A general existence theorem for symmetric floating drops

An existence theorem for floating drops due to Elcrat, Neel, and Siegel is generalized. The theorem applies to all radially symmetric domains, and to both light and heavy floating drops, and utilizes new results in annular capillary theory.     

A general theorem of existence of quasi absolutely minimal Lipschitz extensions

In this paper we consider a wide class of generalized Lipschitz extension problems and the corresponding problem of finding absolutely minimal Lipschitz extensions. We prove that if a minimal Lipschitz extension exists, then under certain other mild conditions, a quasi absolutely minimal Lipschitz extension must exist as well. Here we use the qualifier “quasi” to indicate that the extending function in question nearly satisfies the conditions of being an absolutely minimal Lipschitz extension, up to several factors that can be made arbitrarily small.     

The fundamental existence theorem on invariant fiber bundles

For discrete dynamical systems the theory of invariant manifolds is well known to be of vital importance. In terms of difference equations this theory is basically concerned with autonomous equations. However, the crucial and currently most difficult questions in this field are related to non-periodic, in particular chaotic motions. Since this topic - even in the autonomous context is an intrinsically time-variant matter. There is and urgent need for a non-autonomous version of invariant manifold theory. In this paper we present we present a very general version of the classical result on stable and unstable manifolds for hyperbolic fixed points of diffeomorphisms. In fact, we drop the assumption of invertibility of the mapping, we consider non-autonomous difference equations rather than mappings In effect, we generalize the notion of invariant manifold to the concept of invariant fiber bundle.     

The computational content of Walras’ existence theorem

In [Y. Tanaka, Undecidability of the Uzawa equivalence theorem and LLPO, Appl. Math. Comput. 201 (2008) 378-383] Yasuhito Tanaka showed that Walras’ existence theorem implies the nonconstructive lesser limited principle of omniscience (LLPO); it follows that Walras’ existence theorem does not admit a constructive proof. We give a constructive proof of an approximate version of Walras’ existence theorem from which the full theorem can be recovered with an application of LLPO. We then push Uzawa’s equivalence theorem to the level of approximate solutions, before considering economies with at most one equilibrium.     

Regularized fuzzy clusterwise ridge regression

Fuzzy clusterwise regression has been a useful method for investigating cluster-level heterogeneity of observations based on linear regression. This method integrates fuzzy clustering and ordinary least-squares regression, thereby enabling to estimate regression coefficients for each cluster and fuzzy cluster memberships of observations simultaneously. In practice, however, fuzzy clusterwise regression may suffer from multicollinearity as it builds on ordinary least-squares regression. To deal with this problem in fuzzy clusterwise regression, a new method, called regularized fuzzy clusterwise ridge regression, is proposed that combines ridge regression with regularized fuzzy clustering in a unified framework. In the proposed method, ridge regression is adopted to estimate clusterwise regression coefficients while handling potential multicollinearity among predictor variables. In addition, regularized fuzzy clustering based on maximizing entropy is utilized to systematically determine an optimal degree of fuzziness in memberships. A simulation study is conducted to evaluate parameter recovery of the proposed method as compared to the extant non-regularized counterpart. The usefulness of the proposed method is illustrated by an application concerning the relationship among the characteristics of used cars.     

Inequality constrained ridge regression estimator

We carry out the idea of inequality constrained least squares (ICLS) estimation of Liew (1976) to the inequality constrained ridge regression (ICRR) estimation. We propose ICRR estimator by reducing the primal–dual relation to the fundamental problem of Dantzig and Cottle, 1967, Cottle and Dantzig, 1974 with Lemke (1962) algorithm. Furthermore, we conduct a Monte Carlo experiment.     

A nonstandard Riemann existence theorem

We study elementary extensions of compact complex spaces and deduce that every complete type of dimension is internal to projective space. This amounts to a nonstandard version of the Riemann Existence Theorem, and answers a question posed by Anand Pillay.

     

An existence theorem for antichains

Let k₁, k₂,…, k_n be given integers, 1 ? k₁ ? k₂ ? … ? k_n, and let S be the set of vectors x = (x₁,…, x_n) with integral coefficients satisfying 0 ? x_i ? k_i, i = 1, 2, 3,…, n. A subset H of S is an antichain (or Sperner family or clutter) if and only if for each pair of distinct vectors x and y in H the inequalities x_i ? y_i, i = 1, 2,…, n, do not all hold. Let |H| denote the number of vectors in H, let K = k₁ + k₂ + … + k_n and for 0 ? l ? K let (l)H denote the subset of H consisting of vectors h = (h₁, h₂,…, h_n) which satisfy h₁ + h₂ + … + h_n = l. In this paper we show that if H is an antichain in S, then there exists an antichain H′ in S for which |(l)H′| = 0 if $\text{l <}\text{K}\text{2}$, $\text{|(}\text{K}\text{2}\text{)H′| = |(}\text{K}\text{2}\text{)H|}$ if K is even and |(l)H′| = |(l)H| + |(K ? l)H| if $\text{l>}\text{K}\text{2}$.     

Improved ridge regression estimators for the logistic regression model

The estimation of the regression parameters for the ill-conditioned logistic regression model is considered in this paper. We proposed five ridge regression (RR) estimators, namely, unrestricted RR, restricted ridge regression, preliminary test RR, shrinkage ridge regression and positive rule RR estimators for estimating the parameters $(\beta )$ when it is suspected that the parameter $\beta $ may belong to a linear subspace defined by $H\beta =h$ . Asymptotic properties of the estimators are studied with respect to quadratic risks. The performances of the proposed estimators are compared based on the quadratic bias and risk functions under both null and alternative hypotheses, which specify certain restrictions on the regression parameters. The conditions of superiority of the proposed estimators for departure and ridge parameters are given. Some graphical representations and efficiency analysis have been presented which support the findings of the paper.     

An eigenvector existence theorem in idempotent analysis

In this paper, we prove an eigenvector existence theorem for linear operators on abstract idempotent spaces in the framework of the algebraic approach. Earlier, an algebraic version of a similar statement was known only for operators in free finite-dimensional semimodules. The corresponding result for compact operators in semimodules of real continuous functions is known in the case of topological semimodules.     

A global existence theorem for the general coagulation–fragmentation equation with unbounded kernels

In this article an existence theorem is proved for the coagulation–fragmentation equation with unbounded kernel rates. Solutions are shown to be in the space X⁺ = {c∈L¹: ∫ (1 + x)∣c(x)∣dx < ∞} whenever the kernels satisfy certain growth properties and the non-negative initial data belong to X⁺. The proof is based on weak L¹ compactness methods applied to suitably chosen approximating equations.     

A stochastic restricted ridge regression estimator

Groß [J. Groß, Restricted ridge estimation, Statistics & Probability Letters 65 (2003) 57–64] proposed a restricted ridge regression estimator when exact restrictions are assumed to hold. When there are stochastic linear restrictions on the parameter vector, we introduce a new estimator by combining ideas underlying the mixed and the ridge regression estimators under the assumption that the errors are not independent and identically distributed. Apart from [J. Groß, Restricted ridge estimation, Statistics & Probability Letters 65 (2003) 57–64], we call this new estimator as the stochastic restricted ridge regression (SRRR) estimator. The performance of the SRRR estimator over the mixed estimator in respect of the variance and the mean square error matrices is examined. We also illustrate our findings with a numerical example. The shrinkage generalized least squares (GLS) and the stochastic restricted shrinkage GLS estimators are proposed.