scholarly journals On the Number of Planar Orientations with Prescribed Degrees

10.37236/801 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Stefan Felsner ◽  
Florian Zickfeld
Keyword(s):  
Lower Bounds ◽  
Spanning Trees ◽  
Upper Bounds ◽  
Perfect Matchings ◽  
Asymptotic Enumeration ◽  
Combinatorial Structures ◽  
Schnyder Woods ◽  
Planar Map ◽  
Planar Maps

We deal with the asymptotic enumeration of combinatorial structures on planar maps. Prominent instances of such problems are the enumeration of spanning trees, bipartite perfect matchings, and ice models. The notion of orientations with out-degrees prescribed by a function $\alpha:V\rightarrow {\Bbb N}$ unifies many different combinatorial structures, including the afore mentioned. We call these orientations $\alpha$-orientations. The main focus of this paper are bounds for the maximum number of $\alpha$-orientations that a planar map with $n$ vertices can have, for different instances of $\alpha$. We give examples of triangulations with $2.37^n$ Schnyder woods, 3-connected planar maps with $3.209^n$ Schnyder woods and inner triangulations with $2.91^n$ bipolar orientations. These lower bounds are accompanied by upper bounds of $3.56^n$, $8^n$ and $3.97^n$ respectively. We also show that for any planar map $M$ and any $\alpha$ the number of $\alpha$-orientations is bounded from above by $3.73^n$ and describe a family of maps which have at least $2.598^n$ $\alpha$-orientations.

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 Bounding the Size and Probability of Epidemics on Networks

2008 ◽  
Vol 45 (2) ◽  
pp. 498-512 ◽  
Author(s):  
Joel C. Miller
Keyword(s):  
Infectious Disease ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Marginal Probability ◽  
Transmission Properties ◽  
Disease Spreading ◽  
Special Case

We consider an infectious disease spreading along the edges of a network which may have significant clustering. The individuals in the population have heterogeneous infectiousness and/or susceptibility. We define the out-transmissibility of a node to be the marginal probability that it would infect a randomly chosen neighbor given its infectiousness and the distribution of susceptibility. For a given distribution of out-transmissibility, we find the conditions which give the upper (or lower) bounds on the size and probability of an epidemic, under weak assumptions on the transmission properties, but very general assumptions on the network. We find similar bounds for a given distribution of in-transmissibility (the marginal probability of being infected by a neighbor). We also find conditions giving global upper bounds on the size and probability. The distributions leading to these bounds are network independent. In the special case of networks with high girth (locally tree-like), we are able to prove stronger results. In general, the probability and size of epidemics are maximal when the population is homogeneous and minimal when the variance of in- or out-transmissibility is maximal.

scholarly journals The Essential Dimension of Stacks of Parabolic Vector Bundles over Curves

2012 ◽  
Vol 10 (3) ◽  
pp. 455-488 ◽  
Author(s):  
Indranil Biswas ◽  
Ajneet Dhillon ◽  
Nicole Lemire
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Vector Bundles ◽  
Upper Bounds ◽  
Essential Dimension ◽  
Parabolic Structure ◽  
Moduli Stack

AbstractWe find upper bounds on the essential dimension of the moduli stack of parabolic vector bundles over a curve. When there is no parabolic structure, we improve the known upper bound on the essential dimension of the usual moduli stack. Our calculations also give lower bounds on the essential dimension of the semistable locus inside the moduli stack of vector bundles of rank r and degree d without parabolic structure.

scholarly journals Finding Cut-Offs in Leaderless Rendez-Vous Protocols is Easy

2021 ◽  
pp. 42-61
Author(s):  
A. R. Balasubramanian ◽  
Javier Esparza ◽  
Mikhail Raskin
Keyword(s):  
Lower Bounds ◽  
Petri Net ◽  
Special Class ◽  
Upper Bounds ◽  
Initial State ◽  
Reachability Problem ◽  
Final State ◽  
Final Configuration ◽  
Finite State ◽  
State Agents

AbstractIn rendez-vous protocols an arbitrarily large number of indistinguishable finite-state agents interact in pairs. The cut-off problem asks if there exists a number B such that all initial configurations of the protocol with at least B agents in a given initial state can reach a final configuration with all agents in a given final state. In a recent paper [17], Horn and Sangnier prove that the cut-off problem is equivalent to the Petri net reachability problem for protocols with a leader, and in "Image missing" for leaderless protocols. Further, for the special class of symmetric protocols they reduce these bounds to "Image missing" and "Image missing" , respectively. The problem of lowering these upper bounds or finding matching lower bounds is left open. We show that the cut-off problem is "Image missing" -complete for leaderless protocols, "Image missing" -complete for symmetric protocols with a leader, and in "Image missing" for leaderless symmetric protocols, thereby solving all the problems left open in [17].

Correlation lower bounds from correlation upper bounds

2016 ◽  
Vol 116 (8) ◽  
pp. 537-540
Author(s):  
Shiteng Chen ◽  
Periklis A. Papakonstantinou
Keyword(s):  
Lower Bounds ◽  
Upper Bounds

scholarly journals General numerical radius inequalities for matrices of operators

2016 ◽  
Vol 14 (1) ◽  
pp. 109-117 ◽  
Author(s):  
Mohammed Al-Dolat ◽  
Khaldoun Al-Zoubi ◽  
Mohammed Ali ◽  
Feras Bani-Ahmad
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
Zeros Of Polynomials ◽  
Numerical Radius

AbstractLet Ai ∈ B(H), (i = 1, 2, ..., n), and $ T = \left[ {\matrix{ 0 & \cdots & 0 & {A_1 } \cr \vdots & {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} & {A_2 } & 0 \cr 0 & {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} & {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} & \vdots \cr {A_n } & 0 & \cdots & 0 \cr } } \right] $ . In this paper, we present some upper bounds and lower bounds for w(T). At the end of this paper we drive a new bound for the zeros of polynomials.

scholarly journals On Gap-Based Lower Bounding Techniques for Best-Arm Identification

Entropy ◽  
2020 ◽  
Vol 22 (7) ◽  
pp. 788
Author(s):  
Lan V. Truong ◽  
Jonathan Scarlett
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
Sample Complexity ◽  
Bandit Problem ◽  
Lower Bounding ◽  
Complexity Lower Bounds

In this paper, we consider techniques for establishing lower bounds on the number of arm pulls for best-arm identification in the multi-armed bandit problem. While a recent divergence-based approach was shown to provide improvements over an older gap-based approach, we show that the latter can be refined to match the former (up to constant factors) in many cases of interest under Bernoulli rewards, including the case that the rewards are bounded away from zero and one. Together with existing upper bounds, this indicates that the divergence-based and gap-based approaches are both effective for establishing sample complexity lower bounds for best-arm identification.

scholarly journals Structural bounds on the dyadic effect

2017 ◽  
Vol 5 (5) ◽  
pp. 694-711 ◽  
Author(s):  
Matteo Cinelli ◽  
Giovanna Ferraro ◽  
Antonio Iovanella
Keyword(s):  
Lower Bounds ◽  
Network Topology ◽  
Dimensional Space ◽  
Upper Bounds ◽  
Feasible Region ◽  
Two Dimensional ◽  
Independent Parameters

AbstractThe dyadic effect is a phenomenon that occurs when the number of links between nodes sharing a common feature is larger than expected if the features are distributed randomly on the network. In this article, we consider the case when nodes are distinguished by a binary characteristic. Under these circumstances, two independent parameters, namely dyadicity and heterophilicity are able to detect the presence of the dyadic effect and to measure how much the considered characteristic affects the network topology. The distribution of nodes characteristics can be investigated within a two-dimensional space that represents the feasible region of the dyadic effect, which is bound by two upper bounds on dyadicity and heterophilicity. Using some network structural arguments, we are able to improve such upper bounds and introduce two new lower bounds, providing a reduction of the feasible region of the dyadic effect as well as constraining dyadicity and heterophilicity within a specific range. Some computational experiences show the bounds effectiveness and their usefulness with regards to different classes of networks.

On the Upper Bounds of the Numbers of Perfect Matchings in Graphs with Given Parameters

2007 ◽  
Vol 23 (1) ◽  
pp. 155-160 ◽  
Author(s):  
Hong Lin ◽  
Xiao-feng Guo
Keyword(s):  
Upper Bounds ◽  
Perfect Matchings

Asymptotic behavior of a multiplexer fed by a long-range dependent process

1999 ◽  
Vol 36 (01) ◽  
pp. 105-118 ◽  
Author(s):  
Zhen Liu ◽  
Philippe Nain ◽  
Don Towsley ◽  
Zhi-Li Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Lower Bounds ◽  
Lognormal Distribution ◽  
Arrival Rate ◽  
Upper Bounds ◽  
Single Server ◽  
Service Capacity ◽  
Single Server Queue ◽  
Input Process ◽  
Server Queue

In this paper we study the asymptotic behavior of the tail of the stationary backlog distribution in a single server queue with constant service capacity c, fed by the so-called M/G/∞ input process or Cox input process. Asymptotic lower bounds are obtained for any distribution G and asymptotic upper bounds are derived when G is a subexponential distribution. We find the bounds to be tight in some instances, e.g. when G corresponds to either the Pareto or lognormal distribution and c − ρ < 1, where ρ is the arrival rate at the buffer.

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