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AbstractWe examine the equation for positive integers y, a ≥ 2 and k ≥ 3. This equation can be expressed as a problem in terms of the Zeckendorf representations of integers. Using bounds on linear forms in logarithms and Baker-Davenport reduction methods, we are able to completely solve the equation for y ≤ 25000 when k = 3, for y ≤ 1000 when k = 4, for y ≤ 40 when k = 5, and for y ≤ 3 when k = 6. [ABSTRACT FROM AUTHOR]

Since the existing tropical cryptographic protocols are either susceptible to the Kotov-Ushakov attack and its generalization, or to attacks based on tropical matrix periodicity and predictive behavior, several attempts have been made to propose protocols that resist such attacks. Despite these attempts, many of the proposed protocols remain vulnerable to attacks targeting the underlying hidden problems, one of which we call the tropical two-sided discrete logarithm with shift. An illustrative case is the tropical Stickel protocol, which, when formulated with a single monomial instead of a polynomial, becomes susceptible to attacks based on solutions of the above mentioned tropical version of discrete logarithm. In this paper we will formally introduce the tropical two-sided discrete logarithm with shift, discuss how it is solved, and subsequently demonstrate an attack on a key exchange protocol based on the tropical semiring of pairs. This particular protocol is compromised due to the existence of efficient (albeit heuristic) solution of the tropical two-sided logarithm problem, and this highlights the ongoing challenges in search of a "good" key exchange protocol in tropical cryptography. [ABSTRACT FROM AUTHOR]

In this article, the authors prove the Chover-type law of the iterated logarithm for weighted partial sums of independent and identically distributed random variables, and also give some classical examples. These conclusions promote and improve the corresponding results from the traditional probability space to the sub-linear expectation space. [ABSTRACT FROM AUTHOR]

Summary: We show how a natural mistake in integration leads to insights about the natural logarithm. [ABSTRACT FROM AUTHOR]

Vehicle miles traveled (VMT) is an essential input for many aspects of transportation engineering, and an accurate estimation of VMT is critical for practicing engineers. Linear regression models are a popular method to estimate VMT as they provide insight into the relationships between VMT and other external factors. In linear regression models the prediction of the response variable has a non-zero probability of resulting in a negative value. For this reason, the natural logarithm of VMT is often used as the response variable to force a positive outcome. However, these log-linear regression (LLR) models provide median VMT estimate instead of the mean estimate. To overcome this limitation of LLR models, this study proposes using heteroskedastic LLR and count data methods to estimate VMT. These methods are found to have better performance than LLR models in terms of data fit and prediction accuracy. [ABSTRACT FROM AUTHOR]

In Eurocrypt'96, the concept of Designated Verifier Signature was proposed. Such signatures can only be verified by a single designated verifier specified during signature creation. However, there exist situations in which the signer, as well as the designated verifier, should be equipped with the ability to check the validity of the signature. At the same time, either of them should be able to help third parties to verify the signature. This is achieved by Directed Signature Schemes. In this paper, we consider directed signatures in the certificateless cryptography setting and propose an efficient pairing-free certificateless directed signature (CLDS) scheme. Then, we prove that the proposed CLDS scheme meets the needed security requirements in the random oracle model and under the assumption of the hardness of discrete logarithm and Gap Diffie-Hellman problems. We also compare the proposed scheme with the related ones to indicate the overall superiority of the proposed CLDS scheme. [ABSTRACT FROM AUTHOR]

Let { X n , n ≥ 1 } be a sequence of NSD random variables which is stochastically dominated by a random variable X satisfying that X is α − tailed, α > 0. Let { a n i , 1 ≤ i ≤ n , n ≥ 1 } be an array of real numbers with ∑ i = 1 n a n i 2 = O (n). In this paper, the author proves the Chover-type law of the iterated logarithm for weighted partial sums of NSD random variables, and also gives some classical examples. The results obtained improve and generalize the known results. [ABSTRACT FROM AUTHOR]

Let { Y n , n ≥ 1 } be sequence of random variables with EYn = 0. Assume that the Kolmogrov type exponential inequality is satisfied for { Y x j = − x I { Y j < − x } + Y j I { | Y j | ≤ x } + x I { Y j > x } , j ≥ 1 } with any x > 0. In this paper, the law of the logarithm and law of the iterated logarithm for a class of random variables satisfying the Kolmogrov type exponential inequality with non-identical distributions are obtained. The main results not only extend the known results from NA case to a class of random variables satisfying Kolmogrov type exponential inequality, but also weaken the moment assumptions. [ABSTRACT FROM AUTHOR]

In this paper, we are devoted to a global W 1 , γ (⋅) -estimate for elliptic obstacle problem with double phase growth for the borderline setting in nonsmooth domain. More precisely, we consider the variational inequality ∫ Ω 〈 A (x , D u) , D v − D u 〉 d x ≥ ∫ Ω 〈 B (x , F) , D v − D u 〉 d x ∀ v ∈ A 0 (Ω) , whose characteristics of ellipticity and growth switch between a type of polynomial and its logarithm, then we prove a variable exponent Calderón-Zygmund estimate with an implication that H (x , F) , H (x , D ψ) ∈ L γ (x) (Ω) ⇒ H (x , D u) ∈ L γ (x) (Ω) provided that the nonlinearity satisfies a small BMO condition while the boundary of the underlying domain is flat in the sense of Reifenberg. [ABSTRACT FROM AUTHOR]

Students’ difficulties with proofs are well documented. To remedy this, it is often recommended that reasoning and proving be focused on in all grades and content areas of school mathematics. However, proofs continue to have a marginal place in many classrooms, or are only given explicit attention in courses in Euclidean geometry. Geometry is also the most common topic for educational research on reasoning and proving. This paper compares what four other topics in secondary school mathematics – logarithms, primitive functions, definite integrals, and combinatorics – can offer in terms of opportunities to learn proof. The types and natures of reasoning in expository sections and students’ tasks in 11 Swedish and Finnish textbooks are analysed in search of similarities and differences between these topics. The results are accounted for with special focus on opportunities for reasoning about general cases. Finally, the findings are discussed in relation to mathematical aspects of the four analysed topics. [ABSTRACT FROM AUTHOR]

In this article, we establish two types of the Hartman–Wintner law of the iterated logarithm under sublinear expectations, especially for pseudo-independent random variables with the finite quadratic Choquet expectations, which generalizes the existing results for independent and identically distributed sequences. But unlike the classical situation, one counterexample is provided to show that the Hartman–Wintner law of the iterated logarithm may fail on the sublinear expectation space only with the finite second moment condition. [ABSTRACT FROM AUTHOR]

This article explores the concept of rearranging conditionally convergent series to converge to different sums. It provides examples of such rearrangements that are suitable for calculus and undergraduate analysis students. The article compares these rearranged series to the alternating harmonic series and shows that they have the same limit. It also discusses Abel's theorem on power series and its application to the convergence of a specific series, demonstrating that it converges to ln(k). The article concludes by discussing the usefulness of these rearrangements in calculus and analysis classes and acknowledges the contributions of the editorial board and a colleague. [Extracted from the article]

During the Spanish eighteenth century, a process of modernization took place in scientific knowledge, partly driven by the circulation and appropriation of new scientific ideas. In this context, the Spanish mathematician Benito Bails (1731–1797) published his course Elementos de Matemática (Elements of Mathematics) consisting of ten volumes (1779–1799), in which, among other subjects, he presented one of the most complete mathematical developments of logarithmic calculation methods of his time, by using the infinity through infinite series. The aim of our article is to demonstrate how algebraic analytical reasoning enabled Bails to obtain new and more efficient infinite algorithms that converge more quickly in the computation of logarithms in any system. We show how Euler's number 'e' is calculated, probably for the first time in Spanish teaching, in an eighteenth century mathematical text. Our analysis concludes that Bails' course constituted an innovation and provides evidence of its creativity, originality and ingenuity. [ABSTRACT FROM AUTHOR]

A case is made for an early introduction of transformation of data in a first course in statistics. The topic is well received by students and helps pedagogically later in the course. Linear, squaring, and logarithmic transformations are emphasized. Impacts on subsequent parts of the course are discussed. [ABSTRACT FROM AUTHOR]

A Markov matrix is embeddable if it can represent a homogeneous continuous-time Markov process. It is well known that if a Markov matrix has real and pairwise-different eigenvalues, then the embeddability can be determined by checking whether its principal logarithm is a rate matrix or not. The same holds for Markov matrices that are close enough to the identity matrix. In this paper we exhibit open sets of Markov matrices that are embeddable and whose principal logarithm is not a rate matrix, thus proving that the principal logarithm test above does not suffice generically. [ABSTRACT FROM AUTHOR]

The security of the most of the existing strong designated verifier signature (SDVS) scheme are based on a single cryptographic hard problem. Although these schemes seem secure today, there is a possibility for an adversary to find a solution for this problem in the future and if this happens, then such SDVS schemes will no longer be secure. We develop a new SDVS scheme based on hybrid problems; factorization and discrete logarithm. The new developed scheme provides a greater security level than those previous designated SDVS schemes and satisfies the security properties and requirements of a SDVS scheme. [ABSTRACT FROM AUTHOR]

We extend the direct pricing framework and propose a new generalized model for VIX futures by adding the time-varying component volatility as well as jump intensity to the logarithm of underlying VIX series. We also switch to the normal jump component on which the simpler and faster analytical filter can be applied. Compared with the nested benchmark models, these extensions are supported by significant pricing performance improvements both in- and out-of-sample. In particular, our generalized model can reduce the pricing errors substantially in the high VIX level period and for long maturity contracts. Our research highlights the importance of those newly added features of the VIX series when pricing VIX derivatives. [ABSTRACT FROM AUTHOR]

We consider the strong consistency of the nonparametric estimation of error density in linear regression with right censored data. The estimator is defined to be the kernel-smoothed estimator of error density, which makes use of the Kaplan-Meier estimator of the error distribution. We establish a point-wise law of the iterated logarithm for kernel-type error density estimator in censored Linear Regression. [ABSTRACT FROM AUTHOR]

The application of fast semiautomatic techniques to magnetic data is very useful for data interpretation. A new automatic method is developed to use for 2D magnetic interpretation based on combinations of analytic signals of the logarithm of different order analytic signals (CALA method). The new method provides three linear equations for estimating the structural index, the horizontal location and the depth of magnetic source independently. The CALA method is sensitive to noise owing to using higher-order derivatives, and the upward continuation technique should be used for smoothing the magnetic anomaly. The CALA method successfully estimates the structural indices and positions of the sources when tested on synthetic noise-free and noise-added magnetic data, and has higher accuracy than the AN-EUL method and the method of the derivatives of the logarithm of the analytic signal amplitude. The satisfactory results are obtained by the new method on two real magnetic data sets. [ABSTRACT FROM AUTHOR]

There are four definitions of the natural logarithm function commonly found in analysis texts: the inverse of the natural exponential, an integral, a limit, and the unique continuous solution of a functional equation. This article provides historical and pedagogical comments about each and establishes the equivalence of these four definitions. [ABSTRACT FROM AUTHOR]

The typical trigonometry, precalculus, or calculus student might not agree that logarithms are hot stuff, but we drew motivation from chili peppers to help students get a better taste for logarithms. The Scoville scale, which ranges from 0 to 16,000,000, has been the sole quantitative metric to measure the pungency (spiciness) of peppers since its creation in 1912. With justification from the Weber–Fechner Law (perception is the logarithm of stimulus), students were challenged to create a more palatable rating scale for chili peppers. The first author devised a simple but effective 0–10 rating scale for peppers, calling this new rating system the "Ferven scale." While the Weber–Fechner Law has been applied to the senses of touch, sight, and hearing, this paper presents the first known application of this law to the sense of taste. [ABSTRACT FROM AUTHOR]

Markov-switching models have become a popular tool in areas ranging from finance to electrical engineering. Determining the number of hidden regimes in such models is a key problem in applications. This paper proposes a strongly consistent estimator of the number of regimes for Markov-switching autoregressive models. By using subadditive ergodic theorem, law of iterated logarithm for martingales, together with results from information theory, we derive sufficient conditions to avoid underestimation as well as overestimation. In particular, we propose a modified information criterion, regime-switching information criterion (RSIC) which generates a simple and consistent model selection procedure. Finally, we conduct a Monte Carlo study to evaluate the efficacy of our procedure in finite sample and also compare the performance of RSIC with popular information criteria including AIC, BIC and HQC. [ABSTRACT FROM AUTHOR]

We provide a sufficient condition for the bounded law of the iterated logarithms for strictly stationary random fields expressable as a functional of i.i.d. random fields when the summation is done on rectangles. The study is done via the control of the moments of an appropriated maximal function. Applications to functionals of linear random fields, functions of a Gaussian linear random field and Volterra process are given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a time-changed version of the space-time fractional Poisson process (STFPP) by time changing it by an independent Lévy subordinator with finite moments of any order. We call the introduced process as the time-changed fractional Poisson process (TCFPP). The probability mass function, probability generating function and a variant of the law of iterated logarithm for the TCFPP are obtained. Several special cases of the TCFPP are considered and their governing differential equations are derived. Later, we consider the STFPP time-changed by inverse of Lévy subordinator and discuss its particular cases. [ABSTRACT FROM AUTHOR]

The present study investigates the relationship between two features of dependencies, namely, dependency distances and dependency frequencies. The study is based on the analysis of a parallel dependency treebank that includes 10 Indo-European languages. Two corresponding random dependency treebanks are generated as baselines for comparison. After computing the values of dependency distances and their frequencies in these treebanks, for each lan-guage, we fit four functions, namely quadratic, exponent, logarithm, and power-law func-tions, to its original and random datasets. The preliminary result shows that there is a rela-tion between the two dependency features for all 10 Indo-European languages. The relation can be further formalized as a power-law function which can distinguish the observed data from randomly generated datasets. [ABSTRACT FROM AUTHOR]

Tropical linear algebra has been recently put forward by Grigoriev and Shpilrain as a promising platform for implementation of protocols of Diffie-Hellman and Stickel type. Based on the CSR expansion of tropical matrix powers, we suggest a simple algorithm for the following tropical discrete logarithm problem: "Given that A = V ⊗ F ⊗ t for a unique t and matrices A, V, F of appropriate dimensions, find this t." We then use this algorithm to suggest a simple attack on a protocol based on the tropical semidirect product. The algorithm and the attack are guaranteed to work in some important special cases and are shown to be efficient in our numerical experiments. [ABSTRACT FROM AUTHOR]

The evidential approach uses likelihood ratios, typically the natural logarithm of the likelihood ratio representing the amount of evidence for one hypothesis versus another. One of the barriers to using the approach is the unavailability of sample size calculations for commonly used statistical tests. The t test is the most common statistical test used in scientific publications. This paper derives the equations necessary to calculate evidential probabilities and hence the required sample size for different types of t tests. Compared with the conventional Neyman-Pearson approach, the evidential approach requires larger sample sizes. This drawback is countered by the fact that users know the probability for obtaining misleading evidence (strong evidence that points to the wrong hypothesis). Even with small sample sizes, this is quite small (around 0.05) and decreases further with increasing sample sizes. The main challenge faced by the evidential researcher is of obtaining sufficiently strong evidence for or against one of the two specified hypotheses. Like the probability of a Type II error, this probability is large with a small sample size and decreases as the sample size increases. Sample size is estimated by achieving a low probability (e.g. <.1) for the combined probability of misleading and weak evidence. [ABSTRACT FROM AUTHOR]

A generalization of the well-known Fibonacci sequence is the k-generalized Fibonacci sequence whose first k terms are 0,... , 0, 1 and each term afterwards is the sum of the preceding k terms. In this paper, by using a lower bound to linear forms in logarithms of algebraic numbers due to Matveev and some argument of the theory of continued fractions, we find all the members of F(k) which are close to a power of 2. This paper continues and extends the previous work of Chern and Cui which investigated the Fibonacci numbers close to a power of 2. [ABSTRACT FROM AUTHOR]

ElGamal Cryptosystem (EC) is a non-deterministic scheme which produces different outputs for the same input, making the cryptosystem more secure. On the other hand, the efficiency of its cryptosystem is low as it produces a 2:1 expansion in size from plaintext to ciphertext, resulting in a delay in execution time. Therefore, this paper presents a Modified ElGamal Cryptosystem (MEC) to increase the efficiency by speeding up the execution time and reducing the expansion rate in the file size after the encryption process. A comparison between the proposed MEC and the traditional EC is carried out using the same programming environment, and the implementation is tested using text data of different sizes. The results show that the performance of the proposed MEC is better than the traditional EC in terms of execution time and expansion rate. Whereas, the security of the proposed MEC is analogous to the traditional EC, which is based on the difficulty of solving the discrete logarithm problem. [ABSTRACT FROM AUTHOR]

Logarithm of a square matrix is a factor that clearly affects the quality or result of something. Logarithms are used to maintain exactitude in many numerical computations than the determinant. At present cloud computing is an emerging topic to perform this kind of operations. Cloud computing qualifies resource-awkward clients to solve their immense computational workloads through a cloud server that provides heavy computational power and it is also less expensive. This rising computing paradigm undoubtedly introduces new challenges and security concerns such as security, correctness, competency and result verifiability. Matrix log-determinant is a quite simple scientific, engineering and machine learning application. It inspired us to plan a protocol that satisfies all the concerns for the outsourcing of LDC to a cloud which is malicious in this paper. The primary concept to save privacy is to put in some changes on the original matrix for encryption and send the encrypted matrix to the cloud. The cloud then uses any LU decomposition algorithm on the encrypted matrix and divide the matrix for calculating the log-determinant. After returning from the cloud some transformations are applied to the returned result to get the accurate log-determinant of the original matrix. [ABSTRACT FROM AUTHOR]

Abraham model correlations are reported for describing the logarithms of water-to-plant cuticle partition coefficients and logarithms of air-to-plant partition coefficients based on experimental values retrieved from the published chemical and environmental literature for 24 different plant species. Data sets containing more than 200 experimental partition coefficients were used in determining each correlation equation. The derived correlations back-calculated the experimental data to within a standard deviation of 0.23 log units. [ABSTRACT FROM AUTHOR]

In 1956, John Kelly formulated an optimal strategy, the so-called 'Kelly criterion', for bidding at each step of a favorable game when the odds and probability of winning are known. The Kelly criterion is used to theoretically maximize long-run return. However, in practical situations, it is impossible to play a game for an unlimited time. In this study, we analyze the return of a game when a player must bid after a finite number of time steps. We demonstrate that the logarithm of the return of a game when bidding the optimal fraction (corresponding to the difference between the player's belief probability of the event occurrence and the proportion of real outcomes) for finite-time steps. Moreover, the maximum of the logarithm of the return is achieved when the above two are equal. Finally, a few simulation experiments are conducted for illustrative purposes. We also present an example of a case in stock trading to demonstrate the practicality of this study. [ABSTRACT FROM AUTHOR]

Two new properties of the kernel distribution function estimator of diverse nature are derived. Firstly, a law of the iterated logarithm is proved for both the integrated absolute error and the integrated squared error of the estimator. Secondly, the maximal smoothing principle in kernel density estimation developed by Terrell is extended to kernel distribution function estimation, which allows, among others, the derivation of an alternative quick-and-simple bandwidth selector. In fact, there is a common link between the two topics: both problems are solved through the use of the same, not-so-standard, methodology. The results based on simulated data and a real data set are also presented. [ABSTRACT FROM AUTHOR]

In the paper, we consider some piecewise deterministic Markov process, whose continuous component evolves according to semiflows, which are switched at the jump times of a Poisson process. The associated Markov chain describes the states of this process directly after the jumps. Certain ergodic properties of these two dynamical systems have been already investigated in our recent papers. We now aim to establish the law of the iterated logarithm for the aforementioned continuous-time process. Moreover, we intend to do this using the already proven properties of the discrete-time system. The abstract model under consideration has interesting interpretations in real-life sciences, such as biology. Among others, it can be used to describe the stochastic dynamics of gene expression. [ABSTRACT FROM AUTHOR]

We show that the only Fibonacci numbers that are concatenations of two repdigits are 13, 21, 34, 55, 89, 144, 233, 377. [ABSTRACT FROM AUTHOR]

Interaction of pectin with starch modified by heat-moisture treatment (HMT) was evaluated. HMT preserved the presence of starch fractions with k1 and k2 (digestion rate constants at rapid and slow phase, respectively) being 4.5 and 0.42 × 10−2 min−1. Compared to the native starch, HMT starch had lower k1 and k2 probably due to the change in crystalline type. HMT increased total digestible starch from 1.02% in native to 27.23% in HMT starch. The decrease in k1 and/or k2 with the increase in pectin might be linked to the effect of viscosity or pectin–amylase interaction, while the decrease in digestible starch at rapid phase (C1∞) could be attributed to pectin–starch interaction. Pectin affected significantly water absorption capacity, syneresis, cohesiveness and gumminess of HMT starch but did not alter those of native starch. The effect of HMT on functional, textural and pasting properties was more remarkable than the effect of pectin. [ABSTRACT FROM AUTHOR]

After establishing the moderate deviation principle by the classical Azencott method, we prove the Strassen's compact law of the iterated logarithm (LIL) for a class of stochastic partial differential equations (SPDEs). As an application, we obtain this type of LIL for two population models known as super-Brownian motion and Fleming-Viot process. In addition, the classical LIL is shown for the class of SPDEs and the two population models. [ABSTRACT FROM AUTHOR]

Time series model selection has been widely studied in recent years. It is of importance to select the best model among candidate models proposed for a series in terms of explaining the procedure that governs the series and providing the most accurate forecast for the future observations. In this study, it is aimed to create an algorithm for order selection in Box–Jenkins models that combines penalized natural logarithm of mutual information among the original series and predictions coming from each candidate. The penalization is achieved by subtracting the number of parameters in each candidate and empirical information the data provide.Simulation studies under various scenarios and applications on real data sets imply that our algorithm offers a promising and satisfactory alternative to its counterparts. [ABSTRACT FROM AUTHOR]

We investigate some topics related to the celebrated Baker-Campbell-Hausdorff Theorem: a non-convergence result and prolongation problems. Given a Banach algebra A with identity I, and given X , Y ∈ A , we study the relationship of different issues: the convergence of the BCH series ∑ n Z n (X , Y) , the existence of a logarithm of e X e Y , and the convergence of the Mercator-type series ∑ n (− 1) n + 1 (e X e Y − I) n / n which provides a selected logarithm of e X e Y . We fix general results, among which we provide a non-convergence result for the BCH series, and (by suitable matrix counterexamples) we show that various pathologies can occur. These are related to some recent results, of interest in physics, on closed formulas for the BCH series: while the sum of the BCH series presents several non-convergence issues, these closed formulas can provide a prolongation for the BCH series when it is not convergent. On the other hand, we show by suitable counterexamples that an analytic prolongation of the BCH series can be singular even if the BCH series itself is convergent. [ABSTRACT FROM AUTHOR]

In this paper, we consider the law of the iterated logarithm for error density estimators in the nonlinear autoregressive models under appropriate assumptions. Moreover we get the law of the iterated logarithm for error density estimators in first-order autoregressive models and threshold models as corollaries. [ABSTRACT FROM AUTHOR]

Mathematics educators view the equals sign as a bidirectional relation symbol, but the authors have observed that students might not have such flexibility in their understanding of the equals sign. The authors have observed that students have a hard time viewing mathematical properties (for example, log (MN) = log (M) + log (N) ) with bidirectional flexibility, and subsequently wondered if the "direction" that students are presented mathematical properties has an impact on their ability to use the properties successfully. The authors studied this phenomenon in 11 undergraduate, introductory-level mathematics courses (College Algebra and Precalculus) by presenting logarithmic properties in different "directions" and seeing what effect that had on students' ability to apply the logarithmic properties successfully. Our results indicate that, despite careful classroom discussion of the meaning of the properties and sustained problem solving activities, the "direction" that the properties are presented to the students does have an effect on their ability to use the properties successfully. [ABSTRACT FROM AUTHOR]

In order to explore tonality outside of the "Pythagorean" paradigm of integer ratios, Robert Schneider introduced a musical scale based on the logarithm function. We seek to refine Schneider's scale so that the difference tones generated by different degrees of the scale are themselves octave equivalents of notes in the scale. In doing so, we prove that a scale which contains all its difference tones in this way must consist solely of integer ratios. With this in mind, we present some methods for producing logarithmic scales which contain many, but not all, of the difference tones they generate. [ABSTRACT FROM AUTHOR]

This paper deals with the statistical studies of the normal tempered stable model defined by Barndorff-Nielsen and Shephard. It represents the natural extension of the normal inverse Gaussian one introduced by Barndorff-Nielsen. We basically use the Monte-Carlo's approximation in order to simulate this distribution. We introduce a linear regression model with normal tempered stable error. We apply this model for the analyzing of the daily logarithm returns data on CAC40 index. The parameters estimation results show that this model better deals with long tailed distribution which is the case for the CAC40 logarithm returns. [ABSTRACT FROM AUTHOR]

We analyze the set of increasingly enumerable additive submonoids of R , for instance, the set of logarithms of the positive integers with respect to a given base. We call them ω-monoids. The ω-monoids for which consecutive elements become arbitrarily close are called tempered monoids. This is, in particular, the case for the set of logarithms. We show that any ω-monoid is either a scalar multiple of a numerical semigroup or a tempered monoid. We will also show how we can differentiate ω-monoids that are multiples of numerical semigroups from those that are tempered monoids by the size and commensurability of their minimal generating sets. All the definitions and results are illustrated with examples from music theory. [ABSTRACT FROM AUTHOR]

We have Derringer and Suich's desirability functions in mind, especially, the two-sided ones in our analysis throughout this study. We propose and develop a finite partitioning procedure of the individual desirability functions over their compact and connected interval which leads to the definition of generalized desirability functions. We call the negative logarithm of an individual desirability function having a max-type structure and including a finite number of nondifferentiable points as a generalized individual desirability function. By introducing continuous selection functions into desirability functions and, especially, employing piecewise max-type functions, it is possible to describe some structural and topological properties of these generalized functions. Our aim with this generalization is to show the mechanism that gives rise to a variation and extension in the structure of functions used in classical desirability approaches. [ABSTRACT FROM AUTHOR]

Let be a polynomial. The Gauss-Lucas theorem states that its critical points, , are contained in the convex hull of its roots. In a recent quantitative version, Totik shows that if almost all roots are contained in a bounded convex domain , then almost all roots of the derivative are in an neighborhood (in a precise sense). We prove a quantitative version: if a polynomial p has n roots in K and roots outside of K, then has at least n−1 roots in. This establishes, up to a logarithm, a conjecture of the first author: we also discuss an open problem whose solution would imply the full conjecture. [ABSTRACT FROM AUTHOR]

This article addresses permutation flowshop scheduling problems with simple linear deterioration. The objectives are to minimize logarithm of the makespan, total logarithm of the completion time, total weighted logarithm of the completion time, and the sum of the quadratic job logarithms of the completion times. Approximation algorithms and their worst-case bounds are presented and analysed. Branch-and-bound algorithms are also proposed to solve the problems optimally. Computational experiments are performed to illustrate the algorithms. [ABSTRACT FROM AUTHOR]

A simple art project which attempts to aid the student in visualizing exponentials and logarithms using areas of quadrilaterals and hexagons. There is an emphasis on learning through play and creativity, and is intended as an aid to revision of the topic. The project is split into two parts, the first dealing with a single base number to powers of whole numbers and the second to several bases to powers of fractions and whole numbers. [ABSTRACT FROM AUTHOR]

This article describes a technique for numerically solving a class of nonlinear boundary integral equations of the second kind with logarithmic singular kernels. These types of integral equations occur as a reformulation of boundary value problems of Laplace's equations with nonlinear Robin boundary conditions. The method uses thin plate splines (TPSs) constructed on scattered points as a basis in the discrete collocation method. The TPSs can be seen as a type of the free shape parameter radial basis functions which establish effective and stable methods to estimate an unknown function. The proposed scheme utilizes a special accurate quadrature formula based on the non-uniform Gauss-Legendre integration rule for approximating logarithm-like singular integrals appeared in the approach. The numerical method developed in the current paper does not require any mesh generations, so it is meshless and independent of the geometry of the domain. The algorithm of the presented scheme is accurate and easy to implement on computers. The error analysis of the method is provided. The convergence validity of the new technique is examined over several boundary integral equations and obtained results confirm the theoretical error estimates. [ABSTRACT FROM AUTHOR]

Purpose: To assess the visual function in patient with myopic foveoschisis (MF). Methods: Fifty-one eyes of 51 subjects were enrolled and divided into myopic foveoschisis group (MFG, N = 16), myopic control group (MCG, N = 18), and normal control group (NCG, N = 17) according to their diagnoses. Measurements of the logarithm of the minimum angle of resolution (LogMAR), the objective scatter index (OSI), the cut-off value of modulation transfer function (COMTF), the Strehl ratio (SR), the foveal threshold (FT), and the amplitude density (AD) in the first central ring of multifocal electroretinography were recorded and compared amongst groups. Results: LogMAR of MFG was the highest (MFG vs. MCG p＜0.001, MFG vs. NCG p＜0.001), and LogMAR of MCG and NCG were similar (p = 0.258). OSI of MFG was higher than that of MCG (p＜0.001), and the latter was higher than OSI of NCG (p＜0.001). COMTF was lowest in MFG, intermediate in MCG and highest in NCG (MFG vs. MCG p = 0.001, MFG vs. NCG p＜0.001, MCG vs. NCG p＜0.001). SR, FT, and AD were all similar between MFG and MCG (p = 0.187, p = 0.401, and p = 0.446, respectively) while highest in NCG (SR MFG vs. NCG p＜0.001, MCG vs. NCG p＜0.001; FT MFG vs. NCG p = 0.003, MCG vs. NCG p = 0.025; AD MFG vs. NCG p＜0.001, MCG vs. NCG p＜0.001). Conclusions: Similarities were found between various functional parameters from different devices, revealing that the visual function in MF patients was impaired. [ABSTRACT FROM AUTHOR]