scholarly journals On the spectrality and spectral expansion of the non-self-adjoint mathieu-hill operator in $ L_{2}(-\infty, \infty) $

2020 ◽  
Vol 19 (3) ◽  
pp. 1537-1562 ◽  
Author(s):  
O. A. Veliev ◽  
Keyword(s):  
Spectral Expansion ◽  
Hill Operator

scholarly journals Scalar correlation functions in de Sitter space from the stochastic spectral expansion

2019 ◽  
Vol 2019 (08) ◽  
pp. 001-001 ◽  
Author(s):  
Tommi Markkanen ◽  
Arttu Rajantie ◽  
Stephen Stopyra ◽  
Tommi Tenkanen
Keyword(s):  
Correlation Functions ◽  
De Sitter Space ◽  
Spectral Expansion ◽  
De Sitter

scholarly journals Multicolor Ramsey Numbers via Pseudorandom Graphs

10.37236/9071 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Xiaoyu He ◽  
Yuval Wigderson
Keyword(s):  
Spectral Expansion ◽  
Free Graph ◽  
Ramsey Numbers ◽  
Positive Integers

A weakly optimal $K_s$-free $(n,d,\lambda)$-graph is a $d$-regular $K_s$-free graph on $n$ vertices with $d=\Theta(n^{1-\alpha})$ and spectral expansion $\lambda=\Theta(n^{1-(s-1)\alpha})$, for some fixed $\alpha>0$. Such a graph is called optimal if additionally $\alpha = \frac{1}{2s-3}$. We prove that if $s_{1},\ldots,s_{k}\ge3$ are fixed positive integers and weakly optimal $K_{s_{i}}$-free pseudorandom graphs exist for each $1\le i\le k$, then the multicolor Ramsey numbers satisfy\[\Omega\Big(\frac{t^{S+1}}{\log^{2S}t}\Big)\le r(s_{1},\ldots,s_{k},t)\le O\Big(\frac{t^{S+1}}{\log^{S}t}\Big),\]as $t\rightarrow\infty$, where $S=\sum_{i=1}^{k}(s_{i}-2)$. This generalizes previous results of Mubayi and Verstra\"ete, who proved the case $k=1$, and Alon and Rödl, who proved the case $s_1=\cdots = s_k = 3$. Both previous results used the existence of optimal rather than weakly optimal $K_{s_i}$-free graphs.

scholarly journals Exploring Analytical Models for Performability Evaluation of Virtualized Servers using Dynamic Resource

2019 ◽  
Vol 14 (5) ◽  
pp. 647
Author(s):  
Yonal Kirsal
Keyword(s):  
Resource Utilization ◽  
Blocking Probability ◽  
Spectral Expansion ◽  
Form Solution ◽  
Analytical Models ◽  
Mean Response Time ◽  
Proposed Model ◽  
Dynamic Resource ◽  
Mean Queue Length ◽  
Markov Reward

Virtualization of resources is a widely accepted technique to optimize resources in recent technologies. Virtualization allows users to execute their services on the same physical machine, keeping these services isolated from each other. This paper proposes the analytical models for performability evaluation of virtualized servers with dynamic resource utilization. The performance and avalability models are considered separately due to the behaviour of the proposed system. The well-known Markov Reward Model (MRM) is used for the solution of the analytical model considered together with an exact spectral expansion and product form solution. The dynamic resource utilization is employed to enhance the QoS of the proposed model which is another major issue in the performance characterization of virtulazilation. In this paper, the performability output parameters, such as mean queue length, mean response time and blocking probability are computed and presented for the proposed model. In addition, the performability results obtained from the analytical models are validated by the simulation (DES) results to show the accuracy and effectiveness of the proposed work. The results indicate the proposed modelling results show good agreement with DES and understand the factors are very important to improve the QoS.

scholarly journals Models And Mining Of On-Line Social Networks

2021 ◽  
Author(s):  
Yanhua Tian
Keyword(s):  
Social Networks ◽  
Power Law ◽  
Degree Distribution ◽  
Small World ◽  
Spectral Expansion ◽  
On Line ◽  
Simulation Results ◽  
Small World Property ◽  
P Type ◽  
Law Degree

Power law degree distribution, the small world property, and bad spectral expansion are three of the most important properties of On-line Social Networks (OSNs). We sampled YouTube and Wikipedia to investigate OSNs. Our simulation and computational results support the conclusion that OSNs follow a power law degree distribution, have the small world property, and bad spectral expansion. We calculated the diameters and spectral gaps of OSNs samples, and compared these to graphs generated by the GEO-P model. Our simulation results support the Logarithmic Dimension Hypothesis, which conjectures that the dimension of OSNs is m = [log N]. We introduced six GEO-P type models. We ran simulations of these GEO-P-type models, and compared the simulated graphs with real OSN data. Our simulation results suggest that, except for the GEO-P (GnpDeg) model, all our models generate graphs with power law degree distributions, the small world property, and bad spectral expansion.

The structure of Taylor's constraint in three dimensions

2008 ◽  
Vol 464 (2100) ◽  
pp. 3149-3174 ◽  
Author(s):  
Philip W Livermore ◽  
Glenn Ierley ◽  
Andrew Jackson
Keyword(s):  
Lorentz Force ◽  
Inner Core ◽  
Rotation Axis ◽  
Analytic Form ◽  
Spectral Expansion ◽  
Three Dimensions ◽  
Necessary Condition ◽  
Dynamo Action ◽  
Azimuthal Component ◽  
Electrically Conducting

In a 1963 edition of Proc. R. Soc. A , J. B. Taylor (Taylor 1963 Proc. R. Soc. A 9 , 274–283) proved a necessary condition for dynamo action in a rapidly rotating electrically conducting fluid in which viscosity and inertia are negligible. He demonstrated that the azimuthal component of the Lorentz force must have zero average over any geostrophic contour (i.e. a fluid cylinder coaxial with the rotation axis). The resulting dynamical balance, termed a Taylor state, is believed to hold in the Earth's core, hence placing constraints on the class of permissible fields in the geodynamo. Such states have proven difficult to realize, apart from highly restricted examples. In particular, it has not yet been shown how to enforce the Taylor condition exactly in a general way, seeming to require an infinite number of constraints. In this work, we derive the analytic form for the averaged azimuthal component of the Lorentz force in three dimensions after expanding the magnetic field in a truncated spherical harmonic basis chosen to be regular at the origin. As the result is proportional to a polynomial of modest degree (simply related to the order of the spectral expansion), it can be made to vanish identically on every geostrophic contour by simply equating each of its coefficients to zero. We extend the discussion to allow for the presence of an inner core, which partitions the geostrophic contours into three distinct regions.

Linear inversion of gravity data using the spectral expansion method

Geophysics ◽  
1985 ◽  
Vol 50 (5) ◽  
pp. 820-824 ◽  
Author(s):  
Rene E. Chavez ◽  
George D. Garland
Keyword(s):  
Inverse Problem ◽  
Gravity Data ◽  
Linear Equations ◽  
Thin Sheet ◽  
Gravity Anomalies ◽  
Expansion Method ◽  
Spectral Expansion ◽  
Uniform Density ◽  
Linear Inversion ◽  
Prismatic Structure

Inversion of gravity anomalies in terms of an anomalous mass distribution with irregular outline but uniform density leads to a nonlinear inverse problem. An alternative approach based on the thin‐sheet approximation can, however, be formulated as a linear inverse problem provided the structures are two‐dimensional. The anomalous mass is represented by a thin sheet, which is located at a depth [Formula: see text] and is divided into M strips with N < M data points. Thus, an underdetermined system of linear equations is obtained which is solved by the spectral expansion method for the surface density distribution of each segment. This set of parameters is then transformed into a prismatic structure with variable depth but uniform density. The modeling procedure involves a noniterative method. A gravity problem is investigated, and the solution obtained compares well with previous interpretations.

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