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.

LOWER BOUNDS FOR THE NORMALIZED HEIGHT AND NON-DENSE SUBSETS OF SUBVARIETIES OF ABELIAN VARIETIES

2010 ◽  
Vol 06 (03) ◽  
pp. 471-499 ◽  
Author(s):  
EVELINA VIADA
Keyword(s):  
Lower Bound ◽  
Elliptic Curve ◽  
Abelian Variety ◽  
Lower Bounds ◽  
Finite Rank ◽  
Height Function ◽  
Algebraic Numbers ◽  
Algebraic Subgroup ◽  
Rank Zero ◽  
The Third

This work is the third part of a series of papers. In the first two, we considered curves and varieties in a power of an elliptic curve. Here, we deal with subvarieties of an abelian variety in general. Let V be a proper irreducible subvariety of dimension d in an abelian variety A, both defined over the algebraic numbers. We say that V is weak-transverse if V is not contained in any proper algebraic subgroup of A, and transverse if it is not contained in any translate of such a subgroup. Assume a conjectural lower bound for the normalized height of V. Then, for V transverse, we prove that the algebraic points of bounded height of V which lie in the union of all algebraic subgroups of A of codimension at least d + 1 translated by the points close to a subgroup Γ of finite rank, are non-Zariski-dense in V. If Γ has rank zero, it is sufficient to assume that V is weak-transverse. The notion of closeness is defined using a height function.

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.

Progress report on 21-cm observations at low latitude between longitudes (l 11) 0° and 66°

1967 ◽  
Vol 31 ◽  
pp. 177-179
Author(s):  
W. W. Shane
Keyword(s):  
Preliminary Report ◽  
Progress Report ◽  
Galactic Centre ◽  
Low Latitude ◽  
The Third ◽  
Introductory Lecture ◽  
Centre Region

In the course of several 21-cm observing programmes being carried out by the Leiden Observatory with the 25-meter telescope at Dwingeloo, a fairly complete, though inhomogeneous, survey of the regionl11= 0° to 66° at low galactic latitudes is becoming available. The essential data on this survey are presented in Table 1. Oort (1967) has given a preliminary report on the first and third investigations. The third is discussed briefly by Kerr in his introductory lecture on the galactic centre region (Paper 42). Burton (1966) has published provisional results of the fifth investigation, and I have discussed the sixth in Paper 19. All of the observations listed in the table have been completed, but we plan to extend investigation 3 to a much finer grid of positions.

The orbit of Pluto over a long interval of time

1966 ◽  
Vol 25 ◽  
pp. 227-229 ◽  
Author(s):  
D. Brouwer
Keyword(s):  
Periodic Orbit ◽  
Numerical Integration ◽  
Restricted Problem ◽  
Three Dimensional ◽  
The Third ◽  
Circular Restricted Problem ◽  
Resonance Relation

The paper presents a summary of the results obtained by C. J. Cohen and E. C. Hubbard, who established by numerical integration that a resonance relation exists between the orbits of Neptune and Pluto. The problem may be explored further by approximating the motion of Pluto by that of a particle with negligible mass in the three-dimensional (circular) restricted problem. The mass of Pluto and the eccentricity of Neptune's orbit are ignored in this approximation. Significant features of the problem appear to be the presence of two critical arguments and the possibility that the orbit may be related to a periodic orbit of the third kind.

Monte Carlo Algorithms for Moments of Transition Arrays

1988 ◽  
Vol 102 ◽  
pp. 79-81
Author(s):  
A. Goldberg ◽  
S.D. Bloom
Keyword(s):  
Monte Carlo ◽  
State Vector ◽  
Basis Vector ◽  
Collective State ◽  
Dispersion Characteristics ◽  
Dipole Strength ◽  
The Third ◽  
Monte Carlo Algorithms ◽  
Third Moment ◽  
Very High

AbstractClosed expressions for the first, second, and (in some cases) the third moment of atomic transition arrays now exist. Recently a method has been developed for getting to very high moments (up to the 12th and beyond) in cases where a “collective” state-vector (i.e. a state-vector containing the entire electric dipole strength) can be created from each eigenstate in the parent configuration. Both of these approaches give exact results. Herein we describe astatistical(or Monte Carlo) approach which requires onlyonerepresentative state-vector |RV> for the entire parent manifold to get estimates of transition moments of high order. The representation is achieved through the random amplitudes associated with each basis vector making up |RV>. This also gives rise to the dispersion characterizing the method, which has been applied to a system (in the M shell) with≈250,000 lines where we have calculated up to the 5th moment. It turns out that the dispersion in the moments decreases with the size of the manifold, making its application to very big systems statistically advantageous. A discussion of the method and these dispersion characteristics will be presented.

Probe size and 5th-order aberration

1987 ◽  
Vol 45 ◽  
pp. 134-135 ◽  
Author(s):  
Zhifeng Shao
Keyword(s):  
Electron Probe ◽  
Spherical Aberration ◽  
Wave Length ◽  
Cylindrical Symmetry ◽  
Electron Wave ◽  
Third Order ◽  
The Third ◽  
Probe Radius ◽  
Small Probe ◽  
Probe Size

A small electron probe has many applications in many fields and in the case of the STEM, the probe size essentially determines the ultimate resolution. However, there are many difficulties in obtaining a very small probe.Spherical aberration is one of them and all existing probe forming systems have non-zero spherical aberration. The ultimate probe radius is given byδ = 0.43Csl/4ƛ3/4where ƛ is the electron wave length and it is apparent that δ decreases only slowly with decreasing Cs. Scherzer pointed out that the third order aberration coefficient always has the same sign regardless of the field distribution, provided only that the fields have cylindrical symmetry, are independent of time and no space charge is present. To overcome this problem, he proposed a corrector consisting of octupoles and quadrupoles.

The effects of Cyclosporin-A (CYA) on the Ultrastructure of Gingival Mast Cells (GMC) in Behcet's disease (BD)

1990 ◽  
Vol 48 (3) ◽  
pp. 358-359
Author(s):  
Oktay Arda ◽  
Ulkü Noyan ◽  
Selgçk Yilmaz ◽  
Mustafa Taşyürekli ◽  
İsmail Seçkin ◽  
...  
Keyword(s):  
Unknown Etiology ◽  
Control Group ◽  
Target Cells ◽  
Gingival Hyperplasia ◽  
Cellular Activity ◽  
Direct Relation ◽  
Immune Disease ◽  
Genital Ulceration ◽  
Functional Changes ◽  
The Third

Turkish dermatologist, H. Beheet described the disease as recurrent triad of iritis, oral aphthous lesions and genital ulceration. Auto immune disease is the recent focus on the unknown etiology which is still being discussed. Among the other immunosupressive drugs, CyA included in it's treatment newly. One of the important side effects of this drug is gingival hyperplasia which has a direct relation with the presence of teeth and periodontal tissue. We are interested in the ultrastructure of immunocompetent target cells that were affected by CyA in BD.Three groups arranged in each having 5 patients with BD. Control group was the first and didn’t have CyA treatment. Patients who had CyA, but didn’t show gingival hyperplasia assembled the second group. The ones displaying gingival hyperplasia following CyA therapy formed the third group. GMC of control group and their granules are shown in FIG. 1,2,3. GMC of the second group presented initiation of supplementary cellular activity and possible maturing functional changes with the signs of increased number of mitochondria and accumulation of numerous dense cored granules next to few normal ones, FIG. 4,5,6.

Effects of Phonological Decoding Training on Children's Word Recognition of CVC, CV, and VC Structures

1996 ◽  
Vol 5 (1) ◽  
pp. 67-78 ◽  
Author(s):  
Kenyatta O. Rivers ◽  
Linda J. Lombardino ◽  
Cynthia K. Thompson
Keyword(s):  
Word Recognition ◽  
Multiple Baseline ◽  
Single Subject ◽  
Group Training ◽  
Complex Structures ◽  
Multiple Baseline Design ◽  
Baseline Design ◽  
Phonological Decoding ◽  
The Third ◽  
Letter Sound

The effects of training in letter-sound correspondences and phonemic decoding (segmenting and blending skills) on three kindergartners' word recognition abilities were examined using a single-subject multiple-baseline design across behaviors and subjects. Whereas CVC pseudowords were trained, generalization to untrained CVC pseudowords, untrained CVC real words, untrained CV and VC pseudowords, and untrained CV and VC real words were assessed. Generalization occurred to all of the untrained constructions for two of the three subjects. The third subject did not show the same degree of generalization to VC pseudowords and real words; however, after three training sessions, this subject read all VC constructions with 100% accuracy. Findings are consistent with group training studies that have shown the benefits of decoding training on word recognition and spelling skills and with studies that have demonstrated the effects of generalization to less complex structures when more complex structures are trained.

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