scholarly journals Exponential Cosmological Solutions with Three Different Hubble-Like Parameters in (1 + 3 + k1 + k2)-Dimensional EGB Model with a Λ-Term

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 250
Author(s):  
K. K. Ernazarov ◽  
V. D. Ivashchuk
Keyword(s):  
Lower Bounds ◽  
Gravitational Constant ◽  
Cosmological Term ◽  
Upper And Lower Bounds ◽  
Symmetry Groups ◽  
Exponential Time ◽  
Scale Factors ◽  
Exponential Expansion ◽  
Cosmological Solutions ◽  
Stable Solutions

A D-dimensional Einstein–Gauss–Bonnet model with a cosmological term Λ , governed by two non-zero constants: α 1 and α 2 , is considered. By restricting the metrics to diagonal ones, we study a class of solutions with the exponential time dependence of three scale factors, governed by three non-coinciding Hubble-like parameters: H > 0 , h 1 , and h 2 , obeying 3 H + k 1 h 1 + k 2 h 2 ≠ 0 and corresponding to factor spaces of dimensions: 3, k 1 > 1 , and k 2 > 1 , respectively, with D = 4 + k 1 + k 2 . The internal flat factor spaces of dimensions k 1 and k 2 have non-trivial symmetry groups, which depend on the number of compactified dimensions. Two cases: (i) 3 < k 1 < k 2 and (ii) 1 < k 1 = k 2 = k , k ≠ 3 , are analyzed. It is shown that in both cases, the solutions exist if α = α 2 / α 1 > 0 and α Λ > 0 obey certain restrictions, e.g., upper and lower bounds. In Case (ii), explicit relations for exact solutions are found. In both cases, the subclasses of stable and non-stable solutions are singled out. Case (i) contains a subclass of solutions describing an exponential expansion of 3 d subspace with Hubble parameter H > 0 and zero variation of the effective gravitational constant G.

Exact exponential cosmological solutions with two factor spaces of dimension m in EGB model with a Λ-term

2019 ◽  
Vol 16 (02) ◽  
pp. 1950025 ◽  
Author(s):  
V. D. Ivashchuk ◽  
A. A. Kobtsev
Keyword(s):  
Time Dependence ◽  
Gravitational Constant ◽  
Cosmological Term ◽  
Exponential Time ◽  
Dimension Formula ◽  
Scale Factors ◽  
All Solutions ◽  
Cosmological Solutions

A [Formula: see text]-dimensional Einstein–Gauss–Bonnet (EGB) model with the cosmological term [Formula: see text] is considered. We find a class of solutions with exponential time dependence of two scale factors, governed by two Hubble-like parameters [Formula: see text] and [Formula: see text], corresponding to two factor spaces of dimension [Formula: see text] and obeying [Formula: see text]. We prove that the solutions, obeying [Formula: see text], are stable (in a class of cosmological solutions with diagonal metrics) for [Formula: see text] and they are unstable for [Formula: see text]. A subclass of solutions with small enough variation of the effective gravitational constant [Formula: see text] is considered. It is shown that all solutions from this subclass are stable.

scholarly journals Exponential cosmological solutions with two factor spaces in EGB model with $$\Lambda = 0$$ revisited

2019 ◽  
Vol 79 (10) ◽  
Author(s):  
V. D. Ivashchuk ◽  
A. A. Kobtsev
Keyword(s):  
Analytical Solutions ◽  
Gravitational Constant ◽  
Cosmological Solution ◽  
Cosmological Term ◽  
Exponential Dependence ◽  
The Real ◽  
Real Solutions ◽  
Scale Factors ◽  
Cosmological Solutions

Abstract We study exact cosmological solutions in D-dimensional Einstein–Gauss–Bonnet model (with zero cosmological term) governed by two non-zero constants: $$\alpha _1$$α1 and $$\alpha _2$$α2 . We deal with exponential dependence (in time) of two scale factors governed by Hubble-like parameters $$H >0$$H>0 and h, which correspond to factor spaces of dimensions $$m >2$$m>2 and $$l > 2$$l>2, respectively, and $$D = 1 + m + l$$D=1+m+l. We put $$h \ne H$$h≠H and $$mH + l h \ne 0$$mH+lh≠0. We show that for $$\alpha = \alpha _2/\alpha _1 > 0$$α=α2/α1>0 there are two (real) solutions with two sets of Hubble-like parameters: $$(H_1, h_1)$$(H1,h1) and $$(H_2, h_2)$$(H2,h2), which obey: $$ h_1/ H_1< - m/l< h_2/ H_2 < 0$$h1/H1<-m/l<h2/H2<0, while for $$\alpha < 0$$α<0 the (real) solutions are absent. We prove that the cosmological solution corresponding to $$(H_2, h_2)$$(H2,h2) is stable in a class of cosmological solutions with diagonal metrics, while the solution corresponding to $$(H_1, h_1)$$(H1,h1) is unstable. We present several examples of analytical solutions, e.g. stable ones with small enough variation of the effective gravitational constant G, for $$(m, l) = (9, l >2), (12, 11), (11,16), (15, 6)$$(m,l)=(9,l>2),(12,11),(11,16),(15,6).

scholarly journals Stable exponential cosmological solutions with three factor spaces in (1 + 3 + 3 + k)-dimensional Einstein–Gauss–Bonnet model with a Λ-term

2019 ◽  
Vol 34 (14) ◽  
pp. 1950111
Author(s):  
K. K. Ernazarov
Keyword(s):  
Cosmological Model ◽  
Gravitational Constant ◽  
Cosmological Solutions ◽  
Stable Solutions ◽  
Three Factor

We consider a (7 + k)-dimensional Einstein–Gauss–Bonnet (EGB) model with the cosmological [Formula: see text]-term. A cosmological model with three factor spaces of dimensions 3, 3 and k, k[Formula: see text]2 is considered. Exact stable solutions with three (non-coinciding) Hubble-like parameters in these models are obtained. Some examples of solutions (e.g. with zero variation of the effective gravitational constant G) are considered in selected dimensions (for k = 5, 6).

scholarly journals On non-exponential cosmological solutions with two factor spaces of dimensions m and 1 in the Einstein–Gauss–Bonnet model with a Λ-term

2017 ◽  
Vol 32 (39) ◽  
pp. 1750202 ◽  
Author(s):  
K. K. Ernazarov
Keyword(s):  
Time Dependence ◽  
Dimensional Subspace ◽  
Accelerated Expansion ◽  
Exponential Dependence ◽  
Isotropic Solution ◽  
Exponential Time ◽  
Scale Factors ◽  
Cosmological Solutions

We consider a [Formula: see text]-dimensional Einstein–Gauss–Bonnet (EGB) model with the cosmological [Formula: see text]-term. We restrict the metrics to be diagonal ones and find for certain [Formula: see text] class of cosmological solutions with non-exponential time dependence of two scale factors of dimensions [Formula: see text] and 1. Any solution from this class describes an accelerated expansion of [Formula: see text]-dimensional subspace and tends asymptotically to isotropic solution with exponential dependence of scale factors.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

scholarly journals Hadamard k-fractional inequalities of Fejér type for GA-s-convex mappings and applications

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hui Lei ◽  
Gou Hu ◽  
Zhi-Jie Cao ◽  
Ting-Song Du
Keyword(s):  
Lower Bounds ◽  
Hypergeometric Functions ◽  
Upper And Lower Bounds ◽  
Weighted Inequalities ◽  
Convex Mappings ◽  
Special Means

Abstract The main aim of this paper is to establish some Fejér-type inequalities involving hypergeometric functions in terms of GA-s-convexity. For this purpose, we construct a Hadamard k-fractional identity related to geometrically symmetric mappings. Moreover, we give the upper and lower bounds for the weighted inequalities via products of two different mappings. Some applications of the presented results to special means are also provided.

scholarly journals On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 512
Author(s):  
Maryam Baghipur ◽  
Modjtaba Ghorbani ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Distance Matrix ◽  
Upper And Lower Bounds ◽  
Laplacian Eigenvalue ◽  
Signless Laplacian ◽  
Graph Parameters ◽  
Simple Connected Graph ◽  
Second Largest Eigenvalue ◽  
Reciprocal Distance Matrix

The signless Laplacian reciprocal distance matrix for a simple connected graph G is defined as RQ(G)=diag(RH(G))+RD(G). Here, RD(G) is the Harary matrix (also called reciprocal distance matrix) while diag(RH(G)) represents the diagonal matrix of the total reciprocal distance vertices. In the present work, some upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters are investigated. Besides, all graphs attaining these new bounds are characterized. Additionally, it is inferred that among all connected graphs with n vertices, the complete graph Kn and the graph Kn−e obtained from Kn by deleting an edge e have the maximum second-largest signless Laplacian reciprocal distance eigenvalue.

scholarly journals Multilevel Monte Carlo methods and lower–upper bounds in initial margin computations

2020 ◽  
Vol 26 (2) ◽  
pp. 131-161
Author(s):  
Florian Bourgey ◽  
Stefano De Marco ◽  
Emmanuel Gobet ◽  
Alexandre Zhou
Keyword(s):  
Monte Carlo ◽  
Lower Bounds ◽  
Asymptotic Optimality ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Independent Random Variables ◽  
Outer Function ◽  
Control Problems ◽  
Multilevel Monte Carlo ◽  
Primal Dual

AbstractThe multilevel Monte Carlo (MLMC) method developed by M. B. Giles [Multilevel Monte Carlo path simulation, Oper. Res. 56 2008, 3, 607–617] has a natural application to the evaluation of nested expectations {\mathbb{E}[g(\mathbb{E}[f(X,Y)|X])]}, where {f,g} are functions and {(X,Y)} a couple of independent random variables. Apart from the pricing of American-type derivatives, such computations arise in a large variety of risk valuations (VaR or CVaR of a portfolio, CVA), and in the assessment of margin costs for centrally cleared portfolios. In this work, we focus on the computation of initial margin. We analyze the properties of corresponding MLMC estimators, for which we provide results of asymptotic optimality; at the technical level, we have to deal with limited regularity of the outer function g (which might fail to be everywhere differentiable). Parallel to this, we investigate upper and lower bounds for nested expectations as above, in the spirit of primal-dual algorithms for stochastic control problems.

scholarly journals A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels and a Provably Optimal Instance-Based Schema

Algorithms ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 164
Author(s):  
Tobias Rupp ◽  
Stefan Funke
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Real World ◽  
Road Network ◽  
Upper And Lower Bounds ◽  
Bounded Degree ◽  
Shortest Path Routing ◽  
Graph Classes ◽  
Speed Up ◽  
To Come

We prove a Ω(n) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular, it is planar and has a bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.

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