scholarly journals Homotopical complexity of a billiard flow on the 3D flat torus with two cylindrical obstacles

2017 ◽  
Vol 39 (4) ◽  
pp. 1071-1081
Author(s):  
CALEB C. MOXLEY ◽  
NANDOR J. SIMANYI
Keyword(s):  
Natural Habitat ◽  
Hyperbolic Group ◽  
Height Function ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Lower And Upper Bounds ◽  
Flat Torus ◽  
Average Speed ◽  
Rotation Vectors ◽  
Rotation Set

We study the homotopical rotation vectors and the homotopical rotation sets for the billiard flow on the unit flat torus with two disjoint and orthogonal toroidal (cylindrical) scatterers removed from it. The natural habitat for these objects is the infinite cone erected upon the Cantor set $\text{Ends}(G)$ of all ‘ends’ of the hyperbolic group $G=\unicode[STIX]{x1D70B}_{1}(\mathbf{Q})$. An element of $\text{Ends}(G)$ describes the direction in (the Cayley graph of) the group $G$ in which the considered trajectory escapes to infinity, whereas the height function $s$ ($s\geq 0$) of the cone gives us the average speed at which this escape takes place. The main results of this paper claim that the orbits can only escape to infinity at a speed not exceeding $\sqrt{3}$ and, in any direction $e\in \text{Ends}(\unicode[STIX]{x1D70B}_{1}({\mathcal{Q}}))$, the escape is feasible with any prescribed speed $s$, $0\leq s\leq 1/(\sqrt{6}+2\sqrt{3})$. This means that the radial upper and lower bounds for the rotation set $R$ are actually pretty close to each other. Furthermore, we prove the convexity of the set $\mathit{AR}$ of constructible rotation vectors, and that the set of rotation vectors of periodic orbits is dense in $\mathit{AR}$. We also provide effective lower and upper bounds for the topological entropy of the studied billiard flow.

scholarly journals Homotopical complexity of 2D billiard orbits

2011 ◽  
Vol 48 (4) ◽  
pp. 540-562
Author(s):  
Lee Goswick ◽  
Nándor Simányi
Keyword(s):  
Fundamental Group ◽  
Smooth Boundary ◽  
Natural Habitat ◽  
Hyperbolic Group ◽  
Height Function ◽  
Upper And Lower Bounds ◽  
Closed Loops ◽  
Rotation Numbers ◽  
Sinai Billiard ◽  
Rotation Set

Traditionally, rotation numbers for toroidal billiard flows are defined as the limiting vectors of average displacements per time on trajectory segments. Naturally, these creatures live in the (commutative) vector space ℝn, if the toroidal billiard is given on the flatn-torus. The billiard trajectories, being curves, often getting very close to closed loops, quite naturally define elements of the fundamental group of the billiard table. The simplest non-trivial fundamental group obtained this way belongs to the classical Sinai billiard, i.e. the billiard flow on the 2-torus with a single, strictly convex obstacle (with smooth boundary) removed. This fundamental group is known to be the groupF2freely generated by two elements, which is a heavily noncommutative, hyperbolic group in Gromov’s sense. We define the homotopical rotation number and the homotopical rotation set for this model, and provide lower and upper estimates for the latter one, along with checking the validity of classically expected properties, like the density (in the homotopical rotation set) of the homotopical rotation numbers of periodic orbits.The natural habitat for these objects is the infinite cone erected upon the Cantor set Ends (F2) of all ŋds" of the hyperbolic groupF2. An element of Ends (F2) describes the direction in (the Cayley graph of) the groupF2in which the considered trajectory escapes to infinity, whereas the height functiont(t≧ 0) of the cone gives us the average speed at which this escape takes place.The main results of this paper claim that the orbits can only escape to infinity at a speed not exceeding\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\sqrt 2 $ \end{document}, and any directione∈ Ends (F2) for the escape is feasible with any prescribed speeds, 0 ≦s≦\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\sqrt 2 $ \end{document}/2. This means that the radial upper and lower bounds for the rotation setRare actually pretty close to each other.

scholarly journals Predicting the unpredictable: how dynamic COVID-19 policies and restrictions challenge model forecasts

2021 ◽  
Author(s):  
Farah Houdroge ◽  
Anna Palmer ◽  
Dominic Delport ◽  
Tom Walsh ◽  
Sherrie Kelly ◽  
...  
Keyword(s):  
Upper Bounds ◽  
Actual Data ◽  
Upper And Lower Bounds ◽  
Baseline Scenario ◽  
Policy Changes ◽  
Lower And Upper Bounds ◽  
Model Accuracy ◽  
Agent Based ◽  
Future Policy ◽  
Post Hoc

Abstract Introduction To retrospectively assess the accuracy of a mathematical modelling study that projected the rate of COVID-19 diagnoses for 72 locations worldwide in 2021, and to identify predictors of model accuracy. Methods Between June and August 2020, an agent-based model was used to project rates of COVID-19 infection incidence and cases diagnosed as positive from 15 September to 31 October 2020 for 72 geographic settings. Five scenarios were modelled: a baseline scenario where no future changes were made to existing restrictions, and four scenarios representing small or moderate changes in restrictions at two intervals. Post hoc, upper and lower bounds for number of diagnosed Covid-19 cases were compared with actual data collected during the prediction window. A regression analysis with 17 covariates was performed to determine correlates of accurate projections. Results The actual data fell within the lower and upper bounds in 27 settings and out of bounds in 45 settings. The only statistically significant predictor of actual data within the predicted bounds was correct assumptions about future policy changes (OR = 15.04; 95%CI 2.20-208.70; p=0.016). Conclusions For this study, the accuracy of COVID-19 model projections was dependent on whether assumptions about future policies are correct. Frequent changes in restrictions implemented by governments, which the modelling team was not always able to predict, in part explains why the majority of model projections were inaccurate compared with actual outcomes and supports revision of projections when policies are changed as well as the importance of policy experts collaborating on modelling projects.

scholarly journals Single source unsplittable flows with arc-wise lower and upper bounds

2021 ◽  
Author(s):  
Sarah Morell ◽  
Martin Skutella
Keyword(s):  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Common Source ◽  
Lower And Upper Bounds ◽  
New Approach ◽  
Math Program ◽  
Unsplittable Flow ◽  
The Common ◽  
Special Case ◽  
Single Path

AbstractIn a digraph with a source and several destination nodes with associated demands, an unsplittable flow routes each demand along a single path from the common source to its destination. Given some flow x that is not necessarily unsplittable but satisfies all demands, it is a natural question to ask for an unsplittable flow y that does not deviate from x by too much, i.e., $$y_a\approx x_a$$ y a ≈ x a for all arcs a. Twenty years ago, in a landmark paper, Dinitz et al. (Combinatorica 19:17–41, 1999) proved that there exists an unsplittable flow y such that $$y_a\le x_a+d_{\max }$$ y a ≤ x a + d max for all arcs a, where $$d_{\max }$$ d max denotes the maximum demand value. Our first contribution is a considerably simpler one-page proof for this classical result, based upon an entirely new approach. Secondly, using a subtle variant of this approach, we obtain a new result: There is an unsplittable flow y such that $$y_a\ge x_a-d_{\max }$$ y a ≥ x a - d max for all arcs a. Finally, building upon an iterative rounding technique previously introduced by Kolliopoulos and Stein (SIAM J Comput 31:919–946, 2002) and Skutella (Math Program 91:493–514, 2002), we prove existence of an unsplittable flow that simultaneously satisfies the upper and lower bounds for the special case when demands are integer multiples of each other. For arbitrary demand values, we prove the weaker simultaneous bounds $$x_a/2-d_{\max }\le y_a\le 2x_a+d_{\max }$$ x a / 2 - d max ≤ y a ≤ 2 x a + d max for all arcs a.

scholarly journals Bounds for the variance of the busy period of the M/G/∞ queue

1984 ◽  
Vol 16 (4) ◽  
pp. 929-932 ◽  
Author(s):  
M. F. Ramalhoto
Keyword(s):  
Distribution Function ◽  
Lower Bounds ◽  
Service Time ◽  
Time Distribution ◽  
Busy Period ◽  
Random Variable ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Service Time Distribution ◽  
Lower And Upper Bounds

Some bounds for the variance of the busy period of an M/G/∞ queue are calculated as functions of parameters of the service-time distribution function. For any type of service-time distribution function, upper and lower bounds are evaluated in terms of the intensity of traffic and the coefficient of variation of the service time. Other lower and upper bounds are derived when the service time is a NBUE, DFR or IMRL random variable. The variance of the busy period is also related to the variance of the number of busy periods that are initiated in (0, t] by renewal arguments.

Cardinality of height function’s range in case of maximally many rectangular islands — computed by cuts

2013 ◽  
Vol 11 (2) ◽  
Author(s):  
Eszter Horváth ◽  
Branimir Šešelja ◽  
Andreja Tepavčević
Keyword(s):  
Height Function ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Combinatorial Properties ◽  
Square Cells

AbstractWe deal with rectangular m×n boards of square cells, using the cut technics of the height function. We investigate combinatorial properties of this function, and in particular we give lower and upper bounds for the number of essentially different cuts. This number turns out to be the cardinality of the height function’s range, in case the height function has maximally many rectangular islands.

scholarly journals Bounds for the variance of the busy period of the M/G/∞ queue

1984 ◽  
Vol 16 (04) ◽  
pp. 929-932
Author(s):  
M. F. Ramalhoto
Keyword(s):  
Distribution Function ◽  
Lower Bounds ◽  
Service Time ◽  
Time Distribution ◽  
Busy Period ◽  
Random Variable ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Service Time Distribution ◽  
Lower And Upper Bounds

Some bounds for the variance of the busy period of an M/G/∞ queue are calculated as functions of parameters of the service-time distribution function. For any type of service-time distribution function, upper and lower bounds are evaluated in terms of the intensity of traffic and the coefficient of variation of the service time. Other lower and upper bounds are derived when the service time is a NBUE, DFR or IMRL random variable. The variance of the busy period is also related to the variance of the number of busy periods that are initiated in (0, t] by renewal arguments.

scholarly journals Assessment of Relevant Physical Phenomena Controlling Thermal Performance of Nanofluids

2006 ◽  
Author(s):  
M. Bahrami ◽  
M. M. Yovanovich ◽  
J. R. Culham
Keyword(s):  
Experimental Data ◽  
Systematic Approach ◽  
Large Degree ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Theoretical Studies ◽  
Lower And Upper Bounds ◽  
Physical Phenomena ◽  
Experimental And Theoretical Studies

This paper provides an overview of the important physical phenomena necessary for the determination of effective thermal conductivity of nanofluids. Through an investigation, a large degree of randomness and scatter has been observed in the experimental data published in the open literature. Given the inconsistency in the data, it is impossible to develop a comprehensive physical-based model that can predict all the trends. This also points out the need for a systematic approach in both experimental and theoretical studies. Upper and lower bounds are developed for steady-state conduction in stationary nanofluids. Comparisons between these bounds and the experimental data indicate that all the data (except for carbon nanotube data) lie between the lower and upper bounds.

scholarly journals Spaces with high topological complexity

2014 ◽  
Vol 144 (4) ◽  
pp. 761-773
Author(s):  
Aleksandra Franc ◽  
Petar Pavešić
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Topological Complexity ◽  
Lower And Upper Bounds ◽  
Cw Complex ◽  
The Difference

By a formula of Farber, the topological complexity TC(X) of a (p − 1)-connected m-dimensional CW-complex X is bounded above by (2m + 1)/p + 1. We show that the same result holds for the monoidal topological complexity TCM(X). In a previous paper we introduced various lower bounds for TCM(X), such as the nilpotency of the ring H*(X × X, Δ(X)), and the weak and stable (monoidal) topological complexity wTCM(X) and σTCM(X). In general, the difference between these upper and lower bounds can be arbitrarily large. In this paper we investigate spaces with topological complexity close to the maximal value given by Farber's formula. We show that in these cases the gap between the lower and upper bounds is narrow and TC(X) often coincides with the lower bounds.

scholarly journals A short note on hyper Zagreb index

2017 ◽  
Vol 37 (2) ◽  
pp. 51-58
Author(s):  
Suresh Elumalai ◽  
Toufik Mansour ◽  
Mohammad Ali Rostami ◽  
Gnanadhass Britto Antony Xavier
Keyword(s):  
Maximum Degree ◽  
Minimum Degree ◽  
Short Note ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Lower And Upper Bounds ◽  
Zagreb Index ◽  
Second Zagreb Index ◽  
Counter Example ◽  
Inverse Degree

In this paper, we present and analyze the upper and lower bounds on the Hyper Zagreb index $\chi^2(G)$ of graph $G$ in terms of the number of vertices $(n)$, number of edges $(m)$, maximum degree $(\Delta)$, minimum degree $(\delta)$ and the inverse degree $(ID(G))$. In addition, we give a counter example on the upper bound  of the second Zagreb index for Theorems 2.2 and  2.4 from \cite{ranjini}. Finally, we present lower and upper bounds on $\chi^2(G)+\chi^2(\overline G)$, where $\overline G$ denotes the complement of $G$.

scholarly journals Lower and Upper Bounds for Hyper-Zagreb Index of Graphs

2020 ◽  
pp. 1401-1406
Author(s):  
G. H. SHIRDEL ◽  
H. REZAPOUR ◽  
R. NASIRI
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Chemical Compounds ◽  
Upper Bounds ◽  
Topological Indices ◽  
Upper And Lower Bounds ◽  
Lower And Upper Bounds ◽  
Zagreb Index ◽  
Simple Connected Graph

The topological indices are functions on the graph that do not depend on the labeling of their vertices. They are used by chemists for studying the properties of chemical compounds.  Let  be a simple connected graph. The Hyper-Zagreb index of the graph ,  is defined as  ,where  and  are the degrees of vertex  and , respectively. In this paper, we study the Hyper-Zagreb index and give upper and lower bounds for .

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