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We investigate two notions of independence—(usual) independence and complete independence—applied to the Peano axioms for the sequence of natural numbers. We review the results that, although they are independent, the Peano axioms are not completely independent. The standard proof that the Peano axioms are not completely independent is algebraic, in the sense that it makes essential reference to the relationship between several mathematical structures that satisfy, or do not satisfy, these axioms. We then present an alternative logical proof, which makes no essential references to the relationship between mathematical structures. There is a completely independent set of axioms for the sequence of natural numbers, but it is based on primitives different from those originally adopted by Peano. Therefore, we present a new completely independent set of axioms based on the same set of primitives as the one originally adopted by Peano. [ABSTRACT FROM AUTHOR]

This article examines the influence of mathematics on early modern and modern moral thought. In particular, I trace the origin and development of the "moral arithmetic" (which has previously been misattributed); that is, the attempt to introduce quantificatory calculation to morality. Tracing the origins and development of both the term and the idea, I show that it was coined by the Third Earl of Shaftesbury in the seventeenth century, and that it contributed to the rise of rationalistic, utilitarian Benthamite thought in the nineteenth century, laying the foundations for the modern discipline of mathematical economics. In particular, I emphasize that, while today we habitually associate ideas of self-interest with "cold, hard" mathematical analyses, the original application of mathematics to moral thought began among the theologians and philosophers of the seventeenth and eighteenth centuries in order to combat ideas of self-interest. I conclude by reflecting on some of the problems with the scope and limits of the history of political thought, and argue for the importance of drawing intellectual history and the history of science, in particular the history of mathematics, closer together. [ABSTRACT FROM AUTHOR]

The Laguerre calculus is widely used for the inversion of differential operators on the Heisenberg group. Applying the Laguerre calculus established on nilpotent Lie groups of step two in Chang et al. [The Laguerre calculus on the nilpotent Lie group of step two. Preprint; 2019. Available from: ], we find the explicit formulas for the heat kernel of sub-Laplace operator and the fundamental solution of power of sub-Laplace operator on nilpotent Lie groups of step two. Calin, Chang and Markina [Generalized Hamilton–Jacobi equation and heat kernel on step two nilpotent Lie groups. In: Gustafsson B, Vasil'ev A, editors. Analysis and mathematical physics. Basel: Birkhüser; 2009 (Trends in mathematics)] also get the formulas for the heat kernel of sub-Laplace operator on nilpotent Lie groups of step two by using the Hamiltonian and Lagrangian formalisms that are related to geometric mechanics. In this paper, we use a totally different method to prove our main results by using the Laguerre calculus, which is more direct from the point of view of Fourier analysis. [ABSTRACT FROM AUTHOR]

Some improvements of the celebrated Schwarz inequality in complex inner product spaces are given. Applications for n-tuples of complex numbers are provided. [ABSTRACT FROM AUTHOR]

The purpose of this article is to study cohomology and representations of Bihom-Lie superalgebras. In particular, trivial representations and adjoint representations of Bihom-Lie superalgebras are studied in detail. Derivations and central extensions of Bihom-Lie superalgebras are also discussed as an application. [ABSTRACT FROM AUTHOR]

Let T and S be two self-adjoint positive invertible operators of a complex Hilbert space H. In this paper, we investigate some inequalities involving the quadratic forms of the weighted arithmetic and harmonic means of T and S. Application for operator entropies is also discussed. [ABSTRACT FROM AUTHOR]

Motivated by Hirano-Tominaga's work on rings for which every element is a sum of two idempotents and by de Seguins Pazzis's results on decomposing every matrix over a field of positive characteristic as a sum of idempotent matrices, we address decomposing every matrix over a commutative ring as a sum of three idempotent matrices and, respectively, as a sum of three involutive matrices. [ABSTRACT FROM AUTHOR]

When A is a density matrix in , Audenaert [J Math Phys. 2007;48] proved an interesting inequality for Schatten q-norms where and stand for the first and second partial trace, respectively, a notion from quantum information theory. As an analogue of this result, Lin [Canad Math Bull. 2016;59] recently presented the following determinantal inequality In this note, we first correct a result of Kuai [Linear Multilinear Algebra. doi:10.1080/03081087.2017.1304521] and then generalize the above two inequalities to a larger class of matrices whose numerical ranges are contained in a sector. [ABSTRACT FROM AUTHOR]

Research has shown that the types of tasks assigned to students affect their learning. Various authors have described desirable features of mathematical tasks or of the activity they initiate. Others have suggested task taxonomies that might be used in classifying mathematical tasks. Drawing on this literature, we propose a set of task types that are deemed appropriate for undergraduate students and which foster mathematical habits of mind. These are: generating examples; analyzing reasoning; evaluating mathematical statements; conjecturing and/or generalizing; visualizing; and using definitions. We give rationales for our choices and examples of each type of task suitable for use in an introductory calculus course. [ABSTRACT FROM AUTHOR]

The order of subgroups with large relative commutativity degrees are determined. Also a relationship between the number of relative commutativity degrees of a group and the length of its subgroup chains is obtained. [ABSTRACT FROM AUTHOR]

Given that the ℓ-rings RL of real-valued continuous functions on completely regular frames L are monoreflective in the category of all archimedean f-rings with unit, one can ask how a sub-ℓ-ring A with unit of some RL has to be related to L to make RL the corresponding reflection of A. This note provides an answer in terms of a uniformity on L naturally determined by A, and then establishes the analogous result for the ℓ-ring ZL of real-valued continuous functions on 0-dimensional frames and the archimedean f-rings with singular unit. [ABSTRACT FROM AUTHOR]

In this paper, for a prime p, we propose some plausible definitions for the notion of Sylow fuzzy p-subgroup of a finite group. We derive a number of results for finite fuzzy groups using one of the proposed definitions. We also discuss some of the relationships between various proposed definitions for suitability, including the crisp case, with some examples. [ABSTRACT FROM AUTHOR]

In this note we address a problem that was suggested by Petra Staynova in [6] regarding quasi-Lindelöf spaces, and continue to study topological properties of quasi-Lindelöf spaces. In particular, we construct an example of a Tychonoff weakly Lindelöf space which is not quasi-Lindelöf. [ABSTRACT FROM AUTHOR]

Two new subclasses Hp, k(λ, A, B) and Qp, k(λ, A, B) of meromorphically multivalent functions starlike with respect to k-symmetric points are introduced. Distortion bounds and inclusion relations for these subclasses are obtained. [ABSTRACT FROM AUTHOR]

This paper deals with extension of analytic covers. We prove topological extension theorems for analytic covers. The main result is an extension theorem which only uses the extension of the ramification divisor. We give also a Thullen-type and a Hartogs-type extensions theorems. [ABSTRACT FROM AUTHOR]

In this paper, for 𝒞 a strict braided monoidal category with tensor product ⊗, we improve the definition of Hom-associative algebra by removing the multiplicativity condition of the automorphismα. After that we state the close connection between the classical notions of (co)algebra, (co)module and Hopf algebra and the corresponding ones in the Hom-world. As a consequence we can study the questions about the Hom-world by passing to the classical one, and the main contribution of this paper is to illustrate the suitable procedure to move from one side to the other. Finally, we apply our techniques to get in the Hom setting some classical results about cleft and Galois extensions. [ABSTRACT FROM PUBLISHER]

Catalan and Pippenger discovered infinite products for various values related to e. Based on these, infinite products for e1/2 and e2/3 were found by Sondow and Yi [3], who furthermore conjectured that the products could be generalized for a power of e1/k . The discoveries in this paper result from attempting to prove this conjecture. By generalizing Sondow and Yi's products, infinite products for powers of e1/k are found. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of sBCI/sBCK/eBCI/eBCK-algebras as a generalization of the notion of BCI/BCK-algebras. This structure is studied in detail. Also we introduce a way to make an eBCK-algebra from a BCK-algebra and vice versa. [ABSTRACT FROM AUTHOR]

A modified Degasperis-Procesi equation is investigated. The local existence and uniqueness of the strong solution for the equation are established in the Sobolev space Hs(R) with s > 3/2. The L¹(R) local stability for the strong solution is obtained under certain assumptions on the initial data. [ABSTRACT FROM AUTHOR]

In clinical studies, the period before an event occurs is frequently considered as an outcome. The general term used for this analytical method in clinical studies is "survival time analysis". The method employs Cox regression to compare hazards between groups. In this paper, we propose an index defined in the Bayesian framework to compare hazards. This index compares the sizes of the hazards directly. Further, we propose two methods to calculate the sizes of the hazards. One is an approximate method, by which we can easily calculate the size, and the other is an exact method. In addition, we apply the proposed index to two actual clinical studies to assess its usability. [ABSTRACT FROM AUTHOR]

In the paper, the authors find the logarithmically complete monotonicity of the Catalan-Qi function related to the Catalan numbers. [ABSTRACT FROM AUTHOR]

In this article, we introduce the Zweier Paranorm I-convergent double sequence spaces 2ZI(q), 2Z0I(q) and 2Z∞I(q) for q = (qij), a sequence of positive real numbers. We study some algebraic and topological properties on these spaces. [ABSTRACT FROM AUTHOR]

This article emerged from an analysis of learners’ responses to a task presented to learners studying Mathematics and Mathematical Literacy (ML) in Gauteng, South Africa. Officially, Mathematics and ML are two separate learning areas. Learners from Grade 10 onwards are supposed to take either one or the other, but not both. This means that by Grade 11, learners would have acquired different kinds of knowledge and problem solving skills depending on which of these they take. The article demonstrates that this possibility may not necessarily happen. Grade 11 learners from both learning areas appeared to have acquired sufficient knowledge and skills and were able to solve similar Mathematics tasks in similar ways, with similar adequacy. We argue that, contrary to common perceptions in the field of Mathematics education (particularly in South Africa), engaging in ML does not and should not make one less mathematically advanced than engagement in pure Mathematics. [ABSTRACT FROM PUBLISHER]

Classification rules with a reserve judgment option provide a way to satisfy constraints on the misclassification probabilities when there is a high degree of overlap among the populations. Constructing rules which maximize the probability of correct classification while satisfying such constraints is a difficult optimization problem. This paper uses the form of the optimal solution to develop a relatively simple and computationally fast method for three populations which has a non parametric quality in controlling the misclassification probabilities. Simulations demonstrate that this procedure performs well. [ABSTRACT FROM AUTHOR]

This article presents a bird's eye view of the celebrated theorems of Picard and their close relatives, with particular emphasis on some surprising developments of the past half century. [ABSTRACT FROM AUTHOR]

Under some weak conditions, we use the moment inequality for-mixing sequence to investigate the Bahadur representation of sample quantiles based on this stochastic sequence. For all, we obtain a rate of Bahadur representation in the form of, which is close to the rate. [ABSTRACT FROM PUBLISHER]

We show that the low rank matrix completion problem can be reduced to the problem of finding the rank of a certain tensor. [ABSTRACT FROM AUTHOR]

The article reviews the main situations one may encounter in constrained optimization of computer experiments, some of the corresponding existing tools and areas open for innovation. Topics discussed include the analytical formulas of the main criterion of L. Gramacy et al for the single constraint problem and the general behavior of the AL strategy and proposed modifications to improve its efficiency.

We are grateful for the many insightful comments provided by the discussants. One team politely pointed out oversights in our literature review and the subsequent omission of a formidable comparator. Another made an important clarification about when a more aggressive variation (the so-called NoMax) would perform poorly. A third team offered enhancements to the framework, including a derivation of closed-form expressions and a more aggressive updating scheme; these enhancements were supported by an empirical study comparing new alternatives with old. The last team suggested hybridizing the statistical augmented Lagrangian (AL) method with modern stochastic search. Here we present our responses to these contributions and detail some improvements made to our own implementations in light of them. We conclude with some thoughts on statistical optimization using surrogate modeling and open-source software. [ABSTRACT FROM AUTHOR]

This paper discusses the estimation of average treatment effects in observational causal inferences. By employing a working propensity score and two working regression models for treatment and control groups, Robins et al. (1994, 1995) introduced the augmented inverse probability weighting (AIPW) method for estimation of average treatment effects, which extends the inverse probability weighting (IPW) method of Horvitz and Thompson (1952); the AIPW estimators are locally efficient and doubly robust. In this paper, we study a hybrid of the empirical likelihood method and the method of moments by employing three estimating functions, which can generate estimators for average treatment effects that are locally efficient and doubly robust. The proposed estimators of average treatment effects are efficient for the given choice of three estimating functions when the working propensity score is correctly specified, and thus are more efficient than the AIPW estimators. In addition, we consider a regression method for estimation of the average treatment effects when working regression models for both the treatment and control groups are correctly specified; the asymptotic variance of the resulting estimator is no greater than the semiparametric variance bound characterized by the theory of Robins et al. (1994, 1995). Finally, we present a simulation study to compare the finite-sample performance of various methods with respect to bias, efficiency, and robustness to model misspecification. [ABSTRACT FROM AUTHOR]

The adhesive contact between a sphere and a longitudinal wavy surface is simulated numerically. A modified simulation method is proposed using the Newton BI-CGSTAB method in a rectangular coordinate. The effective Tabor parameter is proposed. It is found that when the amplitude of the wavy surface is larger, the contact area is smaller and the pull-off force is smaller. Jump-in from noncontact phenomena occurs when the Tabor parameter is large. Jumping from one ridge to the next ridge occurs when the effect of the Tabor parameter is large and the amplitude of the wavy surface is not too small. Jumping from noncontact to full contact is affected by the amplitude and the wave number of the wavy surface and is also affected by the Tabor parameter. [ABSTRACT FROM PUBLISHER]

Letbe the algebra of allreal matrices. Anreal matrixis even doubly stochastic if it is a convex combination of even permutation matrices. For,is said to be even majorized by(written as), if there exists an even doubly stochastic matrixsuch that. In this paper, the concept of even majorization is investigated and then the linear preservers and strong linear preservers of this concept are characterized on. [ABSTRACT FROM PUBLISHER]

We obtain a containment region for a Fischer-type determinantal ratio for matrices with given angular numerical range less than or equal to a right angle. [ABSTRACT FROM PUBLISHER]

In this paper, we present the necessary and sufficient conditions for which the given pair of closed subspaces has a pair of non-trivial principal invariant subspaces. Using this we show that every pairof closed subspaces withorhas a pair of non-trivial principal invariant subspaces. Also we prove that the pairof closed subspaces withhas a pair of non-trivial principal invariant subspaces if and only ifSimilar results are proved for non-degenerate principal invariant subspaces. [ABSTRACT FROM PUBLISHER]

A matrixover a field is minimal if for every matrixthat commutes with, the centralizer ofis a subset of the centralizer of. In this paper, we study the minimal matrices over the field with two elements. We characterize the minimal matrices with their minimal polynomial of the form, whereis an irreducible polynomial and. We also characterize all minimal matrices with spectrum in. [ABSTRACT FROM AUTHOR]

Letdenote an algebraically closed field of characteristic zero and letdenote an even at least. Letandbebymatrices. Thenis a Leonard pair onof Bannai/Ito type. We determine all the matricessuch thatform a Leonard triple on. [ABSTRACT FROM AUTHOR]

Number processing disorder is an acquired deficit in mathematical skills commonly observed in Alzheimer’s disease (AD), usually as a consequence of neurological dysfunction. Common impairments include syntactic errors (800012 instead of 8012) and intrusion errors (8 thousand and 12 instead of eight thousand and twelve) in number transcoding tasks. This study aimed to understand the characterization of AD-related number processing disorder within an alphabetic language (English) and ideographical language (Chinese), and to investigate the differences between alphabetic and ideographic language processing. Chinese-speaking AD patients were hypothesized to make significantly more intrusion errors than English-speaking ones, due to the ideographical nature of both Chinese characters and Arabic numbers. A simplified number transcoding test derived from EC301 battery was administered to AD patients. Chinese-speaking AD patients made significantly more intrusion errors (p = 0.001) than English speakers. This demonstrates that number processing in an alphabetic language such as English does not function in the same manner as in Chinese. The impaired inhibition capability likely contributes to such observations due to its competitive lexical representation in brain for Chinese speakers. [ABSTRACT FROM PUBLISHER]