Some independence results related to the arc-sine law

AbstractIn the absence of the axiom of choice, the set-theoretic status of many natural statements about metrizable compact spaces is investigated. Some of the statements are provable in $$\mathbf {ZF}$$ ZF , some are shown to be independent of $$\mathbf {ZF}$$ ZF . For independence results, distinct models of $$\mathbf {ZF}$$ ZF and permutation models of $$\mathbf {ZFA}$$ ZFA with transfer theorems of Pincus are applied. New symmetric models of $$\mathbf {ZF}$$ ZF are constructed in each of which the power set of $$\mathbb {R}$$ R is well-orderable, the Continuum Hypothesis is satisfied but a denumerable family of non-empty finite sets can fail to have a choice function, and a compact metrizable space need not be embeddable into the Tychonoff cube $$[0, 1]^{\mathbb {R}}$$ [ 0 , 1 ] R .

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Independence of algebraic monodromy groups in compatible systems

Selecta Mathematica ◽

10.1007/s00029-021-00657-y ◽

2021 ◽

Vol 27 (4) ◽

Author(s):

Federico Amadio Guidi

Keyword(s):

Galois Representations ◽

Global Function ◽

Irreducible Decomposition ◽

Global Function Fields ◽

Normal Varieties ◽

Independence Result ◽

Monodromy Groups ◽

Independence Results ◽

Lisse Sheaves ◽

General Method

AbstractIn this paper we develop a general method to prove independence of algebraic monodromy groups in compatible systems of representations, and we apply it to deduce independence results for compatible systems both in automorphic and in positive characteristic settings. In the abstract case, we prove an independence result for compatible systems of Lie-irreducible representations, from which we deduce an independence result for compatible systems admitting what we call a Lie-irreducible decomposition. In the case of geometric compatible systems of Galois representations arising from certain classes of automorphic forms, we prove the existence of a Lie-irreducible decomposition. From this we deduce an independence result. We conclude with the case of compatible systems of Galois representations over global function fields, for which we prove the existence of a Lie-irreducible decomposition, and we deduce an independence result. From this we also deduce an independence result for compatible systems of lisse sheaves on normal varieties over finite fields.

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Further independence results

Lecture Notes in Mathematics - Proper Forcing ◽

10.1007/bfb0096600 ◽

1982 ◽

pp. 388-393

Keyword(s):

Independence Results

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Algebraic Independence Results Related to 〈q, r〉-Number Systems

Monatshefte für Mathematik ◽

10.1007/s00605-005-0320-5 ◽

2006 ◽

Vol 147 (4) ◽

pp. 319-335 ◽

Cited By ~ 3

Author(s):

Shin-ichiro Okada ◽

Iekata Shiokawa

Keyword(s):

Algebraic Independence ◽

Number Systems ◽

Independence Results

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Alternate current measurement

Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character ◽

10.1098/rspa.1908.0029 ◽

1908 ◽

Vol 80 (539) ◽

pp. 310-352

Keyword(s):

Alternate Current ◽

Current Theory ◽

High Ratio ◽

Quadratic Functions ◽

Wave Form ◽

Numerical Comparison ◽

Sine Law ◽

The Magnetic Field ◽

Exciting Voltage ◽

Careful Design

The lack of precision of measurements with alternate currents, as compared with those using direct currents, is mainly due to the relative sensitiveness of the instruments available for such tests. The fact that the turning moment acting on the moving system depends in one case on the square of the current and in the other on the first power of the current, readily explains the high ratio between the currents needed to cause the minimum measurable deflection in the two cases, but this ratio is, nevertheless, most striking when a numerical comparison is actually made on some fair basis. The only likely way at present of improving alternate current instruments is to use iron cored electromagnets to increase the strength of the magnetic field. I have found that the difficulties due to varying permeability and hysteresis of the iron can be avoided by exciting the electromagnet in shunt. It proves possible, with careful design, to construct an electromagnet whose flux is connected with the exciting voltage by a strict mathematical law involving no variable physical properties like permeability, etc. Such an electromagnet is eminently suited for measuring purposes. The theoretical and experimental study of instruments constructed on this principle has brought out certain novel points which are set forth in the present paper. The first part discusses the mathematical relations of cyclic quantities having a common fundamental period, and constitutes a development of a method already published. This method is the only one known to me which is independent of assumptions in regard to the wave form of the quantities dealt with. The usual methods, which are based on the erroneous assumptions of sine law wave form, are not any simpler in working, and are most unsatisfactory when the accuracy of new results has to be critically examined. All alternate current measurements refer to mean squares or to mean products, and the natural method of obtaining the connections between such squares and products is to study the properties of quadratic functions of the variables. The earliest instance of this in alternate current theory was in connection with the “three voltmeter method.” Such processes lead to a very simple form of calculus appropriate to cyclic quantities.

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Field Independence and Resistance to Reversal of Perspective

Perceptual and Motor Skills ◽

10.2466/pms.1966.22.2.543 ◽

1966 ◽

Vol 22 (2) ◽

pp. 543-546 ◽

Cited By ~ 20

Author(s):

Frank Haronian ◽

A. Arthur Sugerman

Keyword(s):

Short Form ◽

Necker Cube ◽

Normal Male ◽

Field Independence ◽

Embedded Figures Test ◽

Following Instructions ◽

Independence Results ◽

Male University Students ◽

Rod And Frame Test ◽

Rod And Frame

The more successful 102 normal male university students were in following instructions to resist fluctuations of the Necker cube, the more field-independently they scored on both Series III of the rod-and-frame test ( r = .28) and on Jackson's short form of the embedded-figures test ( r = .24). Under neutral instructions, the correlations were nil. Results support prior findings that a small but significant portion of the variance of Necker cube fluctuations under instructions to control the rate of shift is related to scores of field independence. Results support Jackson's finding that ability to control the rate of shift is not related to intelligence.

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