scholarly journals Asymptotic behavior of spanning forests and connected spanning subgraphs on two-dimensional lattices

2020 ◽  
Vol 34 (27) ◽  
pp. 2050249
Author(s):  
Shu-Chiuan Chang ◽  
Robert Shrock
Keyword(s):  
Asymptotic Behavior ◽  
Exponential Growth ◽  
Upper Bounds ◽  
Great Length ◽  
Two Dimensional ◽  
Lower And Upper Bounds ◽  
Spanning Subgraphs ◽  
Spanning Forests ◽  
Growth Constants ◽  
Limiting Values

We calculate exponential growth constants [Formula: see text] and [Formula: see text] describing the asymptotic behavior of spanning forests and connected spanning subgraphs on strip graphs, with arbitrarily great length, of several two-dimensional lattices, including square, triangular, honeycomb, and certain heteropolygonal Archimedean lattices. By studying the limiting values as the strip widths get large, we infer lower and upper bounds on these exponential growth constants for the respective infinite lattices. Since our lower and upper bounds are quite close to each other, we can infer very accurate approximate values for these exponential growth constants, with fractional uncertainties ranging from [Formula: see text] to [Formula: see text]. We show that [Formula: see text] and [Formula: see text] are monotonically increasing functions of vertex degree for these lattices.

scholarly journals Exponential growth constants for spanning forests on Archimedean lattices: Values and comparisons of upper bounds

2021 ◽  
Vol 35 (06) ◽  
pp. 2150085
Author(s):  
Shu-Chiuan Chang ◽  
Robert Shrock
Keyword(s):  
Asymptotic Behavior ◽  
Exponential Growth ◽  
Upper Bounds ◽  
Growth Constant ◽  
Spanning Forests ◽  
Growth Constants

We compare our upper bounds on the exponential growth constant [Formula: see text] characterizing the asymptotic behavior of spanning forests on Archimedean lattices [Formula: see text] with recently derived upper bounds.Our upper bounds on [Formula: see text], which are very close to the respective values of [Formula: see text] that we have calculated, are shown to be significantly better for these lattices than the new upper bounds.

scholarly journals ACYCLIC ORIENTATIONS ON THE SIERPINSKI GASKET

2012 ◽  
Vol 26 (24) ◽  
pp. 1250128 ◽  
Author(s):  
SHU-CHIUAN CHANG
Keyword(s):  
Sierpinski Gasket ◽  
Upper Bounds ◽  
Two Dimensional ◽  
Sierpiński Gasket ◽  
Asymptotic Behaviors ◽  
Asymptotic Growth ◽  
Acyclic Orientations ◽  
Growth Constants

We study the number of acyclic orientations on the generalized two-dimensional Sierpinski gasket SG 2,b(n) at stage n with b equal to two and three, and determine the asymptotic behaviors. We also derive upper bounds for the asymptotic growth constants of SG 2,b and d-dimensional Sierpinski gasket SG d.

scholarly journals Asymptotic Behavior of the Steklov Eigenvalues For the p−Laplace Operator

2007 ◽  
Vol 7 (3) ◽  
Author(s):  
Juan Pablo Pinasco
Keyword(s):  
Asymptotic Behavior ◽  
Laplace Operator ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Asymptotic Bounds ◽  
Steklov Eigenvalues

AbstractIn this paper we study the asymptotic behavior of the Steklov eigenvalues of the p- Laplacian. We show the existence of lower and upper bounds of a Weyl-type expansion of the function N(λ) which counts the number of eigenvalues less than or equal to λ, and we derive from them asymptotic bounds for the eigenvalues.

scholarly journals Spanning forests on the Sierpinski gasket

2008 ◽  
Vol Vol. 10 no. 2 (Combinatorics) ◽  
Author(s):  
Shu-Chiuan Chang ◽  
Lung-Chi Chen
Keyword(s):  
Sierpinski Gasket ◽  
Upper Bounds ◽  
Sierpiński Gasket ◽  
Asymptotic Behaviors ◽  
Asymptotic Growth ◽  
Spanning Forests ◽  
International Audience ◽  
Growth Constants

Combinatorics International audience We study the number of spanning forests on the Sierpinski gasket SGd(n) at stage n with dimension d equal to two, three and four, and determine the asymptotic behaviors. The corresponding results on the generalized Sierpinski gasket SGd;b(n) with d = 2 and b = 3 ; 4 are obtained. We also derive upper bounds for the asymptotic growth constants for both SGd and SG2,b.

Collapse Strength of Redundant Beams After Lateral Buckling

1957 ◽  
Vol 24 (2) ◽  
pp. 283-288
Author(s):  
E. F. Masur ◽  
K. P. Milbradt
Keyword(s):  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Postbuckling Behavior ◽  
Collapse Strength ◽  
Neutral Equilibrium ◽  
Slender Beams ◽  
End Restraint ◽  
Limiting Values ◽  
Increasing Load ◽  
Good Agreement

Abstract According to classical linear theory, slender beams buckle laterally under vertical loads which remain constant as the buckling amplitude increases. Unless prior yielding takes place, the loads corresponding to neutral equilibrium represent therefore the collapse strength of the beam. However, the inclusion of nonlinear terms in the strain-displacement relations modifies the predicted postbuckling behavior of redundant beams. If continued elasticity is postulated, increasing amplitudes are associated with increasing load magnitudes, which generally approach limiting values. These “ultimate loads” may be estimated by means of two principles establishing lower and upper bounds. Tests performed on a single-span beam of varying degrees of end restraint show good agreement with the proposed theory.

scholarly journals Sharp bounds for the modified multiplicative Zagreb indices of graphs with vertex connectivity at most k

Filomat ◽  
2019 ◽  
Vol 33 (14) ◽  
pp. 4673-4685
Author(s):  
Haiying Wang ◽  
Shaohui Wang ◽  
Bing Wei
Keyword(s):  
Molecular Graph ◽  
Upper Bounds ◽  
Two Dimensional ◽  
Quantitative Structure ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
Vertex Connectivity ◽  
Zagreb Indices ◽  
Molecular Chirality ◽  
Structure Activity

Zagreb indices and their modified versions of a molecular graph originate from many practical problems such as two dimensional quantitative structure-activity (2D QSAR) and molecular chirality. Nowadays, they have become important invariants which can be used to characterize the properties of graphs from different aspects. LetVk n (or Ek n respectively) be a set of graphs of n vertices with vertex connectivity (or edge connectivity respectively) at most k. In this paper, we explore some properties of the modified first and second multiplicative Zagreb indices of graphs in Vkn and Ekn. By using analytic and combinatorial tools, we obtain some sharp lower and upper bounds for these indices of graphs in Vk n and Ekn. In addition, the corresponding extremal graphs which attain the lower or upper bounds are characterized. Our results enrich outcomes on studying Zagreb indices and the methods developed in this paper may provide some new tools for investigating the values on modified multiplicative Zagreb indices of other classes of graphs.

scholarly journals Bounds for eigenvalues of the Dirichlet problem for the logarithmic Laplacian

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Huyuan Chen ◽  
Laurent Véron
Keyword(s):  
Asymptotic Behavior ◽  
Fourier Transform ◽  
Dirichlet Problem ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Laplacian Operator ◽  
Lower And Upper Bounds

Abstract We provide bounds for the sequence of eigenvalues { λ i ⁢ ( Ω ) } i {\{\lambda_{i}(\Omega)\}_{i}} of the Dirichlet problem L Δ ⁢ u = λ ⁢ u ⁢  in  ⁢ Ω , u = 0 ⁢  in  ⁢ ℝ N ∖ Ω , L_{\Delta}u=\lambda u\text{ in }\Omega,\quad u=0\text{ in }\mathbb{R}^{N}% \setminus\Omega, where L Δ {L_{\Delta}} is the logarithmic Laplacian operator with Fourier transform symbol 2 ⁢ ln ⁡ | ζ | {2\ln\lvert\zeta\rvert} . The logarithmic Laplacian operator is not positively defined if the volume of the domain is large enough. In this article, we obtain the upper and lower bounds for the sum of the first k eigenvalues by extending the Li–Yau method and Kröger’s method, respectively. Moreover, we show the limit of the quotient of the sum of the first k eigenvalues by k ⁢ ln ⁡ k {k\ln k} is independent of the volume of the domain. Finally, we discuss the lower and upper bounds of the k-th principle eigenvalue, and the asymptotic behavior of the limit of eigenvalues.

scholarly journals Induced and Non-induced Forbidden Subposet Problems

10.37236/4644 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Balázs Patkós
Keyword(s):  
Asymptotic Behavior ◽  
Boolean Lattice ◽  
Maximum Size ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Extra Condition ◽  
Multi Level

The problem of determining the maximum size $La(n,P)$ that a $P$-free subposet of the Boolean lattice $B_n$ can have, attracted the attention of many researchers, but little is known about the induced version of these problems. In this paper we determine the asymptotic behavior of $La^*(n,P)$, the maximum size that an induced $P$-free subposet of the Boolean lattice $B_n$ can have for the case when $P$ is the complete two-level poset $K_{r,t}$ or the complete multi-level poset $K_{r,s_1,\dots,s_j,t}$ when all $s_i$'s either equal 4 or are large enough and satisfy an extra condition. We also show lower and upper bounds for the non-induced problem in the case when $P$ is the complete three-level poset $K_{r,s,t}$. These bounds determine the asymptotics of $La(n,K_{r,s,t})$ for some values of $s$ independently of the values of $r$ and $t$.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

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