Length of stay as a stochastic process: A general approach and application to hospitalization for schizophrenia†

A general approach to the study of length of stay (LOS) for hospitalization is presented. Data on first hospitalization for schizophrenia from the Maryland Psychiatric Case Register are applied to discussions of the Life Table and seven stochastic models of the LOS process. As far as possible, prior applications of the various models to this process are reviewed, and the models are conceptualized on the individual and aggregate level. The models are the exponential, mixed exponential, type XI, Weibull, gamma, lognormal and Inverse Gaussian. The lognormal and Inverse Gaussian show the best fits to the data in terms of the maximum absolute deviation. However, the Inverse Gaussian is superior due to its attractive statistical characterization. Special attention is given to the relatively new Inverse Gaussian, and there is a brief section on LOS and theory verification. Recommendations are made for future LOS research.     

Queueing approximation of suprema of spectrally positive Lévy process

Let W=sup _0≤t<∞(X(t)−β t), where X is a spectrally positive Lévy process with expectation zero and 0<β<∞. One of the main results of the paper says that for such a process X, there exists a sequence of M/GI/1 queues for which stationary waiting times converge in distribution to W. The second result shows that condition (III) of Proposition 2 in the paper is not implied by all other conditions.     

The distribution of the spine of a Fleming–Viot type process

We show uniqueness of the spine of a Fleming–Viot particle system under minimal assumptions on the driving process. If the driving process is a continuous time Markov process on a finite space, we show that asymptotically, when the number of particles goes to infinity, the distribution of the spine converges to that of the driving process conditioned to stay alive forever, the branching rate for the spine is twice that of a generic particle in the system, and every side branch has the distribution of the unconditioned generic branching tree.     

Asymptotic behavior of the convex hull of a stationary Gaussian process?

Let X = {X(t), t ?? T} be a stationary centered Gaussian process with values in ? ^d , where the parameter set T equals ? or ?+. Let ?? _t = Cov(X ₀ ,X _t ) be the covariance function of X, and (??,?, P) be the underlying probability space. We consider the asymptotic behavior of convex hulls W _t = conv{X _u , u ?? T ?? [0, t]} as t ?? +?? and show that under the condition ??t ?? 0, t????, the rescaled convex hull (2 ln t) ^?1/2 W _t converges almost surely (in the sense of Hausdorff distance) to an ellipsoid ? associated to the covariance matrix ?? ₀. The asymptotic behavior of the mathematical expectations E f(W _t ), where f is a homogeneous function, is also studied. These results complement and generalize in some sense the results of Davydov [Y. Davydov, On convex hull of Gaussian samples, Lith. Math. J., 51(2): 171?C179, 2011].     

Educational aspirations as a markovian process: A panel study of college and non‐college plans during high school†

An initial test of the discrete‐time Markov model in the study of educational aspirations throughout high school was carried out. The design of the study permitted testing for sex differences and order effects The results indicate a good fit between the data and the model across several cohorts of students. Order effects were apparent, but sex differences in the transition probabilities were not found. Future change in aspirations appears least likely for students with a history of stable college plans, while it is most likely for those who start with non‐college aspirations and change to college plans.     

Dimensions of students’ views of themselves as learners of mathematics

Students’ views of themselves as learners of mathematics are a decisive parameter for their engagement and success in school. We are interested in students’ experiences with mathematics encompassing cognitive, emotional and motivational aspects. In particular, we focus on capturing the structural properties of affect related to mathematics. Participants in our study were 1,436 randomized chosen students of secondary schools from overall Finland. In the Finnish upper secondary school, there are two different syllabi for mathematics: the general and the advanced one. Schools were invited to organize the survey by one of their year 2 general syllabus courses and one of their year 2 advanced syllabus courses in grade 11. By means of factor analysis, we obtained seven dimensions in which students’ hold beliefs and emotions about mathematics partly intertwined with their motivational orientations. These dimensions are described by reliable scales, which allow outlining an average image of Finnish students’ views of themselves as learners of mathematics. Moreover, we analyzed relations between the seven dimensions and what kind of structure they generate. Thereby, a core of three high correlating dimensions could be identified, yielding different accentuations with regard to course choice.     

The density evolution of the killed McKean–Vlasov process

ABSTRACT

The density evolution of McKean–Vlasov stochastic differential equations in the presence of an absorbing boundary is analysed where the solution to such equations corresponds to the dynamics of partially killed large populations. By using a fixed point theorem, we show that the density evolution is characterized as the solution of an integro-differential Fokker–Planck equation with Cauchy–Dirichlet data. This problem arises naturally within mean field game theory.     

Determinants of Cauchy-Riemann operators as τ-functions

The -functions introduced by Sato, Miwa, and Jimbo for the deformation theory associated with the Riemann-Hilbert problem on P ¹ is shown to be a determinant for a singular Cauchy-Riemann operator whose domain incorporates functions with prescribed branching behavior. The analysis relies heavily on previous work of Malgrange on monodromy preserving deformation theory.     

How small are the increments of a Wiener process?

Let $\text{γ}{}_{\mathrm{т}}\text{=(8(}\text{logTa}{}^{-1}{}_{\mathrm{T}}\text{+log log T)}\text{π}{}^2\text{a}{}_{\mathrm{T}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, 0<a_T?T<∞, and {W(t);0?t<∞} be a standard Wiener process. This exposition studies the almost sure behaviour of

, under varying conditions on a_T and T/a_T. The following analogue of Lévy's modulus of continuity of a Wiener Process is also given:

$\text{lim}{}_{\mathrm{h\to 0}}\text{inf}{}_{\mathrm{0?t?1}}\text{sup}{}_{\mathrm{0?s?h}}\text{(}\text{8 log h}{}^{-1}\text{π}{}^2\text{h}\text{)}\text{1}\text{2}\text{|W(t+s)?W(t)| =}{}^{\mathrm{a.s.}}\text{1.}$

and this may be viewed as the exact “modulus of non-differentiability” of a Wiener Process.     

The mathematics textbook at tertiary level as curriculum material – exploring the teacher's decision-making process

This paper reports on a study about how the mathematics textbook was perceived and used by the teacher in the context of a calculus part of a basic mathematics course for first-year engineering students. The focus was on the teacher's choices and the use of definitions, examples and exercises in a sequence of lectures introducing the derivative concept. Data were collected during observations of lectures and an interview, and informal talks with the teacher. The introduction and the treatment of the derivative as proposed by the teacher during the lectures were analysed in relation to the results of the content text analysis of the textbook. The teacher's decisions were explored through the lens of intended learning goals for engineering students taking the mathematics course. The results showed that the sequence of concepts and the formal introduction of the derivative as proposed by the textbook were closely followed during the lectures. The examples and tasks offered to the students focused strongly on procedural knowledge. Although the textbook proposes both examples and exercises that promote conceptual knowledge, these opportunities were not fully utilized during the observed lectures. Possible reasons for the teacher's choices and decisions are discussed.     

Conditions of LU as M, H-matrix

ＣｏｎｄｉｔｉｏｎｓｏｆＬＵａｓＭ，Ｈ－ｍａｔｒｉｘ￥ＷａｎｇＨｕａ；ＬｉＱｉｎｇ（ＩｎｆｏｒｍａｔｉｏｎＥｎｇｉｎｅｅｒｉｎｇＩｎｓｔｉｔｕｔｅｏｆＰＬＡ）（ＣｏｌｌｅｇｅｏｆＥｎｇｉｎｅｅｒｉｎｇＹａｎｂｉａｎＵｎｉｖｅｒｓｉｔｙ）ＷａｎｇＣｈｕ...     

The Geometry of Grassmann Manifolds as Submanifolds

As is well known, grassmann manifolds play an important role in geometry,their aspect of differential geometry has been studied intensely (cf. [3],[4]etc.). In this paper we regard the oriented Grassmann manifolds as submanifoldsisometrically imbedded in unit spheres, and observe their behaviour as submani-folds. Let G_(mp)p be the set of all oriented linear subspaces of dimension m in     

Detection of cellular aging in a Galton–Watson process

We consider the bifurcating Markov chain model introduced by Guyon to detect cellular aging from cell lineage. To take into account the possibility for a cell to die, we use an underlying super-critical binary Galton–Watson process to describe the evolution of the cell lineage. We give in this more general framework a weak law of large number, an invariance principle and thus fluctuation results for the average over all individuals in a given generation, or up to a given generation. We also prove that the fluctuations over each generation are independent. Then we present the natural modifications of the tests given by Guyon in cellular aging detection within the particular case of the auto-regressive model.     

On convergence of a generalized Pál interpolation process

Let f∈C _[−1,1] ^″ (r≥1) and R_n(f,α,β,x) be the generalized Pál interpolation polynomials satisfying the conditions R_n(f,α,β,x_k)=f(x_k),R_n ^′(f,α,β,x_k)=f′(x_k)(k=1,2,…,n), where {x_k} are the roots of n-th Jacobi polynomial P_n(α,β,x),α,β>−1 and {x _k ^″ } are the roots of (1−x²)P_n″(α,β,x). In this paper, we prove that holds uniformly on [0,1]. In Memory of Professor M. T. Cheng Supported by the Science Foundation of CSBTB and the Natural Science Foundatioin of Zhejiang.     

The asymptotic variance rate of the output process of finite capacity birth-death queues

We analyze the output process of finite capacity birth-death Markovian queues. We develop a formula for the asymptotic variance rate of the form λ ^*+∑v _i where λ ^* is the rate of outputs and v _i are functions of the birth and death rates. We show that if the birth rates are non-increasing and the death rates are non-decreasing (as is common in many queueing systems) then the values of v _i are strictly negative and thus the limiting index of dispersion of counts of the output process is less than unity. In the M/M/1/K case, our formula evaluates to a closed form expression that shows the following phenomenon: When the system is balanced, i.e. the arrival and service rates are equal, is minimal. The situation is similar for the M/M/c/K queue, the Erlang loss system and some PH/PH/1/K queues: In all these systems there is a pronounced decrease in the asymptotic variance rate when the system parameters are balanced. Research supported in part by Israel Science Foundation Grant 249/02 and 454/05 and by European Network of Excellence Euro-NGI.     

A geometric interpretation of the transition density of a symmetric Lévy process

We study for a class of symmetric Lévy processes with state space R n the transition density pt（x） in terms of two one-parameter families of metrics, （dt）t>0 and （δt）t>0. The first family of metrics describes the diagonal term pt（0）; it is induced by the characteristic exponent ψ of the Lévy process by dt（x, y） = ^1/2tψ（x-y）. The second and new family of metrics δt relates to ^1/2tψ through the formulawhere F denotes the Fourier transform. Thus we obtain the following "Gaussian" representation of the transition density: pt（x） = pt（0）e- δ2t （x,0） where pt（0） corresponds to a volume term related to tψ and where an "exponential" decay is governed by δ2t . This gives a complete and new geometric, intrinsic interpretation of pt（x）.     

Some interesting processes arising as heavy traffic limits in an M/M/∞ storage process

Recent works by Newell [13] and Coffman. [2] have studied a queuing or storage model which is most easily visualized as the process of parkins cars in a parking lot where customers park as close as they can to some fixed point. This paper describes several space-time processes which arise as heavy-traffic limits in this model, and which seem interesting in their own right.     

How big are the increments of a multi-parameter Wiener process?

Summary Let _T = (2a _T(log Ta _T ^–1 +log log T))^–1/2, 0< a _T T< and let R^* be the set of sub-rectangles of the square [0, T ^1/2]x[0, T^1/2], having an area a _T . This paper studies the almost sure limiting behaviour of as T, where W is a two-time parameter Wiener process. With a _T =T, our results give the well-known law of iterated logarithm and a generalization of the latter is also attained. The multi-time parameter analogues of our twotime parameter Wiener process results are also stated in the text.Research partially supported by a Canadian NRC grant and a Canada Council Leave Fellowship     

Dragging maintaining symmetry: can it generate the concept of inclusivity as well as a family of shapes?

This article describes a project using Design Based Research methodology to ascertain whether a pedagogical task based on a dynamic figure designed in a Dynamic Geometry Software (DGS) program could be instrumental in developing students’ geometrical reasoning. A dragging strategy which I have named ‘Dragging Maintaining Symmetry’ (DMS) was shown to be important for the making of mathematical meanings in the context of Dynamic Geometry. In particular, it encouraged students’ development of the concept of inclusive relations between shapes generated from the dynamic figure, especially the rhombus as a special case of the kites. This development was not automatic and in addition to their work with the dynamic figure the students were shown an animation of the figure under DMS. Watching the animation allowed the students to attend to the continuous nature of the changing figure and proved to be the catalyst for moving their reasoning towards perceiving inclusive relations between the rhombus and kite.