Random sums of exchangeable variables and actuarial applications

In this paper we study the accumulated claim in some fixed time period, skipping the classical assumption of mutual independence between the variables involved. Two basic models are considered: Model 1 assumes that any pair of claims are equally correlated which means that the corresponding square-integrable sequence is exchangeable one. Model 2 states that the correlations between the adjacent claims are the same. Recurrence and explicit expressions for the joint probability generating function are derived and the impact of the dependence parameter (correlation coefficient) in both models is examined. The Markov binomial distribution is obtained as a particular case under assumptions of Model 2.     

The correlation matrix problem in insurance and banking

In insurance, the analyst is often faced with a large number of inter-related variables for which correlations need to be estimated. Clearly, all correlations lie in the interval [?1,?1], but the numbers cannot be assigned independently. Here, the choices left to the analyst are considered from both a geometric and a probabilistic viewpoint. In practice, the underwriter or risk manager may fix some of the correlations and this paper considers, from both these viewpoints, what effect this has on the analyst's choice of correlations between the remaining variables.     

Enhance-synergism and suppression effects in multiple regression

Relations between pairwise correlations and the coefficient of multiple determination in regression analysis are considered. The conditions for the occurrence of enhance-synergism and suppression effects when multiple determination becomes bigger than the total of squared correlations of the dependent variable with the regressors are discussed. It is shown that such effects can occur just for stochastic relations among the variables that have non-transitive signs of pairwise correlations. Consideration of these problems facilitates better understanding the properties of regression.     

Remote control of quantum correlations in a two-qubit receiver by a three-qubit sender

We study remote control of quantum correlations (discord) in a subsystem of two qubits (receiver) via parameters of the initial state of another three-qubit subsystem (sender) connected to the receiver by an inhomogeneous spin s = 1/2 chain. We propose two parameters characterizing the creatable correlations. The first parameter is the discord between the receiver and the remainder of the spin s = 1/2 chain, and it concerns the mutual correlations between these two subsystems. The second parameter is the discord between the two nodes of the receiver and describes the inner correlations of the receiver. We study the dependence of these two discords on the inhomogeneity parameter of the spin chain.     

Profile of Random Exponential Binary Trees

We introduce the random exponential binary tree (EBT) and study its profile. As customary, the tree is extended by padding each leaf node (considered internal), with the appropriate number of external nodes, so that the outdegree of every internal node is made equal to 2. In a random EBT, at every step, each external node is promoted to an internal node with probability p, stays unchanged with probability 1 - p, and the resulting tree is extended. We study the internal and external profiles of a random EBT and get exact expectations for the numbers of internal and external nodes at each level. Asymptotic analysis shows that the average external profile is richest at level \(\frac {2p}{p+1}n\), and it experiences phase transitions at levels a n, where the a’s are the solutions to an algebraic equation. The rates of convergence themselves go through an infinite number of phase changes in the sublinear range, and then again at the nearly linear levels.     

Energy aware proactive optimized link state routing in mobile ad-hoc networks

In this paper we develop a novel energy aware routing approach for mobile ad hoc network (MANET) problems. The approach is based on using Optimized Link State Routing Protocol. Our Energy Aware OLSR labeled as OLSR_EA measures and predicts per-interval energy consumptions using the well-known Auto-Regressive Integrated Moving Average time series method. We develop a composite energy cost, by considering transmission power consumption and residual energy of each node, and use this composite energy index as the routing metric. Our extensive ns2 simulation experiments show that OLSR_EA substantially prolongs the network lifetime and saves total energy used in MANET. In our experiments we considered different scenarios considering a variety of traffic loads, node mobilities, homogeneous power consumption, and heterogeneous power consumption. Simulation results also confirm that OLSR_EA improves the traffic balance between nodes, and packet delivery ratio in higher node speed. We further develop characteristics of OLSR_EA in power-wise heterogeneous MANET to achieve efficient energy preserving performance.     

Exact identification of critical nodes in sparse networks via new compact formulations

Critical node detection problems aim to optimally delete a subset of nodes in order to optimize or restrict a certain metric of network fragmentation. In this paper, we consider two network disruption metrics which have recently received substantial attention in the literature: the size of the remaining connected components and the total number of node pairs connected by a path. Exact solution methods known to date are based on linear 0–1 formulations with at least $\varTheta (n^3)$ entities and allow one to solve these problems to optimality only in small sparse networks with up to 150 nodes. In this work, we develop more compact linear 0–1 formulations for the considered types of problems with $\varTheta (n^2)$ entities. We also provide reformulations and valid inequalities that improve the performance of the developed models. Computational experiments show that the proposed formulations allow finding exact solutions to the considered problems for real-world sparse networks up to 10 times larger and with CPU time up to 1,000 times faster compared to previous studies.     

Locating flow-capturing units on a network with multi-counting and diminishing returns to scale

In the paper we consider the problem of locating flow-capturing units (facilities) on a transportation network, where the level of consuming service by customers depends on the number of facilities that they encounter on their pre-planned tour (the effect of multi-counting). Two location problems are considered: Problem 1 — minimizing the number of facilities required to ensure the maximal level of consumption, and Problem 2 — maximizing the total consumption given a restriction on the number of facilities. Both problems are NP-hard on general networks. Integer programming formulations of the problems are given. For Problem 2, a heuristic with worst-case analysis is presented. It is shown that Problem 2 is NP-hard even on a tree (and even without multi-counting). For Problem 1 on a tree a polynomial algorithm is presented. If it is required additionally that at most one facility can be located at each node and locations are restricted to nodes, then both problems are NP-hard on trees.     

The probabilistic 1-maximal covering problem on a network with discrete demand weights

We discuss the probabilistic 1-maximal covering problem on a network with uncertain demand. A single facility is to be located on the network. The demand originating from a node is considered covered if the shortest distance from the node to the facility does not exceed a given service distance. It is assumed that demand weights are independent discrete random variables. The objective of the problem is to find a location for the facility so as to maximize the probability that the total covered demand is greater than or equal to a pre-selected threshold value. We show that the problem is NP-hard and that an optimal solution exists in a finite set of dominant points. We develop an exact algorithm and a normal approximation solution procedure. Computational experiment is performed to evaluate their performance.     

The problem of identification of parameters by the distribution of the maximum random variable

Suppose that X₁, X₂,…, X_n are independently distributed according to certain distributions. Does the distribution of the maximum of {X₁, X₂,…, X_n} uniquely determine their distributions? In the univariate case, a general theorem covering the case of Cauchy random variables is given here. Also given is an affirmative answer to the above question for general bivariate normal random variables with non-zero correlations. Bivariate normal random variables with nonnegative correlations were considered earlier in this context by T. W. Anderson and S. G. Ghurye.     

Some properties of canonical correlations and variates in infinite dimensions

In this paper the notion of functional canonical correlation as a maximum of correlations of linear functionals is explored. It is shown that the population functional canonical correlation is in general well defined, but that it is a supremum rather than a maximum, so that a pair of canonical variates may not exist in the spaces considered. Also the relation with the maximum eigenvalue of an associated pair of operators and the corresponding eigenvectors is not in general valid. When the inverses of the operators involved are regularized, however, all of the above properties are restored. Relations between the actual population quantities and their regularized versions are also established. The sample functional canonical correlations can be regularized in a similar way, and consistency is shown at a fixed level of the regularization parameter.     

Unified Representations of Nonlinear Splines

Golomb and Jerome's framework is modified and extended. The new framework is more general since it also handles interpolants which are not allowed to “slide” at the nodes. The space of interpolants of variable length is shown to be a smooth manifold. If the length is fixed, and there are no nodes, then the space of interpolants is a manifold. When there is at least one node, and at least one node is not on the line segment between the endpoints, then the space of interpolants of fixed length is a smooth manifold. Sufficient conditions are given which ensure the space of interpolants continues to be a smooth manifold in the presence of additional constraints such as clamping and pinning. A new fundamental finite-dimensional equation is derived. When it is solved it yields all nonlinear splines, and every nonlinear spline appears in this way. An important feature is that the same symbolic equation is used for all possible combinations of the constraints considered. It is shown how to take the solutions of the fundamental equation and use them to express the corresponding nonlinear splines in terms of a pair of elliptic functions. An inequality is derived that specifies which elliptic function appears along each section of the spline. The nonlinear splines are in a unified way shown to beC²for all possible combinations of the constraints considered.     

Numerical performance of approximate queuing formulae with application to flexible manufacturing systems

This paper calls attention to two of the more successful queuing approximation formulae — one by Kramer and one by Marchal. The analytic solution of a range of single server Erlang cases is compared to the two approximation formulae. Then a family of H₂/M/1 cases is similarly considered. Maximum errors are seen to be about three percent. The Kramer formula seems to be better when the interarrival coefficient of variation is less than 0.66 and the Marchal formula is better for larger interarrival coefficients of variation. Finally, a multiserver refinement function (the ratio of G/G/1 results to M/M/1 results) is proposed to scale M/M/s as an approximation for G/G/s. In most of these multiple channel cases, the maximum error is less than six percent. The last section of this paper presents a simple, representative FMS. It is modelled as an open queuing network. Then the approximation procedure is applied node by node to illustrate the estimation of system performance measures such as machine utilizations and throughput.     

CONNECTIVITY OF CARTESIAN PRODUCT DIGRAPHS AND FAULT-TOLERANT ROUTINGS OF GENERALIZED HYPERCUBE

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Stochastic Geometric Models with Non-stationary Spatial Correlations in Lagrangian Fluid Flows

Inspired by spatiotemporal observations from satellites of the trajectories of objects drifting near the surface of the ocean in the National Oceanic and Atmospheric Administration’s “Global Drifter Program”, this paper develops data-driven stochastic models of geophysical fluid dynamics (GFD) with non-stationary spatial correlations representing the dynamical behaviour of oceanic currents. Three models are considered. Model 1 from Holm (Proc R Soc A 471:20140963, 2015) is reviewed, in which the spatial correlations are time independent. Two new models, called Model 2 and Model 3, introduce two different symmetry breaking mechanisms by which the spatial correlations may be advected by the flow. These models are derived using reduction by symmetry of stochastic variational principles, leading to stochastic Hamiltonian systems, whose momentum maps, conservation laws and Lie–Poisson bracket structures are used in developing the new stochastic Hamiltonian models of GFD.     

基于复杂网络的中国股市房地产板块股票强相关性研究

为分析中国股市房地产板块股票的强相关特性,以101只股票为结点,以近17年股票对数回报的相关系数为加权边,建立复杂网络模型,通过对网络拓扑参数计算,发现该网络为无尺度网络,结点度分布P(s)~s-|s|δ,在不同相关系数阈值下,δ值介于0.8~1.6之间.网络平均集聚系数为0.53.文章也对网络中心性进行测量和子群划分,发现代码为000592和601588的结点在网络中具有很高的中介性,网络中大部分结点都受其影响较大.     

A branch-and-cut-and-price algorithm for one-dimensional stock cutting and two-dimensional two-stage cutting

The one-dimensional cutting stock problem (1D-CSP) and the two-dimensional two-stage guillotine constrained cutting problem (2D-2CP) are considered in this paper. The Gilmore–Gomory models of these problems have very strong continuous relaxations providing a good bound in an LP-based solution approach. In recent years, there have been several efforts to attack the one-dimensional problem by LP-based branch-and-bound with column generation (called branch-and-price) and by general-purpose Chvátal–Gomory cutting planes. In this paper we investigate a combination of both approaches, i.e., the LP relaxation at each branch-and-price node is strengthened by Chvátal–Gomory and Gomory mixed-integer cuts. The branching rule is that of branching on variables of the Gilmore–Gomory formulation. Tests show that, for 1D-CSP, general-purpose cuts are useful only in exceptional cases. However, for 2D-2CP their combination with branching is more effective than either approach alone and mostly better than other methods from the literature.     

Relationship between the molecular structure and certain mechanical properties of oriented polymers

The correlations between the molecular structure and intermolecular energy and the fatigue and elastic properties of fibers are considered. It is shown that the maximum of these properties corresponds to an optimal value of the intermolecular energy.Leningrad Branch of the All-Union Scientific-Research Institute of Synthetic Fibers. Translated from Mekhanika Polimerov, No. 4, pp. 734–735, July–August, 1969.     

An extended shortest path problem: A data envelopment analysis approach

A special and important network structured linear programming problem is the shortest path problem. Classical shortest path problems assume that there are unit of shipping cost or profit along an arc. In many real occasions, various attributes (various costs and profits) are usually considered in a shortest path problem. Because of the frequent occurrence of such network structured problems, there is a need to develop an efficient procedure for handling these problems. This paper studies the shortest path problem in the case that multiple attributes are considered along the arcs. The concept of relative efficiency is defined for each path from initial node to final node. Then, an efficient path with the maximum efficiency is determined.     

Depth in bucket recursive trees with variable capacities of buckets

We consider bucket recursive trees of sizen consisting of all buckets with variable capacities1,2,...,b and with a specifc stochastic growth rule.This model can be considered as a generalization of random recursive trees like bucket recursive trees introduced by Mahmoud and Smythe where all buckets have the same capacities.In this work,we provide a combinatorial analysis of these trees where the generating function of the total weights satisfes an autonomous frst order diferential equation.We study the depth of the largest label（i.e.,the number of edges from the root node to the node containing label n）and give a closed formula for the probability distribution.Also we prove a limit law for this quantity which is a direct application of quasi power theorem and compute its mean and variance.Our results for b=1 reduce to the previous results for random recursive trees.