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Soil salinization is an important stress factor that limits plant growth and yield. Increased salinization is projected to affect more than 50% of all arable land by 2050. In addition, the growing demand for food, together with the increase in the world population, forces the need to seek salt-tolerant crops. Quinoa (Chenopodium quinoa Willd.) is an Andean crop of high importance, due to its nutritional characteristics and high tolerance to different abiotic stresses. The aim of this work is to determine the physiological, anatomical, and biochemical salt-tolerance mechanisms of a salt-tolerant (Vikinga) and a salt-sensitive (Regalona) quinoa variety. Plants were subjected to salinity stress for 15 days, starting at 100 mM NaCl until progressively reaching 400 mM NaCl. Physiological, anatomical, and biochemical parameters including growth, chlorophyll content, quantum yield of PSII (ϕPSII), gas exchange, stomatal density, size, and lipid peroxidation (via malondialdehyde, MDA) were measured. Results show that chlorophyll content, ϕPSII, and MDA were not significantly reduced under saline stress in both varieties. The most stress-affected process was the CO2 net assimilation, with an up to 60% reduction in both varieties, yet Vikinga produced higher dry weight than Regalona due to the number of leaves. The stomatal densities increased under salinity for both varieties, with Regalona the one showing higher values. The averaged stomatal size was also reduced under salinity in both varieties. The capacity of Vikinga to generate higher dry weight is a function of the capacity to generate greater amounts of leaves and roots in any condition. The stomatal control is a key mechanism in quinoa's salinity tolerance, acquiring higher densities with smaller sizes for efficient management of water loss and carbon assimilation. These findings highlight the potential of Vikinga for cultivation in temperate salinized environments during winter, such as Deltas and lowlands where rice is grown during summer. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new family of Lucas quaternions by using finite operators. We call these quaternions as Lucas finite operator quaternions. We give some properties and identities of Lucas finite operator quaternions such as Binet-like formula, generating function, exponential generating function, Catalan's identity, Cassini's identity, d'Ocagne's identity and many binomial-sum identities. As an application, we generate Cassini's identity in another form by matrix representations. [ABSTRACT FROM AUTHOR]

Modified third-order Jacobsthal quaternion sequence is defined in this study. Some properties involving this sequence, including the Binet-style formula and the generating function are presented. [ABSTRACT FROM AUTHOR]

This paper presents exact derivations of the first two moments of the total order picking time in a warehouse for three routing heuristics, under the assumption of random storage. The analysis is done for general order size distributions and provides exact closed expressions in terms of the probability generating function of the order size distribution. We also indicate how the methods and insights in this paper extend to different storage policies and multi-block warehouses. The exact results derived in this paper are used to investigate effects of routing heuristics, order size distributions and layouts on warehouse efficiency. As illustration, we model the warehouse as a queueing system. By using approximations of the average order-lead time in terms of the first two moments of the order picking time, we are able to find optimal warehouse layouts and batch pick sizes. [ABSTRACT FROM AUTHOR]

Transmission matrix measurements of multimode fibers are now routinely performed in numerous laboratories, enabling control of the electric field at the distal end of the fiber and paving the way for the potential application to ultrathin medical endoscopes with high resolution. The same concepts are applicable to other areas, such as space division multiplexing, targeted power delivery, fiber laser performance, and the general study of the mode coupling properties of the fiber. However, the process of building an experimental setup and developing the supporting code to measure the fiber's transmission matrix remains challenging and time consuming, with full details on experimental design, data collection, and supporting algorithms spread over multiple papers or lacking in detail. Here, we outline a complete and self-contained description of the specific experiment we use to measure fully polarization-resolved transmission matrices, which enable full control of the electric field, in contrast to the more common scalar setups. Our exact implementation of the full polarization experiment is new and is easy to align while providing flexibility to switch between full-polarization and scalar measurements if desired. We utilize a spatial light modulator to measure the transmission matrix using linear phase gratings to generate the basis functions and measure the distal electric field using phase-shifting interferometry with an independent reference beam derived from the same laser. We introduce a new method to measure and account for the phase and amplitude drift during the measurement using a Levenberg–Marquardt nonlinear fitting algorithm. Finally, we describe creating distal images through the multimode fiber using phase-to-amplitude shaping techniques to construct the correct input electric field through a superposition of the basis functions with the phase-only spatial light modulator. We show that results are insensitive to the choice of phase-to-amplitude shaping technique as quantified by measuring the contrast of a razor blade at the distal end of the fiber, indicating that the simplest but most power efficient method may be the best choice for many applications. We also discuss some of the possible variations on the setup and techniques presented here and highlight the details that we have found key in achieving high fidelity distal control. Throughout the paper, we discuss applications of our setup and measurement process to a variety of specialty fibers, including fibers with harsh environment coatings, coreless fibers, rectangular core fibers, pedestal fibers, and a pump–signal combiner based on a tapered fiber bundle. This demonstrates the usefulness of these techniques across a variety of application areas and shows the flexibility of our setup in studying various fiber types. [ABSTRACT FROM AUTHOR]

Quantum N -toroidal algebras are generalization of the quantum toroidal algebras. In this paper, we construct two types of level-one vertex representations of the quantum N -toroidal algebras in type B , which also realize the quantum toroidal algebras of type B as special cases. [ABSTRACT FROM AUTHOR]

The expected spacing, or average difference between consecutive order statistics, is known for uniform and exponential random variates. For other distributions, we can estimate it using the derivative of the inverse cumulative density (quantile) function, since passing a uniformly drawn value, whose spacing we know, through this function generates a random value from the distribution, and the difference between two such uniform values approximates the derivative. We calculate the spacing for two new distributions, the logistic and Gumbel, and show the estimator is exact for the first and approximate for the second. Comparing the estimators for six other distributions to numeric simulations shows they are also approximations, best in the middle of the order statistics with an error that goes inversely with the square of the sample size, but degrading in the tails. [ABSTRACT FROM AUTHOR]

In this research paper, we introduce a new three-parameter model called Marshall-Olkin Extended Gamma Lindley, which provides greater flexibility for modeling lifetime data. We investigate some of its structural properties, including moments and moment generating function. Furthermore, we prove that this model is identifiable. We corroborate that the Marshall-Olkin Extended Gamma Lindley distribution is characterized by its truncated moments of order statistics. Relying on this characterization and the least squares, we set forward a new method in order to estimate the unknown parameters of the proposed model. A comparative study is conducted between this approach and different classical methods of estimation. A real data set is applied to show the flexibility of the suggested model against other models. [ABSTRACT FROM AUTHOR]

Assume that $Y$ Y is a random variable whose moment generating function exists in a neighbourhood of the origin. In this paper, we study the probabilistic degenerate central Bell polynomials associated with $Y$ Y , as probabilistic extension of the degenerate central Bell polynomials. In addition, we investigate the probabilistic degenerate central factorial numbers of the second kind associated with $Y$ Y and the probabilistic degenerate central Fubini polynomials associated with $Y$ Y. The aim of this paper is to derive some properties, explicit expressions, certain identities and recurrence relations for those polynomials and numbers. [ABSTRACT FROM AUTHOR]

Assume that is Y a random variable whose moment generating function exists in a neighbourhood of the origin. The aim of this paper is to study probabilistic poly-Bernoulli numbers associated with Y, as probabilistic extensions of poly-Bernoulli numbers. We derive explicit expressions, some related identities and a symmetric relation for those numbers. We also investigate explicit expressions for the modified probabilisitc Bernoulli numbers associated with Y, which are slightly different from probabilisitic Bernoulli numbers associated with Y. As special cases of Y, we treat the Poisson, gamma and Bernoulli random variables. [ABSTRACT FROM AUTHOR]

In this paper, we delve into the intricate properties of degenerate Poisson random variables, exploring their moment generating function, the law of large numbers, and the central limit theorem. Our study into the law of large numbers unveils the convergence patterns of degenerate Poisson random variables, enlightening the stability and predictability of their expected values and variances. Further enriching our study, we extend our focus to the central limit theorem, unravelling the interesting results that emerge as the sample size approaches infinity. [ABSTRACT FROM AUTHOR]

This study introduces a new approach to the development of generalized 1-parameter, 2-variable Hermite–Frobenius–Euler polynomials, which are characterized by their generating functions, series definitions and summation formulae. Additionally, the research utilizes a factorization method to establish recurrence relations, shift operators and various differential equations, including differential, integro-differential and partial differential equations. The framework elucidates the fundamental properties of these polynomials by utilizing generating functions, series definitions and summation formulae. The results of the study contribute to the understanding of the properties of these polynomials and their potential applications. [ABSTRACT FROM AUTHOR]

Recently, many authors have studied degenerate Bernoulli and degenerate Euler polynomials. Let $Y$ Y be a random variable whose moment generating function exists in a neighbourhood of the origin. The aim of this paper is to introduce and study the probabilistic extension of degenerate Bernoulli and degenerate Euler polynomials, namely the probabilistic degenerate Bernoulli polynomials associated with $Y$ Y and the probabilistic degenerate Euler polynomials associated with $Y$ Y. Also, we intoduce the probabilistic degenerate $r$ r -Stirling numbers of the second associated with $Y$ Y and the probabilistic degenerate two variable Fubini polynomials associated with $Y$ Y. We obtain some properties, explicit expressions, recurrence relations and certain identities for those polynomials and numbers. As special cases of $Y$ Y , we treat the gamma random variable with parameters $\alpha ,\beta \gt 0$ α , β > 0 , the Poisson random variable with parameter $\alpha \gt 0$ α > 0 , and the Bernoulli random variable with probability of success $p$ p. [ABSTRACT FROM AUTHOR]

The aim of this paper is to further study probabilistic versions of the degenerate Stirling polynomials of the second kind, namely the probabilistic degenerate Stirling polynomials of the second kind associated with $Y$ Y , which are also degenerate versions of the probabilistic Stirling polynomials of the second kind investigated earlier. Here $Y$ Y is a random variable whose moment generating function exists in some neighbourhood of the origin. We derive some properties, explicit expressions and certain identities for those polynomials. In addition, we apply our results to the Bernoulli random variable with probability of success $p$ p , the normal random variable with parameters $\mu = 0,{\sigma ^2} = 1$ μ = 0 , σ 2 = 1 , and to the uniform random variable on $(0,1)$ (0 , 1). [ABSTRACT FROM AUTHOR]

Many properties of special polynomials, such as recurrence relations, sum formulas, integral transforms and symmetric identities, have been studied in the literature with the help of generating functions and their functional equations. In this paper, we introduce hybrid forms of q-Mittag-Leffler functions. The q-Mittag-Leffler–Bessel and q-Mittag-Leffler–Tricomi functions are constructed using a q-symbolic operator. The generating functions, series definitions, q-derivative formulas and q-recurrence formulas for q-Mittag-Leffler–Bessel and q-Mittag-Leffler–Tricomi functions are obtained. The N q -transforms and N q -transforms of q-Mittag-Leffler–Bessel and q-Mittag-Leffler–Tricomi functions are obtained. These hybrid q-special functions are also studied by plotting their graphs for specific values of the indices and parameters. [ABSTRACT FROM AUTHOR]

We give a new proof of a theorem of Bell and Coons ['Transcendence tests for Mahler functions', Proc. Amer. Math. Soc. 145 (3) (2017), 1061–1070] on the leading order radial asymptotics of Mahler functions that are the generating functions of regular sequences. Our method allows us to provide a description of the oscillations whose existence was shown by Bell and Coons. This extends very recent results of Poulet and Rivoal ['Radial behavior of Mahler functions', Int. J. Number Theory , to appear]. [ABSTRACT FROM AUTHOR]

We propose generating functions, $\textrm {RGF}_p(x)$ , for the quotients of numerical semigroups which are related to the Sylvester denumerant. Using MacMahon's partition analysis, we can obtain $\textrm {RGF}_p(x)$ by extracting the constant term of a rational function. We use $\textrm {RGF}_p(x)$ to give a system of generators for the quotient of the numerical semigroup $\langle a_1,a_2,a_3\rangle $ by p for a small positive integer p , and we characterise the generators of ${\langle A\rangle }/{p}$ for a general numerical semigroup A and any positive integer p. [ABSTRACT FROM AUTHOR]

In this study, we explore the impact of stochastic resetting on the dynamics of random walks on a T-fractal network. By employing the generating function technique, we establish a recursive relation between the generating function of the first passage time (FPT) and derive the relationship between the mean first passage time (MFPT) with resetting and the generating function of the FPT without resetting. Our analysis covers various scenarios for a random walker reaching a target site from the starting position; for each case, we determine the optimal resetting probability γ * that minimizes the MFPT. We compare the results with the MFPT without resetting and find that the inclusion of resetting significantly enhances the search efficiency, particularly as the size of the network increases. Our findings highlight the potential of stochastic resetting as an effective strategy for the optimization of search processes in complex networks, offering valuable insights for applications in various fields in which efficient search strategies are crucial. [ABSTRACT FROM AUTHOR]

Quantitative bisimulations between weighted finite automata are defined as solutions of certain systems of matrix-vector inequalities and equations. In the context of fuzzy automata and max-plus automata, testing the existence of bisimulations and their computing are performed through a sequence of matrices that is built member by member, whereby the next member of the sequence is obtained by solving a particular system of linear matrix-vector inequalities and equations in which the previously computed member appears. By modifying the systems that define bisimulations, systems of matrix-vector inequalities and equations with k unknowns are obtained. Solutions of such systems, in the case of existence, witness to the existence of a certain type of partial equivalence, where it is not required that the word functions computed by two WFAs match on all input words, but only on all input words whose lengths do not exceed k. Solutions of these new systems represent finite sequences of matrices which, in the context of fuzzy automata and max-plus automata, are also computed sequentially, member by member. Here we deal with those systems in the context of WFAs over the field of real numbers and propose a different approach, where all members of the sequence are computed simultaneously. More precisely, we apply a simultaneous approach in solving the corresponding systems of matrix-vector equations with two unknowns. Zeroing neural network (ZNN) neuro-dynamical systems for approximating solutions of heterotypic bisimulations are proposed. Numerical simulations are performed for various random initial states and comparison with the Matlab, linear programming solver linprog, and the pseudoinverse solution generated by the standard function pinv is given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel class of graphs referred to as the Horadam–Lucas cubes. This class extends the concept of Lucas cubes and retains numerous desirable properties associated with them. Horadam–Lucas cubes can also be viewed as a companion graph family of the Horadam cubes, in a similar way the Lucas cubes relate to Fibonacci cubes or the Lucas-run graphs relate to Fibonacci-run graphs. As special cases, they also give rise to new graph families, such as Pell–Lucas cubes and Jacobsthal–Lucas cubes. We derive the several metric and enumerative properties of these cubes, including their diameter, periphery, radius, fundamental decomposition, number of edges, cube polynomials, and generating function of the cube polynomials. [ABSTRACT FROM AUTHOR]

A generalized n-site Asymmetric Simple Inclusion Process (ASIP) network is studied, where gate-opening instants are determined by a renewal process and arrivals occur to all sites. Various types of batch particle movements between sites are analyzed: (i) unidirectional probabilistic forward movements; (ii) probabilistic forward movements combined with feedback to the first site; and (iii) general probabilistic multidirectional movements. In contrast to the tedious successive substitution method used in previous ASIP studies, an efficient matrix approach is applied to derive the multidimensional probability generating function (PGF) of site occupancies right after gate opening instants. The complexity of the ASIP processes allows us to obtain explicit PGF results for small-size networks only, while for larger networks, a formula to calculate the mean site occupancies is derived for all types of movements. In movement case (i) the means are directly and explicitly calculated. For movement case (ii), where the network is homogeneous with equal probabilities of forward movements from site i to downstream sites j ≥ i , we show that the ratio between the mean occupancies of consecutive sites approaches a constant when the network becomes large, and calculate this ratio. Finally, we investigate an n-site network where at gate opening instants all gates open simultaneously, and particles move in all directions. Numerical examples are presented. [ABSTRACT FROM AUTHOR]

Mohsin, Abbas and Khan [Stochastic Environmental Research and Risk Assessment, 2024, doi: 10.1007/s00477–024-02787-z] used a generalized Pareto-exponential distribution to model extreme events involving exponentially decaying variables. In this letter, we show that the maximum likelihood procedure used in simulation studies as well as the data application is not correct. We derive the corrected version. We also derive simpler expressions for mathematical properties of the generalized Pareto-exponential distribution. [ABSTRACT FROM AUTHOR]

Most pricing methods for VIX futures and European VIX options rely on the existence of the squared VIX moment generating function. Yet this function does not exist for some state-of-the-art option pricing models, which prevents their widespread use. This letter presents a semi-closed-form solution for European VIX option prices based on the squared VIX characteristic function. When coupled with a new, alternative VIX futures pricing formula, the VIX option pricing method requires only the evaluation of a single integral, making its implementation very efficient. These pricing formulas are applicable to a wide class of models—virtually all exponentially affine models in the literature, among others—as the characteristic function always exists. We also test our newly proposed pricing methodologies against usual benchmarks in the literature and report that they lead to more efficient and accurate prices. [ABSTRACT FROM AUTHOR]

We consider a two‐dimensional time‐inhomogeneous birth‐death process to model the time‐evolution of fake news in a population. The two components of the process represent, respectively: (i) the number of individuals (say spreaders) who know the rumor and intend to spread it and (ii) the number of individuals (say inactives) who have forgotten the rumor previously received. We employ the probability generating function‐based approach to obtain the moments and the covariance of the two‐dimensional process. We also analyze a new adimensional index to study the correlation between the two components. Some special cases are considered in which both the expected numbers of spreaders and inactives are equal to suitable sigmoidal curves, which are often adopted in modeling growth phenomena. Finally, we provide an application based on real data related to the diffusion of fake news, in which the optimal choice of the sigmoidal curve that fit the datasets is based on the minimization of the mean square error and of the relative absolute error. [ABSTRACT FROM AUTHOR]

We discuss non-equilibrium thermodynamics of the mean-field Ising model from a geometric perspective, focusing on the thermodynamic limit. When the number of spins is finite, the Gibbs equilibria form a smooth Legendrian submanifold in the thermodynamic phase space whose points describe the stable macroscopic states of the system. We describe the convergence of these smooth Legendrian submanifolds, as the number of spins goes to infinity, to a singular Legendrian submanifold, admitting an analytic continuation that contains both the stable and metastable states. We also discuss the relaxation to a Gibbs equilibrium when the physical parameters are changed abruptly. The relaxation is defined via the gradient flow of the free energy with respect to the Wasserstein metric on microscopic states, that is, in the geometric language, via the gradient flow of the generating function of the equilibrium Legendrian with respect to the ghost variables. This leads to a discrete Fokker-Planck equation when the number of spins is finite. We show that in the thermodynamic limit this description is closely related to the seminal model of relaxation proposed by Glauber. Finally, we find a special range of parameters where such relaxation happens instantaneously, along the Reeb chords connecting the initial and the terminal Legendrian submanifolds. [ABSTRACT FROM AUTHOR]

In this paper, we define a novel family of arithmetic sequences associated with the Fibonacci numbers. Consider the ordinary Fibonacci sequence { f n } n ∈ N 0 having initial terms f 0 = 0 , and f 1 = 1 and recurrence relation f n = f n − 1 + f n − 2 (n ≥ 2) . In many studies, authors worked on the generalizations of integer sequences in different ways, some by preserving the initial terms, others by preserving the recurrence relation, and some for numeric sets other than positive integers. Here, we will follow the third path. So, in this article, we study a new extension t f n ∗ , with initial terms t f 0 ∗ = (f 0 ∗ , f 1 ∗ , f 2 ∗) and t f 1 ∗ = (f 1 ∗ , f 2 ∗ , f 3 ∗) , which is generated by the recurrence relation t f n ∗ = t f n − 1 ∗ + t f n − 2 ∗ (n ≥ 2) , the Fibonacci-type sequence. The aim of this paper is to define Tricomplex Fibonacci numbers as an extension of the Fibonacci sequence and to examine some of their properties such as the recurrence relation, summation formula and generating function, and some classical identities. [ABSTRACT FROM AUTHOR]

Many prominent combinatorial sequences, such as the Fibonacci, Lucas, Pell, Jacobsthal and Tribonacci sequences, are defined by homogeneous linear recurrence relations with constant coefficients. These sequences are often referred to as C-finite sequences, and a variety of representations have been employed throughout the literature, largely influenced by the author's background and the specific application under consideration. Beyond the representation through recurrence relations, other approaches include those based on generating functions, explicit formulas, matrix exponentiation, the method of undetermined coefficients and several others. Among these, the generating function approach is particularly prevalent in enumerative combinatorics due to its versatility and widespread use. The primary objective of this work is to introduce an alternative representation grounded in the theory of Riordan arrays. This representation provides a general formula expressed in terms of the vectors of constants and initial conditions associated with any recurrence relation of a given order, offering a new perspective on the structure of such sequences. [ABSTRACT FROM AUTHOR]

Given an integer n ≥ 4, how many inequivalent quadrilaterals with ordered integer sides and perimeter n are there? Denoting such number by Q(n), the answer is given by the following closed formula: ... . [ABSTRACT FROM AUTHOR]

In this paper, we define quadra Fibona-Pell hybrid numbers by using the definition of hybrid numbers. After that, we introduce quadra Fibona-Pell hybrinomials and investigate some properties of quadra Fibona-Pell hybrid numbers and hybrinomials. Finally, we show the matrix representation of quadra Fibona-Pell hybrinomials. [ABSTRACT FROM AUTHOR]

Özdemir [1] introduced Hybrid numbers as a generalization of complex, hyperbolic and dual numbers. Hybrinomial sequence is the combination of hybrid numbers and polynomial sequence. In this paper we consider special kind of hybrinomial sequence, namely the Van Der Laan hybrinomial sequence. Binet-Like Formula, generating function and exponential generating function of this sequence are shown in this paper. Properties and some summation identities for Van Der Laan polynomial sequence and hybrinomial sequence are represented in this paper. In addition, some interesting summation identities of Van Der Laan hybrid numbers are obtained. [ABSTRACT FROM AUTHOR]

We show that if a group contains ℤ n × F m as a finite-index subgroup, then its cogrowth series is the diagonal of a rational function for every generating set. This answers a question of Pak and Soukup on the cogrowth of virtually abelian groups; and generalizes a result by Elder, Rechnitzer, Janse van Rensburg, and Wong on the cogrowth series of the Baumslag–Solitar groups BS (N , N). [ABSTRACT FROM AUTHOR]

In this paper, we derive some nonlinear differential equations from generating function of generalized harmonic numbers and give some identities involving generalized harmonic numbers and special numbers by using these differential equations. For example, for any positive integers N, n, r, a and any integer m ≥2, ... where S1 (n, k) is Stirling number of the first kind. [ABSTRACT FROM AUTHOR]

Special functions and special polynomials have been used and studied widely in the context of Geometric function theory of Complex Analysis by the many authors. Here, in our present investigations, making use of the convolution, we first introduce a new subclass of Bazilevivč bi-univalent functions involving the Bernoulli Polynomials. We then find the first two Taylor-Maclaurin coefficient | a 2 | and | a 3 | for our defined functions class. We then calculate the bounds for the Fekete Szegö inequality for the defined function class. Also some known consequences of our main results are highlighted in the form of Corollaries. [ABSTRACT FROM AUTHOR]

This paper aims to introduce a new class of special polynomials called q-modified Laguerre-Appell polynomials. Some definitions and concepts related to this class of polynomials, including generating function and series definition are explored. Also, certain interesting properties related to this class and some of its members are investigated. Furthermore, the class of 2D q-modified Laguerre-Appell polynomials is introduced. Finally, we present certain graphical representations related to some members of these new classes. [ABSTRACT FROM AUTHOR]

With over 1.6 million traffic deaths in 2016, automated vehicles equipped with automated driving systems (ADSs) have the potential to increase traffic safety by assuming human driving tasks within the operational design domain (ODD). However, safety validation is challenging due to the open-context problem. Current strategies, such as pure driving and requirement-based testing, are insufficient. Scenario-based testing offers a solution but necessitates appropriate scenario selection, testing methods, and evaluation criteria. This paper builds upon a method to calculate the covered ODD using tested scenarios generated from logical scenarios, considering parameter discretisation uncertainty. Acceptance criteria for the safety argumentation are proposed based on parameter space coverage and variance introduced via discretisation, thus contributing to quantifying the residual risks of safety validation. The approach is demonstrated through two logical scenarios with probability density functions of the parameters generated using a trajectory dataset. These criteria can serve as risk acceptance criteria, providing comparability and explainable results. By developing a robust scenario-based testing approach, ADS safety can be validated, leading to increased traffic safety and reduced fatalities. Since ADSs incorporate AI models, this proposed validation strategy can be extended to AI systems across multiple domains for the respective assurance argument required for deployment. [ABSTRACT FROM AUTHOR]

Given an integer $k\ge 2$ , let $\omega _k(n)$ denote the number of primes that divide n with multiplicity exactly k. We compute the density $e_{k,m}$ of those integers n for which $\omega _k(n)=m$ for every integer $m\ge 0$. We also show that the generating function $\sum _{m=0}^\infty e_{k,m}z^m$ is an entire function that can be written in the form $\prod _{p} \bigl (1+{(p-1)(z-1)}/{p^{k+1}} \bigr)$ ; from this representation we show how to both numerically calculate the $e_{k,m}$ to high precision and provide an asymptotic upper bound for the $e_{k,m}$. We further show how to generalize these results to all additive functions of the form $\sum _{j=2}^\infty a_j \omega _j(n)$ ; when $a_j=j-1$ this recovers a classical result of Rényi on the distribution of $\Omega (n)-\omega (n)$. [ABSTRACT FROM AUTHOR]

In this paper, constraint-oriented coordination control of a first-order multi-robot system has been considered in a leaderless consensus where rectangular velocity components of participating robots are subject to constraints while also avoiding inter-robots collisions. The desired formation control of different robots and their inter-robots collision avoidance in a team of robots are achieved by solving an optimization problem based on control barrier functions whereas graph theory concepts are used to represent the interaction relations among robots. A devised quadratic programme subject to the velocity and safety conditions, minimizes the desired cost function encoded in the control barrier function and generates separate control inputs for each robot. The rectangular components of the velocity against each robot are kept constrained during the entire operation of the formation control. The optimization-technique-based decentralized controllers were simulated in MATLAB and the corresponding results were recorded. The robots in the team successfully attained the desired formation in a leaderless consensus, deploying themselves in a plane under constrained rectangular velocities without colliding with each other. Several simulation examples with different values of velocity constraints have been shown to illustrate the operation of constrained controllers while ensuring that the desired leaderless-consensus-based formation remains attainable in a safe manner. [ABSTRACT FROM AUTHOR]

To circumvent reporting spurious correlations, species distribution models often explicitly account for spatial autocorrelation, for example by including spatially structured random effects. The validity of statistical inference derived from such models has been tested by simulations using null environmental predictors that do not have any causal dependency with the response. Such null environmental predictors can be obtained by permutations of the original predictors or by simulating spatial structures resembling the original predictors. In such approaches, it is important that the permuted or simulated predictors reflect the nature of spatial variation present in the original predictors.Here we present a novel approach for generating realistic null predictors by a shift‐&‐rotate (S&R) approach: we extract environmental variables after randomly translating and rotating the sampling area within a window of defined environmental layers. In this way, the null environmental variables have fully realistic spatial variation and covariation, but no relationship to the response variable. We implement the S&R approach to three main R‐functions and demonstrate with a simulation study how they can be used to untangle causal versus non‐causal relationships within species distribution modelling.These methods allow us to quantify the predictive power attributed within the models due to non‐causal correlations generated by the realistic structure of the environmental covariates. In our case study, we identify when a model incorrectly estimates parameter values, yet still has high predictive power due to the structured nature of the predictor variables.The use of null models is imperative in ecological modelling for testing the accuracy of statistical inference in complex ecological systems and the choice of these null models is far from trivial. Here we provide R functions for generating spatially realistic null models to use in species distribution modelling as well as other spatially explicit fields such as landscape genetics. [ABSTRACT FROM AUTHOR]

Motivated by real-world engineering systems, this paper presents an in-depth exploration of a novel reliability model for a complex multi-state series system (MSSS) that incorporates a performance sharing mechanism. Specifically, the MSSS is composed of m distinct subsystems featuring a k-out-of-n: G structure. The ith subsystem consists of ni elements, such that a minimum of ki functioning elements is required for normal operation. Transmission devices (TDs) are present between adjacent subsystems to share surplus performance. Both element performance and subsystem demand are treated as random variables, and the performance of all elements in each subsystem is cumulated to meet its individual random demand. Subsystem failure can result from either an inability to meet performance demands or an insufficient number of functioning elements. The surplus performance of a subsystem can only be transferred to its adjacent subsystem via the TD. The entire MSSS will fail if at least one subsystem does not work properly by using performance sharing mechanism. To analyze the reliability indexes of the MSSS, a new algorithm based on the generalized universal generating function (GUGF) has been developed. Numerical examples are provided to illustrate the accuracy and effectiveness of the proposed model and method. [ABSTRACT FROM AUTHOR]

This paper presents a generalization of the sequence defined by the third-order recurrence relation Vn (aj, Pj) = ³Σj=1PjVn-j, n ≥ 4, P3 ≠ 0 with initial terms Vj = aj, where ajand pj j = 1,2,3, are any non-zero real numbers. The generating function and Binet's formula are derived for this generalized tribonacci sequence. Classical second-order generalized Fibonacci sequences and other existing sequences based on second-order recurrence relations are implicitly included in this analysis. These derived sequences are discussed as special cases of the generalization. A pictorial representation is provided, illustrating the growth and variation of tribonacci numbers for different initial terms aj and coefficients pj. Additionally, the tribonacci constant is examined and visually represented. It is observed that the constant is influenced solely by the coefficients pj of the recurrence relation and is unaffected by the initial terms aj. [ABSTRACT FROM AUTHOR]

The empirical wavelet transform is a fully adaptive time-scale representation that has been widely used in the last decade. Inspired by the empirical mode decomposition, it consists of filter banks based on harmonic mode supports. Recently, it has been generalized to build the filter banks from any generating function using mappings. In practice, the harmonic mode supports can have a low-constrained shape in 2D, leading to numerical difficulties to estimate mappings adapted to the construction of empirical wavelet filters. This work aims to propose an efficient numerical scheme to compute empirical wavelet coefficients using the demons registration algorithm. Results show that the proposed approach is robust, accurate, and continuous wavelet filters permitting reconstruction with a low signal-to-noise ratio. An application for texture segmentation of scanning tunneling microscope images is also presented. [ABSTRACT FROM AUTHOR]

Iron (Fe) is an essential micronutrient for plants. Due to the low Fe bioavailability in cultivated soils, Fe deficiency is a widespread agricultural problem. In this study, we present the functional characterization of a petunia (Petunia hybrida) basic-helix-loop-helix transcription factor PhbHLH121 in response to Fe shortage. Real-time PCR revealed that the expression of PhbHLH121 in petunia roots and shoots was downregulated under Fe-limited conditions. CRISPR/Cas9-edited phbhlh121 mutant plants were generated to investigate the functions of PhbHLH121 in petunia. Loss-of-function of PhbHLH121 enhanced petunia tolerance to Fe deficiency. Further investigations revealed that the expression level of several structural genes involved in Fe uptake in petunia, such as IRT1 and FRO2, was higher in phbhlh121 mutants compared to that in wild-type under Fe-limited conditions, and the expression level of several genes involved in Fe storage and Fe transport, such as VTL2, FERs and ZIF1, was lower in phbhlh121 mutants compared to that in wild-type under Fe-deficient conditions. Yeast one-hybrid assays revealed that PhbHLH121 binds to the G-box element in the promoter of genes involved in Fe homeostasis. Yeast two-hybrid assays revealed that PhbHLH121 interacts with petunia bHLH IVc proteins. Taken together, PhbHLH121 plays an important role in the Fe deficiency response in petunia. [ABSTRACT FROM AUTHOR]

In the 1970s, Nicolas proved that the partition function p(n) is log-concave for n > 25 . In Heim et al. (Ann Comb 27(1):87–108, 2023), a precise conjecture on the log-concavity for the plane partition function pp (n) for n > 11 was stated. This was recently proven by Ono, Pujahari, and Rolen. In this paper, we provide a general picture. We associate to double sequences { g d (n) } d , n with g d (1) = 1 and 0 ≤ g d n - n d ≤ g 1 n n - 1 d - 1 , polynomials { P n g d (x) } d , n given by ∑ n = 0 ∞ P n g d (x) q n : = exp x ∑ n = 1 ∞ g d (n) q n n = ∏ n = 1 ∞ 1 - q n - x f d (n) . We recover p (n) = P n σ 1 (1) and pp n = P n σ 2 (1) , where σ d (n) : = ∑ ℓ ∣ n ℓ d and f d (n) = n d - 1 . Let n ≥ 6 . Then the sequence { P n σ d (1) } d is log-concave for almost all d if and only if n is divisible by 3. Let id (n) = n . Then P n id (x) = x n L n - 1 (1) (- x) , where L n α x denotes the α -associated Laguerre polynomial. In this paper, we investigate Turán inequalities Δ n g d (x) : = P n g d (x) 2 - P n - 1 g d (x) P n + 1 g d (x) ≥ 0. Let n ≥ 6 and 0 ≤ x < 2 - 12 n + 4 . Then n is divisible by 3 if and only if Δ n g d (x) ≥ 0 for almost all d. Let n ≥ 6 and n ≢ 2 (mod 3) . Then the condition on x can be reduced to x ≥ 0 . We determine explicit bounds. As an analogue to Nicolas' result, we have for g 1 = id that Δ n id (x) ≥ 0 for all x ≥ 0 and all n. [ABSTRACT FROM AUTHOR]

We have introduced a new continuous-time network evolution model. We have described cooperation, so we have considered the cliques of nodes. The evolution of the network was based on cliques of nodes of the network and was governed by a branching process. The basic properties of the evolution process were described. Asymptotic theorems were proved for the number of cliques having a fixed size and the degree of a fixed node. The generating function was calculated, and then the probability of extinction was obtained. For the proof, advanced results of multi-type branching processes were used. Besides precise mathematical proofs, simulation examples also supported our results. [ABSTRACT FROM AUTHOR]

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Our motivation for this study is to define two new and particular sequences. The most essential feature of these sequences is that they are spinor sequences. In this study, these new spinor sequences obtained using spinor representations of Pell and Pell-Lucas quaternions are expressed. Moreover, some formulas such that Binet formulas, Cassini formulas and generating functions of these spinor sequences, which are called as Pell and Pell-Lucas spinor sequences, are given. Then, some relationships between Pell and Pell-Lucas spinor sequences are obtained. Therefore, an easier and more interesting representations of Pell and Pell-Lucas quaternions, which are a generalization of Pell and Pell-Lucas number sequences, are obtained. We believe that these new spinor sequences will be useful and advantageable in many branches of science, such as geometry, algebra and physics. [ABSTRACT FROM AUTHOR]

Multi-level thresholding for image segmentation is one of the key techniques in image processing. Although numerous methods have been introduced, it remains challenging to achieve stable and satisfactory thresholds when segmenting images with various unknown properties. This paper proposes an equilibrium optimizer algorithm to find the optimal multi-level thresholds for grayscale images. The proposed algorithm AEO (advanced equilibrium optimizer) uses two sub-populations to balance exploration and exploitation during the multi-level threshold search process. Two mutation schemes are proposed for the sub-populations to prevent them from being trapped in local optima. AEO offers a repair function to avoid generating duplicate thresholds. The performance of AEO is evaluated on multiple benchmark images. Experimental results demonstrate that AEO has an outstanding ability for multi-level threshold image segmentation in terms of cross-entropy, signal-to-noise ratio (PSNR), structural similarity index measure (SSIM), and feature similarity index (FSIM). [ABSTRACT FROM AUTHOR]

Reversible Data Hiding (RDH) in the encrypted domain has been extensively investigated because of its myriad of applications in different fields and security and privacy concerns of secret messages and covers. However, most of the methods mainly suffer due to low embedding capacity, communication overhead, security of secret messages and overflow/underflow issues. In order to address these issues, an efficient separable RDH method employing bit-pair differences has been proposed. Embedding at bit pairs level not only achieves high secret data embedding capacity but also enhances the security and privacy of the secret message and cover. The proposed method employs the average and floor function to generate the interpolated cover and hence, avoids the overflow/underflow problem. The proposed separable method supports blind data extraction from the marked cover thereby being computationally inexpensive. In this, secret messages and covers are recovered independently on the basis of available keys rather than sequence order. This method is computationally inexpensive as the embedding algorithm only consideres simple rules based data hiding and data retrieval is achieved through simple XOR operation which overcomes the communication overhead. The performance of the proposed method is compared with the state-of- the-art methods and results reveal significant improvements in the embedding capacity. THe maximum embedding capacity of 4 bits per pixel(bpp) with an average PSNR value of 34.99 dB is achieved irrespective of the cover. [ABSTRACT FROM AUTHOR]

This paper presents and examines computationally convenient goodness-of-fit tests for the family of generalized Poisson distributions, which encompasses notable distributions such as the Compound Poisson and the Katz distributions. The tests are consistent against fixed alternatives and their null distribution can be consistently approximated by a parametric bootstrap. The goodness of the bootstrap estimator and the power for finite sample sizes are numerically assessed through an extensive simulation experiment, including comparisons with other tests. In many cases, the novel tests either outperform or match the performance of existing ones. Real data applications are considered for illustrative purposes. [ABSTRACT FROM AUTHOR]

In this paper, a multi-objective prediction and optimization integration method combining response surface method (RSM) and non-dominated sorting genetic algorithm-II (NSGA-II) was proposed to improve the overhead low-power pulsed laser-MAG (metal active gas) hybrid welding quality. A four-factor with a five-level experiment matrix considering the weld current(I), weld speed(V), laser power(P), and assembly clearance(C) was established based on te central composite design method. The relationship between weld parameters and back weld width (BW) and weld reinforcement (WR) was approximated by RSM. The significance and adequacies were validated by the analysis of variance and validation experiments, and the average error of validation experiments between predict and actual value is less than 5%, indicating that the model exhibits high predictive accuracy. The individual effect of a single parameter and the interaction of multiple parameters on BW and WR were studied, and the weld current and assembly clearance are the most significant parameters influencing BW and WR, respectively. NSGA-II was used for multi-objective optimization taking the constructed RSM models as objective functions and generating the Pareto-optimal front composed of optimal solutions. Finally, the verification experiments show that the optimal solutions of different fronts can obtain the desired welding morphology without any defect. [ABSTRACT FROM AUTHOR]