scholarly journals Positive Polynomials and Boundary Interpolation with Finite Blaschke Products

2021 ◽  
Author(s):  
Sergei Kalmykov ◽  
Béla Nagy
Keyword(s):  
Unit Circle ◽  
Specific Form ◽  
The Other ◽  
Geometric Representation ◽  
Polynomial Equations ◽  
Blaschke Products ◽  
Positive Polynomials ◽  
Boundary Interpolation ◽  
Finite Blaschke Products ◽  
System Of Polynomial Equations

AbstractThe famous Jones–Ruscheweyh theorem states that n distinct points on the unit circle can be mapped to n arbitrary points on the unit circle by a Blaschke product of degree at most $$n-1$$ n - 1 . In this paper, we provide a new proof using real algebraic techniques. First, the interpolation conditions are rewritten into complex equations. These complex equations are transformed into a system of polynomial equations with real coefficients. This step leads to a “geometric representation” of Blaschke products. Then another set of transformations is applied to reveal some structure of the equations. Finally, the following two fundamental tools are used: a Positivstellensatz by Prestel and Delzell describing positive polynomials on compact semialgebraic sets using Archimedean module of length N. The other tool is a representation of positive polynomials in a specific form due to Berr and Wörmann. This, combined with a careful calculation of leading terms of occurring polynomials finishes the proof.

Boundary Interpolation by Finite Blaschke Products

2006 ◽  
Vol 27 (1) ◽  
pp. 75-98 ◽  
Author(s):  
Pamela Gorkin ◽  
Robert C. Rhoades
Keyword(s):  
Blaschke Products ◽  
Boundary Interpolation ◽  
Finite Blaschke Products

scholarly journals A Discontinuity of the Energy of Quantum Walk in Impurities

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1134
Author(s):  
Kenta Higuchi ◽  
Takashi Komatsu ◽  
Norio Konno ◽  
Hisashi Morioka ◽  
Etsuo Segawa
Keyword(s):  
Unit Circle ◽  
Stationary State ◽  
Discrete Time ◽  
Quantum Walk ◽  
Time Limit ◽  
The Other ◽  
Unitary Matrix ◽  
Initial State ◽  
Time Quantum ◽  
Long Time

We consider the discrete-time quantum walk whose local dynamics is denoted by a common unitary matrix C at the perturbed region {0,1,⋯,M−1} and free at the other positions. We obtain the stationary state with a bounded initial state. The initial state is set so that the perturbed region receives the inflow ωn at time n(|ω|=1). From this expression, we compute the scattering on the surface of −1 and M and also compute the quantity how quantum walker accumulates in the perturbed region; namely, the energy of the quantum walk, in the long time limit. The frequency of the initial state of the influence to the energy is symmetric on the unit circle in the complex plain. We find a discontinuity of the energy with respect to the frequency of the inflow.

On the presence of Anchistioides willeyi (Borradaile, 1899) (Decapoda: Caridea: Anchistioididae) in the Red Sea

Zootaxa ◽  
2009 ◽  
Vol 2243 (1) ◽  
pp. 53-56 ◽  
Author(s):  
IVAN MARIN
Keyword(s):  
Red Sea ◽  
Type Species ◽  
Specific Form ◽  
The Other ◽  
Valid Species ◽  
Caribbean Region ◽  
Western Atlantic ◽  
Abdominal Somite ◽  
Single Genus ◽  
Pacific Species

The palaemonoid family Anchistioididae Borradaile, 1915 includes a single genus Anchistioides Paulson, 1875 with four known valid species: Anchistioides compressus Paulson, 1875 (type species), A. willeyi (Borradaile, 1899), A. australiensis (Balss, 1921) and A. antiguensis (Schmitt, 1924). Borradaile (1915) suggested two more species within the genus Amphipalaemon Nobili, 1901 (a junior synonym of Anchisitioides Paulson), Amphipalaemon gardineri Borradaile, 1915 (= Anchistioides gardineri) and Amphipalaemon cooperi Borradaile, 1915 (= Anchistioides cooperi) which were later synonomyzed with Anchisitioides willeyi by Gordon (1935), who also suggested their conspecificity with Anchistioides australiensis. At the present time, Anchistioides australiensis is a valid species (Bruce, 1971; Chace & Bruce, 1993) based on specific morphological features such as the presence of sharp postorbital tooth, oblique distal lamela of scaphocerite and sharply produced spines on posterodorsal angles of sixth abdominal somite (see Bruce, 1971: fig. 9). The other Indo-Pacific species, Anchistioides compressus and A. willeyi, can be clearly identified by specific form of scaphocerite, the presence of a well marked blunt postorbital tubercle in A. willeyi which is absent in A. compressus (e.g., Bruce, 1971) and the number of ventral rostral teeth (3-4 large ventral rostral teeth present in A. willeyi while up to 8 small ventral rostral teeth in A. compressus (Paulson, 1875; Gordon, 1935)). Anchistioides antiguensis is clearly separated geographically being known only from the tropical Western Atlantic and Caribbean region (Schmitt, 1924; Holthuis, 1951; Wheeler & Brown, 1968; Martinez-Iglesias, 1986; Markham et al, 1990; Ramos-Porto et al, 1998; Cardoso, 2006).

scholarly journals A Numerical Local Dimension Test for Points on the Solution Set of a System of Polynomial Equations

2009 ◽  
Vol 47 (5) ◽  
pp. 3608-3623 ◽  
Author(s):  
Daniel J. Bates ◽  
Jonathan D. Hauenstein ◽  
Chris Peterson ◽  
Andrew J. Sommese
Keyword(s):  
Solution Set ◽  
Polynomial Equations ◽  
Local Dimension ◽  
System Of Polynomial Equations

scholarly journals Bivariate systems of polynomial equations with roots of high multiplicity

2021 ◽  
Author(s):  
I. Nikitin
Keyword(s):  
High Multiplicity ◽  
Polynomial Equations ◽  
Support Sets ◽  
Multiplicity Formula ◽  
System Of Polynomial Equations ◽  
Systems Of Polynomial Equations

Given a bivariate system of polynomial equations with fixed support sets [Formula: see text] it is natural to ask which multiplicities its solutions can have. We prove that there exists a system with a solution of multiplicity [Formula: see text] for all [Formula: see text] in the range [Formula: see text], where [Formula: see text] is the set of all integral vectors that shift B to a subset of [Formula: see text]. As an application, we classify all pairs [Formula: see text] such that the system supported at [Formula: see text] does not have a solution of multiplicity higher than [Formula: see text].

Nancian virtual doubts about ‘Leformal’ democracy

2011 ◽  
Vol 37 (9) ◽  
pp. 999-1010 ◽  
Author(s):  
Ignaas Devisch
Keyword(s):  
Specific Form ◽  
The Other ◽  
Political Regime ◽  
Berlin Wall ◽  
Global Democracy ◽  
Claude Lefort ◽  
Other Hand ◽  
Contemporary Politics ◽  
To Come ◽  
The Many

French philosopher Jean-Luc Nancy is acting uneasily when it comes to contemporary politics. There is a sort of agitation in his work in relation to this question. At several places we read an appeal to deal thoroughly with this question and ‘ qu’il y a un travail à faire’, that there is still work to do. From the beginning of the 1980s with the ‘Centre de Recherches Philosophiques sur le Politique’ and the two books resulting out of that, until the many, rather short texts he published on this topic during the last years of the century, the question of politics crosses very clearly Nancy’s work. He not only fulminates against the contemporary philosophical ‘content’ with democracy. Instead of defending a political regime, he wants to think the form of politics in the most critical and sceptical way. To Nancy, the worst thing we can do in thinking contemporary politics, is taking it for granted that we know what politics is about today, given the evidence of the global democracy. So to him, we almost have to be at unease when it comes to politics. On the other hand, in thinking contemporary democracy, the work of Claude Lefort is undeniably the main reference. Long before the collapse of the Berlin Wall and the upsurge of an all-too-easy anti-Marxism, Lefort articulated in a nuanced way the formal differences between totalitarianism and democracy. According to Lefort, the specific ‘form’ of democracy is that it never becomes an accomplished and fulfilled form as such. In a certain sense, the only ‘form’ of democracy is formlessness, a form without form. In a democracy, the place of power becomes literally ‘ infigurable’ as Lefort says. Democracy stands for formlessness or the relation to a void. Nancy objects so to say against a ‘Leformal’ conception of democracy – the empty place, the formless, the ‘ infigurable’ or ‘ sans figure’, the ever yet to come. … This conception of democracy would still be caught in the infinite metaphysical, dialectical horizon of immanentism, while it pretends to have already left that horizon behind it, presenting itself as the finite alternative to the infinite totalitarian politics. Democracy as formlessness is indeed no longer based on a metaphysical Idea, Figure, or Truth. We want to clear up the philosophical sky of Nancy’s remarks by confronting them with some thoughts of Lefort.

Sign in / Sign up

Export Citation Format

Share Document