scholarly journals On a generalization of the Levin-May Theorem

2014 ◽  
Vol 30 (1) ◽  
pp. 55-62
Author(s):  
JAN CERMAK ◽  
◽  
JIRI JANSKY ◽  
Keyword(s):  
Unit Circle ◽  
Practical Importance ◽  
Complete List ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Alternate Proof ◽  
Necessary And Sufficient

The paper discusses a distribution of the zeros of the polynomial ...with respect to the unit circle. This problem is of theoretic as well as practical importance which motivated S.A. Levin and R. May to formulate a necessary and sufficient condition guaranteeing the location of all the zeros of p(λ) inside the unit circle. We give a simple alternate proof of their criterion and, as the main result, present a complete list of all possible zero distributions of p(λ) with respect to this circle.

scholarly journals QKSpaces on the Unit Circle

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Jizhen Zhou
Keyword(s):  
Sobolev Space ◽  
Unit Circle ◽  
Measurable Function ◽  
Weight Functions ◽  
Sufficient Condition ◽  
General Criterion ◽  
Necessary And Sufficient Condition ◽  
New Space ◽  
Necessary And Sufficient

We introduce a new spaceQK(∂D)of Lebesgue measurable functions on the unit circle connecting closely with the Sobolev space. We obtain a necessary and sufficient condition onKsuch thatQK(∂D)=BMO(∂D), as well as a general criterion on weight functionsK1andK2,K1≤K2, such thatQK1(∂D)QK2(∂D). We also prove that a measurable function belongs toQK(∂D)if and only if it is Möbius bounded in the Sobolev spaceLK2(∂D). Finally, we obtain a dyadic characterization of functions inQK(∂D)spaces in terms of dyadic arcs on the unit circle.

scholarly journals The complex geometry of Blaschke products of degree 3 and associated ellipses

Filomat ◽  
2017 ◽  
Vol 31 (1) ◽  
pp. 61-68
Author(s):  
Masayo Fujimuraa
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Complex Geometry ◽  
Sufficient Condition ◽  
Blaschke Products ◽  
Necessary And Sufficient Condition ◽  
Geometrical Properties ◽  
Natural Extensions ◽  
Necessary And Sufficient

A bicentric polygon is a polygon which has both an inscribed circle and a circumscribed one. For given two circles, the necessary and sufficient condition for existence of bicentric triangle for these two circles is known as Chapple?s formula or Euler?s theorem. As one of natural extensions of this formula, we characterize the inscribed ellipses of a triangle which is inscribed in the unit circle. We also discuss the condition for the ?circumscribed? ellipse of a triangle which is circumscribed about the unit circle. For the proof of these results, we use some geometrical properties of Blaschke products on the unit disk.

scholarly journals Compressions of Multiplication Operators and Their Characterizations

2020 ◽  
Vol 75 (4) ◽  
Author(s):  
M. Cristina Câmara ◽  
Kamila Kliś–Garlicka ◽  
Bartosz Łanucha ◽  
Marek Ptak
Keyword(s):  
Unit Circle ◽  
Toeplitz Operator ◽  
Toeplitz Operators ◽  
Multiplication Operators ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Truncated Toeplitz Operator ◽  
Independent Variable ◽  
Necessary And Sufficient

AbstractDual truncated Toeplitz operators and other restrictions of the multiplication by the independent variable $$M_z$$ M z on the classical $$L^2$$ L 2 space on the unit circle are investigated. Commutators are calculated and commutativity is characterized. A necessary and sufficient condition for any operator to be a dual truncated Toeplitz operator is established. A formula for recovering its symbol is stated.

A NECESSARY AND SUFFICIENT CONDITION TO HAVE r PAIRS OF COMPLEX MULTIPLIERS WITH MODULUS EQUAL TO UNITY IN THE CASE OF AN n-DIMENSIONAL MAP

1995 ◽  
Vol 05 (04) ◽  
pp. 1193-1204 ◽  
Author(s):  
JEAN-PIERRE CARCASSES
Keyword(s):  
Unit Circle ◽  
Periodic Point ◽  
Parameter Plane ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

Considering a cycle (or periodic point) of an n-dimensional map, a bifurcation occurs when at least one of its multipliers crosses the unit circle in the complex.plane. This paper presents a necessary and sufficient condition for the bifurcations occurring when r pairs of complex multipliers cross the unit circle. Among this kind of bifurcations, the most known is the Neimark's bifurcation corresponding to the case when r is equal to one. From the established condition, an algorithm to trace out these bifurcation curves in a parameter plane of the considered map is displayed.

A Proposed Framework Emphasizing Auditor Reliability over Auditor Independence

2003 ◽  
Vol 17 (3) ◽  
pp. 257-266 ◽  
Author(s):  
Mark H. Taylor ◽  
F. Todd DeZoort ◽  
Edward Munn ◽  
Martha Wetterhall Thomas
Keyword(s):  
Public Interest ◽  
Auditor Independence ◽  
Public Confidence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Accounting Profession ◽  
The Public ◽  
Necessary And Sufficient

This paper introduces an auditor reliability framework that repositions the role of auditor independence in the accounting profession. The framework is motivated in part by widespread confusion about independence and the auditing profession's continuing problems with managing independence and inspiring public confidence. We use philosophical, theoretical, and professional arguments to argue that the public interest will be best served by reprioritizing professional and ethical objectives to establish reliability in fact and appearance as the cornerstone of the profession, rather than relationship-based independence in fact and appearance. This revised framework requires three foundation elements to control subjectivity in auditors' judgments and decisions: independence, integrity, and expertise. Each element is a necessary but not sufficient condition for maximizing objectivity. Objectivity, in turn, is a necessary and sufficient condition for achieving and maintaining reliability in fact and appearance.

The Power of Public Positions

2018 ◽  
Author(s):  
Thomas Sinclair
Keyword(s):  
Detailed Analysis ◽  
Political Authority ◽  
The State ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
State Institutions ◽  
State Of Nature ◽  
Necessary And Sufficient

The Kantian account of political authority holds that the state is a necessary and sufficient condition of our freedom. We cannot be free outside the state, Kantians argue, because any attempt to have the “acquired rights” necessary for our freedom implicates us in objectionable relations of dependence on private judgment. Only in the state can this problem be overcome. But it is not clear how mere institutions could make the necessary difference, and contemporary Kantians have not offered compelling explanations. A detailed analysis is presented of the problems Kantians identify with the state of nature and the objections they face in claiming that the state overcomes them. A response is sketched on behalf of Kantians. The key idea is that under state institutions, a person can make claims of acquired right without presupposing that she is by nature exceptional in her capacity to bind others.

scholarly journals Lorentz Boosts and Wigner Rotations: Self-Adjoint Complexified Quaternions

Physics ◽  
2021 ◽  
Vol 3 (2) ◽  
pp. 352-366
Author(s):  
Thomas Berry ◽  
Matt Visser
Keyword(s):  
Point Of View ◽  
Sufficient Condition ◽  
Use Of Self ◽  
Necessary And Sufficient Condition ◽  
Wigner Rotation ◽  
Inertial Frames ◽  
Explicit Formulae ◽  
Specific Implementation ◽  
Necessary And Sufficient ◽  
Lorentz Boosts

In this paper, Lorentz boosts and Wigner rotations are considered from a (complexified) quaternionic point of view. It is demonstrated that, for a suitably defined self-adjoint complex quaternionic 4-velocity, pure Lorentz boosts can be phrased in terms of the quaternion square root of the relative 4-velocity connecting the two inertial frames. Straightforward computations then lead to quite explicit and relatively simple algebraic formulae for the composition of 4-velocities and the Wigner angle. The Wigner rotation is subsequently related to the generic non-associativity of the composition of three 4-velocities, and a necessary and sufficient condition is developed for the associativity to hold. Finally, the authors relate the composition of 4-velocities to a specific implementation of the Baker–Campbell–Hausdorff theorem. As compared to ordinary 4×4 Lorentz transformations, the use of self-adjoint complexified quaternions leads, from a computational view, to storage savings and more rapid computations, and from a pedagogical view to to relatively simple and explicit formulae.

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