Let k be an algebraically closed field of characteristic 0 and let D be a division algebra whose center F contains k. We shall say that D can be reduced to r parameters if we can write D ≃ D0 ⊗F 0 F , where D0 is a division algebra, the center F0 of D0 contains k and trdeg k (F0) = r. We show that every division algebra of odd degree n ≥ 5 can be reduced to ≤ 1 2 (n − 1)(n − 2) parameters. More...

It was observed long ago (see e.g., [32] or [20], page 62) that Roth’s theorem [28] and its p-adic extension established by Ridout [27] can be used to prove the transcendence of real numbers whose expansion in some integer base contains repetitive patterns. This was properly written only in 1997, by Ferenczi and Mauduit [21], who adopted a point of view from combinatorics on words before applyi...

Let k be an algebraically closed field of characteristic zero, L its finitely generated extension of transcendence degree ≥ 2, and L another finitely generated extension of k. It is a result of Bogomolov [B2] that any isomorphism between Gal(L/L) and Gal(L′/L) is induced by an isomorphism of fields L −→ L′ identifying L with L. If the transcendence degree of L over k is one, the group Gal(L/L) ...

Let K be a field and v a valuation on K. We discuss some properties of extensions w of v to the field K(X1, . . . , Xn) of rational functions in n variables over K, for which the residue field of w has transcendence degree n over the residue field of v. 2000 Mathematics Subject Classification: 12J20, 12F20. 1.Introduction LetK be a field and v a valuation onK. Residual transcendental extensions...

The authors address the diagnosis problem in nonlinear systems by using the concept of the differential transcendence degree of a differential field extension. The authors also consider the algebraic observability concept of the variable which models the failure presence for the solvability of the diagnosis problem. The construction of a reduced-order uncertainty observer (unknown-input observe...

Is it possible to distinguish algebraic from transcendental real numbers by considering the b-ary expansion in some base b 2? In 1950, É. Borel suggested that the answer is no and that for any real irrational algebraic number x and for any base g 2, the g-ary expansion of x should satisfy some of the laws that are shared by almost all numbers. For instance, the frequency where a given finite se...

Transcendental numbers form a fascinating subject: so little is known about the nature of analytic constants that more research is needed in this area. Even when one is interested only in numbers like π and eπ that are related to the classical exponential function, it turns out that elliptic functions are required (so far, this should not last forever!) to prove transcendence results and get a ...

This is a survey article on recent progress concerning transcendence problems over function fields in positive characteristic. We are interested in some special values that occur in the following two ways. One is the special values of certain special transcendental functions, e.g., Carlitz ζ-values at positive integers, which are specialization of Goss’ two-variable ζ-function, arithmetic (resp...

In this first investigation of genetic and environmental influences on children's values, 271 German twin pairs (50.2% boys) reported their values at ages 7-11 years using the Portrait Values Questionnaire (Schwartz & Rubel, 2005). We distinguished between gender-neutral (conservation vs. openness to change) and gender-typed (self-transcendence vs. self-enhancement) values. Boys differed from g...

Research has overwhelmingly shown that spending time in nature can be beneficial. Yet the field is dominated by studies which compare built and manicured natural environments. Therefore relatively little is known about experiences in wild or untamed natural environments and what personal factors may affect these experiences. This study compares visitor experiences, measured as affective apprais...