This paper generalizes some results on pivotal functions of maximum likelihood estimators of location and scale parameters and the related ancillary statistics obtained by Antle and Bain, and Fisher. It shows that the maximum likelihood estimator of the parameter in an invariant statistical model is an essentially equivariant estimator or a transformation variable in a structural model. In the latter case, ancillary statistics in the sense of Fisher used in conjunction with the maximum likelihood estimators can be easily recognized. It is also remarked that the values of maximum likelihood estimators from samples having the same “complexion” are simply related to those of other, perhaps simpler, transformation variables. In the development it also points out the importance of using the correct definition of the likelihood function originally proposed by Fisher.

Consider the extended hull of a weak model set together with its natural shift action. Equip the extended hull with the Mirsky measure, which is a certain natural pattern frequency measure. It is known that the extended hull is a measure-theoretic factor of some group rotation, which is called the underlying torus. Among other results, in the article Periods and factors of weak model sets, we showed that the extended hull is isomorphic to a factor group of the torus, where certain periods of the window of the weak model set have been factored out. This was proved for weak model sets having a compact window. In this note, we argue that the same results hold for arbitrary measurable and relatively compact windows. Our arguments crucially rely on Moody’s work on uniform distribution in model sets. We also discuss implications for the diffraction of such weak model sets and discuss a new class of examples which are generic for the Mirsky measure.

Let F be a relatively closed subset of a domain G in the complex plane. Let f be a function that is the limit, in the Lip α norm on F, of functions which are holomorphic or meromorphic on G (0 < α < 1). We prove that, under the same conditions that give Lip α-approximation (0 < α < 1 ) on relatively closed subsets of G, it is possible to choose the approximating function m in such a way that f — m can be extended to a function belonging to lip

The following conjecture has been investigated recently by Mordell [1]. “Let z1 , …, zn be any set of n complex numbers. Then

1

the equality sign being necessary in the case when the z's are at the vertices of a regular n-sided polygon with center at the origin.”

The extended completeness theorem of the predicate calculas of the first order. In section 12, we developed a deductive theory of the first order predicate calculus, while in section II we dealt with the semantic theory of that calculus. We now have to consider the connection between these two theories. We recall that a sentence X can be satisfied by a structure M only if X is defined in M. Given a sentence X (a set of sentences K) we shall say that the structure M is a model of X (of K) if X is (all the sentences of K are) satisfied by M.

In this note, we patch up the proof of a Theorem due to Handel on the characterization of homotopy epimorphisms ([6], 2.2) and generalize a Theorem due to Ibisch on attaching disk-bundles to Dold fibrations ([7], Satz 1).

We work throughout in the category TopB of spaces over B for some fixed topological space B.

In this note we present a simple alternative proof for the Bernstein problem in the three dimensional Heisenberg group $\text{Ni}{{\text{l}}_{3}}$ by using the loop group technique. We clarify the geometric meaning of the two-parameter ambiguity of entire minimal graphs with prescribed Abresch-Rosenberg differential.

The purpose of this paper is to consider a representation for the elements of a linear topological space in the form of a σ-integral over a linearly ordered subset of V; this ordered subset is what will be called an L basis. The formal definition of an L basis is essentially an abstraction from ideas used, often tacitly, in proofs of many of the theorems concerning integral representations for continuous linear functionals on function spaces.

The L basis constructed in this paper differs in several basic ways from the integral basis considered by Edwards in [5]. Since the integrals used here are of Hellinger type rather than Radon type one has in the approximating sums for the integral an immediate and natural analogue to the partial sum operators of summation basis theory.

A ring R (associative ring) is said to be fully ordered provided that R is a linearly ordered set under a relation such that for any a, b and c in R, implies that and if c ε 0 then and . We say a subset K of R is convex provided that if a, b ε K such that then the interval [a, b] is a subset of K. Obviously an additive subgroup K of R is convex if and only if b ε K and b > 0 implies [a, b] ⊆ K.

Let $q\,=\,{{p}^{r}}$ with $p$ an odd prime, and ${{\mathbf{F}}_{q}}$ denote the finite field of $q$ elements. Let $\text{Tr}\,:\,{{\mathbf{F}}_{q}}\,\to \,{{\mathbf{F}}_{p}}$ be the usual trace map and set ${{\zeta }_{p}}\,=\,\exp (2\pi i/p)$. For any positive integer $e$, define the (modified) Gauss sum ${{g}_{r}}(e)$ by

$${{g}_{r}}(e)=\underset{x\in {{\mathbf{F}}_{q}}}{\mathop{\sum }}\,\zeta _{p}^{\text{Tr}{{x}^{e}}}$$

Recently, Evans gave an elegant determination of ${{g}_{1}}(12)$ in terms of ${{g}_{1}}(3),\,{{g}_{1}}(4)$ and ${{g}_{1}}(6)$ which resolved a sign ambiguity present in a previous evaluation. Here I generalize Evans' result to give a complete determination of the sum ${{g}_{r}}(12)$.

The article introduces and studies Hausdorff–Berezin operators on the unit ball in a complex space. These operators are a natural generalization of the Berezin transform. In addition, the class of such operators contains, for example, the invariant Green potential, and some other operators of complex analysis. Sufficient and necessary conditions for boundedness in the space of p – integrable functions with Haar measure (invariant with respect to involutive automorphisms of the unit ball) are given. We also provide results on compactness of Hausdorff–Berezin operators in Lebesgue spaces on the unit ball. Such operators have previously been introduced and studied in the context of the unit disc in the complex plane. Present work is a natural continuation of these studies.

In this paper, we prove a noncommutative (nc) analog of Schwarz lemma for the nc Schur–Agler class and prove that the regular nc Schur–Agler class and the regular free Herglotz–Agler class are homeomorphic. Moreover, we give a characterization of regular free Herglotz–Agler functions. As an application, we will show that any regular free Herglotz–Agler functions can uniformly be approximated by regular Herglotz–Agler free polynomials.

Statistics in the 20th century has been enlivened by a passionate, occasionally bitter, and still vibrant debate on the foundations of statistics and in particular on Bayesian vs. frequentist approaches to inference. In 1975 D. V. Lindley predicted a Bayesian 21st century for statistics. This prediction has often been discussed since, but there is still no consensus on the probability of its correctness. Recent developments in the asymptotic theory of statistics are, surprisingly, shedding new light on this debate, and may have the potential to provide a common middle ground.

We study the existence of solutions of the semilinear equations (1) in which the non-linearity g may grow superlinearly in u in one of directions u → ∞ and u → −∞, and (2) −Δu + g(x, u) = h, in which the nonlinear term g may grow superlinearly in u as |u| → ∞. The purpose of this paper is to obtain solvability theorems for (1) and (2) when the Landesman-Lazer condition does not hold. More precisely, we require that h may satisfy are arbitrarily nonnegative constants, . The proofs are based upon degree theoretic arguments.

We characterize two important notions of amenability and compactness of a locally compact quantum group $\mathbb{G}$ in terms of certain homological properties. For this, we show that $\mathbb{G}$ is character amenable if and only if it is both amenable and co-amenable. We finally apply our results to Arens regularity problems of the quantum group algebra ${{L}^{1}}\left( \mathbb{G} \right)$. In particular, we improve an interesting result by Hu, Neufang, and Ruan.