scholarly journals A PSEUDOEXPONENTIAL-LIKE STRUCTURE ON THE ALGEBRAIC NUMBERS

2015 ◽  
Vol 80 (4) ◽  
pp. 1339-1347
Author(s):  
VINCENZO MANTOVA
Keyword(s):  
Algebraic Varieties ◽  
Algebraic Numbers ◽  
Exponential Functions ◽  
Existentially Closed ◽  
Existential Closure ◽  
Independent Functions ◽  
Schanuel's Conjecture

AbstractPseudoexponential fields are exponential fields similar to complex exponentiation which satisfy the Schanuel Property, i.e., the abstract statement of Schanuel’s Conjecture, and an adapted form of existential closure.Here we show that if we remove the Schanuel Property and just care about existential closure, it is possible to create several existentially closed exponential functions on the algebraic numbers that still have similarities with complex exponentiation. The main difficulties are related to the arithmetic of algebraic numbers, and they can be overcome with known results about specialisations of multiplicatively independent functions on algebraic varieties.

The emergence of middle voice structures with and without agents

2014 ◽  
Vol 31 (2) ◽  
Author(s):  
Antonio Fábregas ◽  
Michael Putnam
Keyword(s):  
Semantic Interpretation ◽  
Modal Operator ◽  
Language Family ◽  
Middle Voice ◽  
Existentially Closed ◽  
Existential Closure ◽  
Grammatical Structures ◽  
Event Variable

AbstractThis article presents evidence that, cross-linguistically or within the same language (family), there appears to be no morphosyntactic properties and/or structures specifically designated for the formation of middle voice constructions. What has been labeled a ‘middle voice construction’ is a semantic interpretation that, crucially, is blocked when an event variable is existentially closed by T. This article focuses on two ways of expressing a middle statement; namely (i) middle voice readings that occur with lexical-s passives, and (ii) adjectival middles – in Mainland Scandinavian, showing that properties such as the availability of an agent in middles pattern with whether an event variable is present (in the structure) or not. These are the result of two equally valid and productive grammatical structures: one where an event variable is present, an agent is projected and a modal operator blocks existential closure of the event variable, and another one where the event variable is not present in the structure, and therefore the operator is not necessary – hence impossible.

scholarly journals Schanuel’s Conjecture and the Transcendence of Power Towers

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 717
Author(s):  
Eva Trojovská ◽  
Pavel Trojovský
Keyword(s):  
Algebraic Numbers ◽  
The Third ◽  
Schanuel's Conjecture

We give three consequences of Schanuel’s Conjecture. The first is that P(e)Q(e) and P(π)Q(π) are transcendental, for any non-constant polynomials P(x),Q(x)∈Q¯[x]. The second is that π≠αβ, for any algebraic numbers α and β. The third is the case of the Gelfond’s conjecture (about the transcendence of a finite algebraic power tower) in which all elements are equal.

Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)

2017 ◽  
Author(s):  
Claire Voisin
Keyword(s):  
Recent Work ◽  
Chow Groups ◽  
Ring Structure ◽  
Complete Intersections ◽  
Algebraic Varieties ◽  
Chow Ring ◽  
Hodge Structures ◽  
Chow Rings ◽  
Geometric Methods ◽  
The Right

This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The book is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by the author. It focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by the author looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

Longitudinal oscillation of a rod with a variable cross section

2019 ◽  
Vol 14 (2) ◽  
pp. 138-141
Author(s):  
I.M. Utyashev
Keyword(s):  
Cross Section ◽  
Natural Frequencies ◽  
Liouville Problem ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Longitudinal Vibrations ◽  
Cross Sectional ◽  
Independent Functions ◽  
The Cross ◽  
The Right

Variable cross-section rods are used in many parts and mechanisms. For example, conical rods are widely used in percussion mechanisms. The strength of such parts directly depends on the natural frequencies of longitudinal vibrations. The paper presents a method that allows numerically finding the natural frequencies of longitudinal vibrations of an elastic rod with a variable cross section. This method is based on representing the cross-sectional area as an exponential function of a polynomial of degree n. Based on this idea, it was possible to formulate the Sturm-Liouville problem with boundary conditions of the third kind. The linearly independent functions of the general solution have the form of a power series in the variables x and λ, as a result of which the order of the characteristic equation depends on the choice of the number of terms in the series. The presented approach differs from the works of other authors both in the formulation and in the solution method. In the work, a rod with a rigidly fixed left end is considered, fixing on the right end can be either free, or elastic or rigid. The first three natural frequencies for various cross-sectional profiles are given. From the analysis of the numerical results it follows that in a rigidly fixed rod with thinning in the middle part, the first natural frequency is noticeably higher than that of a conical rod. It is shown that with an increase in the rigidity of fixation at the right end, the natural frequencies increase for all cross section profiles. The results of the study can be used to solve inverse problems of restoring the cross-sectional profile from a finite set of natural frequencies.

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