On the isoperimetric problem for radial log-convex densities

Given a smooth, radial, uniformly log-convex density e ^V on ${\mathbb{R}^n}$ , n ≥ 2, we characterize isoperimetric sets E with respect to weighted perimeter ${\int_{\partial E}e^{V} d \mathcal{H}^{n-1}}$ and weighted volume m = ∫ _E e ^V as balls centered at the origin, provided ${m \in [0, m_0)}$ for some (potentially computable) m ₀>0; this affirmatively answers conjecture (Rosales et al. Calc Var Part Differ Equat 31(1):27–46, 2008, Conjecture 3.12) for such values of the weighted volume parameter. We also prove that the set of weighted volumes such that this characterization holds true is open, thus reducing the proof of the full conjecture to excluding the possibility of bifurcation values of the weighted volume parameter. Finally, we show the validity of the conjecture when V belongs to a C ²-neighborhood of c|x|² (c> 0).     

The likelihood ratio for densities with singularities

Let t_n be the Bayesian estimator of the parameter constructed from independent observations in the case of infinite information and probability distribution density with some singularities. It is shown that under certain conditions on the behavior of the densities near their singularities, the normalizing factor (n) ensuring that _n=(n) (t_n-) has a nontriviallimiting distribution for n is regularly varying in Karamata's sense. The limiting distribution of _n is determined.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 74, pp. 66–82, 1977.I would like to acknowledge the guidance of I. A. Ibragimov in the course of this research.     

The eigenvalue gap for vibrating strings with symmetric densities

We establish several comparison results on the eigenvalue gap for vibrating strings with symmetric single-well densities or symmetric double-well densities.      

Geometry of manifolds with densities

We study the geometry of complete Riemannian manifolds endowed with a weighted measure, where the weight function is of quadratic growth. Assuming the associated Bakry–Émery curvature is bounded from below, we derive a new Laplacian comparison theorem and establish various sharp volume upper and lower bounds. We also obtain some splitting type results by analyzing the Busemann functions. In particular, we show that a complete manifold with nonnegative Bakry–Émery curvature must split off a line if it is not connected at infinity and its weighted volume entropy is of maximal value among linear growth weight functions.     

The Cauchy-Nicoletti problem with poles

The Cauchy-Nicoletti boundary value problem for a system of ordinary differential equations with pole-type singularities is investigated. The conditions of the existence, uniqueness, and nonuniqueness of a solution in the class of continuously differentiable functions are given. The classical Banach contraction principle is combined with a special transformation of the original problem.     

The Ramanujan sum and Chebotarev densities

The Ramanujan Journal - In this short note, we show an analogue of one of Alladi’s and Dawsey’s formulas with respect to the Ramanujan sum $$c_n(m)$$ for $$m\geqslant 1$$ . Their...     

The newsvendor problem with unknown distribution

Newsvendor theory assumes that the decision-maker faces a knowndistribution. But in real-life situations, demand distribution is not alwaysknown. In the experimental study which this paper presents, half of theparticipants assuming the newsvendor role were unaware of the underlying demanddistribution, while the other half knew the demand distribution. Participantshad to decide how many papers to order each day (for 100 days). The experimentalfindings indicate that subjects who know the demand distribution behavedifferently to those who do not. However, interestingly enough, knowing thedemand distribution does not necessarily lead the subject closer to the optimalsolution or improve profits. It was found that supply surplus at a certainperiod strongly affects the order quantity towards the following period, despitethe knowledge of the demand distribution.     

多项式分裂可行问题

本文研究多项式分裂可行问题,即由多项式不等式定义的分裂可行问题,包括凸与非凸、可行与不可行的问题;给出多项式分裂可行问题解集的半定松弛表示;研究其半定松弛化问题的性质;并基于这些性质建立求解多项式分裂可行问题的半定松弛算法.本文在较为一般的条件下证明了,如果分裂可行问题有解,则可通过本文建立的算法求得一个解点;如果问题...     

An explicit solution of the linear filtering problem for certain classes of nonrational spectral densities

We present an explicit solution of the problem of optimal linear filtering: the recovery of the useful signal(s) at the instantt+, (>0,<0, or=0) from known values of the received signal(s)=(s)+(s) in the past, i.e., at the instantt–s, s0. In doing so we assume the random processes(s) and /gr(s) are stationary and jointly stationary, while the stationary process of noise (s) with zero mean is assumed to be mutually correlated and jointly stationary with the process(s) under the assumption that there exists a common spectral densityf() for these processes.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 83–91, 1986.     

The knapsack problem with neighbour constraints

We study a constrained version of the knapsack problem in which dependencies between items are given by the adjacencies of a graph. In the 1-neighbour knapsack problem, an item can be selected only if at least one of its neighbours is also selected. In the all-neighbours knapsack problem, an item can be selected only if all its neighbours are also selected.We give approximation algorithms and hardness results when the vertices have both uniform and arbitrary weight and profit functions, and when the dependency graph is directed and undirected.     

The max-flow problem with parametric capacities

In this paper we present a comprehensive analysis of the max-flow problem with n parametric capacities, and give the basis for an algorithm to solve it. In particular we give a method for finding the max-flow value as a function of the parameters, and max-flows for all parameter points, in terms of max-flow values to problems at certain key parameter points. In the problem with nonzero lower bounds on the arc flows, we derive a set of linear constraints whose solution set is identical to the set of all feasible parameter points.The intrinsic difficulty of the problem is compared with that of the general multiparametric linear programming problem, and thus light is shed on the difficulty of the latter problem, whose complexity is currently unknown.     

The principal-agent problem with adaptive players

In this study we show predictions made by the standard principal-agent theory may not hold when knowlege assumptions are relaxed. Conventional principal-agent models assume players are completely rational: they know their own and other player's utilities and probabilities of all states of nature. In reality, players must make decisions without such knowledge.We define a simple version of the principal-agent game and examine it using object-oriented computer simulation. Player learning is modeled with a statistical learning model. Our results show that even this simple game combined with standard learning assumptions results in complex behavior. Expectations of both the principal and the agents are crucial in determining the system outcomes. Expectations and lack of prior knowledge make it possible for the principal to converge on suboptimal behavior or not converge on a consistent behavior at all. The same attributes in the agents make it possible for the principal to drive expectations down and thus get higher effort for lower reward.This study contributes a more robust understanding of the principal-agent model and its application to incentive design.     

The Weber problem with regional demand

This paper is devoted to the study of the Regional Weber Problem, an extension of the Weber problem which allows the demand not be concentrated onto a finite set of points.The most serious drawback of this formulation, from a resolution viewpoint, is the high computational cost involved in the evaluation of the objective function. A new approach is proposed, which requires a low amount of computation and where it is possible to control the error on the approximation.This approximation suggests a new methodology to solve the problem. This methodology is compared with the existing ones, showing its relevance from a practical point of view.     

The embedding problem with metabelian kernel

The relationship between the solvability conditions for the embedding problem with metabelian kernel over a p-extension of fields and the associated embedding problem of the first kind with a maximal p-group is studied. It is proved that these problems are equivalent. Bibliography: 1title.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 236, 1997, pp. 97–99.The present paper was supported by the Russian Foundation for Basic Research, grant No. 96-01-00854.