit two old and apparently little known papers by Basuev [2] [3] and show that the results contained there yield strong improvements on current lower bounds of the convergence radius of the Mayer series for continuous particle systems interacting via a very large class of stable and tempered potentials which includes the Lennard-Jones type potentials. In particular we analyze the case of the classical Lennard-Jones gas under the light of the Basuev scheme and, using also some new results [33] on this model recently obtained by one of us, we provide a new lower bound for the Mayer series convergence radius of the classical Lennard-Jones gas which improves by a factor of the order $10^5$ on the current best lower bound recently obtained in [17]., Comment: Final version as will appear in Comm. Math. Phys

Off‐nucleus isotropic magnetic shielding (σ iso(r)) and multi‐points nucleus independent chemical shift (NICS(0‐2 Å)) index were utilized to find the impacts of the isomerization of gas‐phase furfuraldehyde (FD) on bonding and aromaticity of FD. Multidimensional (1D to 3D) grids of ghost atoms (bqs) were used as local magnetic probes to evaluate σ iso(r) through gauge‐including atomic orbitals (GIAO) at density functional theory (DFT) and B3LYP functional/6‐311+G(d,p) basis set level of theory. 1D σ iso(r) responses along each bond of FD were examined. Also, a σ iso(r) 2D‐scan was performed to obtain σ iso(r) behavior at vertical heights of 0–1 Å above the FD plane in its cis, transition state (TS) and trans forms. New techniques for evaluating 2D σ iso(r) cross‐sections are also included. Our findings showed that bonds in cyclic and acyclic parts of FD are dissimilar. Unlike the C−O bond of furanyl, the C=O bond of the formyl group is magnetically different. C−C and C−H bonds in furanyl are found similar to those in aromatic rings. A unique σ iso(r) trend was observed for the C2−C6 bond during FD isomerization. Based on NICS(0‐2 Å) values, the degree of aromaticity follows the order of cis FD, Isomerization, bonding and aromaticity of furfuraldehyde (FD) have been evaluated through multidimensional off‐nucleus isotropic magnetic shieldings at the DFT‐GIAO/6‐311+G(d,p) level of theory. Significant changes in magnetic behaviour were observed mainly for the rotated C−C bond that linked the furanyl with the formyl group. Other C−C bonds in furanyl behave like C−C bonds of aromatic rings. The transition state FD has more aromaticity than the cis or trans isomers.

Introduction: Refractive surgery procedures, transepithelial photorefractive keratectomy (T- PRK) and femtosecond laser in situ keratomileusis (Fs-LASIK) are regarded as safe and efficacious methods for correcting myopia and myopic astigmatism. These two methods do not have much differences in results when treating spherical myopia, while some differences does exist in treatment of myopic astigmatism. Vector analysis presents powerful tool to show the real differences between these two methods regarding higher order ocular aberrations and central corneal thickness of treated eyes. Aim: The aim of the study is to investigate changes in higher order ocular aberrations (HOAs) and central corneal thickness (CCT) following treatment of myopia and myopic astigmatism above -5.00DS and up to -2.00DC after either T-PRK or Fs-LASIK. Methods: Patients (30 eyes per group) underwent T- PRK (group I) or Fs-LASIK (group II) procedure using Schwind Amaris 750S laser. HOAs (3mm&5mm pupil) and CCT were measured objectively at pre-, 1, 3 & 6 months postop in each case. Results: Key results at 6 months were: i) mean values of trefoil (5mm pupil) were 0.092μm (sd, 0.055, 95% CI 0.072 to 0.112) & 0.126μm (sd, 0.078, 95% CI 0.098 to 0.154) in group I, and 0.088μm (sd, 0.058, 95% CI 0.067 to 0.109) & 0.064μm (sd, 0.034, 95% CI 0.052 to 0.076) in group II (P=0.001 at 6 months) ; ii) Changes in CCT (ΔCCT) and best spherical equivalent correction (ΔBSE) was significant in group II (ΔCCT=-26.55[ΔBSE]-14.06, R=0.486, P=0.006) but not in group I (p=0.034). Conclusions: After T-PRK trefoil is worse than Fs- LASIK. The predictability of corneal changes is better following Fs-LASIK.

The identification of the right order of the equation in applied fractional modeling plays an important role. In this paper we consider an inverse problem for determining the order of time fractional derivative in a subdiffusion equation with an arbitrary second order elliptic differential operator. We prove that the additional information about the solution at a fixed time instant at a monitoring location, as "the observation data", identifies uniquely the order of the fractional derivative., Comment: Version 1

In this short paper, we generalize Ismail–Zhang’s q-2D Hermite polynomials (Trans Am Math Soc 369:6779–6821 (2017), [14]) with an extra parameter and prove that if an analytic function in several variables satisfies a set of partial differential equations of second order, then it can be expanded in terms of the product of the generalized q-2D Hermite polynomials. In addition, we give some generating functions as applications.

In order to solve the Sylvester equations more efficiently, a new four parameters positive and skew-Hermitian splitting (FPPSS) iterative method is proposed in this paper based on the previous research of the positive and skew-Hermitian splitting (PSS) iterative method. We prove that when coefficient matrix A and B satisfy certain conditions, the FPPSS iterative method is convergent in the parameter&rsquo, s value region. The numerical experiment results show that compared with previous iterative method, the FPPSS iterative method is more effective in terms of iteration number IT and runtime.

Motivation Log-rank test is a widely used test that serves to assess the statistical significance of observed differences in survival, when comparing two or more groups. The log-rank test is based on several assumptions that support the validity of the calculations. It is naturally assumed, implicitly, that no errors occur in the labeling of the samples. That is, the mapping between samples and groups is perfectly correct. In this work, we investigate how test results may be affected when considering some errors in the original labeling. Results We introduce and define the uncertainty that arises from labeling errors in log-rank test. In order to deal with this uncertainty, we develop a novel algorithm for efficiently calculating a stability interval around the original log-rank P-value and prove its correctness. We demonstrate our algorithm on several datasets. Availability and implementation We provide a Python implementation, called LoRSI, for calculating the stability interval using our algorithm https://github.com/YakhiniGroup/LoRSI. Supplementary information Supplementary data are available at Bioinformatics online.

Motivation Position-specific probability matrices (PPMs, also called position-specific weight matrices) have been the dominating model for transcription factor (TF)-binding motifs in DNA. There is, however, increasing recent evidence of better performance of higher order models such as Markov models of order one, also called adjacent dinucleotide matrices (ADMs). ADMs can model dependencies between adjacent nucleotides, unlike PPMs. A modeling technique and software tool that would estimate such models simultaneously both for monomers and their dimers have been missing. Results We present an ADM-based mixture model for monomeric and dimeric TF-binding motifs and an expectation maximization algorithm MODER2 for learning such models from training data and seeds. The model is a mixture that includes monomers and dimers, built from the monomers, with a description of the dimeric structure (spacing, orientation). The technique is modular, meaning that the co-operative effect of dimerization is made explicit by evaluating the difference between expected and observed models. The model is validated using HT-SELEX and generated datasets, and by comparing to some earlier PPM and ADM techniques. The ADM models explain data slightly better than PPM models for 314 tested TFs (or their DNA-binding domains) from four families (bHLH, bZIP, ETS and Homeodomain), the ADM mixture models by MODER2 being the best on average. Availability and implementation Software implementation is available from https://github.com/jttoivon/moder2. Supplementary information Supplementary data are available at Bioinformatics online.

This paper applies a powerful scheme, namely Bernoulli sub-equation function method, to some partial differential equations with high non-linearity. Many new travelling wave solutions, such as mixed dark-bright soliton, exponential and complex domain, are reported. Under a suitable choice of the values of parameters, wave behaviours of the results obtained in the paper – in terms of 2D, 3D and contour surfaces – are observed.

The current paper deals with the transmission of MERS-CoV model between the humans populace and the camels, which are suspected to be the primary source for the infection. The effect of time MERS-CoV disease transmission is explored using a non-linear fractional order in the sense of Caputo operator in this paper. The considered model is analyzed for the qualitative theory, uniqueness of the solution are discussed by using the Banach contraction principle. Stability analysis is investigated by the aid of Ulam-Hyres (UH) and its generalized version. Finally, we show the numerical results with the help of generalized Adams-Bashforth-Moulton Method (GABMM) are used for the proposed model, for supporting our analytical work.

In this paper, we derive the explicit analytical solution of incommensurate fractional differential equation systems with fractional order $ 1 < \alpha, \beta < 2 $. The derivation is extended from a recently published paper by Huseynov et al. in [1], which is limited for incommensurate fractional order $ 0 < \alpha, \beta < 1 $. The incommensurate fractional differential equation systems were first converted to Volterra integral equations. Then, the Mittag-Leffler function and Picard's successive approximations were used to obtain the analytical solution of incommensurate fractional order systems with $ 1 < \alpha, \beta < 2 $. The solution will be simplified via some combinatorial concepts and bivariate Mittag-Leffler function. Some special cases will be discussed, while some examples will be given at the end of this paper.

This paper deals with the problem of characterizing all functions f : [ 0 , 1 ] → R + such that C f ( x , y ) = x y f ( ( 1 − x ) ( 1 − y ) ) , x , y ∈ [ 0 , 1 ] is a bivariate copula. We provide a complete characterization for the two cases: (i) when C f is, in addition, totally positive of order 2 (TP2) and (ii) when f is twice continuously differentiable. In general, the function f need only be twice differentiable Lebesgue almost everywhere as shown by investigating necessary conditions for C f to be a copula. The paper also contains numerous examples illustrating obtained results and connections to known facts from the literature. Moreover, several properties of such copulas are described.

This paper deals with stability in the Liapunov sense of the 4-th order quasilinear differential equations with quasiderivatives. An asymptotic stability criterion is derived. An illustrative example is added.

It was shown by Mukai that the maximum order of a finite group acting faithfully and symplectically on a K3 surface is $960$ and that the group is isomorphic to the group $M\_{20}$. Then Kondo showed that the maximum order of a finite group acting faithfully on a K3 surface is $3\,840$ and this group contains the Mathieu group $M\_{20}$ with index four. Kondo also showed that there is a unique K3 surface on which this group acts faithfully, which is the Kummer surface $\Km(E\_i\times E\_i)$. In this paper we describe two more K3 surfaces admitting a big finite automorphism group of order $1\,920$, both groups contains $M\_{20}$ as a subgroup of index 2. We show moreover that these two groups and the two K3 surfaces are unique. This result was shown independently by S. Brandhorst and K. Hashimoto in a forthcoming paper, with the aim of classifying all the finite groups acting faithfully on K3 surfaces with maximal symplectic part., 15 pages

Suppose an oracle knows a string $S$ that is unknown to us and that we want to determine. The oracle can answer queries of the form "Is $s$ a substring of $S$?". In 1995, Skiena and Sundaram showed that, in the worst case, any algorithm needs to ask the oracle $\sigma n/4 -O(n)$ queries in order to be able to reconstruct the hidden string, where $\sigma$ is the size of the alphabet of $S$ and $n$ its length, and gave an algorithm that spends $(\sigma-1)n+O(\sigma \sqrt{n})$ queries to reconstruct $S$. The main contribution of our paper is to improve the above upper-bound in the context where the string is compressible. We first present a universal algorithm that, given a (computable) compressor that compresses the string to $\tau$ bits, performs $q=O(\tau)$ substring queries; this algorithm, however, runs in exponential time. For this reason, the second part of the paper focuses on more time-efficient algorithms whose number of queries is bounded by specific compressibility measures. We first show that any string of length $n$ over an integer alphabet of size $\sigma$ with $rle$ runs can be reconstructed with $q=O(rle (\sigma + \log \frac{n}{rle}))$ substring queries in linear time and space. We then present an algorithm that spends $q \in O(\sigma g\log n)$ substring queries and runs in $O(n(\log n + \log \sigma)+ q)$ time using linear space, where $g$ is the size of a smallest straight-line program generating the string., Comment: Accepted for publication in Theoretical Computer Science

In this paper, we carry out in an abstract order context some real subset combinatorial problems. Specifically, let [Formula: see text] be a finite poset, where [Formula: see text] is an order-reversing and involutive map such that [Formula: see text] for each [Formula: see text]. Let [Formula: see text] be the Boolean lattice with two elements and [Formula: see text] the family of all the order-preserving 2-valued maps [Formula: see text] such that [Formula: see text] if [Formula: see text] for all [Formula: see text]. In this paper, we build a family [Formula: see text] of particular subsets of [Formula: see text], that we call [Formula: see text]-bases on [Formula: see text], and we determine a bijection between the family [Formula: see text] and the family [Formula: see text]. In such a bijection, a [Formula: see text]-basis [Formula: see text] on [Formula: see text] corresponds to a map [Formula: see text] whose restriction of [Formula: see text] to [Formula: see text] is the smallest 2-valued partial map on [Formula: see text] which has [Formula: see text] as its unique extension in [Formula: see text]. Next we show how each [Formula: see text]-basis on [Formula: see text] becomes, in a particular context, a sub-system of a larger system of linear inequalities, whose compatibility implies the compatibility of the whole system.

In a prior paper (Boukerrou et al. IACR Trans. Symmetric Cryptol. 2020(1), 331–362 2020), Boukerrou et al. introduced the Feistel Boomerang Connectivity Table (FBCT). FBCT is an important cryptanalytic technique on Feistel ciphers. In fact, the coefficients of FBCT are actually related to the second-order zero differential spectra of functions in even characteristic. In this paper, we push further the study initiated in Boukerrou et al. (IACR Trans. Symmetric Cryptol. 2020(1), 331–362 2020). Almost perfect nonlinear (APN) functions and the inverse function are interesting in cryptography and coding theory. In Boukerrou et al. (IACR Trans. Symmetric Cryptol. 2020(1), 331–362 2020), Boukerrou et al. determined the second-order zero differential spectra of APN functions and the inverse function in even characteristic. In order to derive further cryptographic properties of APN functions and the inverse function in odd characteristic, we calculate the second-order zero differential spectra of some APN functions and the inverse function in odd characteristic. In addition, these APN functions and the inverse function have low second-order zero differential uniformity.

Under consideration in this paper are two nonlinear evolution models: One is the (1 + 1)-dimensional generalized Broer–Kaup (gBK) system derived by Zhang et al. (Appl Math Comput 219:5837–5848, 2013), and the other is the (2 + 1)-dimensional gBK system reported for the first time. Based on the bilinear forms given in this paper, novel N-soliton solutions of these two gBK systems are obtained by using Hirota’s bilinear method. As a comparison, the obtained two-soliton solutions of the (1 + 1)-dimensional gBK system are taken to demonstrate the difference from the known ones constructed by Darboux transformation. In order to understand the nonlinear dynamics localized in the gBK systems, local structures of the obtained one-, two-, three- and four-soliton solutions are shown. This paper reveals that each pair of the obtained N-soliton solutions of the gBK systems couple bell and kink soliton dynamics.

In this paper, in the light of weighted sharing of sets, we investigate the possible uniqueness of meromorphic function of restricted hyper order with its linear c-shift operator. Our first two theorems improve a number of earlier results. Our last theorem together with a corollary improves and extends a result due to Li, Lu and Xu [14]. Most importantly, our another corollary deducted from the last theorem not only provides an answer to an open question posed by Liu [16] but also noticeably improves two results of Chen and Chen [4]. A number of examples have been exhibited by us pertinent with the content of the paper. At the penultimate section which is also the application part of our result, we extend a recent result of Liu, Ma and Zhai [17]. Finally, presenting two examples, we conjecture that one of our result may hold for a larger class of functions and we place it as an open question for future investigations.

We study the Steiner k-eccentricity on trees, which generalizes the previous one in the paper [On the average Steiner 3-eccentricity of trees, arXiv:2005.10319 ]. We achieve much stronger properties for the Steiner k-ecc tree than that in the previous paper. Based on this, a linear time algorithm is devised to calculate the Steiner k-eccentricity of a vertex in a tree. On the other hand, lower and upper bounds of the average Steiner k-eccentricity index of a tree on order n are established based on a novel technique which is quite different and also much easier to follow than the earlier one.

This paper is devoted to exact (in the sense of the order of smallness) estimates of the best rational approximations of functions with derivative of generalized finite variation on a finite segment of a straight line in uniform and integral metrics. The obtained results were announced in the authors' paper in 2014. They are analogous to the results of the first author, where A. Khatamov establishes exact (in the sense of the order of smallness) estimates of the best spline approximations of functions with derivative of generalized finite variation on a finite segment of a straight line in uniform and integral metrics. Results announced by the authors in 2014 generalize those obtained by N.Sh. Zagirov in 1982, namely, exact (in the sense of the order of smallness) estimates of rational approximations of functions with generalized finite variation in the integral metric, to the best rational approximations of functions with derivative of generalized finite variation on a finite segment in uniform and integral metrics. Generally speaking, the calculation of exact (in the sense of the order of smallness) estimates for the best approximations for any class of functions in any metric is a difficult problem.

PurposeMulti-term variable-order fractional differential equations (VO-FDEs) are powerful tools in accurate modeling of transient-regime real-life problems such as diffusion phenomena and nonlinear viscoelasticity. In this paper the Chebyshev polynomials of the fourth kind is employed to obtain a numerical solution for those multi-term VO-FDEs.Design/methodology/approachTo this end, operational matrices for the approximation of the VO-FDEs are obtained using the Fourth kind Chebyshev Wavelets (FKCW). Thus, the VO-FDE is condensed into an algebraic equation system. The solution of the system of those equations yields a coefficient vector, the coefficient vector in turn yields the approximate solution.FindingsSeveral examples that we present at the end of the paper emphasize the efficacy and preciseness of the proposed method.Originality/valueThe value of the paper stems from the exploitation of FKCWs for the numerical solution of multi-term VO-FDEs. The method produces accurate results even for relatively small collocation points. What is more, FKCW method provides a compact mapping between multi-term VO-FDEs and a system of algebraic equations given in vector-matrix form.

In the paper mentioned in the title, it is proved the boundedness of the Riesz potential operator of variable order 𝛼(x) from variable exponent Morrey space to variable exponent Campanato space, under certain assumptions on the variable exponents p(x) and 𝜆(x) of the Morrey space. Assumptions on the exponents were different depending on whether 𝛼(x)p(x)−n+𝜆(x) p(x) takes or not the critical values 0 or 1. In this note, we improve those results by unifying all the cases and covering the whole range 0 ⩽ 𝛼(x)p(x)−n+𝜆(x) p(x) ⩽ 1. We also provide a correction to some minor technicality in the proof of Theorem 2 in the aforementioned paper. info:eu-repo/semantics/publishedVersion

Up to now, the problem of synchronization of mutual coupled integer order systems has been tackled by several researchers. However, few works have been done to investigate such context on fractional order systems. In this context, this paper presents a complete methodology to solve this problem. On the other hand, to the best of our knowledge, among all the existing works dealing with the synchronization of mutual coupled fractional order systems, no paper has treated the special class of one-sided Lipschitz systems. In order to validate the theoretical results, two numerical examples are studied in the simulation section.

PurposeThe paper aims to build the relationship between an entire function of restricted hyper-order with its linear c-shift operator.Design/methodology/approachStandard methodology for papers in difference and shift operators and value distribution theory have been used.FindingsThe relation between an entire function of restricted hyper-order with its linear c-shift operator was found under the periphery of sharing a set of two small functions IM (ignoring multiplicities) when exponent of convergence of zeros is strictly less than its order. This research work is an improvement and extension of two previous papers.Originality/valueThis is an original research work.

In this paper we study a generalisation of the Igusa-Todorov functions which gives rise to a vast class of algebras satisfying the finitistic dimension conjecture. This class of algebras is called Lat-Igusa-Todorov and includes, among others, the Igusa-Todorov algebras (defined by J. Wei) and the self-injective algebras which in general are not Igusa-Todorov algebras. Finally, some applications of the developed theory are given in order to relate the different homological dimensions which have been discussed through the paper., 18 pages, submitted to a peer-reviewed journal

In this paper we study the maximal possible difference N of values of a quasi-copula at two different points of the unit square. This study enables us to give upper and lower bounds, called constrained bounds, for quasi-copulas with fixed value at a given point in the unit square, thus extending an earlier result from copulas to quasi-copulas. It turns out that the two bounds are actually copulas. Difference N is also the main tool in exhibiting two new characterizations of quasi-copulas, a major result of this paper, which sheds new light on the subject of copulas as well. Significant applications of our results are also given in the imprecise probability theory, one of the more important non-standard approaches to probability. After a full-scale bivariate Sklar's theorem has been proven under this approach, we want to establish the tightness of its background before moving to the more general multivariate scene. We present an extension of the partial order on quasi-distributions used in the said theorem, i.e., pointwise order with fixed margins, using again the difference N as a main tool. A careful study of the interplay between the order on quasi-distributions and the order on corresponding quasi-copulas that represent them is also given. Due to a recent result that the quasi-copulas obtained via Sklar's theorem in the imprecise setting are exactly the same as the ones in the standard setting, it is not surprising that results on quasi-copulas can shed some light both on open questions in the standard probability theory and in the imprecise probability theory at the same time.

Riordan matrices are infinite lower triangular matrices corresponding to the certain operators in the space of formal power series. The nth descending diagonal of the ordinary Riordan matrix and the nth descending diagonal of the exponential Riordan matrix have the generating functions respectively g n ( φ x ) / ( 1 − φ x ) n + 1 and h n ( φ x ) / ( 1 − φ x ) 2 n + 1 , where g n ( x ) , h n ( x ) are polynomials of degree ≤n. We will call these polynomials the numerator polynomials of Riordan matrices. General properties of these polynomials were considered in separate paper. In this paper, we will consider numerator polynomials of the Riordan matrices associated with the family of series a ( β ) ( x ) = a ( x ( β ) a β ( x ) ) . The matrices of transformations, in which these polynomials participate, have the form A n E n β A n − 1 , where A n is the certain matrix of order n + 1 , E is the matrix of the shift operator. The main focus is on studying the properties of these matrices.

Let be a connected graph with vertices set and edges set . The ordinary distance between any two vertices of is a mapping from into a nonnegative integer number such that is the length of a shortest path. The maximum distance between two subsets and of is the maximum distance between any two vertices and such that belong to and belong to . In this paper, we take a special case of maximum distance when consists of one vertex and consists of vertices, . This distance is defined by: where is the order of a graph . In this paper, we defined – polynomials based on the maximum distance between a vertex in and a subset that has vertices of a vertex set of and – index. Also, we find polynomials for some special graphs, such as: complete, complete bipartite, star, wheel, and fan graphs, in addition to polynomials of path, cycle, and Jahangir graphs. Then we determine the indices of these distances.

In this paper, we consider the Schnakenberg model with heterogeneity on the interval ( − 1 , 1 ) . We first construct stationary solutions which concentrate at a suitable point by using the Liapunov-Schmidt reduction method. Moreover, by investigating the associated linearized eigenvalue problem, we establish the linear stability of the solutions above. Iron, Wei, and Winter (2004) established the existence and stability of multi-peak symmetric stationary solutions in non-heterogeneity case. In their work, the one-peak solution is always stable. For the symmetric heterogeneity case, Ishii and Kurata (2019) gave the analysis of one-peak symmetric solutions in details and revealed a destabilization effect of the heterogeneity. In this paper, we reveal that the mechanism which the location of the concentration point and the stability with respect to eigenvalues of order o ( 1 ) are determined by the interaction of the heterogeneity with the associated Green's function. In particular, we not suppose that the heterogeneity is symmetric. Also, by a typical example, we performed several numerical simulations to illustrate our main results.

In this paper, we study the forced oscillatory theory for higher order fractional differential equations with damping term via $\Psi$-Hilfer fractional derivative. We get sufficient conditions which ensure the oscillation of all solutions and give an illustrative example for our results. The $\Psi$-Hilfer fractional derivative according to the choice of the $\Psi$ function is a generalization of the different fractional derivatives defined earlier. The results obtained in this paper are a generalization of the known results in the literature, and present new results for some fractional derivatives.

In the present paper, we consider the linear wave equation with the scale-invariant damping and mass. It is known that the global behavior of the solution depends on the size of the coefficients in front of the damping and mass at initial time t = 0 t=0 . Indeed, the solution satisfies the similar decay estimate to that of the corresponding heat equation if it is large and to that of the modified wave equation if it is small. In our previous paper, we obtained the scattering result and its asymptotic order for the data in the energy space H 1 × L 2 H^1\times L^2 when the coefficients are in the wave regime. In fact, the threshold of the coefficients relies on the spatial decay of the initial data. Namely, it varies depending on r r when the initial data is in L r L^r ( 1 ≤ r > 2 1\leq r > 2 ). In the present paper, we will show the scattering result and the asymptotic order in the wave regime for L r L^r -data, which is wider than the wave regime for the data in the energy space. Moreover, we give an improvement of the asymptotic order obtained in our previous paper for the data in the energy space.

In this paper, our aim is to consider the new class of log-convex fuzzy-interval-valued function known as log-s-convex fuzzy-interval-valued functions (log-s-convex fuzzy-IVFs). By this concept, we have introduced Hermite–Hadamard inequalities (HH-inequalities) by means of fuzzy order relation. This fuzzy order relation is defined level-wise through Kulisch–Miranker order relation defined on interval space. Moreover, some new Hermite–Hadamard–Fejér inequalities (HH–Fejér-inequalities) and Jensen’s inequalities via log-s-convex fuzzy-IVFs are also established and verified with the support of useful examples. Some special cases are also discussed which can be viewed as applications of fuzzy-interval HH-inequalities. The concepts and approaches of this paper may be the starting point for further research in this area.

The importance of industrial control networks security is growing, but the intrusion detection research of industrial control networks is seriously restricted by the existing attack samples of the business dataset, especially the quantity and quality. In order to solve the problem of the scarcity of attack industrial control datasets, this paper proposes an attack sample generation algorithm. Firstly, based on the weight and degree of membership distribution, calculate the value of membership distance between dimensions, and the data association is strong when the membership distance of dimensions is small. Then, divide dimensions which have small distance into a group, so as to realize the association grouping of the original data. The data association of dimensions in an association group is strong when the association group appears frequently. According to the frequency of the association group, all the association groups are divided into strong association group and weak association group. Attack all the dimensions of one strong association group in the original data by false data injection attack, realized attack sample generation algorithm in the original data. Finally, expand the attack sample into a large amount of attack sample industrial control dataset by the Generative Adversarial Network. In this paper, the attack samples are generated by the BATADAL dataset and the business dataset of an oil depot, and the data is expanded by 100 times through the algorithm. Compared with the attack samples provided by the BATADAL dataset, the coincidence degree and fitting degree of generated data is improved by 38.20%–42.94% and 98.22%–98.36%, respectively. The classification results of XGBoost and SVM are 100% and 98.01%, which is close to the classification result of attack samples provided by BATADAL dataset.

In this paper, we investigate convex semigroups on Banach lattices with order continuous norm, having $L^p$-spaces in mind as a typical application. We show that the basic results from linear $C_0$-semigroup theory extend to the convex case. We prove that the generator of a convex $C_0$-semigroup is closed and uniquely determines the semigroup whenever the domain is dense. Moreover, the domain of the generator is invariant under the semigroup; a result that leads to the well-posedness of the related Cauchy problem. In a last step, we provide conditions for the existence and strong continuity of semigroup envelopes for families of $C_0$-semigroups. The results are discussed in several examples such as semilinear heat equations and nonlinear integro-differential equations., The manuscript has been split into two parts. The second part of the paper can be found under arXiv:2010.04594. 24 pages

The paper focuses on the new Multipoint meshless finite difference method, following the original Collatz higher order multipoint concept and the essential ideas of the Meshless FDM. The method was formulated, developed, and tested for various boundary value problems. Generalization of the multipoint method application to nonlinear analysis is the purpose of this research. The first attempt of the multipoint technique application to the geometrically nonlinear problems was successfully done recently. The case of physically nonlinear problem is considered in this paper. Several benefits of the proposed approach are highlighted, numerical algorithm and selected results are presented, and application of the multipoint method to nonlinear analysis is summarized.

Any process of technical product development may involve dependability testing. If in the course of operation, the recovery of an entity after a failure is the norm, then test plans of types NRect, NRecR, NNoRect и NNoRecR are normally used, where N is the number of tested same-type entities; t is the testing time of each of the N entities; R is the number of failures; Rec (NoRec) is the characteristic of the plan that indicates that the entity’s operability after each failure within the testing time is recovered (not recovered). Normally, NRect and NRecR indicate that, in the process of testing, failures are recovered immediately. In order not to confuse plans NRect, NRecR, NNoRect and NNoRecR with test plans with long recovery times, let us denote the latter as NRec!t, NRec!R, NNoRec!t and NNoRec!R respectively. Let us simplify the problem description and require, for test plans of types NRec!t, NRec!R, NNoRec!t and NNoRec!R, the fulfilment of condition D = R, where D is the number of recoveries, i.e. after the conclusion of testing, at the moment of time t, the recovery of entities continues until the last of R failed entities is recovered. We will denote such test plans NRec!t(D=R), NRec!R(D=R), NNoRec!t(D=R) and NNoRec!R(D=R). As the dependability model, an exponential distribution is adopted. Steady-state availability is normally defined as the composite dependability indicator of recoverable entities. Finding efficient estimates is one of the primary goals of the dependability theory. Since the 1960s, Russian scientific literature has featured next to no research dedicated to the properties of steady-state availability estimates. The best known work in the steady-state availability estimates for a NRecR test plan is in the book: Beletsky B.R. [Dependability theory of radio engineering systems (mathematical foundations). Study guide for colleges]. Moscow: Sovetskoye radio; 1978. This paper makes up for this deficiency. In order to identify the efficient steady-state availability estimate out of infinite many, first, an estimate efficiency comparison criterion is to be constructed. The paper Aims to construct a simple criterion of steady-state availability estimation for test plans with long recovery times and identify the efficient estimate out of the available ones using the constructed criterion. Methods of research. The efficient estimate was found using integral numerical characteristics of the accuracy of estimate, i.e., the sum square of the displacement of the expected realization of an estimate from the considered parameters of the distribution laws. Conclusions. The authors constructed simple criteria of efficiency of steady-state availability estimation for test plans with long recovery time (case of N≥1). Estimate G 3 =(1+VR/S(R1)) -1 is bias-efficient out of those available for test plans of types NRec!t(D=R) and NNoRec!t(D=R). Conventional estimate G 1 =(1+ V/S) -1 is bias-efficient out of those available for test plans of types NRec!R(D=R) and NNoRec!R(D=R).

Coronavirus Disease 2019 has become global pandemic in the world. Since its appearance, many researchers in world try to understand the disease, including mathematics researchers. In mathematics, many approaches are developed to study the disease. One of them is to understand the spreading of the disease by constructing an epidemiology model. In this approach, a system of differential equations is formed to understand the spread of the disease from a population. This is achieved by using the SIR model to solve the system, two numerical methods are used, namely Euler Method and 4th order Runge-Kutta. In this paper, we study the performance and comparison of both methods in solving the model. The result in this paper that in the running process of solving it turns out that using the euler method is faster than using the 4th order Runge-Kutta method and the differences of solutions between the two methods are large.

In this paper, several oscillation criteria for a class of higher order dynamic equations with superlinear neutral term are established. The proposed results provide a unified platform that adequately covers both discrete and continuous equations and further sufficiently comments on oscillatory behavior of more general class of equations than the ones reported in the literature. We conclude the paper by demonstrating illustrative examples.