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A radio k -labeling of a graph G is an assignment of non-negative integers (labels) to the vertices of G such that the absolute value of the difference between the labels of any two distinct vertices u and v is not less than k + 1 − d ( u , v ) , where d ( u , v ) denotes the distance between u and v . The radio k -labeling number of G is the minimum span of a radio k -labeling of G . For a given set of integers D = { d 1 , d 2 , … , d t } , the distance graph G ( Z , D ) , denoted also by D ( d 1 , d 2 , … , d t ) , has the set of integers Z as its vertex set, while two distinct vertices i , j ∈ Z are adjacent in G ( Z , D ) if and only if | i − j | ∈ D . Radio k -labelings of three families of distance graphs are considered. Some improved theoretical lower and upper bounds on the radio k -labeling number are presented and some computer-based methods for computing radio k -labelings of these families of graphs are proposed.

For q an odd prime power, A = F q [ T ] , and F = F q ( T ) , let ψ : A → F { τ } be a Drinfeld A-module over F of rank 2 and without complex multiplication, and let p = p A be a prime of A of good reduction for ψ, with residue field F p . We study the growth of the absolute value | Δ p | of the discriminant of the F p -endomorphism ring of the reduction of ψ modulo p and prove that, for all p , | Δ p | grows with | p | . Moreover, we prove that, for a density 1 of primes p , | Δ p | is as close as possible to its upper bound | a p 2 − 4 μ p p | , where X 2 + a p X + μ p p ∈ A [ X ] is the characteristic polynomial of τ deg p .

The energy management (EM) of active distribution network (ADN) under uncertainties is a stochastic, nonconvex and nonlinear problem, which cannot be solved by traditional algorithms in acceptable time. In this paper, we decouple this computationally intractable problem into a series of subproblems which are easier to handle, and then solve them successively according to an improved approximate dynamic programming (IADP) algorithm. Different from the existing approximate dynamic programming (ADP) algorithms, which need to update value functions iteratively, IADP obtains approximate value functions directly using Galerkin method. Such that the time of updating approximate value functions can be omitted. Furthermore, the influence of each basis function on the approximate value function is evaluated according to the absolute value inequality principle. Then the unimportant basis functions are removed from the basis function set to speed up the algorithm. The historical data can also be embedded into IADP to facilitate the online decision-making and reduce the dependency of real-time forecast information. Numerical simulations on two modified IEEE test systems and a real 682-bus ADN are given to illustrate the effectiveness of the proposed approach.

In this paper, we consider a new system of absolute value variational inclusions. Some interesting and extensively problems such as absolute value equations, difference of monotone operators, absolute value complementarity problem and hemivariational inequalities as special case. It is shown that variational inclusions are equivalent to the fixed point problems. This alternative formulation is used to study the existence of a solution of the system of absolute value inclusions. New iterative methods are suggested and investigated using the resolvent equations, dynamical system and nonexpansive mappings techniques. Convergence analysis of these methods is investigated under monotonicity. Some special cases are discussed as applications of the main results.

The autocorrelation properties of Boolean functions are closely related to the Shannon’s concept of diffusion and can be accompanied with other cryptographic criteria (such as high nonlinearity and algebraic degree) for ensuring an overall robustness to various cryptanalytic methods. In a series of recent articles [14,9,15], the design methods of n-variable balanced Boolean functions n is strictly even) with small absolute indicator Δ f n ⩾ 46 , a recent approach [15] has introduced a generic design framework achieving Δf n ⩾ 22 . Based on a suitable modification of the method of Rothaus, used to construct new bent functions from known ones, we provide a generic iterative framework for designing balanced functions satisfying the condition Δf n ⩾ 12 . Even though the problem of specifying functions having Δf n ⩾ 48 satisfying n ≡ 0 mod 4 and n ⩾ 54 with n ≡ 2 mod 4. In the latter case, our nonlinearity bound is better than the one presented in [14].

This research studies novel analytical solutions of an Atangana conformable fractional Lotka–Volterra (LV) model system arising in ecology by means of three systematic schemes (the extended simplest equation method, modified Kudryashov method, and the sech–tanh expansion method). Then we use the obtained solutions to evaluate the boundary and initial conditions that helps to apply the B-spline collection numerical schemes to evaluate the value of absolute error. The investigation aims also to explain the accuracy of the analytical solutions by evaluating absolute value of errors between exact and numerically obtained solutions. In order to better explain the obtained solutions, some relevant sketches are given in three different types. The novelty of the work done is attempted to explain by comparing our results with some available and relevant results. Further, relevant strengths, usefulness, practical applications, and the methods’ ability for applying in various nonlinear evolution equations have been explored.

Simpson inequalities for differentiable convex functions and their fractional versions have been studied extensively. Simpson type inequalities for twice differentiable functions are also investigated. More precisely, Budak et al. established the first result on fractional Simpson inequality for twice differentiable functions. In the present article, we prove a new identity for twice differentiable functions. In addition to this, we establish several fractional Simpson type inequalities for functions whose second derivatives in absolute value are convex. This paper is a new version of fractional Simpson type inequalities for twice differentiable functions.

Let A ∈ Zm×n be an integral matrix and a, b, c ∈ Z satisfy a ≥ b ≥ c ≥ 0. The question is to recognize whether A is {a, b, c}-modular, i.e., whether the set of n×n subdeterminants of A in absolute value is {a, b, c}. We will succeed in solving this problem in polynomial time unless A possesses a duplicative relation, that is, A has nonzero n × n subdeterminants k1 and k2 satisfying 2·|k1|=|k2|. This is an extension of the well-known recognition algorithm for totally unimodular matrices. As a consequence of our analysis, we present a polynomial time algorithm to solve integer programs in standard form over {a, b, c}-modular constraint matrices for any constants a, b and c., Vietnam Journal of Mathematics, 50 (2), ISSN:2305-2228, ISSN:0866-7179, ISSN:2305-221X

The aim of this study was to produce a mathematics worksheet on absolute value material based on the Realistic Mathematical Education (RME) approach. The location of this research is Tribhuwana Tunggadewi University. The development model in this research is designed based on ADDIE, which includes: (A) analysis, which is about curriculum analysis and needs analysis; (D) design, containing the preparation of the worksheet; (D) development, intended in the process of developing an RME-based absolute value worksheet; (I) implementation, namely product testing conducted for small groups and for large groups with the aim of obtaining practical data and data on mathematical thinking skills; (E) evaluation, which is carried out in data analysis refers to the implementation stage. The results of product validation through a google form questionnaire conducted by media experts were 92%, material experts 87%, small group test 91%, large group test 88%, and students' mathematical thinking ability test results reached 83%. Based on these data, it shows that the designed worksheet is feasible and practical in improving the mathematical thinking skills of Tribhuwana Tunggadewi University students.

Individual differences (IDs) may reduce the detection-accuracy of drowsiness-driving by influencing measurements’ drowsiness-detection performance (MDDP). The purpose of this paper is to propose a model that can quantify the effects of IDs on MDDP and find measurements with less impact by IDs to build drowsiness-detection models. Through field experiments, drivers’ naturalistic driving data and subjective-drowsiness levels were collected, and drowsiness-related measurements were calculated using the double-layer sliding time window. In the model, MDDP was represented by |Z-statistics| of the Wilcoxon-test. First, the individual driver’s measurements were analysed by Wilcoxon-test. Next, drivers were combined in pairs, measurements of paired-driver combinations were analysed by Wilcoxon-test, and measurement’s IDs of paired-driver combinations were calculated. Finally, linear regression was used to fit the measurements’ IDs and changes of MDDP that equalled the individual driver’s |Z-statistics| minus the paired-driver combination’s |Z-statistics|, and the slope’s absolute value (|k|) indicated the effects of ID on the MDDP. As a result, |k| of the mean of the percentage of eyelid closure (MPECL) is the lowest (4.95), which illustrates MPECL is the least affected by IDs. The results contribute to the measurement selection of drowsiness-detection models considering IDs.

Recently, generative adversarial networks began to be actively studied in the field of medical imaging. These models are used for augmenting the variation of images to improve the accuracy of computer-aided diagnosis. In this paper, we propose an alternative new image generative model based on transformer decoder blocks and verify the performance of our model in generating SPECT images that have characteristics of Parkinson’s disease patients. Firstly, we proposed a new model architecture that is based on a transformer decoder block and is extended to generate slice images. From few superior slices of 3D volume, our model generates the rest of the inferior slices sequentially. Our model was trained by using [123I]FP-CIT SPECT images of Parkinson's disease patients that originated from the Parkinson’s Progression Marker Initiative database. Pixel values of SPECT images were normalized by the specific/nonspecific binding ratio (SNBR). After training the model, we generated [123I]FP-CIT SPECT images. The transformation of images of the healthy control case SPECT images into PD-like images was also performed. Generated images were visually inspected and evaluated using the mean absolute value and asymmetric index. Our model was successfully generated and transformed into PD-like SPECT images. The mean absolute SNBR was mostly less than 0.15 in absolute value. The variation of the obtained dataset images was confirmed by the analysis of the asymmetric index. These results showed the potential ability of our new generative approach for SPECT images that the generative model based on the transformer realized both generation and transformation by a single model.

The distribution of ranked heights of excursions of a Brownian bridge is given by Pitman and Yor (2001). In this work, we consider excursions of a Brownian excursion above a random level x , where x is the value of the excursion at an independent uniform time on [0,1]. We study the maximum heights of these excursions as Pitman and Yor did for excursions of a Brownian bridge. In particular, the probability functions and the moments of the sum of the j th highest maximum over an excursion above x and the absolute value of the k th lowest minimum over an excursion below x , including j = 1 and k = 2 , are computed in this paper.

This article deals with a class of variational inequalities known as absolute value variational inequalities. Some new merit functions for the absolute value variational inequalities are established. Using these merit functions, we derive the error bounds for absolute value variational inequalities. Since absolute value variational inequalities contain variational inequalities, absolute value complementarity problem and system of absolute value equations as special cases, the findings presented here recapture various known results in the related domains. The conclusions of this paper are more comprehensive and may provoke futuristic research.

For an $${\text {L}}_2$$ -bounded martingale starting at 0 and having final variance $$\sigma ^2$$ , the expected local time at $$a \in \text {R}$$ is at most $$\sqrt{\sigma ^2+a^2}-|a|$$ . This sharp bound is attained by Standard Brownian Motion stopped at the first exit time from the interval $$(a-\sqrt{\sigma ^2+a^2},a+\sqrt{\sigma ^2+a^2})$$ . In particular, the maximal expected local time anywhere is at most $$\sigma $$ , and this bound is sharp. Sharp bounds for the expected maximum, maximal absolute value, maximal diameter and maximal number of upcrossings of intervals have been established by Dubins and Schwarz (Societe Mathematique de France, Asterisque 157(8), 129–145 1988), by Dubins et al. (Ann Probab 37(1), 393–402 2009) and by the authors (2018).

The article deals with the homogeneous Stokes resolvent system in a 3D exterior domain, under inhomogeneous Dirichet boundary conditions. Solutions to this boundary value problem are estimated in L^p-norms, with the bounds in these estimates depending on the absolute value of the resolvent parameter λ in an explicit way. Two types of boundary data are considered, that is, L^p-and W^{2−1/p, p}-data. It is shown in particular that the L^p-norm of the velocity outside a vicinity of the boundary, after subtraction of the gradient of a certain harmonic function, is bounded by a constant times |λ| \| b\|p. This estimate carries over to the Oseen resolvent system , leading to a result which has applications in the theory of spatial asymptotics of solutions to the 3D time-dependent Navier-Stokes system with Oseen term.

In this paper, the authors introduce the (p,q)-trapezoidal integral inequalities, which are the (p,q)-analogues of the recently introduced q-trapezoidal integral inequalities. We derive a new (p,q)-integral identity for twice (p,q)-differentiable function. Utilizing this as an auxiliary result, we establish several new (p,q)-trapezoidal type integral inequalities for the function whose absolute value of twice (p,q)-derivative is (η1,η2)-convex functions. Some special means of real numbers are also given. At the end, we give brief conclusion. It is expected that this method which is very useful, accurate, and versatile will open a new venue for the real-world phenomena of special relativity and quantum theory.

The main objective of this paper is basically to acquire some new extensions of Ostrowski type inequalities for the function whose first derivatives' absolute value are $s$--type $p$--convex. We initially presented a new auxiliary definition namely $s$--type $p$--convex function. Some beautiful algebraic properties and examples related to the newly introduced definition are discussed. We additionally investigated some beautiful cases that can be derived from the novel refinements of the paper. These new results yield us some generalizations of the prior results. We trust that the techniques introduced in this paper will further motivate intrigued researchers.

In this paper, we extend some estimates of a Hermite-Hadamard type inequality for functions whose absolute values of the first derivatives are $p$-convex. By means of the obtained inequalities, some bound functions involving beta functions and hypergeometric functions are derived as applications. Also, we suggest an upper bound for error in numerical integration of $p$-convex functions via composite trapezoid rule.

In this paper, the problem of the finite-time synchronization is addressed for a kind of fractional-order memristive neural networks (FMNNs). First, a new power law inequality with fractional-order and two finite-time fractional differential inequalities are established by means of L’Hospital rule, Laplace transform, and reduction to absurdity, which greatly extend some existing results. In addition, unlike the traditional maximum absolute value-based method to propose memristive synaptic weights, by introducing some transformations, FMNNs are translated to a type of fractional-order systems with uncertain parameters. Furthermore, the finite-time synchronization of FMNNs is investigated by designing a discontinuous control scheme and several criteria are derived based on the developed fractional inequalities and M-matrix theory. Note that in addition to the traditional Lyapunov function with absolute value form, a more general Lyapunov function is constructed to deal with the finite-time synchronization, which makes the derived criteria more flexible and less conservative. Lastly, the derived theoretical results are verified via numerical simulations.

For safe visual navigation, areas with high localization errors should be concentrated and could be further refined by additional mapping operations. Given an environment map, we propose to predict the visual localization error and hence to either improve the navigation performance or call an additional mapping to refine the built map. Previous work adopts the uncertainty of landmarks for the error prediction. In our work, we take into account both the spatial distribution of visual landmarks and the uncertainty of landmarks. Our main idea is that standing at one place, a good spatial distribution of landmarks means a sufficient enough visible landmarks from all views at that place, i.e., enough landmarks under arbitrary view-direction. Combining the spatial distribution and the uncertainty of landmarks, we propose a new framework to predict the error of visual localization. Furthermore, we show that additional mapping in the area with high predicted error can significantly improve the visual localization precision. Experimental results show that there is a strong relationship between the predicted error and the real error, of which the absolute value of correlation coefficient is between 0.707 to 0.915. We apply our method to conduct an optimal refining policy on the built map and the experimental results show the improved localization precision. Applications on navigation test verify the superiority of our proposed method.

Conway and Gordon proved that for every spatial complete graph on six vertices, the sum of the linking numbers over all of the constituent two-component links is odd, and Kazakov and Korablev proved that for every spatial complete graph with arbitrary number of vertices greater than six, the sum of the linking numbers over all of the constituent two-component Hamiltonian links is even. In this paper, we show that for every spatial complete graph whose number of vertices is greater than six, the sum of the square of the linking numbers over all of the two-component Hamiltonian links is determined explicitly in terms of the sum over all of the triangle–triangle constituent links. As an application, we show that if the number of vertices is sufficiently large then every spatial complete graph contains a two-component Hamiltonian link whose absolute value of the linking number is arbitrary large. Some applications to rectilinear spatial complete graphs are also given.

Background. Even though the six-minute walking test is a simple and widely available tool for the evaluation of the functional capacity of cardiac patients, its interpretation is associated with some difficulties and contradictions. Aims: To evaluate the dynamics of tolerance to physical activity during outpatient rehabilitation of patients with ischemic heart disease using predicted values of distance in the six-minute walking test. Materials and methods. 97 patients (70 men and 27 women, average age 59.6 [50; 60] years) after acute coronary syndrome and after myocardial revascularization were included. The six-minute walking test was performed at the beginning of the 3-weeks stage of cardiac rehabilitation and before the patients’ discharge. The results of the test were reported as an absolute value, a change in absolute value, and the percentage of predicted values, estimated with the reference equation by Enright and colleagues. Results. The absolute value of distance in the six-minute walking test was increased significantly from 418 [385; 465] m to 485 [440; 525] m ( p

The p-adic completion ℚ p of the rational numbers induces a different absolute value |⋅| p than the typical | ⋅| we have on the real numbers. In this paper we compare and contrast functions f : ℝ+ → ℝ+, for which the composition with the p-adic metric dp generated by |⋅| p is still a metric on ℚ p , with the usual metric preserving functions and the functions that preserve the Euclidean metric on ℝ. In particular, it is shown that f ∘ d p is still an ultrametric on ℚ p if and only if there is a function g such that f ∘ d p = g ∘ d p and g ∘ d is still an ultrametric for every ultrametric d. Some general variants of the last statement are also proved.

In this paper, the single-machine scheduling problem is studied by simultaneously considering due-date assignment and group technology (GT). The objective is to determine the optimal sequence of groups and jobs within groups and optimal due-date assignment to minimize the weighted sum of the absolute value in lateness and due-date assignment cost, where the weights are position dependent. For the common (CON) due-date assignment, slack (SLK) due-date assignment, and different (DIF) due-date assignment, an O n log n time algorithm is proposed, respectively, to solve the problem, where n is the number of jobs.

It is well known that any finite commutative ring is isomorphic to a direct product of local rings via the Chinese remainder theorem. Hence, there is a great significance to the study of character sums over local rings. Character sums over finite rings have applications that are analogous to the applications of character sums over finite fields. In particular, character sums over local rings have many applications in algebraic coding theory. In this paper, we firstly present an explicit description on additive characters and multiplicative characters over a certain local ring. Then we study Gaussian sums, hyper Eisenstein sums and Jacobi sums over a certain local ring and explore their properties. It is worth mentioning that we are the first to define Eisenstein sums and Jacobi sums over this local ring. Moreover, we present a connection between hyper Eisenstein sums over this local ring and Gaussian sums over finite fields, which allows us to give the absolute value of hyper Eisenstein sums over this local ring. As an application, several classes of codebooks with new parameters are presented.

We introduce a Milnor metric on the determinant line of the cohomology of the underlying closed manifold with coefficients in a flat vector bundle, by means of interactions between the fixed points and the closed orbits of a Morse-Smale flow. This allows us to generalise the notion of the absolute value at zero point of the Ruelle dynamical zeta function, even in the case where this value is not well defined in the classical sense. We give a formula relating the Milnor metric and the Ray-Singer metric. An essential ingredient of our proof is Bismut-Zhang's Theorem., Final version

In this paper, we first establish a version of multidimensional analogues of the refined Bohr’s inequality. Then we establish two versions of multidimensional analogues of improved Bohr’s inequality with initial coefficient being zero. Finally we establish two versions of multidimensional analogues of improved Bohr’s inequality with the initial coefficient being replaced by absolute value of the function, and to prove that most of the results are sharp.

Several phenomena encountered in nature are characterized by very localized events occurring randomly at given times. Random pulses are an appropriate modelling tool for such events. Usually, the impulses are hidden in the noise due to unwanted convolution. In some cases, the problem is more complex because of the short time lag between the pulses. Considering these problems, the resulting signal is unclear and can lead to an erroneous analysis. Hence the need for deconvolution to restore the pulsed signal in order to obtain a more accurate diagnosis. The main objective of this study is to propose a new algorithm called orthogonal least absolute value. The particularity of this algorithm lies in its selection criterion. The algorithm iteratively selects the atom minimizing the absolute value of the approximation error. This allows the proposed algorithm to outperform classical greedy algorithms when the peaks are very close to each other. Numerical and experimental simulations are performed to study the proposed algorithm and compare its behavior to other greedy algorithms in deconvolution framework. Simulations results prove the performance of the proposed algorithm, especially when the impulses are very close to each other.

An (n, d, λ)-graph is a d regular graph on n vertices in which the absolute value of any nontrivial eigenvalue is at most λ. For any constant d ≥ 3, ϵ > 0 and all sufficiently large n we show that there is a deterministic poly(n) time algorithm that outputs an (n, d, λ)-graph (on exactly n vertices) with $$\lambda \le 2\sqrt {d - 1} + $$ . For any d=p + 2 with p ≡ 1 mod 4 prime and all sufficiently large n, we describe a strongly explicit construction of an (n, d, λ)-graph (on exactly n vertices) with $$\lambda \le \sqrt {2\left({d - 1} \right)} + \sqrt {d - 2} + o\left(1 \right)\left({< \left({1 + \sqrt 2} \right)\sqrt {d - 1} + o\left(1 \right)} \right)$$ , with the o(1) term tending to 0 as n tends to infinity. For every ϵ> 0, d> d0(ϵ) and n>n0(d, ϵ) we present a strongly explicit construction of an (m, d, λ)-graph with $$\lambda

An efficient solution of calculating the spherical surface integral of a Gauss function defined as $$h\left( {s,{\mathbf{Q}}} \right) = \int_{0}^{2\pi } {\int_{0}^{\pi } {\left( {{\mathbf{s}} + {\mathbf{Q}}} \right)_{x}^{i} \left( {{\mathbf{s}} + {\mathbf{Q}}} \right)_{y}^{j} \left( {{\mathbf{s}} + {\mathbf{Q}}} \right)_{z}^{k} e^{{ - \gamma \left( {{\mathbf{s}} + {\mathbf{Q}}} \right)^{2} }} } } \sin \theta d\theta d\varphi$$ is provided, where $$\gamma \ge 0$$ , and i, j, k are nonnegative integers. A computationally concise algorithm is proposed for obtaining the expansion coefficients of polynomial terms when the coordinate system is transformed from cartesian to spherical. The resulting expression for $$h\left( {s,{\mathbf{Q}}} \right)$$ includes a number of cases of elementary integrals, the most difficult of which is $$II\left( {n,\mu } \right) = \int_{0}^{\pi } {\cos^{n} \theta e^{ - \mu \cos \theta } } d\theta$$ , with a nonnegative integer n and positive μ. This integral can be formed by linearly combining modified Bessel functions of the first kind $$B(n,\mu ) = \frac{1}{\pi }\int\limits_{0}^{\pi } {e^{\mu \cos \theta } \cos \left( {n\theta } \right)d\theta }$$ , with a nonnegative integer n and negative μ. Direct applications of the standard approach using Mathematica and GSL are found to be inefficient and limited in the range of the parameters for the Bessel function. We propose an asymptotic function for this expression for n = 0,1,2. The relative error of asymptotic function is in the order of 10−16 with the first five terms of the asymptotic expansion. At last, we give a new asymptotic function of $$B\left( {n,\mu } \right)$$ based on the expression for $$e^{ - \mu } II\left( {n,\mu } \right)$$ when n is an integer and μ is real and large in absolute value.

In this work, we establish the concept of a new class of non-convex functions, namely hyperbolic type preinvex functions. Secondly few algebraic properties of this class are obtained. Further Hermite-Hadamard type integral inequalities are established for this class. We also derive several new inequalities for the functions for which absolute value of first derivative, with exponent greater or equal to one is hyperbolic type preinvexity. The results are obtained by using both the Hölder's inequality and Hölder-Iscan inequality and compared at the end. Several special cases are discussed as applications of the results.

An upper estimate for the absolute value of the sum of the Spitzer series is obtained. This estimate depends explicitly on the distribution in terms of which the Spitzer series is defined.

We investigate the evolution of population density vector, u = ( u 1 , ⋯ , u k ) , of k-species whose diffusion is controlled by its absolute value | u | . More precisely, we study the properties and asymptotic large time behaviour of solution u = ( u 1 , ⋯ , u k ) of degenerate parabolic system ( u i ) t = ∇ ⋅ ( | u | m − 1 ∇ u i ) for m > 1 and i = 1 , ⋯ , k . Under some regularity assumptions, we prove that the component u i which describes the population density of i-th species with population M i converges to M i | M | B | M | in space with two different approaches where B | M | is the Barenblatt solution of the standard porous medium equation with L 1 mass | M | = M 1 2 + ⋯ + M k 2 . As an application of the asymptotic behaviour, we establish a suitable Harnack type inequality which makes the spatial average of u i under control by the value of u i at one point. We also find 1-directional travelling wave type solutions and the properties of solutions which has travelling wave behaviour at infinity.

We continue our researches concerning the generalization and improvement of R.T.Turganaliev’s result that states an asymptotic formula for the mean value of the Riemann zeta function in the critical strip with power factor saving in the remainder term.We find an asymptotic for the mean value of Dirichlet L-function in the critical strip. This assertion improves R.T.Turganaliev’s theorem for zeta-function in the whole interval $$(1/2 < Re 𝑠 ≤ 1)$$. Our result is based on the special use of the estimation of exponential sums by second derivative test.

The gravity is used to solve geodesy's primary tasks, such as determining geoid and defining the height and gravimetric reference networks of different scales, from national to global. Knowledge of gravity is of great importance for both metrology and geodetic metrology. In addition to the historical overview of absolute gravimeters, this paper presents the theoretical basis of the most commonly used method for determining the absolute value of gravity. The principle of operation of the absolute gravimeter FG5 and the importance of international comparison of absolute gravimeters are briefly presented. An overview of the gravimetric reference systems is given, emphasizing the establishment of the International Reference Gravimetric System. The previous works concerning the absolute determination of acceleration due to Earth's gravity field in Serbia are presented. Finally, the importance of determining the absolute value of gravity from the geodetic and metrological perspective is pointed out. Both national and international significance of determining absolute gravity in defining gravimetric reference systems and the importance of absolute gravity in monitoring global phenomena are emphasized.

Understanding the concept is one aspect that is needed and must be owned by students in learning mathematics. This research aims to make learning media assisted by hawgent dynamic mathematics software on understanding the absolute value in grade 10 Senior high school to help teachers explain the concepts and assist students in finding and understanding the basic concepts of absolute value topic. The development model used in this research is ADDIE development model. ADDIE is an abridgment of Analyze, Design, Develop and Evaluation. Researchers the learning media Hawgent can help students to understand and find the concept of absolute value. Based on several aspects, clearly and attractively with an average of 78.9% in the excellent category. It is concluded that the development of learning media Hawgent dynamic mathematics software can be used on the subject of understanding the concept of absolute value and with the help of this software it can help teachers explain the concept of absolute value and the students are also very interested in this ICT-based learning media. This conclusion is related to the validation results of media experts and material experts, where the validation results from media experts on the use of learning media are in good categories.

The absolute value equation appears in various fields of applied mathematics such as operational research. Here we consider its generalized version AX + B | X | = C , where A , B , C ∈ C n × n are given, | X | = ( | x i , j | ) and X ∈ C n × n is an unknown matrix that must be determined. In this investigation, based on the Picard matrix splitting iteration method, we applied a matrix splitting method for solving it. We will see that under the condition σ min ( A ) > n σ max ( | B | ) , this method is convergent, where σ max ( | B | ) denotes the largest singular value of matrix | B | and σ min ( A ) denotes the smallest singular value of matrix A. Then we give some convergence theorems for our new method and analyze this procedure in detail. Then we consider a p-step iteration method for solving this equation and analyze this procedure. Numerical experiment results show the efficiency of the method.

In this paper, we study the stability of the zero equilibrium and the occurrence of flip bifurcation on the following system of difference equations: \[x_{n+1} =a_1\frac{y_n}{b_1+y_n} +c_1\frac{x_ne^{k_1-d_1x_n}}{1+e^{k_1-d_1x_n}},\]\\ \[y_{n+1} =a_2\frac{z_n}{b_2+z_n} +c_2\frac{y_ne^{k_2-d_2y_n}}{1+e^{k_2-d_2y_n}},\]\\ \[z_{n+1} =a_3\frac{x_n}{b_3+x_n} +c_3\frac{z_ne^{k_3-d_3z_n}}{1+e^{k_3-d_3z_n}}\] where $a_i$, $b_i$, $c_i$, $d_i$, $k_i$, for $i=1,2,3$, are real constants and the initial values $x_0$, $y_0$ and $z_0$ are real numbers. We study the stability of this system in the special case when one of the eigenvalues is equal to -1 and the remaining eigenvalues have absolute value less than 1, using center manifold theory.

We consider wavelet decompositions of Haar type spaces on arbitrary nonuniform grids by methods of the nonclassical theory of wavelets. The number of nodes of the original (nonuniform) grid can be arbitrary, and the main grid can be any subset of the original one. We proposie decomposition algorithms that take into account the character of changes in the original numerical flow. The number of arithmetical operations is proportional to the length of the original flow, and successive real-time processing is possible for the original flow. We propose simple decomposition and reconstruction algorithms leading to formulas where the coefficients are independent of the grid and are equal to 1 in absolute value.

Our aim in the this research is to introduce the notion of algebra fuzzy absolute value in order to introduce new type of fuzzy normed space called algebra fuzzy normed space after that some examples is introduced to illustrate these notions. Then some properties of algebra fuzzy absolute value space are proved after that basic of algebra fuzzy normed space also proved.

Given a prime number p we consider C p the topological completion of the algebraic closure of the field of p-adic numbers with respect to the usual p-adic absolute value. In this paper, we point ou...

Let $$\mathbb K$$ be an algebraically closed field of characteristic zero, complete for a non-Archimedean absolute value. In this paper, we give a new class of unique range sets for meromorphic functions on $$\mathbb K.$$ We also show the existence of a $$ bi-URS $$ for $$\mathcal M(\mathbb K)$$ of the form $$(\{a_1,a_2, a_3, a_4,a_5,\infty\}),$$ which is different from A. Boutabaa-A. Escassut’s [3].

In the article, we introduce the generalized exponentially μ-preinvex function, derive a new q-integral identity for second order q-differentiable function, and establish several new q-trapezoidal type integral inequalities for the function whose absolute value of second q-derivative is exponentially μ-preinvex.

When monic integral polynomials of degree $n \geq 2$ are ordered by the maximum of the absolute value of their coefficients, the Hilbert irreducibility theorem implies that asymptotically 100% are irreducible and have Galois group isomorphic to $S_n$. In particular, amongst such polynomials whose coefficients are bounded by $B$ in absolute value, asymptotically $(1+o(1))(2B+1)^n$ are irreducible and have Galois group $S_n$. When $G$ is a proper transitive subgroup of $S_n$, however, the asymptotic count of polynomials with Galois group $G$ has been determined only in very few cases. Here, we show that if there are strong upper bounds on the number of degree $n$ fields with Galois group $G$, then there are also strong bounds on the number of polynomials with Galois group $G$. For example, for any prime $p$, we show that there are at most $O(B^{3 - \frac{2}{p}} (\log B)^{p - 1})$ polynomials with Galois group $C_p$ and coefficients bounded by $B$.

Motivation of this study is to check the sensitivity of dosimetric tool gamma with 2D detector array combination when unexpected errors occur while transferring intensity-modulated radiation therapy treatment plans from planning system to treatment unit. This study consists of 17 head and neck cancer patient’s treatment plans. Nine types of verification plans are created for all 17 clinically approved treatment plans by consecutively deleting different segments (up to eight) one by one from each field of the plan. Decrement factor (χ) is introduced in our study which illustrated the degree of decay of gamma passing rate when intentional errors are introduced. We analyzed the data by two different methods—one without selecting the region of interest (ROI) in dose distributions and the other by selecting the region of interest. By linear regression, the absolute value of slopes is 0.025, 0.024 and 0.015 without ROI and 0.030, 0.027 and 0.015 with ROI for 2%/2 mm, 3%/3 mm and 5%/5 mm criteria, respectively. The higher absolute value of the fitted slope indicates the higher sensitivity of this method to identify erroneous plan in treatment unit. The threshold value for 2%/2 mm equivalent to 95% passing criteria in 3%/3 mm used in clinical practice is obtained as 83.44%. The 2D detector array with dosimetric tool gamma is less sensitive in detecting errors when unprecedented errors of segment deletion occur within the treatment plans.

Band surgery is an operation relating pairs of knots or links in the three-sphere. We prove that if two quasi-alternating knots $K$ and $K'$ of the same square-free determinant are related by a band surgery, then the absolute value of the difference in their signatures is either 0 or 8. This obstruction follows from a more general theorem about the difference in the Heegaard Floer $d$-invariants for pairs of L-spaces that are related by distance one Dehn fillings and satisfy a certain condition in first homology. These results imply that $T(2, 5)$ is the only torus knot $T(2, m)$ with $m$ square-free that admits a chirally cosmetic banding, i.e. a band surgery operation to its mirror image. We conclude with a discussion on the scarcity of chirally cosmetic bandings., Comment: The main theorem has been strengthened. This version accepted for publication in the Bulletin of the London Mathematical Society

We prove a central limit theorem for the real and imaginary part and the absolute value of the Riemann zeta-function sampled along a vertical line in the critical strip with respect to an ergodic transformation similar to the Boolean transformation. This result complements a result by Steuding who has proven a strong law of large numbers for the same system. As a side result we state a general central limit theorem for a class of unbounded observables on the real line over the same ergodic transformation. The proof is based on the transfer operator method., Comment: 20 pages