Existence of the golden ratio in Tanjavur Brihadeeshwarar temple
OMS-Vol. 4 (2020), Issue 1, pp. 211 – 219 Open Access Full-Text PDF
C. Velmurugan, R. Kalaivanan
Abstract: In this study, we discussed the existence of golden ratio in Brihadeeshwarar temple, Tanjavur, Tamil Nadu, India, built in 1010 AD. It is listed on the UNESCO’s world heritage site of the Chola temples in southern India. This temple represents an outstanding creative achievement in the architectural idea of the pure form of the Dravida temples. Golden ratio has great influence in architecture, mathematics and art. We analyzed existence of the Golden ratio in structural design of Tanjavur Brihadeeshwarar temple prakaram. We used the Phi Grid and Phi Spiral software to measure the golden ratio and verified our result.
On a generalization of KU-algebras pseudo-KU algebras
OMS-Vol. 4 (2020), Issue 1, pp. 200 – 210 Open Access Full-Text PDF
Daniel A. Romano
Abstract: As a generalization of KU-algebras, the notion of pseudo-KU algebras is introduced in 2020 by the author (D. A. Romano. Pseudo-UP algebras, An introduction. Bull. Int. Math. Virtual Inst., 10(2)(2020), 349-355). Some characterizations of pseudo-KU algebras are established in that article. In addition, it is shown that each pseudo-KU algebra is a pseudo-UP algebra. In this paper it is a concept developed of pseudo-KU algebras in more detail and it has identified some of the main features of this type of universal algebras such as the notions of pseudo-subalgebras, pseudo-ideals, pseudo-filters and pseudo homomorphisms. Also, it has been shown that every pseudo-KU algebra is a pseudo-BE algebra. In addition, a congruence was constructed on a pseudo-KU algebra generated by a pseudo-ideal and shown that the corresponding factor-structure is and pseudo-KU algebra as well.
Intuitionistic fuzzy subgroups with respect to norms (\(T,S\))
EASL-Vol. 3 (2020), Issue 2, pp. 40 – 53 Open Access Full-Text PDF
Rasul Rasuli
Abstract: The purpose of this paper is introduce the notion of intuitionistic fuzzy subgroups with respect to norms (\(t\)-norm \(T\) and \(s\)-norm \(S\)). Also we introduce intersection and normality of them and investigate some properties of them. Finally, we provide some results of them under group homomorphisms.
On Caputo fractional derivatives via exponential \((s,m)\)-convex functions
EASL-Vol. 3 (2020), Issue 2, pp. 32 – 39 Open Access Full-Text PDF
Saad Ihsan Butt, Mehroz Nadeem, Ghulam Farid
Abstract: In this paper, we establish several integral inequalities including Caputo fractional derivatives for exponential \((s,m)\)-convex functions. By using convexity for exponential \((s,m)\)-convex functions of any positive integer order differentiable function some novel results are obtained.
New perspectives on internet electricity use in 2030
EASL-Vol. 3 (2020), Issue 2, pp. 19 – 31 Open Access Full-Text PDF
Anders S.G. Andrae
Abstract: The main problems with several existing Information and Communication Technology (ICT) power footprint investigations are: too limited (geographical and temporal) system boundary, overestimation of power saving potential in the next decade, assume that historical power use can predict future global power use in the next decade despite unprecedented data traffic growth, assume that Moore´s law relation to digital circuitry can continue “forever” and that no problems with extra cooling power will occur for several decades. The highly variable outlooks for the future power consumptions depend on “starting values”, disruptions, regional differences and perceptual estimations of electricity intensity reductions and data traffic increase. A hugely optimistic scenario – which takes into account 20% annual improvement of the J/bit in data centers and networks until 2030 is presented. However, the electric power consumption of the present ICT scope will be significant unless great efforts are put into power saving features enabling such improvements of J/bit. Despite evident risks, it seems though that planned power saving measures and innovation will be able to keep the electricity consumption of ICT and the World under some kind of control. The major conclusion is based on several simulations in the present study – that future consumer ICT infrastructure cannot slow its overall electricity use until 2030 and it will use more than today. Data traffic may not be the best proxy metric for estimating computing electricity. Operations and J/operation seem more promising for forecasting and scaling of bottom-up models.
Turing instability for a attraction-repolsion chemotaxis system with logistic growth
OMA-Vol. 4 (2020), Issue 1, pp. 98 – 118 Open Access Full-Text PDF
Abdelhakam Hassan Mohammed, Shengmao Fu
Abstract: In this paper, we investigate the nonlinear dynamics for an attraction-repulsion chemotaxis Keller-Segel model with logistic source term
\(u_{1t}=d_{1}\Delta{u_{1}}-\chi \nabla (u_{1}\nabla{u_{2}})+ \xi{ \nabla (u_{1}\nabla{u_{3}})}+\mathbf g(u),{\mathbf x}\in\mathbb{T}^{d}, t>0,\)
\( u_{2t}=d_{2}\Delta{u_{2}}+\alpha u_{1}-\beta u_{2},{\mathbf x}\in\mathbb{T}^{d}, t>0,\)
\(u_{3t}=d_{3}\Delta{u_{3}}+\gamma u_{1}- \eta u_{3},{\mathbf x}\in\mathbb{T}^{d}, t>0,\)
\( \frac{\partial{u_{1}}}{\partial{x_{i}}}=\frac{\partial{u_{2}}}{\partial{x_{i}}}=\frac{\partial{u_{3}}}{\partial{x_{i}}}=0,x_{i}=0,\pi, 1\leq i\leq d,\)
\( u_{1}(x,0)=u_{10}(x), u_{2}(x,0)=u_{20}(x), u_{3}(x,0)=u_{30}(x), {\mathbf x}\in\mathbb{T}^{d} (d=1,2,3).\)
Under the assumptions of the unequal diffusion coefficients, the conditions of chemotaxis-driven instability are given in a \(d\)-dimensional box \(\mathbb{T}^{d}=(0,\pi)^{d} (d=1,2,3)\). It is proved that in the condition of the unique positive constant equilibrium point \({\mathbf w_{c}}=(u_{1c},u_{2c},u_{3c})\) of above model is nonlinearly unstable. Moreover, our results provide a quantitative characterization for the early-stage pattern formation in the model.
\(u_{1t}=d_{1}\Delta{u_{1}}-\chi \nabla (u_{1}\nabla{u_{2}})+ \xi{ \nabla (u_{1}\nabla{u_{3}})}+\mathbf g(u),{\mathbf x}\in\mathbb{T}^{d}, t>0,\)
\( u_{2t}=d_{2}\Delta{u_{2}}+\alpha u_{1}-\beta u_{2},{\mathbf x}\in\mathbb{T}^{d}, t>0,\)
\(u_{3t}=d_{3}\Delta{u_{3}}+\gamma u_{1}- \eta u_{3},{\mathbf x}\in\mathbb{T}^{d}, t>0,\)
\( \frac{\partial{u_{1}}}{\partial{x_{i}}}=\frac{\partial{u_{2}}}{\partial{x_{i}}}=\frac{\partial{u_{3}}}{\partial{x_{i}}}=0,x_{i}=0,\pi, 1\leq i\leq d,\)
\( u_{1}(x,0)=u_{10}(x), u_{2}(x,0)=u_{20}(x), u_{3}(x,0)=u_{30}(x), {\mathbf x}\in\mathbb{T}^{d} (d=1,2,3).\)
Under the assumptions of the unequal diffusion coefficients, the conditions of chemotaxis-driven instability are given in a \(d\)-dimensional box \(\mathbb{T}^{d}=(0,\pi)^{d} (d=1,2,3)\). It is proved that in the condition of the unique positive constant equilibrium point \({\mathbf w_{c}}=(u_{1c},u_{2c},u_{3c})\) of above model is nonlinearly unstable. Moreover, our results provide a quantitative characterization for the early-stage pattern formation in the model.
Modeling the movement of particles in tilings by Markov chains
OMA-Vol. 4 (2020), Issue 1, pp. 84 – 97 Open Access Full-Text PDF
Zirhumanana Balike, Arne Ring, Meseyeki Saiguran
Abstract: This paper studies the movement of a molecule in two types of cell complexes: the square tiling and the hexagonal one. This movement from a cell \(i\) to a cell \(j\) is referred to as an homogeneous Markov chain. States with the same stochastic behavior are grouped together using symmetries of states deduced from groups acting on the cellular complexes. This technique of lumpability is effective in forming new chains from the old ones without losing the primitive properties and simplifying tedious calculations. Numerical simulations are performed using R software to determine the impact of the shape of the tiling and other parameters on the achievement of the equilibrium. We start from small square tiling to small hexagonal tiling before comparing the results obtained for each of them. In this paper, only continuous Markov chains are considered. In each tiling, the molecule is supposed to leave the central cell and move into the surrounding cells.
Exponential growth of solution with \(L_p\)-norm for class of non-linear viscoelastic wave equation with distributed delay term for large initial data
OMA-Vol. 4 (2020), Issue 1, pp. 76 – 83 Open Access Full-Text PDF
Abdelbaki Choucha, Djamel Ouchenane, Khaled Zennir
Abstract: In this work, we are concerned with a problem for a viscoelastic wave equation with strong damping, nonlinear source and distributed delay terms. We show the exponential growth of solution with \(L_{p}\)-norm, i.e., \(\lim\limits_{t\rightarrow \infty}\Vert u\Vert_p^p \rightarrow \infty\).
A new recursion for Bressoud’s polynomials
ODAM-Vol. 3 (2020), Issue 2, pp. 23 – 29 Open Access Full-Text PDF
Helmut Prodinger
Abstract: A new recursion in only one variable allows very simple verifications of Bressoud’s polynomial identities, which lead to the Rogers-Ramanujan identities. This approach might be compared with an earlier approach due to Chapman. Applying the \(q\)-Chu-Vandermonde convolution, as suggested by Cigler, makes the computations particularly simple and elementary. The same treatment is also applied to the Santos polynomials and perhaps more polynomials from a list of Rogers-Ramanujan like polynomials [1].
Reinterpreting the middle-levels theorem via natural enumeration of ordered trees
ODAM-Vol. 3 (2020), Issue 2, pp. 8 – 22 Open Access Full-Text PDF
Italo Jose Dejter
Abstract: Let \(0< k\in\mathbb{Z} \). A reinterpretation of the proof of existence of Hamilton cycles in the middle-levels graph \(M_k\) induced by the vertices of the \((2k+1)\)-cube representing the \(k\)- and \((k+1)\)-subsets of \(\{0,\ldots,2k\}\) is given via an associated dihedral quotient graph of \(M_k\) whose vertices represent the ordered (rooted) trees of order \(k+1\) and size \(k\).