The forced convection flow of a uniform stream over a flat surface with a convective surface boundary condition

The forced convection heat transfer resulting from the flow of a uniform stream over a flat surface on which there is a convective boundary condition is considered. In previous papers [5], [6], [7], [8] it was assumed that the convective heat transfer parameter h_f associated with the hot surface depended on x, where x measures distance along the surface, so that problem could be reduced to similarity form. Here it is assumed that this heat transfer parameter h_f is a constant, with the result that the temperature profiles and overall heat transfer characteristics evolve as the solution develops from the leading edge. The heat transfer near the leading edge (small x), which we find to be dominated by the surface heat flux, the solution at large distances along the surface (large x), which dominated by the surface temperature, are discussed. A numerical solution to the full problem is then obtained for a range of values of the Prandtl number to join these two solution regimes.     

Fractional parts of the function <Emphasis Type="Italic">x/n</Emphasis>

Asymptotic formulas for sums of values of some class of smooth functions of fractional parts of numbers of the form x/n, where the parameter x increases unboundedly and the integer n ranges over various subsets of the interval [1, x], are obtained.     

Approximations and Mittag-Leffler conditions the tools

Mittag-Leffler modules occur naturally in algebra, algebraic geometry, and model theory, [20], [14], [19]. If R is a non-right perfect ring, then it is known that in contrast with the classes of all projective and flat modules, the class of all flat Mittag-Leffler modules is not deconstructible [16], and it does not provide for approximations when R has cardinality ≤ ?₀, [8]. We remove the cardinality restriction on R in the latter result. We also prove an extension of the Countable Telescope Conjecture [23]: a cotorsion pair (A, B) is of countable type whenever the class B is closed under direct limits.In order to prove these results, we develop new general tools combining relative Mittag-Leffler conditions with set-theoretic homological algebra. They make it possible to trace the above facts to their ultimate, countable, origins in the properties of Bass modules. These tools have already found a number of applications: e.g., they yield a positive answer to Enochs’ problem on module approximations for classes of modules associated with tilting [4], and enable investigation of new classes of flat modules occurring in algebraic geometry [26]. Finally, the ideas from Section 3 have led to the solution of a long-standing problem due to Auslander on the existence of right almost split maps [22].     

The stability of the equilibrium ellipsoids of a rotating liquid

The stability of the Riemann equilibrium ellipsoids, in the class of perturbations which satisfy the Dirichlet assumptions, is investigated using Rumyantsev's method [1]. It is shown that the equations of motion of the Dirichlet liquid ellipsoid are Hamiltonian on each combined level of the moment and circulation integrals, which corresponds to well-known results [2], although it is not a consequence of them. This fact provides, generally speaking, additional possibilities for solving the problem of determining the instability region. In the parameter space of each of the two families [3, 4] of Riemann ellipsoids, the region U for which almost all the equilibrium ellipsoids belonging to it are unstable is determined in explicit analytical form. It is shown that the stability region can be specified in explicit analytical form in both cases.     

The direct product of a hopfian group with a group with cyclic ccnter

In this paper we continue our study of hopficity begun in [1], [2], [3], [4] and [5]. LetA be hopfian and letB have a cyclic center of prime power order. We improve Theorem 4 of [2] by showing that ifB has finitely many normal subgroups which form a chain (we sayB isn-normal), thenAxB is hopfian. We then consider the case whenB is ap-group of nilpotency class 2 and show that in certain casesAxB is hopfian.     

Spinor genera of binary quadratic forms

Spinor genera are defined for binary quadratic forms with integer coefficients in such a way that the theory fits in with the Gaussian theory of genera. It is shown that spinor generic characters exist which distinguish the various spinor genera in the principal genus, and how they can be determined. It is known that each ambiguous class contains exactly two forms of the type [a, 0, c] or [a, a, c], each with its associate [c, 0, a], [4c ? a, 4c ? a, c]. Since the principal class contains such a form with a = 1, it is an interesting question whether one can predict the second form (not counting associates). This question includes that of Dirichlet about the representability of ?1 by the principal class. Methods are given for evaluating the spinor-generic characters of ambiguous forms in the principal genus for variable discriminants d, and are carried through in the eleven cases where d is fundamental, there are two or four genera, and two spinor genera in the principal genus. The problem of determining the “second form” is thus completely solved except when there is more than one ambiguous class in the principal spinor genus.     

Bayesian model selection approach to the one way analysis of variance under homoscedasticity

An objective Bayesian model selection procedure is proposed for the one way analysis of variance under homoscedasticity. Bayes factors for the usual default prior distributions are not well defined and thus Bayes factors for intrinsic priors are used instead. The intrinsic priors depend on a training sample which is typically a unique random vector. However, for the homoscedastic ANOVA it is not the case. Nevertheless, we are able to illustrate that the Bayes factors for the intrinsic priors are not sensitive to the minimal training sample chosen; furthermore, we propose an alternative pooled prior that yields similar Bayes factors. To compute these Bayes factors Bayesian computing methods are required when the sample sizes of the involved populations are large. Finally, a one to one relationship—which we call the calibration curve—between the posterior probability of the null hypothesis and the classical $p$ value is found, thus allowing comparisons between these two measures of evidence. The behavior of the calibration curve as a function of the sample size is studied and conclusions relating both procedures are stated.     

Bayesian estimation of bandwidth in semiparametric kernel estimation of unknown probability mass and regression functions of count data

This work takes advantage of semiparametric modelling which improves significantly in many situations the estimation accuracy of the purely nonparametric approach. Herein for semiparametric estimations of probability mass function (pmf) of count data, and an unknown count regression function (crf), the kernel used is a binomial one and the bandiwdth selection is investigated by developing Bayesian approaches. About the latter, Bayes local and global bandwidth approaches are used to establish data-driven selection procedures in semiparametric framework. From conjugate beta prior distributions of the smoothing parameter and under the squared errors loss function, Bayes estimate for pmf is obtained in closed form. This is not available for the crf which is computed by the Markov Chain Monte Carlo technique. Simulation studies demonstrate that both proposed methods perform better than the classical cross-validation procedures, in particular the smoothing quality and execution times are optimized. All applications are made on real data sets.     

Values and cores of fuzzy games with infinitely many players

The problem of the existence ofvalues (FA-valued, linear, positive, symmetric and efficient operators) on symmetric spaces of “fuzzy games” (that is, ideal set functions of bounded variation) arises naturally from [8], [18], [23] and [2], [3], [4] where it is implicitely approached for technical purposes. In our present work, this problem is approached in itself for the main reason that it is essentially related with the problem of the existence of significant countable additive measures lying in the cores of the “market games”. In fact, it is shown here that there exists a continuous value on the closed subspacebv′ICA ofIBV spanned by thebv′ functions of “fuzzy probability measures” ([9]), this values is “diagonal” onpICA, the closed subspace ofbv′ICA spanned by the natural powers of the fuzzy measures and this is used to prove the main result stating that the cooperative markets contained inpICA have unique fuzzy measures in their cores which are exactly the corresponding diagonal values. This result is of interest because it is providing a tool of determiningCA measures lying in the cores of large classes of games which are not necessarily “non-atomic” and, specially, because it is opening a way toward a new approach of the “Value Equivalence Principle” for differentiable markets with a continuum of traders which are not “perfectly competitive”.     

Hierarchical Bayes variable selection and microarray experiments

Hierarchical and empirical Bayes approaches to inference are attractive for data arising from microarray gene expression studies because of their ability to borrow strength across genes in making inferences. Here we focus on the simplest case where we have data from replicated two colour arrays which compare two samples and where we wish to decide which genes are differentially expressed and obtain estimates of operating characteristics such as false discovery rates. The purpose of this paper is to examine the frequentist performance of Bayesian variable selection approaches to this problem for different prior specifications and to examine the effect on inference of commonly used empirical Bayes approximations to hierarchical Bayes procedures. The paper makes three main contributions. First, we describe how the log odds of differential expression can usually be computed analytically in the case where a double tailed exponential prior is used for gene effects rather than a normal prior, which gives an alternative to the commonly used B-statistic for ranking genes in simple comparative experiments. The second contribution of the paper is to compare empirical Bayes procedures for detecting differential expression with hierarchical Bayes methods which account for uncertainty in prior hyperparameters to examine how much is lost in using the commonly employed empirical Bayes approximations. Third, we describe an efficient MCMC scheme for carrying out the computations required for the hierarchical Bayes procedures. Comparisons are made via simulation studies where the simulated data are obtained by fitting models to some real microarray data sets. The results have implications for analysis of microarray data using parametric hierarchical and empirical Bayes methods for more complex experimental designs: generally we find that the empirical Bayes methods work well, which supports their use in the analysis of more complex experiments when a full hierarchical Bayes analysis would impose heavy computational demands.     

Symmetry in Generalized Quadrangles

In this paper, we describe some aspects of a Lenz(-Barlotti)-type classification of finite generalized quadrangles, which is being prepared by the author. Some new points of view are given. We also prove that each span-symmetric generalized quadrangle of order s > 1 with s even is isomorphic to $ \mathcal{Q} $ (4, s), without using the canonical connection (obtained by S. E. Payne in [15] between groups of order s ³ ? s with a 4-gonal basis and span-symmetric generalized quadrangle of order s. (The latter result was obtained for general s independently by W. M. Kantor in [10], and the author in [30] Finally, we obtain a classification program for all finite translation generalized quadrangles, which is suggested by the main results of [27], [30], [32], [35], [38] and [37].     

Spinning gas clouds: Liouville integrable cases

We consider the class of ellipsoidal gas clouds expanding into a vacuum [1, 2] which has been shown to be a Liouville integrable Hamiltonian system [3]. This system presents several interesting features, such as the Painlevé property [4, 5], the existence of Bäcklund transformations and the separability of variables, all shown to be present in at least several sub-cases. A remarkable result that emerged from the study of the cases of rotation around a fixed principal axis, was that the Liouville torus, which is the locus of trajectories of the representative point of the cloud when all the constants of motion are fixed, could be assimilated with a quartic surface presenting 16 conic point singularities. The geometry of such surfaces is entirely determined by the datum of a 6th degree polynomial in one variable, and the consideration of the corresponding natural coordinate system then led to the separation of variables for these cases [6]. Further, the equation of the surface takes the form of a 4×4 determinant, which constitutes a generalization of Stieltjes 4 × 4 determinant formulation of the addition formula for elliptic functions; and the corresponding matrix also defines the system of the equations of motion; so that it can be said that the differential system is completely determined by the surface’s geometry. Forsaking now the assumption of a fixed rotation axis, in cases where the energy constant takes its minimum value compatible with the other constants of motion, we found that the Liouville torus was still reducible to the form of a quartic surface, presenting 15 conic points only instead of 16 (16 conic points were indeed present originally, but one of them had to disappear in the process of reducing the surface to the 4th degree). The geometry of these surfaces is entirely determined by the datum of a plane unicursal quartic (which is the transformed version of the missing conic point). The system can be reduced to the form of a differential equation of second degree, the coefficients of which are polynomials of degree 7, which are determined by the surface’s geometry, except for their quadratic dependence on a single free parameter, z. Defining u the (time-like) independent variable, and Φ the integration constant (which are functions defined on the Liouville torus), it is found that Φ depends linearly on the parameter: Φ = Φ(z) and then u may be taken to coincide with Φ(z′), for any value of z′ distinct from z. Solving the system for one particular value of z therefore also solves it for all other values of the parameter. It appears that the geometry alone does not specify in this case any particular value of z, but then any two values lead to differential systems which (although their solutions differ) turn out to be equivalent. It may also be worth pointing out that changing z may be viewed as exchanging the roles of u and Φ. Finally, in degenerate cases the Liouville torus presents a double line of self-intersection, and the separation of variables can be achieved. Sections by planes through the double line are conic sections, which may be labeled by a parameter w, say. Denoting α the eccentric anomaly on the conic, the differential system in fact takes a remarkably simple form: da/dw = f(w), and involves an elliptic integral.     

The Rohde–Schramm theorem via the Gaussian free field

The Rohde–Schramm theorem states that Schramm–Loewner Evolution with parameter κ (or SLE_κ for short) exists as a random curve, almost surely, if κ ≠ 8. Here we give a new and concise proof of the result, based on the Liouville quantum gravity coupling (or reverse coupling) with a Gaussian free field. This transforms the problem of estimating the derivative of the Loewner flow into estimating certain correlated Gaussian free fields. While the correlation between these fields is not easy to understand, a surprisingly simple argument allows us to recover a derivative exponent first obtained by Rohde and Schramm [14], subsequently shown to be optimal by Lawler and Viklund [17], which then implies the Rohde–Schramm theorem.     

On Bayes estimates

In this paper the author tries to give general conditions for the existence of Bayes estimates and for the consistency of sequences of Bayes estimates.In Section 3 we prove existence theorems for Bayes estimates, which contain those of DeGroot and Rao [3], as a special case. The proof is based on a theorem of Landers [5].Section 4 gives a characterization of Bayes estimates with convex loss and linear decision space. This theorem is also a generalization of a similar theorem of DeGroot and Rao [3].In Section 5 we generalize the theory of minimum contrast estimates (the foundations of which were laid by Huber [4], cf. Pfanzagl [6]) in such a way that we can apply it to the theory of Bayes estimates.Section 6 tries to give a general theory of consistency for Bayes estimates using the martingale argument of Doob [1] and the theory of minimum contrast estimates. Confer in this connection the results of Schwartz [8].Section 7 contains some auxiliary results.     

Bootstrap of deviation probabilities with applications

We show that under different moment bounds on the underlying variables, bootstrap approximation to the large deviation probabilities of standardized sample sum, based on independent random variables, is valid for a wider zone of n, the sample size, compared to the classical normal tail probability approximation. As an application, different notions of efficiency for statistical tests are considered from Bayesian point of view. In particular, efficiency due to Pitman (1938) [11], Chernoff (1952) [1], and Bayes risk efficiency due to Rubin and Sethuraman (1965) [12] turn out to be special cases with the choice of the weight function; i.e., prior density times loss.     

Estimation of a non-negative location parameter with unknown scale

For a vast array of general spherically symmetric location-scale models with a residual vector, we consider estimating the (univariate) location parameter when it is lower bounded. We provide conditions for estimators to dominate the benchmark minimax MRE estimator, and thus be minimax under scale invariant loss. These minimax estimators include the generalized Bayes estimator with respect to the truncation of the common non-informative prior onto the restricted parameter space for normal models under general convex symmetric loss, as well as non-normal models under scale invariant \(L^p\) loss with \(p>0\) . We cover many other situations when the loss is asymmetric, and where other generalized Bayes estimators, obtained with different powers of the scale parameter in the prior measure, are proven to be minimax. We rely on various novel representations, sharp sign change analyses, as well as capitalize on Kubokawa’s integral expression for risk difference technique. Several properties such as robustness of the generalized Bayes estimators under various loss functions are obtained.     

Locally averaged risk

Summary A heuristic method of reducing a class of admissible or Bayes decision rules is given. A new risk function is defined which is called the locally averaged risk. Bayes and admissible rules with respect to the new risk function are calledG-Bayes andG-admissible, respectively. It is shown under general assumptions that the class ofG-Bayes decision rules is a subset of the class of Bayes decision rules and the class ofG-admissible decision rules is a subset of the class of admissible decision rules. Some examples are considered, showing that the usual estimates of the parameter of a distribution with squared error as loss function, which are known to be admissible, are alsoG-admissible. This work was supported in part by NASA Grant-NGR 15-003-064 and NSF Grant-GP 7496 at Indiana University.     

On the lower bounds for mean square error of empirical bayes estimators

The usual empirical Bayes setting is considered with θ being a shift or a scale parameter. A class of empirical Bayes estimators of a function b(θ) is proposed. The properties of the estimates are studied and mean square errors are calculated. The lower bounds are constructed for mean square errors of the empirical Bayes estimators over the class of all empirical Bayes estimators of b(θ). The results are applied to the case b(θ)=θ. The examples of the upper and lower bounds for mean square error are presented for the most popular families of conditional distributions. Added to the English translaion.     

Mlinex损失函数下一类分布族参数的Bayes估计

考虑分布函数形如F(x;θ)=1-[g(x)]~θ或[1—g(x)]~θ,A≤x≤B,θ0的分布族,其中g(x)是关于x单调递减的可微函数,且g(A)=1,g(B)=0.在Mlinex损失函数下,给出了其中参数θ的Bayes估计及其容许性,并对分布的一个充分统计量的逆线性形式的容许性进行讨论.最后通过蒙特卡洛模拟说明Bayes估计在小样本情形时的优良表现.