Belief structures,weight generating functions and decision-making

We describe the Dempster–Shafer belief structure and provide some of its basic properties. We introduce the plausibility and belief measures associated with a belief structure. We note that these are not the only measures that can be associated with a belief structure. We describe a general approach for generating a class of measures that can be associated with a belief structure using a monotonic function on the unit interval, called a weight generating function. We study a number of these functions and the measures that result. We show how to use weight-generating functions to obtain dual measures from a belief structure. We show the role of belief structures in representing imprecise probability distributions. We describe the use of dual measures, other then plausibility and belief, to provide alternative bounding intervals for the imprecise probabilities associated with a belief structure. We investigate the problem of decision making under belief structure type uncertain. We discuss two approaches to this decision problem. One of which is based on an expected value of the OWA aggregation of the payoffs associated with the focal elements. The second approach is based on using the Choquet integral of a measure generated from the belief structure. We show the equivalence of these approaches.     

Algorithmic aspects of mean–variance optimization in Markov decision processes

We consider finite horizon Markov decision processes under performance measures that involve both the mean and the variance of the cumulative reward. We show that either randomized or history-based policies can improve performance. We prove that the complexity of computing a policy that maximizes the mean reward under a variance constraint is NP-hard for some cases, and strongly NP-hard for others. We finally offer pseudopolynomial exact and approximation algorithms.     

项链李代数的性质

研究项链李代数的性质,给出了其中心元的表示形式,证明了项链李代数非半单、非可解,通过构造项链李代数的可解非幂零子代数,证明了当箭图中有长度大于1的循环时,项链李代数非幂零.还给出了没有圈的箭图上项链李代数的分解.     

United States and South Korean citizens’ interpretation and assessment of COVID-19 quantitative data

We investigate United States and South Korean citizens’ mathematical schemes and how these schemes supported or hindered their attempts to assess the severity of COVID-19. We selected web and media-based COVID-19 data representations that we hypothesized citizens would interpret differently depending on their mathematical schemes. We included items that we conjectured would be easier or more difficult to interpret with schemes that prior research had reported were more or less productive, respectively. We used the representations during clinical interviews with 25 United States and seven South Korean citizens. We illustrate that citizens’ mathematical schemes (as well as their beliefs) impacted how they assessed the severity of COVID-19. We present vignettes of citizens’ schemes that inhibited interpreting representations of COVID-19 in ways compatible with the displayed quantitative data, schemes that aided them in assessing the severity of COVID-19, and beliefs about the reliability of scientific data that overrode their mathematical conclusions.     

Computable Linearizations of Well-partial-orderings

We analyze results on well-partial-orderings from the viewpoint of computability theory, and we answer a question posed by Diana Schmidt. We obtain the following results. De Jongh and Parikh showed that every well-partial-order has a linearization of maximal order type. We show that such a linearization can be found computably. We also show that the process of finding such a linearization is not computably uniform, not even hyperarithmetically.     

Constructions Over Tournaments

We investigate tournaments that are projective in the variety that they generate, and free algebras over partial tournaments in that variety. We prove that the variety determined by three-variable equations of tournaments is not locally finite. We also construct infinitely many finite, pairwise incomparable simple tournaments.     

A Theory of Complexity for Continuous Time Systems

We present a model of computation with ordinary differential equations (ODEs) which converge to attractors that are interpreted as the output of a computation. We introduce a measure of complexity for exponentially convergent ODEs, enabling an algorithmic analysis of continuous time flows and their comparison with discrete algorithms. We define polynomial and logarithmic continuous time complexity classes and show that an ODE which solves the maximum network flow problem has polynomial time complexity. We also analyze a simple flow that solves the Maximum problem in logarithmic time. We conjecture that a subclass of the continuous P is equivalent to the classical P.     

An analysis of a widely used version of the CUSUM tracking signal

Tracking signals are used to determine whether a sequence of forecasts is unbiased. We analyse a version of the CUSUM tracking signal that is based on the unsmoothed average mean absolute deviation of forecasting errors. We make the common assumption that demand values in different periods are independently, identically, and normally distributed. We show that, when forecast errors in different periods are also normal iid, this version of the tracking signal can indicate that the sequence is not unbiased even when it is unbiased. We suggest a simple modification to the tracking signal that serves to eliminate this flaw.     

Scale transformations in phase space and stretched states of a harmonic oscillator

We consider scale transformations (q, p) → (λq, λp) in phase space. They induce transformations of the Husimi functions H(q, p) defined in this space. We consider the Husimi functions for states that are arbitrary superpositions of n-particle states of a harmonic oscillator. We develop a method that allows finding so-called stretched states to which these superpositions transform under such a scale transformation. We study the properties of the stretched states and calculate their density matrices in explicit form. We establish that the density matrix structure can be described using negative binomial distributions. We find expressions for the energy and entropy of stretched states and calculate the means of the number-ofstates operator. We give the form of the Heisenberg and Robertson–Schrödinger uncertainty relations for stretched states.     

Duality in nonlinear programming

In this paper are defined new first- and second-order duals of the nonlinear programming problem with inequality constraints. We introduce a notion of a WD-invex problem. We prove weak, strong, converse, strict converse duality, and other theorems under the hypothesis that the problem is WD-invex. We obtain that a problem with inequality constraints is WD-invex if and only if weak duality holds between the primal and dual problems. We introduce a notion of a second-order WD-invex problem with inequality constraints. The class of WD-invex problems is strictly included in the class of second-order ones. We derive that the first-order duality results are satisfied in the second-order case.     

Ground States and Phase Transition of the λ Model on the Cayley Tree

We consider the λ model, a generalization of the Potts model, with spin values {1, 2, 3} on the order-two Cayley tree. We describe the model ground states and prove that translation-invariant Gibb measures exist, which means that a phase transition exists. We establish that two-periodic Gibbs measures exist.     

Towards higher topology

We categorify the adjunction between locales and topological spaces, this amounts to an adjunction between (generalized) bounded ionads and topoi. We show that the adjunction is idempotent. We relate this adjunction to the Scott adjunction, which was discussed from a more categorical point of view in [21]. We hint that 0-dimensional adjunction inhabits the categorified one.     

Semigroups strongly Morita equivalent to monoids

Periodica Mathematica Hungarica - We consider semigroups strongly Morita equivalent to a fixed monoid. We prove that such semigroups are precisely the enlargements of that monoid. We also show that...     

Annuitization and asset allocation with borrowing constraint

We generalize the result of Yaari (1965) on annuitization with borrowing constraint. We show that inability to borrow against future labor income has a significant influence on an individual’s consumption and asset allocation strategies. We also show that there exists a certain threshold of wealth for annuitization. We find that the wealth threshold is lower in the presence of borrowing constraint than in its absence, implying the individual’s earlier retirement.     

Insensitive,maximum stable allocations converge to proportional fairness

We describe a queuing model where service is allocated as a function of queue sizes. We consider allocations policies that are insensitive to service requirements and have a maximal stability region. We take a limit where the queuing model becomes congested. We study how service is allocated under this limit. We demonstrate that the only possible limit allocation is one that maximizes a proportionally fair optimization problem.     

Deformation Quantization of Symplectic Fibrations

A symplectic fibration is a fibre bundle in the symplectic category (a bundle of symplectic fibres over a symplectic base with a symplectic structure group). We find the relation between the deformation quantization of the base and the fibre, and that of the total space. We consider Fedosov's construction of deformation quantization. We generalize the Fedosov construction to the quantization with values in a bundle of algebras. We find that the characteristic class of deformation of a symplectic fibration is the weak coupling form of Guillemin, Lerman, and Sternberg. We also prove that the classical moment map could be quantized if there exists an equivariant connection.     

On the Hamiltonicity Gap and doubly stochastic matrices

We consider the Hamiltonian cycle problem embedded in singularly perturbed (controlled) Markov chains. We also consider a functional on the space of stationary policies of the process that consists of the (1,1)‐entry of the fundamental matrices of the Markov chains induced by these policies. We focus on the subset of these policies that induce doubly stochastic probability transition matrices which we refer to as the “doubly stochastic policies.” We show that when the perturbation parameter, ε, is sufficiently small, the minimum of this functional over the space of the doubly stochastic policies is attained at a Hamiltonian cycle, provided that the graph is Hamiltonian. We also show that when the graph is non‐Hamiltonian, the above minimum is strictly greater than that in a Hamiltonian case. We call the size of this difference the “Hamiltonicity Gap” and derive a conservative lower bound for this gap. Our results imply that the Hamiltonian cycle problem is equivalent to the problem of minimizing the variance of the first hitting time of the home node, over doubly stochastic policies. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

A distance-based,misclassification rate adjusted classifier for multiclass,high-dimensional data

In this paper, we consider a scale adjusted-type distance-based classifier for high-dimensional data. We first give such a classifier that can ensure high accuracy in misclassification rates for two-class classification. We show that the classifier is not only consistent but also asymptotically normal for high-dimensional data. We provide sample size determination so that misclassification rates are no more than a prespecified value. We propose a classification procedure called the misclassification rate adjusted classifier. We further develop the classifier to multiclass classification. We show that the classifier can still enjoy asymptotic properties and ensure high accuracy in misclassification rates for multiclass classification. Finally, we demonstrate the proposed classifier in actual data analyses by using a microarray data set.     

The G-biliaison class of symmetric determinantal schemes

We consider a family of schemes, that are defined by minors of a homogeneous symmetric matrix with polynomial entries. We assume that they have maximal possible codimension, given the size of the matrix and of the minors that define them. We show that these schemes are G-bilinked to a linear variety of the same dimension. In particular, they can be obtained from a linear variety by a finite sequence of ascending G-biliaisons on some determinantal schemes. We describe the biliaisons explicitly in the proof of Theorem 2.3. In particular, it follows that these schemes are glicci.