Bounding the Size and Probability of Epidemics on Networks

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.

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

Monte Carlo Methods and Applications ◽

10.1515/mcma-2020-2062 ◽

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.

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The Essential Dimension of Stacks of Parabolic Vector Bundles over Curves

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is011008026jkt192 ◽

2012 ◽

Vol 10 (3) ◽

pp. 455-488 ◽

Cited By ~ 1

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.

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

Lecture Notes in Computer Science - Foundations of Software Science and Computation Structures ◽

10.1007/978-3-030-71995-1_3 ◽

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].

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On Mixed Codes with Covering Radius $1$ and Minimum Distance $2$

The Electronic Journal of Combinatorics ◽

10.37236/969 ◽

2007 ◽

Vol 14 (1) ◽

Author(s):

Wolfgang Haas ◽

Jörn Quistorff

Keyword(s):

Lower Bounds ◽

Minimum Distance ◽

Covering Radius ◽

Minimal Cardinality ◽

Finite Sets ◽

Open Problems ◽

Latin Rectangle ◽

Mixed Codes ◽

Special Case

Let $R$, $S$ and $T$ be finite sets with $|R|=r$, $|S|=s$ and $|T|=t$. A code $C\subset R\times S\times T$ with covering radius $1$ and minimum distance $2$ is closely connected to a certain generalized partial Latin rectangle. We present various constructions of such codes and some lower bounds on their minimal cardinality $K(r,s,t;2)$. These bounds turn out to be best possible in many instances. Focussing on the special case $t=s$ we determine $K(r,s,s;2)$ when $r$ divides $s$, when $r=s-1$, when $s$ is large, relative to $r$, when $r$ is large, relative to $s$, as well as $K(3r,2r,2r;2)$. Some open problems are posed. Finally, a table with bounds on $K(r,s,s;2)$ is given.

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Correlation lower bounds from correlation upper bounds

Information Processing Letters ◽

10.1016/j.ipl.2016.03.012 ◽

2016 ◽

Vol 116 (8) ◽

pp. 537-540

Author(s):

Shiteng Chen ◽

Periklis A. Papakonstantinou

Keyword(s):

Lower Bounds ◽

Upper Bounds

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Lower bounds for cutting planes proofs with small coefficients

Journal of Symbolic Logic ◽

10.2307/2275569 ◽

1997 ◽

Vol 62 (3) ◽

pp. 708-728 ◽

Cited By ~ 55

Author(s):

Maria Bonet ◽

Toniann Pitassi ◽

Ran Raz

Keyword(s):

Lower Bound ◽

Lower Bounds ◽

Cutting Planes ◽

Communication Complexity ◽

Proof System ◽

Small Weight ◽

Deduction Rule ◽

Special Case ◽

Exponential Lower Bound

AbstractWe consider small-weight Cutting Planes (CP*) proofs; that is, Cutting Planes (CP) proofs with coefficients up to Poly(n). We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of CP* proofs, for a family of tautologies based on the clique function. Because Resolution is a special case of small-weight CP, our method also gives a new and simpler exponential lower bound for Resolution.We also prove the following two theorems: (1) Tree-like CP* proofs cannot polynomially simulate non-tree-like CP* proofs. (2) Tree-like CP* proofs and Bounded-depth-Frege proofs cannot polynomially simulate each other.Our proofs also work for some generalizations of the CP* proof system. In particular, they work for CP* with a deduction rule, and also for any proof system that allows any formula with small communication complexity, and any set of sound rules of inference.

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General numerical radius inequalities for matrices of operators

Open Mathematics ◽

10.1515/math-2016-0011 ◽

2016 ◽

Vol 14 (1) ◽

pp. 109-117 ◽

Cited By ~ 3

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.

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On Some Issues in Shakedown Analysis

Journal of Applied Mechanics ◽

10.1115/1.1379368 ◽

2001 ◽

Vol 68 (5) ◽

pp. 799-808 ◽

Cited By ~ 35

Author(s):

G. Maier

Keyword(s):

Heterogeneous Media ◽

Direct Methods ◽

Cost Effective ◽

Upper Bounds ◽

Saturated Porous Media ◽

Safety Factors ◽

Shakedown Analysis ◽

Time Stepping ◽

Kinematic Theorem ◽

Special Case

Shakedown analysis, and its more classical special case of limit analysis, basically consists of “direct” (as distinct from time-stepping) methods apt to assess safety factors for variable repeated external actions and procedures which provide upper bounds on history-dependent quantities. The issues reviewed and briefly discussed herein are: some recent engineering-oriented and cost-effective methods resting on Koiter’s kinematic theorem and applied to periodic heterogeneous media; recent extensions (after the earlier ones to dynamics and creep) to another area characterized by time derivatives, namely poroplasticity of fluid-saturated porous media. Links with some classical or more consolidated direct methods are pointed out.

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Polarization of Separating Invariants

Canadian Journal of Mathematics ◽

10.4153/cjm-2008-027-2 ◽

2008 ◽

Vol 60 (3) ◽

pp. 556-571 ◽

Cited By ~ 20

Author(s):

Jan Draisma ◽

Gregor Kemper ◽

David Wehlau

Keyword(s):

Finite Group ◽

Finite Groups ◽

Group Actions ◽

Upper Bounds ◽

Weyl’S Theorem ◽

Finite Group Actions ◽

Separating Invariants ◽

The Difference ◽

Special Case

AbstractWe prove a characteristic free version of Weyl’s theorem on polarization. Our result is an exact analogue ofWeyl’s theorem, the difference being that our statement is about separating invariants rather than generating invariants. For the special case of finite group actions we introduce the concept of cheap polarization, and show that it is enough to take cheap polarizations of invariants of just one copy of a representation to obtain separating vector invariants for any number of copies. This leads to upper bounds on the number and degrees of separating vector invariants of finite groups.

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