Search results for: generating function

Authors: Ladznar S. Laja, Rosalio G. Artes, Jr.

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This paper introduced a graph polynomial relating convexity concepts. A graph polynomial is a polynomial representing a graph given some parameters. On the other hand, a subgraph H of a graph G is said to be convex in G if for every pair of vertices in H, every shortest path with these end-vertices lies entirely in H. We define the convex subgraph polynomial of a graph G to be the generating function of the sequence of the numbers of convex subgraphs of G of cardinalities ranging from zero to the order of G. This graph polynomial is monic since G itself is convex. The convex index which counts the number of convex subgraphs of G of all orders is just the evaluation of this polynomial at 1. Relationships relating algebraic properties of convex subgraphs polynomial with graph theoretic concepts are established.

Keywords: convex subgraph, convex index, generating function, polynomial ring

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Authors: Shumaila Ehtisham

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In Statistical Distribution Theory, considering an additional parameter to classical distributions is a usual practice. In this study, a new distribution referred to as α-Power Pareto distribution is introduced by including an extra parameter. Several properties of the proposed distribution including explicit expressions for the moment generating function, mode, quantiles, entropies and order statistics are obtained. Unknown parameters have been estimated by using maximum likelihood estimation technique. Two real datasets have been considered to examine the usefulness of the proposed distribution. It has been observed that α-Power Pareto distribution outperforms while compared to different variants of Pareto distribution on the basis of model selection criteria.

Keywords: α-power transformation, maximum likelihood estimation, moment generating function, Pareto distribution

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Authors: Serge B. Provost, Yishan Zhang

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A moment-based adjustment to the saddlepoint approximation is introduced in the context of density estimation. First applied to univariate distributions, this methodology is extended to the bivariate case. It then entails estimating the density function associated with each marginal distribution by means of the saddlepoint approximation and applying a bivariate adjustment to the product of the resulting density estimates. The connection to the distribution of empirical copulas will be pointed out. As well, a novel approach is proposed for estimating the support of distribution. As these results solely rely on sample moments and empirical cumulant-generating functions, they are particularly well suited for modeling massive data sets. Several illustrative applications will be presented.

Keywords: empirical cumulant-generating function, endpoints identification, saddlepoint approximation, sample moments, density estimation

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Authors: S. B. Provost, Susan Sheng

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An alternative bivariate density estimation methodology is introduced in this presentation. The proposed approach involves estimating the density function associated with the marginal distribution of each of the two variables by means of the saddlepoint approximation technique and applying a bivariate polynomial adjustment to the product of these density estimates. Since the saddlepoint approximation is utilized in the context of density estimation, such estimates are determined from empirical cumulant-generating functions. In the univariate case, the saddlepoint density estimate is itself adjusted by a polynomial. Given a set of observations, the coefficients of the polynomial adjustments are obtained from the sample moments. Several illustrative applications of the proposed methodology shall be presented. Since this approach relies essentially on a determinate number of sample moments, it is particularly well suited for modeling massive data sets.

Keywords: density estimation, empirical cumulant-generating function, moments, saddlepoint approximation

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Authors: Bahare Fatemi, Seyed Mehran Kazemi, Nazanin Mehrasa

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Sudoku is a logic-based combinatorial puzzle game which people in different ages enjoy playing it. The challenging and addictive nature of this game has made it a ubiquitous game. Most magazines, newspapers, puzzle books, etc. publish lots of Sudoku puzzles every day. These puzzles often come in different levels of difficulty so that all people, from beginner to expert, can play the game and enjoy it. Generating puzzles with different levels of difficulty is a major concern of Sudoku designers. There are several works in the literature which propose ways of generating puzzles having a desirable level of difficulty. In this paper, we propose a method based on constraint satisfaction problems to evaluate the difficulty of the Sudoku puzzles. Then, we propose a hill climbing method to generate puzzles with different levels of difficulty. Whereas other methods are usually capable of generating puzzles with only few number of difficulty levels, our method can be used to generate puzzles with arbitrary number of different difficulty levels. We test our method by generating puzzles with different levels of difficulty and having a group of 15 people solve all the puzzles and recording the time they spend for each puzzle.

Keywords: constraint satisfaction problem, generating Sudoku puzzles, hill climbing

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Authors: M. Tiachachat, M. Mihoubi

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The Bell polynomials are special polynomials in combinatorial analysis that have a wide range of applications in mathematics. They have interested many authors. The exponential partial Bell polynomials have been well reduced to some special combinatorial sequences. Numerous researchers had already been interested in the above polynomials, as evidenced by many articles in the literature. Inspired by this work, in this work, we propose a family of special polynomials named after the 2-successive partial Bell polynomials. Using the combinatorial approach, we prove the properties of these numbers, derive several identities, and discuss some special cases. This family includes well-known numbers and polynomials such as Stirling numbers, Bell numbers and polynomials, and so on. We investigate their properties by employing generating functions

Keywords: 2-associated r-Stirling numbers, the exponential partial Bell polynomials, generating function, combinatorial interpretation

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Authors: M. H. Yee, J. Md Yunos, W. Othman, R. Hassan, T. K. Tee, M. M. Mohamad

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The generation of an idea that goes through several phases is affected by individual factors, interests, preferences and motivation. The purpose of this research was to analyze the difference in difficulties of generating ideas according to individual learning styles. A total of 375 technical students from four technical universities in Malaysia were randomly selected as samples. The Kolb Learning Styles Inventory and a set of developed questionnaires were used in this research. The results showed that the most dominant learning style is among technical students is Doer. A total of 319 (85.1%) technical students faced difficulties in solving individual assignments. Most of the problem faced by technical students is the difficulty of generating ideas for solving individual assignments. There was no significant difference in difficulties of generating ideas according to students’ learning styles. Therefore, students need to learn higher order thinking skills enabling students to generate ideas and consequently complete assignments.

Keywords: difference, difficulties, generating idea, learning styles, Kolb Learning Styles Inventory

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Authors: Mariam Vardiashvili

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The economic value of the asset impairment process is quite large. Impairment reflects the reduction of future economic benefits or service potentials itemized in the asset. The assets owned by public sector entities bring economic benefits or are used for delivery of the free-of-charge services. Consequently, they are classified as cash-generating and non-cash-generating assets. IPSAS 21 - Impairment of non-cash-generating assets, and IPSAS 26 - Impairment of cash-generating assets, have been designed considering this specificity.  When measuring impairment of assets, it is important to select the relevant methods. For measurement of the impaired Non-Cash-Generating Assets, IPSAS 21 recommends three methods: Depreciated Replacement Cost Approach, Restoration Cost Approach, and  Service Units Approach. Impairment of Value in Use of Cash-Generating Assets (according to IPSAS 26) is measured by discounted value of the money sources to be received in future. Value in use of the cash-generating asserts (as per IPSAS 26) is measured by the discounted value of the money sources to be received in the future. The article provides classification of the assets in the public sector  as non-cash-generating assets and cash-generating assets and, deals also with the factors which should be considered when evaluating  impairment of assets. An essence of impairment of the non-financial assets and the methods of measurement thereof evaluation are formulated according to IPSAS 21 and IPSAS 26. The main emphasis is put on different methods of measurement of the value in use of the impaired Cash-Generating Assets and Non-Cash-Generation Assets and the methods of their selection. The traditional and the expected cash flow approaches for calculation of the discounted value are reviewed. The article also discusses the issues of recognition of impairment loss and its reflection in the financial reporting. The article concludes that despite a functional purpose of the impaired asset, whichever method is used for measuring the asset, presentation of realistic information regarding the value of the assets should be ensured in the financial reporting. In the theoretical development of the issue, the methods of scientific abstraction, analysis and synthesis were used. The research was carried out with a systemic approach. The research process uses international standards of accounting, theoretical researches and publications of Georgian and foreign scientists.

Keywords: cash-generating assets, non-cash-generating assets, recoverable (usable restorative) value, value of use

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Authors: Gamze Ozel, Selen Cakmakyapan

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We introduce a new model called the Marshall-Olkin Rayleigh distribution which extends the Rayleigh distribution using Marshall-Olkin transformation and has increasing and decreasing shapes for the hazard rate function. Various structural properties of the new distribution are derived including explicit expressions for the moments, generating and quantile function, some entropy measures, and order statistics are presented. The model parameters are estimated by the method of maximum likelihood and the observed information matrix is determined. The potentiality of the new model is illustrated by means of real life data set.

Keywords: Marshall-Olkin distribution, Rayleigh distribution, estimation, maximum likelihood

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Authors: K. A. Adepoju, O. I. Shittu, A. U. Chukwu

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In this paper, a new four-parameter generalized version of the Fisher Snedecor distribution called Beta- F distribution is introduced. The comprehensive account of the statistical properties of the new distributions was considered. Formal expressions for the cumulative density function, moments, moment generating function and maximum likelihood estimation, as well as its Fisher information, were obtained. The flexibility of this distribution as well as its robustness using cancer remission time data was demonstrated. The new distribution can be used in most applications where the assumption underlying the use of other lifetime distributions is violated.

Keywords: fisher-snedecor distribution, beta-f distribution, outlier, maximum likelihood method

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Authors: Nur Rokhman

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A mathematical function gives relationship between the variables composing the function. Interpolation can be viewed as a process of finding mathematical function which goes through some specified points. There are many interpolation methods, namely: Lagrange method, Newton method, Spline method etc. For some specific condition, such as, big amount of interpolation points, the interpolation function can not be written explicitly. This such function consist of computational steps. The solution of equations involving the interpolation function is a problem of solution of non linear equation. Newton method will not work on the interpolation function, for the derivative of the interpolation function cannot be written explicitly. This paper shows the use of Secton method to determine the numerical solution of the function involving the interpolation function. The experiment shows the fact that Secton method works better than Newton method in finding the root of Lagrange interpolation function.

Keywords: Secton method, interpolation, non linear function, numerical solution

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Authors: Hazem M. Al-Mofleh

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In this paper, a new four-parameter univariate continuous distribution called the Normal-Generalized Hyperbolic Secant Distribution (NGHS) is defined and studied. Some general and structural distributional properties are investigated and discussed, including: central and non-central n-th moments and incomplete moments, quantile and generating functions, hazard function, Rényi and Shannon entropies, shapes: skewed right, skewed left, and symmetric, modality regions: unimodal and bimodal, maximum likelihood (MLE) estimators for the parameters. Finally, two real data sets are used to demonstrate empirically its flexibility and prove the strength of the new distribution.

Keywords: bimodality, estimation, hazard function, moments, Shannon’s entropy

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Authors: Meziane Rachid, Boufala Seddik, Hamzi Amar, Amara Mohamed

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Today’s energy industry seeks maximum benefit with maximum reliability. In order to achieve this goal, design engineers depend on reliability optimization techniques. This work uses a harmony search algorithm (HS) meta-heuristic optimization method to solve the problem of wind-Solar-Gas power systems design optimization. We consider the case where redundant electrical components are chosen to achieve a desirable level of reliability. The electrical power components of the system are characterized by their cost, capacity and reliability. The reliability is considered in this work as the ability to satisfy the consumer demand which is represented as a piecewise cumulative load curve. This definition of the reliability index is widely used for power systems. The proposed meta-heuristic seeks for the optimal design of series-parallel power systems in which a multiple choice of wind generators, transformers and lines are allowed from a list of product available in the market. Our approach has the advantage to allow electrical power components with different parameters to be allocated in electrical power systems. To allow fast reliability estimation, a universal moment generating function (UMGF) method is applied. A computer program has been developed to implement the UMGF and the HS algorithm. An illustrative example is presented.

Keywords: reliability optimization, harmony search optimization (HSA), universal generating function (UMGF)

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Authors: Faisel Eltuhami Alzaalik, Omar Imhemed Alramli, Ahmed Mohamed Elaieb

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The IEEE 802.11 defines two modes of MAC, distributed coordination function (DCF) and point coordination function (PCF) mode. The first sub-layer of the MAC is the distributed coordination function (DCF). A contention algorithm is used via DCF to provide access to all traffic. The point coordination function (PCF) is the second sub-layer used to provide contention-free service. PCF is upper DCF and it uses features of DCF to establish guarantee access of its users. Some papers and researches that have been published in this technology were reviewed in this paper, as well as talking briefly about the distributed coordination function (DCF) technology. The simulation of the PCF function have been applied by using a simulation program called network simulator (NS2) and have been found out the throughput of a transmitter system by using this function.

Keywords: DCF, PCF, throughput, NS2

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Authors: Yilmaz Simsek

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In this paper, we study the Bernstein-type basis functions with their generating functions. We give various properties of these polynomials with the aid of their generating functions. These polynomials and generating functions have many valuable applications in mathematics, in probability, in statistics and also in mathematical physics. By using the Bernstein-Galerkin and the Bernstein-Petrov-Galerkin methods, we give some applications of the Bernstein-type polynomials for solving high even-order differential equations with their numerical computations. We also give Bezier-type curves related to the Bernstein-type basis functions. We investigate fundamental properties of these curves. These curves have many applications in mathematics, in computer geometric design and other related areas. Moreover, we simulate these polynomials with their plots for some selected numerical values.

Keywords: generating functions, Bernstein basis functions, Bernstein polynomials, Bezier curves, differential equations

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Authors: Mandeep Kaur, Fatehgarh Sahib

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The purpose of economic load dispatch is to allocate the required load demand between the available generation units such that the cost of operation is minimized. It is an optimization problem to find the most economical schedule of the generating units while satisfying load demand and operational constraints. The multiobjective optimization problem in which the engineer’s goal is to maximize or minimize not a single objective function but several objective functions simultaneously. The purpose of multiobjective problems in the mathematical programming framework is to optimize the different objective functions. Many approaches and methods have been proposed in recent years to solve multiobjective optimization problems. Weighting method has been applied to convert multiobjective optimization problems into scalar optimization. MATLAB 7.10 has been used to write the code for the complete algorithm with the help of genetic algorithm (GA). The validity of the proposed method has been demonstrated on a three-unit power system.

Keywords: economic load dispatch, genetic algorithm, generating units, multiobjective optimization, weighting method

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Authors: Yaxin Bi

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Accurate prediction and understanding of the evolution mechanisms of earthquakes remain challenging in the fields of geology, geophysics, and seismology. This study leverages Long Short-Term Memory (LSTM) networks and Generative Adversarial Networks (GANs), a generative model tailored to time-series data, for generating synthetic time series data based on Swarm satellite data, which will be used for detecting seismic anomalies. LSTMs demonstrated commendable predictive performance in generating synthetic data across multiple countries. In contrast, the GAN models struggled to generate synthetic data, often producing non-informative values, although they were able to capture the data distribution of the time series. These findings highlight both the promise and challenges associated with applying deep learning techniques to generate synthetic data, underscoring the potential of deep learning in generating synthetic electromagnetic satellite data.

Keywords: LSTM, GAN, earthquake, synthetic data, generative AI, seismic precursors

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Authors: Shatha S. Alhily

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The purpose of this research paper is to show all the possible values of the pth power of the integrable function which make the integral means of the derivative of univalent function existing and finite.

Keywords: derivative, integral means, self conformal maps, univalent function

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Authors: Alexandre Marin, Alexandra Bac, Laurent Astart

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New numerical methods for PDE resolution (such as Finite Volumes (FV) or Virtual Elements Method (VEM)) open new needs in terms of meshing of domains of interest, and in particular, polyhedral meshes have many advantages. One way to build such meshes consists of constructing Restricted Voronoi Diagrams (RVDs) whose boundaries respect the domain of interest. By minimizing a function defined for RVDs, the shapes of cells can be controlled, e.g., elongated according to user-defined directions or adjusted to comply with given aspect ratios (anisotropy) and density variations. In this paper, our contribution is threefold: First, we introduce a new gradient formula for the Voronoi tessellation energy under a continuous anisotropy field. Second, we describe a meshing algorithm based on the optimisation of this function that we validate against state-of-the-art approaches. Finally, we propose a hierarchical approach to speed up our meshing algorithm.

Keywords: anisotropic Voronoi diagrams, meshes for numerical simulations, optimisation, volumic polyhedral meshing

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Authors: Chul H. Jo, Do Y. Kim, Bong K. Cho, Myeong J. Kim

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The exhaustion of oil and the environmental pollution from the use of fossil fuel are increasing. Tidal current power (TCP) has been proposed as an alternative energy source because of its predictability and reliability. By applying a duct and single point mooring (SPM) system, a TCP device can amplify the generating power and keep its position properly. Because the generating power is proportional to cube of the current stream velocity, amplifying the current speed by applying a duct to a TCP system is an effective way to improve the efficiency of the power device. An SPM system can be applied at any water depth and is highly cost effective. Simple installation and maintenance procedures are also merits of an SPM system. In this study, we designed an SPM system for a duct-type TCP device for use in shallow water. Motions of the duct are investigated to obtain the response amplitude operator (RAO) as the magnitude of the transfer function. Parameters affecting the stability of the SPM system such as the fairlead departure angle, current velocity, and the number of clamp weights are analyzed and/or optimized. Wadam and OrcaFlex commercial software is used to design the mooring line.

Keywords: mooring design, parametric analysis, RAO (Response Amplitude Operator), SPM (Single Point Mooring)

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Authors: Prat Yotyawilai, Taratip Suwannasart

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Business Process Model and Notation (BPMN) is more important in the business process and creating functional models, and is a standard for OMG, which becomes popular in various organizations and in education. Researches related to software testing based on models are prominent. Although most researches use the UML model in software testing, not many researches use the BPMN Model in creating test cases. Therefore, this research proposes a design of a tool for generating test cases from the BPMN. The model is analyzed and the details of the various components are extracted before creating a flow graph. Both details of components and the flow graph are used in generating test cases.

Keywords: software testing, test case, BPMN, flow graph

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Authors: F. U. Rahman, R. Q. Zhang

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This work aims to develop a systematic numerical technique which can be easily extended to many-body problem. The Lippmann Schwinger equation (integral form of the Schrodinger wave equation) is solved for the nonrelativistic radial wave of hydrogen atom using iterative integration scheme. As the unknown wave function appears on both sides of the Lippmann Schwinger equation, therefore an approximate wave function is used in order to solve the equation. The Green’s function is obtained by the method of Laplace transform for the radial wave equation with excluded potential term. Using the Lippmann Schwinger equation, the product of approximate wave function, the Green’s function and the potential term is integrated iteratively. Finally, the wave function is normalized and plotted against the standard radial wave for comparison. The outcome wave function converges to the standard wave function with the increasing number of iteration. Results are verified for the first fifteen states of hydrogen atom. The method is efficient and consistent and can be applied to complex systems in future.

Keywords: Green’s function, hydrogen atom, Lippmann Schwinger equation, radial wave

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Authors: Zhongzhi Hu, Jie Shen, Jiqiang Wang

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A high precision aeroengine model is needed when developing the engine control system. Compared with other main components, the axial compressor is the most challenging component to simulate. In this paper, a compressor map optimizing tool based on the introduction of a modifiable β function is developed for FWorks (FADEC Works). Three parameters (d density, f fitting coefficient, k₀ slope of the line β=0) are introduced to the β function to make it modifiable. The comparison of the traditional β function and the modifiable β function is carried out for a certain type of compressor. The interpolation errors show that both methods meet the modeling requirements, while the modifiable β function can predict compressor performance more accurately for some areas of the compressor map where the users are interested in.

Keywords: beta function, compressor map, interpolation error, map optimization tool

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Authors: Leszek Chybowski, Artur Bejger, Katarzyna Gawdzińska

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Subversion analysis is a tool used in the TRIZ (Theory of Inventive Problem Solving) methodology. This article introduces the history and describes the process of subversion analysis, as well as function analysis and analysis of the resources, used at the design stage when generating possible undesirable situations. The article charts the course of subversion analysis when applied to a fuel injection nozzle of a marine engine. The work describes the fuel injector nozzle as a technological system and presents principles of analysis for the causes of a cracked tip of the nozzle body. The system is modelled with functional analysis. A search for potential causes of the damage is undertaken and a cause-and-effect analysis for various hypotheses concerning the damage is drawn up. The importance of particular hypotheses is evaluated and the most likely causes of damage identified.

Keywords: complex technical system, fuel injector, function analysis, importance analysis, resource analysis, sabotage analysis, subversion analysis, TRIZ (Theory of Inventive Problem Solving)

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Authors: Sang Beom Kim, Kyung Ju Kim, Myung Hoon Cho, Ji Hoon Kim, Soo Hyung Kim

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In this study, we systematically investigated the effect of nanoscale energetic materials in formulations of aluminum nanoparticles (Al NPs; heat source)/copper oxide nanoparticles (CuO NPs; oxidizer) on the combustion and gas-generating properties of sodium azide microparticles (NaN3 MPs; gas-generating agent) for potential applications in gas generators. The burn rate of the NaN3 MP/CuO NP composite powder was only ~0.3 m/s. However, the addition of Al NPs to the NaN3 MP/CuO NP matrix caused the rates to reach ~5.3 m/s, respectively. In addition, the N2 gas volume flow rate generated by the ignition of the NaN3 MP/CuO NP composite powder was only ~0.6 L/s, which was significantly increased to ~3.9 L/s by adding Al NPs to the NaN3 MP/CuO NP composite powder. This suggested that the highly reactive NPs, with the assistance of CuO NPs, were effective heat-generating sources enabling the complete thermal decomposition of NaN3 MPs upon ignition. Al NPs were highly effective in the gas generators because of the increased reactivity induced by the reduced particle size. Finally, we successfully demonstrated that a homemade airbag with a specific volume of ~140 mL could be rapidly and fully inflated by the thermal activation of nanoscale energetic material-added gas-generating agents (i.e., NaN3 MP/Al NP/CuO NP composites) within the standard time of ~50 ms for airbag inflation.

Keywords: nanoenergetic materials, aluminum nanoparticles, copper oxide nanoparticles, gas generators

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Authors: Haojie Wang, Xun Li, Rui Yin

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In order to solve the memorization overfitting in the meta-learning MAML algorithm, a method of generating mutually exclusive tasks based on data augmentation is proposed. This method generates a mutex task by corresponding one feature of the data to multiple labels, so that the generated mutex task is inconsistent with the data distribution in the initial dataset. Because generating mutex tasks for all data will produce a large number of invalid data and, in the worst case, lead to exponential growth of computation, this paper also proposes a key data extraction method, that only extracts part of the data to generate the mutex task. The experiments show that the method of generating mutually exclusive tasks can effectively solve the memorization overfitting in the meta-learning MAML algorithm.

Keywords: data augmentation, mutex task generation, meta-learning, text classification.

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Authors: Haojie Wang, Xun Li, Rui Yin

Abstract:

In order to solve the memorization overfitting in the model-agnostic meta-learning MAML algorithm, a method of generating mutually exclusive tasks based on data augmentation is proposed. This method generates a mutex task by corresponding one feature of the data to multiple labels so that the generated mutex task is inconsistent with the data distribution in the initial dataset. Because generating mutex tasks for all data will produce a large number of invalid data and, in the worst case, lead to an exponential growth of computation, this paper also proposes a key data extraction method that only extract part of the data to generate the mutex task. The experiments show that the method of generating mutually exclusive tasks can effectively solve the memorization overfitting in the meta-learning MAML algorithm.

Keywords: mutex task generation, data augmentation, meta-learning, text classification.

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Authors: Zsolt Lipcsey, Sampson Marshal Imeh

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We investigate a certain characterisation for rank of a semigroup by Howie and Ribeiro (1999), to ascertain the relevance of the concept of independence. There are cases where the concept of independence fails to be useful for this purpose. One would expect the basic element to be the maximal independent subset of a given semigroup. However, we construct examples for semigroups where finite basis exist and the basis is larger than the number of independent elements.

Keywords: generating sets, independent set, rank, cyclic semigroup, basis, commutative

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Authors: David Holton, Michelle Dickinson, Giovanni Carta

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The principle of geological disposal is to isolate higher-activity radioactive wastes deep inside a suitable rock formation to ensure that no harmful quantities of radioactivity reach the surface environment. To achieve this, wastes will be placed in an engineered underground containment facility – the geological disposal facility (GDF) – which will be designed so that natural and man-made barriers work together to minimise the escape of radioactivity. Internationally, various multi-barrier concepts have been developed for the disposal of higher-activity radioactive wastes. High-heat-generating wastes (HLW, spent fuel and Pu) provide a number of different technical challenges to those associated with the disposal of low-heat-generating waste. Thermal management of the disposal system must be taken into consideration in GDF design; temperature constraints might apply to the wasteform, container, buffer and host rock. Of these, the temperature limit placed on the buffer component of the engineered barrier system (EBS) can be the most constraining factor. The heat must therefore be managed such that the properties of the buffer are not compromised to the extent that it cannot deliver the required level of safety. The maximum temperature of a buffer surrounding a container at the centre of a fixed array of heat-generating sources, arises due to heat diffusing from neighbouring heat-generating wastes, incrementally contributing to the temperature of the EBS. A range of strategies can be employed for managing heat in a GDF, including the spatial arrangements or patterns of those containers; different geometrical configurations can influence the overall thermal density in a disposal facility (or area within a facility) and therefore the maximum buffer temperature. A semi-analytical thermal dimensioning tool and methodology have been applied at a generic stage to explore a range of strategies to manage the disposal of high-heat-generating waste. A number of examples, including different geometrical layouts and chequer-boarding, have been illustrated to demonstrate how these tools can be used to consider safety margins and inform strategic disposal options when faced with uncertainty, at a generic stage of the development of a GDF.

Keywords: buffer, geological disposal facility, high-heat-generating waste, spent fuel

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Authors: Slobodan B. Tričković

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We present the results concerned with trigonometric series that include sine and cosine functions with a parameter appearing in the denominator. We derive two types of closed-form formulas for trigonometric series. At first, for some integer values, as we know that Riemann’s ζ function and Dirichlet η, λ equal zero at negative even integers, whereas Dirichlet’s β function equals zero at negative odd integers, after a certain number of members, the rest of the series vanishes. Thus, a trigonometric series becomes a polynomial with coefficients involving Riemann’s ζ function and Dirichlet η, λ, β functions. On the other hand, in some cases, one cannot immediately replace the parameter with any positive integer because we shall encounter singularities. So it is necessary to take a limit, so in the process, we apply L’Hospital’s rule and, after a series of rearrangements, we bring a trigonometric series to a form suitable for the application of Choi-Srivastava’s theorem dealing with Hurwitz’s zeta function and Harmonic numbers. In this way, we express a trigonometric series as a polynomial over Hurwitz’s zeta function derivative.

Keywords: Dirichlet eta lambda beta functions, Riemann's zeta function, Hurwitz zeta function, Harmonic numbers

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