Journal of Applied and Industrial Mathematics最新文献

The paper studies a model of a planar beam structure described by a fourth-orderboundary value problem on a geometric graph. Elastic-hinge joint conditions are posed at theinterior vertices of the graph. We study the properties of the spectral points of the correspondingspectral problem, prove upper bounds for the eigenvalue multiplicities, and show that theeigenvalue multiplicities depend on the graph structure (the number of boundary vertices, cycles,etc.). We give an example showing that our estimates are sharp.

A system of( m ) Boolean equations can be solved by a sequential sampling method using an( m )-step algorithm, where at the( i )th step the values of all variables essential for the first( i ) equations are sampled and false solutions are rejected based on theright-hand sides of the equations,( i=1,dots ,m ). The estimate of the complexity of the method depends on the structure ofthe sets of essential variables of the equations and attains its minimum after some permutation ofthe system equations. For the optimal permutation of equations we propose an algorithm thatminimizes the average computational complexity of the algorithm under natural probabilisticassumptions. In a number of cases, the construction of such a permutation is computationallydifficult; in this connection, other permutations are proposed which are computed in a simpler waybut may lead to nonoptimal estimates of the complexity of the method. The results implyconditions under which the sequential sampling method degenerates into the exhaustive searchmethod. An example of constructing an optimal permutation is given.

We consider the additive differential probabilities of functions( x oplus y ) and( (x oplus y) lll r ), where( x, y in mathbb {Z}_2^n ) and( 1 leq r < n ). The probabilities are used for the differential cryptanalysis of ARX ciphersthat operate only with addition modulo( 2^n ), bitwise XOR (( oplus )), and bit rotations (( lll r )). A complete characterization of differentials whose probability exceeds( 1/4 ) is obtained. All possible values of their probabilities are( 1/3 + 4^{2 - i} / 6 ) for( i in {1, dots , n} ). We describe differentials with each of these probabilities and calculate thenumber of these values. We also calculate the number of all considered differentials. It is( 48n - 68 ) for( x oplus y ) and( 24n - 30 ) for( (x oplus y) lll r ), where( n geq 2 ). We compare differentials of both mappings under the given constraint.

The aim of the present work is to create a tool for comparing algorithms for estimatingthe dynamics of the abundance of individual fish species in Lake Baikal based on experimentalcatching. For each instance of a randomly selected fish, its age is determined using a specialtechnology. From the obtained data on the numbers of fish of different ages in the sample, theparameters of the given laws of age distribution are estimated. This serves as a basis for theformation of ideas about the dynamics of mortality and changes in the abundance of fish of thisspecies in previous years. Various algorithms can be used to estimate the parameters ofdistribution laws, sometimes leading to considerably different results.

The discussed technique for comparing parameter estimation algorithms is basedon multiple computational experiments using the Monte Carlo method to simulate randomsamples of fish. The proposed method analyzes several algorithms for estimating the parametersof a truncated exponential distribution law for different sample sizes. As an example, the problemof estimating the mortality dynamics of the Baikal oilfish (Comephorus), the major biomass fish ofLake Baikal, is considered.

The eternal vertex cover problem is a version of the graph vertex cover problem that canbe represented as a dynamic game between two players (the Attacker and the Defender) withan infinite number of steps. At each step, there is an arrangement of guards over the vertices ofthe graph forming a vertex cover. When the Attacker attacks one of the graph’s edges, theDefender must move the guard along the attacked edge from one vertex to the other. In addition,the Defender can move any number of other guards from their current vertices to some adjacentones to obtain a new vertex cover. The Attacker wins if the Defender cannot build a new vertexcover after the attack.

In this paper, we propose a procedure that allows us to answer the questionwhether there exists a winning Defender strategy that permits protecting a given vertex cover fora given finite number of steps. To construct the Defender strategy, the problem is represented asa dynamic Stackelberg game in which at each step the interaction of the opposing sides isformalized as a two-level mathematical programming problem. The idea of the procedure is torecursively check the 1-stability of vertex covers obtained as a result of solving lower-levelproblems and to use some information about the covers already considered.

We construct three-dimensional dynamical systems with block-linear discontinuousright-hand side that simulate the simplest molecular oscillators. The phase portrait of each ofthese systems contains a unique equilibrium point and a cycle lying in the complement of thebasin of attraction of this point. There are no other equilibrium points in these phase portraits.

We consider the problem of constructing a single processor task schedule withminimization of peak resource usage. An example of the resource is the main memory of the targetcomputer. Task set to be scheduled is represented as a directed acyclic graph every node of whichis marked with the amount of resource used by the corresponding task. The resource allocated toa task is released on completion of the last (according to the schedule) immediate successor of thistask in the graph. Correctness constraint on the schedule is the partial order specified by the taskgraph. Task duration values are not considered. The formal statement of the problem is provided.To solve the problem, we propose an ant colony algorithm modified so that the pheromone matrixreflects the desirability of pairwise order in the schedule for every pair of tasks, not only for pairs ofadjacent tasks. During the schedule construction, for every task the algorithm chooses its positionin the schedule, in contrast to existing ant colony scheduling algorithms that construct schedule inincreasing order of positions (left-to-right) choosing a task for every next position. Experimentalevaluation of the algorithm was conducted on two sets of task graphs. The first set containsgraphs generated in such a way that the estimation for the optimum value of the goal function isknown a priori. Graphs from the second set are “layered,” and their structure corresponds to thestructure of multistage data processing applications. In both sets, the graphs are generatedrandomly with respect to specified generation parameters and constraints on the graph structure.The experiments indicate high precision and stability of the proposed ant colony algorithm.

We consider a sequence of block-separable convex programming problems describing theresource allocation in multiagent systems. We construct several iterative algorithms for setting theresource prices. Under various assumptions about the utility functions and resource constraints,we obtain estimates for the average deviation (regret) of the objective function from the optimalvalue and the constraint residuals. Here the average is understood as the expectation forindependent identically distributed data and as the time average in the online optimizationproblem. The analysis of the problem is carried out by online optimization methods and dualitytheory. The algorithms considered use the information concerning the difference between the totaldemand and supply that is generated by agents’ reactions to prices and corresponds to theconstraint residuals.

The paper considers the following problem. Given a set of Euclidean vectors, find severalclusters with a restriction on the maximum scatter of each cluster so that the size of the minimumcluster would be maximum. Here the scatter is the sum of squared distances from the clusterelements to its centroid. The NP-hardness of this problem is proved in the case where thedimension of the space is part of the input.

The aim of this paper is to acquaint applied mathematicians interested in the possibilitiesof applying methods for solving inverse problems in mathematical modeling in natural sciences andengineering to economic problems with our papers in this field. These papers refer to the problemof verifying the market demand theory, developed by the first author based on the revision of theunrealistic axiomatic neoclassical theory of individual demand within the framework of generalscientific methodology. At the same time, the artificial object of individual theory—arational and independent individual who maximizes his/her utility function—was replacedby a “statistical ensemble of consumers” of the market under study, and the postulates ofindividual theory became scientific hypotheses of the new theory. The verification of this theoryconsists in elucidating the question of rationalizing the statistical market demand by the collectiveutility function. This problem refers to the inverse problems of mathematical theories of realphenomena, which are usually ill posed and have many solutions. The solution of such problemsconsists in their regularization with involvement of additional information about the desiredsolution. Our method for verifying the market demand theory is a development of thenonparametric Afriat–Varian demand analysis with using “economic indices” of market demand asadditional information, which allows obtaining solutions with various substantive properties.