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A main drawback of classical Tikhonov regularization is that often the parameters required to apply theoretical results, e.g., the smoothness of the sought-after solution and the noise level, are unknown in practice. In this paper we investigate in new detail the residuals in Tikhonov regularization viewed as functions of the regularization parameter. We show that the residual carries, with some restrictions, the information on both the unknown solution and the noise level. By calculating approximate solutions for a large range of regularization parameters, we can extract both parameters from the residual given only one set of noisy data and the forward operator. The smoothness in the residual allows to revisit parameter choice rules and relate a-priori, a-posteriori, and heuristic rules in a novel way that blurs the lines between the classical division of the parameter choice rules. All results are accompanied by numerical experiments.

In this paper, we consider a special class of four-dimensional real division algebras called transitive division algebras, and its subclass – transitive-diagonal division algebras. We explicitly de...

We propose an algorithm that allows calculating the remainder and the quotient of division between polynomials over commutative coefficient rings, without polynomial long division. We use the previ...

We examined the type of errors on multiplication and division computation problems of 326 rising fifth graders enrolled in four elementary schools in Northern Portugal. We further examined whether there was a difference in the number of errors across age and whether there was an association between students' performance on number knowledge and multiplication and division computation problems. Error analysis of students' responses indicated that miscalculation and no attempt to solve the problem were the two most frequent error types. We found that older students made more errors compared to younger students. We argue that knowledge of individual student error types is critical to making sound instructional decisions. Based on the results of the present study, we discuss implications for future research and classroom practice., This research was sponsored by a research award to the second author from the Fulbright U.S. Scholar Program.

Dickson’s commutative semifields are an important class of finite division algebras. We generalize Dickson’s construction over any base field by doubling not just finite field extensions (which wou...

The ratio content error estimators recommended by the United Nations Statistics Division for censuses do not cover content errors resulting from erroneous enumerations and omissions; thus they unde...

The present study aims to examine the pedagogical effectiveness of a Chinese mathematical dialogue-based intelligent tutoring system used for teaching mathematics. The mathematical unit ‘multiplica...

This article investigates the cohomological kernels ker(H2n+1F→H2n+1E) of field extensions E/F in characteristic two where separable part of E is a quadratic extension F(α) and E is quadratic or qu...

We propose some generalizations of the classical Division Algorithm for polynomials over coefficient rings (possibly non-commutative). These results provide a generalization of the Remainder Theore...

We propose a divisibility criterion for elements of a generic Unique Factorization Domain. As a consequence, we obtain a general divisibility criterion for polynomials over Unique Factorization Dom...

This study focuses on proportional division, a specific problem type of proportions and proportional relationships, which embeds a part-part-whole relationship. We investigated how students in grad...

We propose a generalization of the classical Remainder Theorem for polynomials over commutative coefficient rings that allows calculating the remainder without using the long division metho...

This paper proposes a new model order reduction method for the large-scale dynamic systems using the factor division approach to allow the preservation of dominant poles. For a given higher order m...

The aim of this paper is to construct a new model order reduction method for linear dynamic systems. In this technique, the denominator polynomial of the reduced order model (ROM) is obtain...

Let k be a field of characteristic different from 2 and 3. In this paper, we study the number of rank-2 k-tori required (in fact exhibit such tori explicitly) for the generation of groups of type A2, G2 and F4 arising from division algebras and subgroups of type D4 of Aut(A), for A an Albert division algebra, over infinite perfect fields.

Probability Proportional to Size (PPS) sampling is one of the most widely used designs for finite populations. We propose modifications to PPS designs with replacement and Rao-Hartley-Cochran design without replacement. These modifications consist of division of the population into two groups. Units in the first group are included in the sample with probability one. Under certain conditions, in both with and without replacement designs, the estimator of the population total based on the modified PPS sampling design is shown to be better than the corresponding estimator based on a PPS design. We illustrate our modification by an example and an application.

In a recent Monthly note [1], Athreya and Aulicino found a closed geodesic on the regular dodecahedron at a vertex not passing through any other vertices. This work has been popularized in a Number...

The paper explores the division of a polygon into equal-area pieces using line segments originating at a common point. The mathematical background of the proposed method is very simple and belongs to secondary school geometry. Simple examples dividing a square into two, four or eight congruent pieces provide a starting point to discovering how to divide a regular polygon into any number of equal-area pieces using line segments originating from the centre. Moreover, it turns out that there are infinite ways to do the division. Discovering the basic invariant involved allows application of the same procedure to divide any tangential polygon, as after suitable adjustment, it can be used also for rectangles and parallelograms. Further generalization offers many additional solutions of the problem, and some of them are presented for the case of an arbitrary triangle and a square. Links to dynamic demonstrations in GeoGebra serve to illustrate the main results.

An understanding of partitive division is foundational for numerous other mathematics topics, including unit rate, slope, and probability. However, research has shown that learners tend to have a l...

The purpose of this paper is to investigate the eight-dimensional non-associative unital division algebras {0}≠A over finite fields, admitting elementary abelian 2-group of automorphisms E≤Aut(A), ...

Let n≥3 be a natural number. Let Mn(𝔻) be the ring of all n×n matrices over a noncommutative division ring 𝔻. In the present paper, we will find the description of all additive mappings G:Mn(𝔻)→Mn(𝔻) such that [G(y),y] = G(y)y−yG(y) = 0 for all rank-1 matrix y. Precisely, we will prove that G(x) = λx+μ(x) for all x∈Mn(𝔻), where λ lies in the center of 𝔻 and μ is a central map.

Let F be a Henselian field such that F¯ is a global field. It is shown that if D is an F-central division algebra such that charF¯∤deg(D) then D contains a cyclic division algebra of prime degree even though D can be a noncrossed product. Here, our approach is based on the Albert’s cyclicity criterion and centralizer theorem.

This paper deals with the two-dimensional satellite module polygon packing problem. Based on the duality of material and space, it regards the polygon packing problem as a space allocation problem, which involves allocating the container space to the given polygons reasonably and efficiently. Ant colony’s labour division is essentially a kind of task allocation. Using this task allocation to achieve the space allocation in polygon packing problems, a flexible labour division approach (FLD) is proposed based on the response threshold model. According to the characteristics of space allocation in polygon packing problems, FLD designs three actions for polygons to occupy the container space. With the interaction between environmental stimulus and response threshold, each polygon takes an appropriate action to complete the space allocation and a layout that meets the requirements of satellite module layout is obtained. The results of standard test instances demonstrate the effectiveness of FLD when compared w...

Acceptance sampling, a major division in Statistical quality control, provides a guaranteed confidence on the product quality. Conventionally, when sampling plans are used all bad lots are rejected...

In this paper, we use the index to investigate the supplier selection problem for one-sided processes with multiple independent characteristics. We first review the existing approach, which we refer to as the division method, then develop a new exact approach called the subtraction method. A two-stage selection procedure is developed based on the subtraction method for practical applications. We then compared the two methods with regard to selection power. The results show that the subtraction method we propose is indeed more powerful than the existing division method. A thin-film transistor type liquid-crystal display (TFT-LCD) application example is provided to illustrate the testing procedure.

An approach based on homotopy continuation algorithm is presented to identify the parameters of a cracked beam. Euler–Bernoulli finite beam element with a fully opened crack model is adopted to establish the dynamic equation of the structural system. In the inverse problem, the homotopy equation is derived from minimizing the error between the calculated and the simulated measured acceleration responses. The range of homotopy parameter is divided into a number of divisions. Newton iterative method is employed to estimate the solution at each of these division points. The solution at the last division point corresponds to the homotopy equation matching the objective function. Numerical simulations with a simply supported beam and two-span beam show that the proposed method is very accurate compared to an existing method for both single and multiple cracks identification. The effects of type of excitation, division of the homotopy parameter and measurement noise on the identified results are discuss...

Let k be a field of characteristic different from 2 and K a noncommutative division k-algebra of finite dimension over k. In the paper, subgroups of the general linear group GL2(K) that contain a subgroup which is conjugate in GL2(K) to the group of matrices with r ∈ k are described.

This study investigated 11 pre-service middle school teachers’ solution strategies for exploring their knowledge of fraction division interpretations. Each participant solved six fraction division problems. The problems were organized into two sets: symbolic problems (involving numbers only) and contextual problems (involving measurement interpretation and the determination of unit rate interpretation). Results showed that most of the participants exhibited a great amount of procedural knowledge as they applied algorithms to obtain the correct answers to the symbolic problems. They also exhibited a great amount of conceptual understanding of how and why they obtained the correct answers to the contextual problems. However, the pre-service middle school teachers neither provided interpretations to the symbolic problems nor accepted that the contextual problems involved fraction division operation. The results suggest that the measurement and rate concepts were often unlinked to fraction division.

We study division by 3 in Jacobians of genus 2 curves over binary fields with a 2-torsion subgroup of rank 1 or 2. We characterize the 3-torsion divisors and provide, for every , a formula for the coordinates of the divisors in the set .

Importance measures are used to identify weak components and/or states in a system based on the component state random variables, which seem to be inadequate to show the corresponding actual situations. By contrast, the performance random variables own significant practical meanings and eliminate the subjectivity and limitation of state division and definition in many actual situations. In this paper, instead of state random variables, the performance stochastic processes are used for modeling all the components and the entire system, and the integrated importance measure (IIM) for the performance random variables are extended. The generalized IIM evaluates the contribution of component performance to the desired level of system performance. A case study of an oil transmission system is used to illustrate the effectiveness of our approach with importance measures.

Among other results of this paper, we single out the following ones. We show that division rings are the only division semirings over which the categories of semimodules are Schreier varieties, i.e., all subsemimodules of free semimodules are free too. We a complete description of division semirings R over which the categories of semimodules ℳ R are p-Schreier varieties, i.e., varieties whose all projective algebras are free. We give a complete description of proper division semirings R whose categories of semimodules ℳ R(X) over the polynomial semirings R(X) over R, in not necessary commuting variables X, are p-Schreier varieties. We show that the categories of semimodules ℳ R(X) over the polynomial semirings R(X) over N-valued semirings R, in particular ℳ N(X), are p-Schreier varieties. We also show that for N-valued semirings S, the semimodule categories ℳ S never are Schreier varieties.

This paper deals with an interval-oriented approach to solve general interval constrained optimization problems. Generally, this type of problems has infinitely many compromise solutions. The aim of this approach is to obtain one of such solutions with higher accuracy and lower computational cost. The proposed algorithm is nothing but a different kind of branch and bound algorithm with multi-section division criterion of the search region (or box). In the proposed technique, the prescribed/accepted region is divided into several distinct subregions and in each feasible subregion the interval objective function value is computed. Then the subregion containing the best objective value is found by applying a specific interval ranking rule defined with respect to the pessimistic decision makers’ point of view. The process is continued until the interval width for each variable in the accepted subregion is negligible. Finally, the algorithm converges to a compromise solution in interval form. To illustrate the method and also to test the efficiency as well as the effectiveness of the proposed algorithm, we have solved some numerical examples.

In this article, we consider the supplier selection problem for two-sided processes with multiple independent characteristics. We review the existing division method and develop a new exact approach called the subtraction method. For practical applications, a two-phase selection procedure is established based on the subtraction method. The decision powers of two methods are compared. We show that the subtraction method we proposed is indeed more powerful than the existing division method. Several figures are presented to display the required sample sizes with various powers and various values of ). For the convenience of practitioners, the required sample sizes for the two methods with various powers and various values of ) are also tabulated.

A fast adaptive frequency calibration (AFC) technique with self-calibration for fast-locking phase-locked loops is presented with frequency-selecting switches. The proposed AFC directly calculates the proper switch states of the voltage-controlled oscillator (VCO). It requires only six clock cycles of the reference oscillator regardless of the number of VCO switches to reach the final switch state in the ideal case. The proposed method counts the number of VCO cycles per reference clock period for the minimum VCO frequency (MIN) and the maximum VCO frequency (MAX) during the first four-clock periods. For the following two-clock periods, the proper states of the VCO switches are set to the calculated value from MIN, MAX and the desired division ratio for a target frequency (EST). A frequency synthesiser with the proposed AFC was implemented on a 0.18 µm CMOS process. The AFC time decreased from 40 to 0.4 µs employing the proposed scheme such that the total lock time is 40 µs with the loop bandwidth of 40 kHz.

Elementary school students learn two types of division scenarios: partitive and quotitive. Previous researchers have assumed that the partitive scenario is easier because it reflects the everyday notion of sharing, whereas the quotitive scenario, which represents grouping, is more difficult and is understood gradually in the course of mathematics learning. However, this assumption has not been adequately investigated in empirical studies. The present study examines the assumption in a cross-sectional study. Participants were 336 elementary school students (98 in Grade 3, 82 in Grade 4, 88 in Grade 5, and 68 in Grade 6) and 70 university students who performed two tasks. In the preference task, they generated a division scenario of any type consistent with a given numerical equation. In the problem-posing task, they generated a division scenario consistent with both a numerical equation and a picture representing a partitive or quotitive scenario. On the preference task, students at all grade levels prefer...

Discrete global grid systems DGGSs are considered to be promising structures for global geospatial information representation. Square and triangular DGGSs have had the advantage over hexagonal ones in geospatial data processing over the past few decades. Despite a significant body of research supporting hexagonal grids as the superior alternative, the application thereof has been hindered partly owing to the lack of a hierarchy. This study presents an original perspective to combine two types of aperture 4 hexagonal discrete grid systems into a hierarchy. Each cell of the hierarchy is assigned a unique code using a linear quadtree that constructs the hexagonal quaternary balanced structure HQBS. The mathematical system described by HQBS addressing and the vector operations, including addition, subtraction, multiplication, and division, are defined. Essential spatial operations for HQBS cell retrieval, transformation between HQBS codes and other coordinate systems, and arrangement of HQBS cells on spherical surfaces were studied and implemented. The accuracy and efficiency of algorithms were validated through experiments. The results indicate that the average efficiency of cell retrieval using the HQBS is higher than that using other schemes, thus proving it to be more efficient.

Previous research has demonstrated the effectiveness of particular instructional practices that support students’ constructions of the partitive unit fraction scheme and measurement concepts for fractions. Another body of research has demonstrated the power of a particular mental operation—the splitting operation—in supporting students’ development of advanced fractional knowledge and algebraic reasoning. Steffe (2010) has hypothesized that students construct splitting through the unification of partitioning and iterating operations contained within the partitive unit fraction scheme. We used written assessments of 49 students, across sixth and seventh grades, to test this hypothesis. Our results show that students who have constructed a partitive unit fraction scheme go on to construct splitting within a relatively short period of time. Conversely, students who have not constructed a partitive unit fraction scheme generally do not construct splitting. We discuss these results and their implications for d...

The golden ratio, one of the most beautiful numbers in all of mathematics, arises in some surprising places. At first glance, we might expect that a General checking his troops’ progress would be nothing more than a basic distance-rate-time problem. However, further exploration reveals a multi-faceted problem, one in which the ratio of rates (rather than the explicit rates) is critical to the solution, and one that lends itself to several methods of solution with increasing levels of precision and complexity. This problem is accessible on several levels: it can provide students in elementary courses with an in-depth exposure to proportional reasoning and modeling, as well as offer students in upper division courses the challenge of finding the generalized solution and several intriguing connections to the Golden Ratio.

In this article, a competent interval-oriented approach is proposed to solve bound-constrained uncertain optimization problems. This new class of problems is considered here as an extension of the classical bound-constrained optimization problems in an inexact environment. The proposed technique is nothing but an imitation of the well-known interval analysis-based branch-and-bound optimization approach. Efficiency of this technique is strongly dependent on division, bounding, selection/rejection and termination criteria. The technique involves a multisection division criterion of the accepted/proposed search region. Then, we have employed the interval-ranking definitions with respect to the pessimistic decision makers’ point of view given by Mahato and Bhunia [Interval-arithmetic-oriented interval computing technique for global optimization, Appl. Math. Res. Express 2006 (2006), pp. 1–19] to compare the interval-valued objectives calculated in each subregion and also to select the subregion containing the...

In this paper, we consider the supplier selection problem, which deals with comparing two one-sided processes and selecting a better one that has a higher capability. We first review two existing approximation approaches, and an exact approach proposed which we refer to as the division method. We then develop a new exact approach called the subtraction method. We compare the two exact methods on the selection power. The results show that the proposed subtraction method is indeed more powerful than the division method. A two-phase selecting procedure is then developed based on the subtraction method for practical applications. Some computational results are tabulated for practitioners’ convenience.

This article presents a lesson that uses prediction items, clickers and visuals via PowerPoint slides to help prospective middle-school teachers address two common misconceptions: multiplication makes bigger and division makes smaller (MMB–DMS). Classroom research was conducted to explore the viability of such a lesson. Results show that the lesson was effective in creating awareness that multiplication does not always make bigger and division does not always makes smaller, uncovering students’ misconceptions, and providing opportunities for students to learn from mistakes. Students liked the activity for various reasons, such as getting to learn certain mathematical ideas, to think about the problems, to work in groups and to have fun. The lesson was implemented slightly differently in two classes. The class with an additional phase involving prediction and voting via clickers in the PowerPoint lesson showed a gain of 36 points (an effect size of 1.3 standard deviations, SDs) from the pre-test to the exi...

Discretisation, as one of the basic data preparation techniques, has played an important role in data mining. This article introduces a new hypercube division-based (HDD) algorithm for supervised discretisation. The algorithm considers the distribution of both class and continuous attributes and the underlying correlation structure in the data set. It tries to find a minimal set of cut points, which divides the continuous attribute space into a finite number of hypercubes, and the objects within each hypercube belong to the same decision class. Finally, tests are performed on seven mix-mode data sets, and the C5.0 algorithm is used to generate classification rules from the discretised data. Compared with the other three well-known discretisation algorithms, the HDD algorithm can generate a better discretisation scheme, which improves the accuracy of classification and reduces the number of classification rules.

In this article, we consider categories of all semimodules over semirings which are p-Schreier varieties, i.e., varieties whose projective algebras are all free. Among other results, we show that over a division semiring R all semimodules are projective iff R is a division ring, prove that categories of all semimodules over proper additively π-regular semirings are not p-Schreier varieties (in particular, this result solves Problem 1 of Katsov [8]), as well as prove that categories of all semimodules over cancellative division semirings are, in contrast, p-Schreier varieties.

We show in this article that in many cases the subfields of a nondegenerate tame semiramified division algebra of prime power degree over a Henselian valued field are inertial field extensions of t...

The aim of this paper is to propose a pedagogical explanation of the Le Cam theorem and to illustrate its use, through a practical application, for temporal cluster detection. This theorem focusses on the interval division by randomly chosen points. The aim of the theorem is to characterize the asymptotic behavior of a certain category of sums of functions applied to the length of successive intervals between points. It is not very intuitive and its understanding needs some deepening. After enouncing the theorem, its different aspects are explained and detailed in a way as pedagogical as possible. Theoretical applications are proposed through the proof of two propositions. Then a very concrete application of this theorem for temporal cluster detection is presented, tested by a power study, and compared with other global cluster detection tests. Finally, this approach is applied to the well-known Knox temporal data set.

In this paper, we present a least significant bit (LSB) hiding scheme that can embed variable-size secrets in LSB planes using the block-based chaotic map, adaptive difference range division, pixel-value differencing and modulus function. In particular, the block-based chaotic map in our framework is used to rearrange the locations of image blocks such that the transformed image can have more complex content than the original image; more precisely, the transformed image's inter-blocks have the low correlations while its intra-blocks have the high correlations. In addition, the adaptive division for difference range and pixel-value differencing are applied to separate smooth areas from edge areas for hiding different amounts of secrets in the pixels with the modulus function. The result of the proposed scheme is that the pixels located in the inter-blocks hide more secrets to enlarge the embedding capacity, and those located in the intra-blocks have fewer embedded secrets to maintain good visual qu...

Capsule The division coefficient is an estimate of the proportion of ringed birds migrating to different destination areas taking into account area‐specific re‐encounter probabilities. Aims To explore precision and bias of the division coefficient method by a simulation study and to compare the approach with multi‐state models. Methods In a simulation study true and estimated division coefficients were compared. The division coefficient method was mathematically compared with the multi‐state model. Results The estimated division coefficients seemed to be unbiased if the assumptions were met. The precision decreased if the bird distribution became similar in both bird groups and when difference between area‐specific re‐encounter probabilities increased. A bootstrap method to assess precision is presented. The estimates from the division coefficient method equal the maximum likelihood estimates in a multi‐state model including only one time interval. Conclusion Before applying the division coefficient metho...

We present an algorithm to compute the subresultant sequence of two polynomials that completely avoids division in the ground domain, generalizing an algorithm given by Abdeljaoued et al. [J. Abdeljaoued, G. Diaz-Toca, and L. Gonzalez-Vega, Minors of Bezout matrices, subresultants and the parameterization of the degree of the polynomial greatest common divisor, Int. J. Comput. Math. 81 (2004), pp. 1223–1238]. We evaluate determinants of slightly manipulated Bezout matrices using the algorithm of Berkowitz. Although the algorithm gives worse complexity bounds than pseudo-division approaches, our experiments show that our approach is superior for input polynomials with moderate degrees if the ground domain contains indeterminates.

In order to achieve an efficient block division plan, an optimization system is developed by integrating product model of ship with a genetic algorithm module, with the help of a graph model module which acts as an intermediate agent. As for the generation of block division plans, ‘Nodal Cut Set Method’ is newly proposed in which various plans are generated based on the chromosome by the use of graph theory and logical operations. Evaluation of the plan is carried out by using the product model data. The validity of the proposed system is shown by executing some optimizations.

We describe a method to extract roots of a reducible quintic over the real field, which makes use of a simple division. A procedure to synthesize such quintics is given and a numerical example is solved to extract the roots of quintic with the proposed method.