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Journal articles on the topic 'Product function'

Author: Grafiati

Published: 4 June 2025

Last updated: 1 August 2025

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1

Hooi Ting, Ding, and Chong Meng Chee. "Beyond product core function: the deterritorialization shift." International Journal of Academic Research 5, no. 6 (2013): 258–61. http://dx.doi.org/10.7813/2075-4124.2013/5-6/b.42.

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2

Lee, Haebin, Muhammad Tufail, and KwanMyung Kim. "CLASSIFICATION OF TRANSFORMABLE PRODUCTS BASED ON CHANGES IN PRODUCT FORM AND FUNCTION." Proceedings of the Design Society 1 (July 27, 2021): 641–50. http://dx.doi.org/10.1017/pds.2021.64.

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AbstractThe sentence “form follows function” is the best expression of the relationship between form and function of a product in modern product design. This implies that the form is essential to implement the function while it should be optimized and minimized for the function. Unlike general products, transformable products are designed with the intention to change the function and form of the product according to the situation. This paper presents the types and characteristics of transformable products determined by Phase Model we introduced. We collected 147 transformable product cases and

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3

Bakouch, Hassan, Miroslav Ristic, E. Sandhya, and S. Satheesh. "Random products and product auto-regression." Filomat 27, no. 7 (2013): 1197–203. http://dx.doi.org/10.2298/fil1307197b.

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The operation of taking random products of random variables and the notions of infinite divisibility (ID) and stability of distributions under this operation are discussed here. Based on this stationary product auto-regressive time series models are introduced. We investigate some properties of the models, like autocorrelation function, spectral density function, multi-step ahead conditional mean and parameter estimation.

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4

Jakubowski, Zbigniew Jerzy, Piotr Liczberski, and Łucja Żywień. "Applications of the Hadamard product in geometric function theory." Mathematica Bohemica 116, no. 2 (1991): 148–59. http://dx.doi.org/10.21136/mb.1991.126141.

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5

Jose, Rani, and D. Susha. "Chromatic M-polynomial of Mobius Function Graphs of Product Groups." Indian Journal Of Science And Technology 17, no. 41 (2024): 4332–37. http://dx.doi.org/10.17485/ijst/v17i41.2959.

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Objective: To investigate the chromatic M-polynomials of the Mobius function graph of product groups, specifically focusing on the direct product of finite groups. Methods: We used algebraic and number theoretical techniques to construct a Mobius function graph of product groups and derived the chromatic M-polynomials using graph theoretic methods. Findings: Chromatic M polynomials of Mobius function graphs are introduced with a focus on deriving their significant properties and providing a practical approach for computation. Novelty: This research introduces a novel approach to studying the c

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6

Zhang, Jinpu, Guozhong Cao, Qingjin Peng, Runhua Tan, Huangao Zhang, and Wei Liu. "A Time Correlation Based Clustering Method for a Design of a Transformable Product." Applied Sciences 10, no. 1 (2020): 406. http://dx.doi.org/10.3390/app10010406.

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A single-function product cannot meet various needs of different users when the product user or use environment changes. A transformable product with multiple functions can meet different needs of users. It is critical to determine product functions that a transformable product should have in the product design. However, it is a challenge to decide required functional components of a transformable product in the design process. A clustering method is proposed in this paper using undirected graphs for segmentations of needs in the time dimension. A need-based function model is built to form pro

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7

Shah, Syed Ali Haider, Shahid Mubeen, Gauhar Rahman, and Jihad Younis. "Relation of Some Known Functions in terms of Generalized Meijer G -Functions." Journal of Mathematics 2021 (November 3, 2021): 1–9. http://dx.doi.org/10.1155/2021/7032459.

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The aim of this paper is to prove some identities in the form of generalized Meijer G -function. We prove the relation of some known functions such as exponential functions, sine and cosine functions, product of exponential and trigonometric functions, product of exponential and hyperbolic functions, binomial expansion, logarithmic function, and sine integral, with the generalized Meijer G -function. We also prove the product of modified Bessel function of first and second kind in the form of generalized Meijer G -function and solve an integral involving the product of modified Bessel function

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8

Ali, Omar. "ON THE UNIFORMLY STARLIKENSS OF THE EXPONENTIAL FUNCTION." International Journal of Novel Research in Interdisciplinary Studies 11, no. 2 (2024): 28–32. https://doi.org/10.5281/zenodo.10997500.

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<strong>Abstract:</strong> We introduce and study a new subclass of uniformly starlike functions by the means of the convolution involving the exponential function in the unit disk. <strong>Keywords:</strong> Uniformly Starlike functions; Hadamard product; Linear operator; Exponential function. <strong>Title:</strong> ON THE UNIFORMLY STARLIKENSS OF THE EXPONENTIAL FUNCTION <strong>Author:</strong> Ali Omar <strong>International Journal of Novel Research in Interdisciplinary Studies</strong> <strong>ISSN 2394-9716</strong> <strong>Vol. 11, Issue 2, March 2024 - April 2024</strong> <strong>Page

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9

Frédéric, Ayant, Kumar Prvindra, and Singh Harendra. "On unified infinite integral involving product of multivariable Gimel-function and others special functions." APPLIED SCIENCE PERIODICAL XXIII, no. 3, August 2021 (2021): 1–16. https://doi.org/10.5281/zenodo.6796914.

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              The multivariable Gimel function [2] is an unified special function, it’s an extension of the multivariable Aleph-function defined by Ayant [1], the multivariable I-function defined by Prasad [9], the multivariable I-function defined by Prathima et al. [11] at a time, of course this function is a generalization of the multivariable H-function. To define this function, we use the multiple Mellin-Barnes integrals contour. 

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10

YOSHIOKA, Akira. "Star product and star function." Tambov University Reports. Series: Natural and Technical Sciences, no. 127 (2019): 281–92. http://dx.doi.org/10.20310/2686-9667-2019-24-127-281-292.

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We give a brief review on star products and star functions [8, 9]. We introduce a star product on polynomials. Extending the product to functions on complex space, we introduce exponential element in the star product algebra. By means of the star exponential functions we can define several functions called star functions in the algebra.We show certain examples.

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11

Nishino, Tomotoshi, and Kouichi Okunishi. "Product Wave Function Renormalization Group." Journal of the Physical Society of Japan 64, no. 11 (1995): 4084–87. http://dx.doi.org/10.1143/jpsj.64.4084.

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12

Gustafson, Michael J. "Uncertainty product scattering function estimation." Journal of the Acoustical Society of America 96, no. 2 (1994): 1205. http://dx.doi.org/10.1121/1.410337.

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13

Polke, Reinhard. "Property Function for Product Development." Particle & Particle Systems Characterization 4, no. 1-4 (1987): 54–62. http://dx.doi.org/10.1002/ppsc.19870040112.

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14

Sharma, Manju. "Evaluation of General Class of Beta Integrals Involving Certain Particular Functions." Innovation The Research Concept 9, no. 1 (2024): E1—E7. https://doi.org/10.5281/zenodo.10673412.

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This paper has been published in Peer-reviewed International Journal "Innovation The Research Concept"                  URL : https://www.socialresearchfoundation.com/new/publish-journal.php?editID=8200 Publisher : Social Research Foundation, Kanpur (SRF International)                  Abstract : In this paper general class of Beta integrals is considered for particular function . This class is further used to evaluate certain integrals for some special functio

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15

Ko, Ya-Hsien. "Influencing Factors on Color and Product-Function Association." Psychological Reports 108, no. 3 (2011): 861–73. http://dx.doi.org/10.2466/01.09.24.pr0.108.3.861-873.

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The associations of age, sex, and matching types with color and product-function were examined in a real-world product scenario (shampoo) among 128 volunteers (M age = 29.3 yr.; SD = 15.6). A pilot study identified eight popular colors and eight product-functions. The association between color and product-function was explored in the main sample. Responses suggested seven pairings of color/product-functions: Red/Hot oil treatment, Yellow/Bright and shiny hair, Green/Herbal extracts, Blue/Deep cleaning, Purple/Soothing, Black/Antiseptic, and White/Anti-dandruff. Analyses indicated that adult pa

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16

Bouzeffour, Fethi. "Interpolation of entire functions, product formula for basic sine function." Journal of Nonlinear Mathematical Physics 13, no. 2 (2006): 293–301. http://dx.doi.org/10.2991/jnmp.2006.13.2.11.

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17

Politze, D. P., S. Dierssen, and K. Wegener. "Function module drivers for assessing the similarity between product functions." CIRP Journal of Manufacturing Science and Technology 5, no. 1 (2012): 33–40. http://dx.doi.org/10.1016/j.cirpj.2011.11.001.

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18

Sitonga, Roland YH, Marla Setiawati, and Theodorus Gratianus. "Assessment of Managerial Innovation in Company X Using Fuzzy AHP." RSF Conference Series: Business, Management and Social Sciences 2, no. 1 (2022): 194–205. http://dx.doi.org/10.31098/bmss.v2i1.532.

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Company X is an Indonesian dairy company with a significantly large market share. The successes of its product innovation are highly linked with its managerial function, which indicates the existence of innovation in its company management. The company should understand the level of its managerial innovation and utilize its potential to produce product innovations continuously. This research aims to determine the company’s level of managerial innovation to support product innovation accomplishments. The study utilizes two methods in determining managerial innovation levels. The first method is

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19

Sitonga, Roland YH, Marla Setiawati, and Theodorus Gratianus. "Assessment of Managerial Innovation in Company X Using Fuzzy AHP." RSF Conference Series: Business, Management and Social Sciences 2, no. 1 (2022): 194–205. http://dx.doi.org/10.31098/bmss.v2i1.532.

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Company X is an Indonesian dairy company with a significantly large market share. The successes of its product innovation are highly linked with its managerial function, which indicates the existence of innovation in its company management. The company should understand the level of its managerial innovation and utilize its potential to produce product innovations continuously. This research aims to determine the company’s level of managerial innovation to support product innovation accomplishments. The study utilizes two methods in determining managerial innovation levels. The first method is

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20

Priyanka, Gupta*1 &. Dr Neelam Pandey2. "INTEGRAL INVOLVING THE MULTIVARIABLE H-FUNCTIONS." GLOBAL JOURNAL OF ENGINEERING SCIENCE AND RESEARCHES 6, no. 5 (2019): 80–85. https://doi.org/10.5281/zenodo.2693875.

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In this paper, the author presented certain integrals involving product of the multivariable H-function with exponential function, Gauss’s hypergeometric function and Fox’s function. The results derived here and basic in natural and many include a number of known and new results as particular cases.

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21

Nurliza, Nurliza, Anita Suharyani, and Aditya Nugraha. "The Product Features, Functions, and Benefits of Seafood Products for Competitive Repositioning." AGRARIS: Journal of Agribusiness and Rural Development Research 7, no. 1 (2021): 91–110. http://dx.doi.org/10.18196/agraris.v7i1.10571.

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The global demand for seafood products increased in a dynamic environment. Still, fails to achieve competitive positioning due to labeling, unattractive and unprofitable targeted segments, and less preferred quality and features. Thus, this study tried to create competitive positioning through features, functions, and benefits of seafood product attributes. The results arranged through consumer characteristics, consumer preference with conjoint analysis, market identification and competition with regression analysis, Multidimensional Scaling, and Correspondence Analysis from 206 respondents. T

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22

Chen, Yuwen, and Janice E. Carrillo. "Single firm product diffusion model for single-function and fusion products." European Journal of Operational Research 214, no. 2 (2011): 232–45. http://dx.doi.org/10.1016/j.ejor.2011.04.030.

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23

Sangelkar, Shraddha, Nicholas Cowen, and Daniel McAdams. "User activity – product function association based design rules for universal products." Design Studies 33, no. 1 (2012): 85–110. http://dx.doi.org/10.1016/j.destud.2011.06.002.

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24

Parajuli, Vinod, and Santosh Ghimire. "Product of Ap Weight Functions." Journal of the Institute of Engineering 11, no. 1 (2016): 116–19. http://dx.doi.org/10.3126/jie.v11i1.14703.

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In this paper, we first define Ap weight functions and then show that finite product of weight functions each raised with some power whose sum is one is also an Ap weight function. Journal of the Institute of Engineering, 2015, 11(1): 116-119

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25

Cheng, Jin Xia. "Multi-Functional Product Design Based on Green Design Concept." Advanced Materials Research 945-949 (June 2014): 527–30. http://dx.doi.org/10.4028/www.scientific.net/amr.945-949.527.

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On the base of green design concept, multi-functional product design is given a new meaning. It is no longer simply a combination of multi-functional functions, but to extend product function on the bases of no extra cost and no waste of materials. Through fuzzy functional design, integrated design of related functions, developing the secondary function of product, as well as enhancing the product spiritual function and other ways, we can achieve multi-functional design of products.

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26

Wang, Nianliang, Kalyan Chakraborty, and Takako Kuzumaki. "On the Product of Zeta-Functions." Mathematics 13, no. 11 (2025): 1900. https://doi.org/10.3390/math13111900.

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In this paper, we study the Bochner modular relation (Lambert series) for the kth power of the product of two Riemann zeta-functions with difference α, an integer with the Voronoĭ function weight Vk. In the case of V1(x)=e−x, the results reduce to Bochner modular relations, which include the Ramanujan formula, Wigert–Bellman approximate functional equation, and the Ewald expansion. The results abridge analytic number theory and the theory of modular forms in terms of the sum-of-divisor function. We pursue the problem of (approximate) automorphy of the associated Lambert series. The α=0 case is

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27

Lu, Chunfu, Hao Chai, Mengyang Tian, Xiang Peng, and Shaofei Jiang. "Product function combination design based on functional redundancy analysis." Concurrent Engineering 25, no. 3 (2017): 229–44. http://dx.doi.org/10.1177/1063293x17705180.

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Function combination designs integrate multiple product functions into a single product carrier. In this article, we proposed a function combination design method based on the functional redundancy analysis of parent products for the conceptual design process. According to the design task, we analyzed each parent product function carrier, established an expression model of functional elements based on the functional basis approach, and constructed a functional structure of the parent product based on relationships among energy, material, and signal flow. Then, we established a product function

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28

Sadygov, Misraddin Allahverdi oglu. "PROPERTIES OF THE BISUBDIFFERENTIAL OF BICONVEX FUNCTIONS." Annali d'Italia 47 (September 20, 2023): 28–41. https://doi.org/10.5281/zenodo.8364901.

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In this work, a number of properties of bisubdifferential and biconjugate functions for biconvex functions are studied. Necessary and sufficient conditions for the equality of the bisubdifferential of the sum and the sum of the bisubdifferentials of two biconvex functions are obtained. A biconvex mathematical programming problem is considered.

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29

Nawaz, Tariq, M. Asif Memon, and Kavikumar Jacob. "Hermite–Hadamard-Type Inequalities for Product of Functions by Using Convex Functions." Journal of Mathematics 2021 (January 27, 2021): 1–9. http://dx.doi.org/10.1155/2021/6630411.

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One of the many techniques to obtain a new convex function from the given functions is to calculate the product of these functions by imposing certain conditions on the functions. In general, the product of two or finite number of convex function needs not to be convex and, therefore, leads us to the study of product of these functions. In this paper, we reframe the idea of product of functions in the setting of generalized convex function to establish Hermite–Hadamard-type inequalities for these functions. We have analyzed different cases of double and triple integrals to derive some new resu

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30

Anastassiou, George A. "Approximation by Max-Product Operators." Fasciculi Mathematici 60, no. 1 (2018): 5–28. http://dx.doi.org/10.1515/fascmath-2018-0001.

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Abstract Here we study the approximation of functions by a great variety of Max-Product operators under differentiability. These are positive sublinear operators. Our study is based on our general results about positive sublinear operators. We produce Jackson type inequalities under initial conditions. So our approach is quantitative by producing inequalities with their right hand sides involving the modulus of continuity of a high order derivative of the function under approximation. We improve known related results which do not use smoothness of functions..

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31

Siregar, Ikhsan, and Kevin Adhinata. "PERANCANGAN PRODUK TEMPAT TISU MULTIFUNGSI DENGAN MENGGUNAKAN QUALITY FUNCTION DEPLOYMENT (QFD)." Jurnal Sistem Teknik Industri 19, no. 2 (2018): 21–29. http://dx.doi.org/10.32734/jsti.v19i2.370.

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Changes in human mindset along with the development of technology, trends and the need for quality products increasingly require the development of more complex product functions to meet the expectations and needs of consumers as users of the product. Merging several functions in a single product is often become the main focus for managers in research and product development departments to produce product design concepts in the present. The success of a product can be measured by how much the existence of the product can be accepted by the consumer so as to increase profit for the company or p

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32

Abbas, Riyam Mohammad. "The Role of Target Costing and Quality Function in Improving Product Value." Journal of Social Sciences and Humanities Research Fundamentals 5, no. 4 (2025): 59–77. https://doi.org/10.55640/jsshrf-05-04-13.

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In light of the economic and competitive transformations currently taking place in the world, companies are constantly striving to improve the value of their products and increase profitability. Among the practical tools for achieving this goal are the target costing technique and the quality function, two means of balancing cost and quality. This study aims to provide theoretical frameworks for applying the target costing technique and deploying the quality function to improve product value and achieve competitive advantage. Data from plastic and woven bag factories was collected, along with

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33

GAO, Fei. "Method of product function module partition." Chinese Journal of Mechanical Engineering 43, no. 05 (2007): 29. http://dx.doi.org/10.3901/jme.2007.05.029.

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34

Kumar, Rupesh, and Venkat Allada. "Function-technology-based product platform formation." International Journal of Production Research 45, no. 24 (2007): 5687–714. http://dx.doi.org/10.1080/00207540600806380.

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35

Joshi, Sanjay M., and Joel M. Morris. "Some results on product-function frames." Signal Processing 80, no. 4 (2000): 737–40. http://dx.doi.org/10.1016/s0165-1684(00)00060-8.

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36

Liftaj, Silvana, and Eriola Sila. "On axiomatic of inner product function." International Journal of Mathematical Analysis 7 (2013): 3011–18. http://dx.doi.org/10.12988/ijma.2013.311211.

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37

Banagl, Markus. "The tensor product of function semimodules." Algebra universalis 70, no. 3 (2013): 213–26. http://dx.doi.org/10.1007/s00012-013-0249-9.

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38

BOYD, C. "Function of cystic fibrosis gene product." Lancet 336, no. 8720 (1990): 938. http://dx.doi.org/10.1016/0140-6736(90)92305-2.

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39

Takeuchi, Tsukasa, Naoko Yoshimi, and Akira Yoshioka. "Star Product Deformation of Gamma Function." Geometry, Integrability and Quantization 26 (2023): 53–63. http://dx.doi.org/10.7546/giq-26-2023-53-63.

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40

Li, Qing Hai, Guo Zhong Cao, Hai Xia Guo, and Run Hua Tan. "Research on Innovation Process Model Based on Function Evolution." Applied Mechanics and Materials 20-23 (January 2010): 1336–41. http://dx.doi.org/10.4028/www.scientific.net/amm.20-23.1336.

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Function design is an important method for high-level innovation, and it is important for corporations to forecast product evolution rapidly and effectively from function angle. Three laws of function organization and eight laws of function improvement are presented to discover radically appearance of new functions and improvement of existing functions. The directions of function evolution are confirmed by function forecasting, and the function evolutionary path is described by function evolution tree. The rationality is evaluated by degree of ideality. The product innovation process model bas

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41

Hamd, Liqaa Ghazi. "The Systems of Mathematical Functions as Input Strategies for Contemporary Industrial Product Design." International Journal of Religion 5, no. 11 (2024): 1411–20. http://dx.doi.org/10.61707/ja2t0t15.

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This research emphasizes the crucial role of functions in elucidating physical relationships and diverse applications across scientific domains, particularly in the realm of design. The study investigates the contemporary and applied nature of design methodologies, employing real mathematical functions in the creation of industrial products within an organizational framework. The primary objective is to delineate the systems of functions (inputs and outputs) in industrial product design based on a well-defined strategy. The theoretical foundation encompasses design strategy and its correlation

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42

Olwamba, Levi Otanga, and Maurice Oduor. "On Pointwise Product Vector Measure Duality." JOURNAL OF ADVANCES IN MATHEMATICS 20 (January 6, 2021): 8–18. http://dx.doi.org/10.24297/jam.v20i.8912.

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This article is devoted to the study of pointwise product vector measure duality. The properties of Hilbert function space of integrable functions and pointwise sections of measurable sets are considered through the application of integral representation of product vector measures, inner product functions and products of measurable sets.

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43

Donsimoni, Marie-Paule, and Jonathan H. Hamilton. "Vertical Product Differentiation and Taste Differences." Recherches économiques de Louvain 57, no. 1 (1991): 21–32. http://dx.doi.org/10.1017/s0770451800029328.

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SummaryThe finiteness condition of vertical product differentiation models is translated into the taste distribution model first analyzed by Mussa and Rosen. For a utility function linear in quality, the necessary and sufficient condition for finiteness is that the cost function with respect to quality is strictly concave. Furthermore, for these cost functions, in duopoly, higher quality always implies a higher market share at the Nash equilibrium in prices. The n-firm case is briefly discussed, and some implications for marketing strategy of new products are presented.

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González, Yero, and Juan Rodríguez-Velázquez. "Roman domination in Cartesian product graphs and strong product graphs." Applicable Analysis and Discrete Mathematics 7, no. 2 (2013): 262–74. http://dx.doi.org/10.2298/aadm130813017g.

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A map f : V ? {0, 1, 2} is a Roman dominating function for G if for every vertex v with f(v) = 0, there exists a vertex u, adjacent to v, with f(u) = 2. The weight of a Roman dominating function is f(V ) = ?u?v f(u). The minimum weight of a Roman dominating function on G is the Roman domination number of G. In this article we study the Roman domination number of Cartesian product graphs and strong product graphs.

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45

Qin, YongZhou, and Wu-Sheng Wang. "A Generalized Nonlinear Sum-Difference Inequality of Product Form." Journal of Applied Mathematics 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/247585.

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We establish a generalized nonlinear discrete inequality of product form, which includes both nonconstant terms outside the sums and composite functions of nonlinear function and unknown function without assumption of monotonicity. Upper bound estimations of unknown functions are given by technique of change of variable, amplification method, difference and summation, inverse function, and the dialectical relationship between constants and variables. Using our result we can solve both the discrete inequality in Pachpatte (1995). Our result can be used as tools in the study of difference equati

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46

Kuzoma, Vitalii, and Vadim Spivak. "Theoretical foundations of circular business models in the agriculture sphere: classification, components, and functions." Ukrainian Journal of Applied Economics and Technology 9, no. 2 (2024): 162–65. http://dx.doi.org/10.36887/2415-8453-2024-2-27.

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The article is dedicated to the practical implications of circular business models in the agro-food sector, classifying their types and explaining their strategic components and functions. We explore resource models, design models, models of extending the product life cycle, platform models, product as a service, and end-of-life models of the product. We also delve into cascade processes, which are circular processes that involve the creation of reverse biological cycles and the use of biomass to produce products with a more significant share of added value. The importance of these models for

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47

Zhang, Peng, Yunpeng Su, Hanrui Niu, Yaru Wang, Yuchen Zhang, and Chuankai Zhang. "The Research of Complex Product Design Process Model under the Concept of Self-Recovery." Applied Sciences 12, no. 20 (2022): 10270. http://dx.doi.org/10.3390/app122010270.

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The working environment of contemporary mechanical products is becoming more complex, and the working conditions are becoming more extreme. This has led to a significant increase in the frequency of problems in mechanical products. In order to reduce the frequency of human repair after problems, the application of the self-recovery concept has become a hot research topic in the area of smart design. However, the current application of the self-recovery concept is mostly limited to the structural and parametric levels, with less research at the functional level, which may lead to a waste of res

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48

Reche, Fernando, María Morales, and Antonio Salmerón. "Construction of Fuzzy Measures over Product Spaces." Mathematics 8, no. 9 (2020): 1605. http://dx.doi.org/10.3390/math8091605.

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In this paper, we study the problem of constructing a fuzzy measure over a product space when fuzzy measures over the marginal spaces are available. We propose a definition of independence of fuzzy measures and introduce different ways of constructing product measures, analyzing their properties. We derive bounds for the measure on the product space and show that it is possible to construct a single product measure when the marginal measures are capacities of order 2. We also study the combination of real functions over the marginal spaces in order to produce a joint function over the product

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49

Martínez, Abel Cabrera, Iztok Peterin, and Ismael G. Yero. "Roman domination in direct product graphs and rooted product graphs." AIMS Mathematics 6, no. 10 (2021): 11084–96. http://dx.doi.org/10.3934/math.2021643.

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<abstract><p>Let $ G $ be a graph with vertex set $ V(G) $. A function $ f:V(G)\rightarrow \{0, 1, 2\} $ is a Roman dominating function on $ G $ if every vertex $ v\in V(G) $ for which $ f(v) = 0 $ is adjacent to at least one vertex $ u\in V(G) $ such that $ f(u) = 2 $. The Roman domination number of $ G $ is the minimum weight $ \omega(f) = \sum_{x\in V(G)}f(x) $ among all Roman dominating functions $ f $ on $ G $. In this article we study the Roman domination number of direct product graphs and rooted product graphs. Specifically, we give several tight lower and upper bounds for

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LI, XIAN-JIN. "ON THE EULER PRODUCT OF THE DEDEKIND ZETA FUNCTION." International Journal of Number Theory 05, no. 02 (2009): 293–301. http://dx.doi.org/10.1142/s1793042109002109.

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It is well known that the Euler product formula for the Riemann zeta function ζ(s) is still valid for ℜ(s) = 1 and s ≠ 1. In this paper, we extend this result to zeta functions of number fields. In particular, we show that the Dedekind zeta function ζk(s) for any algebraic number field k can be written as the Euler product on the line ℜ(s) = 1 except at the point s = 1. As a corollary, we obtain the Euler product formula on the line ℜ(s) = 1 for Dirichlet L-functions L(s, χ) of real characters.

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