On two relations characterizing the golden ratio

2021 ◽  
Vol 21 (2) ◽  
pp. 194-202
Author(s):  
A.A. Zhukova ◽  
◽  
A.V. Shutov ◽  
Keyword(s):  
Elementary Proof ◽  
Golden Ratio ◽  
Finite Sets

V.G. Zhuravlev found two relations associated with the golden ratio: $\tau=\frac{1+\sqrt{5}}{2}$: $[([i\tau]+1)\tau]=[i\tau^2]+1$ and $[[i\tau]\tau]+1=[i\tau^2]$. We give a new elementary proof of these relations and show that they give a characterization of the golden ratio. Further we consider satisfability of our relations for finite sets of $i$-s and establish some forcing property for this situation.

A proof-theoretic characterization of the primitive recursive set functions

1992 ◽  
Vol 57 (3) ◽  
pp. 954-969 ◽  
Author(s):  
Michael Rathjen
Keyword(s):  
Set Theory ◽  
Elementary Proof ◽  
Weak Version ◽  
Definable Set ◽  
Set Functions ◽  
Theoretic Characterization ◽  
Universe Of Sets ◽  
Set Function ◽  
Primitive Recursive

AbstractLet KP− be the theory resulting from Kripke-Platek set theory by restricting Foundation to Set Foundation. Let G: V → V (V ≔ universe of sets) be a Δ0-definable set function, i.e. there is a Δ0-formula φ(x, y) such that φ(x, G(x)) is true for all sets x, and V ⊨ ∀x∃!yφ(x, y). In this paper we shall verify (by elementary proof-theoretic methods) that the collection of set functions primitive recursive in G coincides with the collection of those functions which are Σ1-definable in KP− + Σ1-Foundation + ∀x∃!yφ(x, y). Moreover, we show that this is still true if one adds Π1-Foundation or a weak version of Δ0-Dependent Choices to the latter theory.

scholarly journals From Fibonacci Sequence to the Golden Ratio

2013 ◽  
Vol 2013 ◽  
pp. 1-3 ◽  
Author(s):  
Alberto Fiorenza ◽  
Giovanni Vincenzi
Keyword(s):  
Characteristic Polynomial ◽  
Golden Ratio ◽  
Fibonacci Sequence ◽  
Maximum Modulus ◽  
Point Of View ◽  
General Point ◽  
Constant Coefficients ◽  
The Difference ◽  
New Perspective

We consider the well-known characterization of the Golden ratio as limit of the ratio of consecutive terms of the Fibonacci sequence, and we give an explanation of this property in the framework of the Difference Equations Theory. We show that the Golden ratio coincides with this limit not because it is the root with maximum modulus and multiplicity of the characteristic polynomial, but, from a more general point of view, because it is the root with maximum modulus and multiplicity of a restricted set of roots, which in this special case coincides with the two roots of the characteristic polynomial. This new perspective is the heart of the characterization of the limit of ratio of consecutive terms of all linear homogeneous recurrences with constant coefficients, without any assumption on the roots of the characteristic polynomial, which may be, in particular, also complex and not real.

scholarly journals A characterization of Banach-star-algebras by numerical range

1971 ◽  
Vol 4 (2) ◽  
pp. 193-200 ◽  
Author(s):  
Brailey Sims
Keyword(s):  
Elementary Proof ◽  
Numerical Range ◽  
Banach Algebras

It is known that in a B*-algebra every self-adjoint element is hermitian. We give an elementary proof that this condition characterizes B*-algetras among Banach*-algebras.

scholarly journals AN ELEMENTARY PROOF OF JAMES’ CHARACTERIZATION OF WEAK COMPACTNESS

2011 ◽  
Vol 84 (1) ◽  
pp. 98-102 ◽  
Author(s):  
WARREN B. MOORS
Keyword(s):  
Integral Representation ◽  
Banach Spaces ◽  
Elementary Proof ◽  
Weak Compactness ◽  
Representation Theorems

AbstractIn this paper we provide an elementary proof of James’ characterization of weak compactness in separable Banach spaces. The proof of the theorem does not rely upon either Simons’ inequality or any integral representation theorems. In fact the proof only requires the Krein–Milman theorem, Milman’s theorem and the Bishop–Phelps theorem.

An elementary proof for the Rao-Rubin characterization of the Poisson distribution

1974 ◽  
Vol 11 (01) ◽  
pp. 211-215 ◽  
Author(s):  
D. N. Shanbhag
Keyword(s):  
Poisson Distribution ◽  
Elementary Proof

This note gives elementary proofs for the characterizations of the Poisson distribution given by Rao and Rubin (1964) and Talwalker (1970) and disproves a conjecture of R. C. and A. B. L. Srivastava (1970).

scholarly journals An elementary proof for a characterization of *-isomorphisms

2005 ◽  
Vol 134 (1) ◽  
pp. 229-234 ◽  
Author(s):  
S. H. Kulkarni ◽  
M. T. Nair ◽  
M. N. N. Namboodiri
Keyword(s):  
Elementary Proof

An extension of watanabe's theorem of characterization of poisson processes over the positive real half line

1975 ◽  
Vol 12 (2) ◽  
pp. 396-399 ◽  
Author(s):  
P. Bremaud
Keyword(s):  
Poisson Process ◽  
Lebesgue Measure ◽  
Elementary Proof ◽  
Characterization Theorem ◽  
Poisson Processes ◽  
Half Line ◽  
Positive Real ◽  
The Mean ◽  
Martingale Characterization

We give an elementary proof of the martingale characterization theorem for Poisson processes over the positive real half line. This theorem is due to Watanabe [8] in the case where the mean measure associated to the Poisson process is the Lebesgue measure.

scholarly journals Bridge and cycle degrees of vertices of graphs

1984 ◽  
Vol 7 (2) ◽  
pp. 351-360
Author(s):  
Gary Chartrand ◽  
Farrokh Saba ◽  
Nicholas C. Wormald
Keyword(s):  
Finite Sets ◽  
Degree Of A Vertex ◽  
Ordered Pairs

The bridge degreebdeg vand cycle degreecdeg vof a vertexvin a graphGare, respectively, the number of bridges and number of cycle edges incident withvinG. A characterization of finite nonempty setsSof nonnegative integers is given for whichSis the set of bridge degrees (cycle degrees) of the vertices of some graph. The bridge-cycle degree of a vertexvin a graphGis the ordered pair(b,c), wherebdeg v=bandcdeg v=c. Those finite setsSof ordered pairs of nonnegative integers for whichSis the set of bridge-cycle degrees of the vertices of some graph are also characterized.

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