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A graphic procedure is presented for studying differential item functioning (DIF) that is designed to provide diagnostic information for psychometricians and educators. The procedure provides a signature of each test item that can be used with a summary statistic to flag items with DIF. Advantages and limitations are noted. (SLD)

Several metanalytic studies of group variability use variance ratios as measures of effect size. Problems with this approach are discussed, including limitations of using means and medians of ratios. Mean logarithms and the geometric mean are not adversely affected by the arbitrary choice of numerator. (SLD)

Discusses the use of the generalized inverse Gaussian-Poisson (GIGP) distribution in bibliometric studies. The main types of size-frequency distributions are described, bibliometric distributions in logarithms are examined; parameter estimation is discussed; and goodness-of-fit tests are considered. Examples of applications are included. (17 references) (LRW)

Described are three activities designed to increase student participation in the classroom. The first uses earthquake reporting to study logarithms. The second discusses student-generated examples to introduce probability. In the third, prepared handouts and study groups are used to promote student interaction and involvement in calculus classes. (MDH)

Presents two methods for replacing a series by one converging more rapidly: regrouping the terms of a series and manipulations of power series. Describes a general algorithm for approximating the natural logarithm of any number. (YP)

Included in this section are brief articles on an algebraic puzzler, periodic decimal fractions with computers, and the arrow method of teaching logarithms. (MNS)

Discusses the construction of logarithmic and arithmetic scales illustrates why they should be used, and demonstrates the difference between them. Offers suggestions for classroom applications of the scales. (FL)

Some concrete examples of the use of historical materials in developing certain topics from precalculus and calculus are presented. Ideas which can be introduced with a reformulated curriculum are discussed in five areas: algorithms, combinatorics, logarithms, trigonometry, and mathematical models. (MNS)

In this section, suggestions are given for working with radian measures, proving logarithmic properties, the law of cosines as seen by Pythagoras, and an alternative proof of a theorem. (MNS)

Provides several algorithms that use extended precision methods to compute large factorials exactly. The programs are written in BASIC and PASCAL. The approach used for computing N considers how large N is, how the built-in limitation on exact integer representation can be bypassed, and how long it takes to compute N. (JN)

Proposes a new definition of the nuclear zone of a Bradford or Leimkuhler curve. The mathematical rationale is outlined, and examples of the application of this technique to citation analysis are given. (15 references) (CLB)

Examines students' problem solving performance on linear and logarithmic estimation tasks using a microcomputer. Results are discussed in terms of performance data and strategy used. Students were able to estimate faster and more efficiently for the linear estimation task but experienced difficulty on a logarithmic scale. Discusses reasons for the results and reviews instructional implications. (ML)

Describes the recall precision trade-off in information retrieval as the deceleration in the proportion of relevant documents retrieved over time. Several mathematical functions sharing this property are examined and it is concluded that the equation that best models recall as a function of time is the logarithm of a quadratic equation. (21 references) (Author/CLB)

Discusses the use of ratio diagrams, which plot the calculations of equilibrium concentrations of the species of the carbonate system. Provides examples to describe how these diagrams can be used to illustrate the behavior systems of interest in volumetric analysis, where absorption or loss of carbon dioxide takes place. (TW)

Provides some significant figure rules used in chemistry including the general theoretical basis; logarithms and antilogarithms; exponentiation (with exactly known exponents); sines and cosines; and the extreme value rule. (YP)

Exponential growth is examined through an example describing the number pattern formed by postage stamp issuance. Number tables and graphs are used to discover the functional relationship between the number of new stamps issued and time. (MDH)

Develops the idea of fractals through a laboratory activity that calculates the fractal dimension of ordinary white bread. Extends use of the fractal dimension to compare other complex structures as other breads and sponges. (MDH)

Discusses the various ways that time is measured and expressed, noting several discrepancies between descriptives and their respective vocabularies. Suggests that a logarithmic time scale be adopted by chemists (such as one used in the experimental sciences) particularly when addressing very fast processes. (TW)

Checks on the logarithmic property of the eye in receiving light, by using estimation methods of how much brighter a light bulb looks than does its image in a window glass. (GA)

In this paper, a special case of the generalized 4-block n-fold IPs is investigated, where B i = B and B has a rank at most 1. Such IPs, called almost combinatorial 4-block n-fold IPs, include the generalized n-fold IPs as a subcase. We are interested in fixed parameter tractable (FPT) algorithms by taking as parameters the dimensions of the blocks and the largest coefficient. For almost combinatorial 4-block n-fold IPs, we first show that there exists some λ ≤ g (γ) such that for any nonzero kernel element g , λ g can always be decomposed into kernel elements in the same orthant whose ℓ ∞ -norm is bounded by g (γ) (while g itself might not admit such a decomposition), where g is a computable function and γ is an upper bound on the dimensions of the blocks and the largest coefficient. Based on this, we are able to bound the ℓ ∞ -norm of Graver basis elements by O (g (γ) n) and develop an O (g (γ) n 3 + o (1) L ^ 2) -time algorithm (here L ^ denotes the logarithm of the largest absolute value occurring in the input). Additionally, we show that the ℓ ∞ -norm of Graver basis elements is Ω (n) . As applications, almost combinatorial 4-block n-fold IPs can be used to model generalizations of classical problems, including scheduling with rejection, bi-criteria scheduling, and a generalized delivery problem. Therefore, our FPT algorithm establishes a general framework to settle these problems. [ABSTRACT FROM AUTHOR]

We prove that multiplicative chaos measures can be constructed from extreme level sets or thick points of the underlying logarithmically correlated field. We develop a method which covers the whole subcritical phase and only requires asymptotics of suitable exponential moments for the field. As an application, we establish that these estimates hold for the logarithm of the absolute value of the characteristic polynomial of a Haar distributed random unitary matrix (CUE), using known asymptotics for Toeplitz determinant with (merging) Fisher–Hartwig singularities. Hence, this proves a conjecture of Fyodorov and Keating concerning the fluctuations of the volume of thick points of the CUE characteristic polynomial. [ABSTRACT FROM AUTHOR]

Experiments have revealed that biological cells exhibit a universal power-law rheology, but the underlying mechanisms remain elusive. Here, we present a minimal model to explain the power-law creep of cell cortex, which is abstracted as chains of crosslinkers with random binding energies. Using this model, we show that when both the load and chain length are small, the logarithm of both the strain and time scales with the fraction of unbound crosslinkers, leading to power-law creep with a constant exponent, as observed in many experiments. Increasing the load alters the latter relationship between time and unbinding fraction, and thus, increases the power-law exponent, explaining the stress-induced nonlinearity in some experiments. Increasing the chain length alters this relationship as well, and as a result, the exponent grows proportionally with the chain length, explaining the crosslinker-density-induced nonlinearity in other experiments. This work provides a mesoscopic explanation for the linear and nonlinear power-law creep of cell cortex and may serve as a basis for understanding the cytoskeletal mechanics. [ABSTRACT FROM AUTHOR]

We study the image of the Hodge–Tate logarithm map (in any cohomological degree), defined by Heuer, in the case of smooth Stein varieties. Heuer, motivated by the computations for the affine space of any dimension, raised the question whether this image is always equal to the group of closed differential forms. We show that it indeed always contains such forms but the quotient can be non-trivial: it contains a slightly mysterious |$\mathbf{Z}_{p}$| -module that maps, via the Bloch–Kato exponential map, to integral classes in the pro-étale cohomology. This quotient is already non-trivial for open unit disks of dimension strictly greater than |$1$|⁠. [ABSTRACT FROM AUTHOR]

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]

We designed an image encryption algorithm based on a non-degeneracy 3D chaotic map and a keyed strong S-Box, which is capable of encrypting color image with any size. First, we constructed a non-degeneracy 3D discrete hyperchaotic map (3D-DHCM), and based on it we constructed a keyed strong S-Box without fixed point, reverse fixed point or short period rings. The map is based on modular exponentiation operation, and its inverse function is the discrete logarithm problem. Finally, we applied blurring techniques to the plain image prior to encryption and employed permutation, confusion, and diffusion techniques to shuffle all pixels. Experimental and analytical results demonstrated that the NPCR and UACI are all close to their ideal values 99.6094070% and 33.4635070%, and the pixels of the cipher image are completely confused, which are all close to the ideal value 8. In addition, the chi-square values for encrypted images are significantly lower than before, with a maximum value of 255.1503. It is showed that the algorithm is superior to others in terms of its ability to encrypt images with any size and withstand common attacks. [ABSTRACT FROM AUTHOR]

A semiparametric methodology for reconstructing Markovian and non-Markovian Langevin equations from time series data using unscented Kalman filtering is introduced and explored. The drift function and the logarithm of the diffusion function are expanded into sets of polynomial basis functions. In contrast to the more common state augmentation approach, the Kalman filter is here used only for state estimation and propagation; the model parameters are determined by maximum likelihood based on the predictive distribution generated by the Kalman filter. Model selection regarding the number of included drift and diffusion basis functions is performed with the Bayesian information criterion. The method is successfully validated on various simulated datasets with known system dynamics; it achieves accurate identification of drift and diffusion functions, also outside the prescribed model class, from datasets of moderate length with medium computational cost. [ABSTRACT FROM AUTHOR]