scholarly journals $f$-vectors of subdivided simplicial complexes (extended abstract)

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Emanuele Delucchi ◽  
Aaron Pixton ◽  
Lucas Sabalka
Keyword(s):  
Power Series ◽  
Recent Result ◽  
Real Number ◽  
Simplicial Complex ◽  
Point Of View ◽  
Barycentric Subdivision ◽  
Geometric Point ◽  
International Audience ◽  
Finite Simplicial Complex ◽  
Formal Power

International audience We take a geometric point of view on the recent result by Brenti and Welker, who showed that the roots of the $f$-polynomials of successive barycentric subdivisions of a finite simplicial complex $X$ converge to fixed values depending only on the dimension of $X$. We show that these numbers are roots of a certain polynomial whose coefficients can be computed explicitly. We observe and prove an interesting symmetry of these roots about the real number $-2$. This symmetry can be seen via a nice realization of barycentric subdivision as a simple map on formal power series. We then examine how such a symmetry extends to more general types of subdivisions. The generalization is formulated in terms of an operator on the (formal) ring on the set of simplices of the complex. On applique un point de vue géométrique à un récent résultat de Brenti et Welker, qui ont montré que les racines des polynômes $f$ de subdivisions barycentriques successives d'un complexe simplicial $X$ convergent vers des valeurs fixes, ne dépendant que de la dimension de $X$. On prouve que ces nombres sont en effet eux-mêmes racines d'un polynôme dont les coefficients peuvent être calculés explicitement. De plus, on observe et on démontre une symétrie particulière de ces nombres autour du numéro $-2$. Cette symétrie se révèle en exprimant l'opération de subdivision barycentrique par une fonction sur des séries de puissances formelles. Une symétrie pareille existe pour des méthodes de subdivision plus générales, où elle s'exprime par des opérateurs sur l'anneau des sommes formelles de simplexes du complexe.

scholarly journals The group of formal power series under substitution

1988 ◽  
Vol 45 (3) ◽  
pp. 296-302 ◽  
Author(s):  
D. L. Johnson
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Binary Operation ◽  
Point Of View ◽  
Theoretical Point ◽  
Formal Power

AbstractThis is a study of formal power series under the binary operation of formal composition from a group-theoretical point of view. Various “large” properties are derived.

Substitution Groups of Formal Power Series

1954 ◽  
Vol 6 ◽  
pp. 325-340 ◽  
Author(s):  
S. A. Jennings
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Formal Power Series ◽  
Point Of View ◽  
Group Operation ◽  
Several Variables ◽  
Special Cases ◽  
Image Position ◽  
Formal Power ◽  
Complex Coefficients

In this paper we are concerned with the group of formal power series of the form,the coefficients being elements of a commutative ring R and the group operation being substitution. Little seems to be known of the properties of groups of this type, except in special cases, although groups of formal power series in several variables with complex coefficients have been investigated from a different point of view by Bochner and Martin (1, chap. I) and Gotô (2).

scholarly journals Arithmetical investigations of particular Wynn power series

2008 ◽  
Vol Volume 31 ◽  
Author(s):  
Peter Bundschuh
Keyword(s):  
Power Series ◽  
Unit Disc ◽  
Algebraic Integer ◽  
Point Of View ◽  
Open Unit ◽  
Open Unit Disc ◽  
Algebraic Numbers ◽  
Arithmetic Properties ◽  
Analytical Point ◽  
International Audience

International audience Using Borwein's simple analytic method for the irrationality of the $q$-logarithm at rational points, we prove a quite general result on arithmetic properties of certain series, where the entering parameters are algebraic numbers. More precisely, our main result says that $\sum_{k\ge1}\beta^k/(1-\alpha q^k)$ is not in $\mathbb{Q}(q)$, if $q$ is an algebraic integer with all its conjugates (if any) in the open unit disc, if $\alpha\in\mathbb{Q}(q)^\times\setminus\{q^{-1},q^{-2},\ldots\}$ satisfies a mild denominator condition (implying $|q|>1$), and if $\beta$ is a unit in $\mathbb{Q}(q)$ with $|\beta|\le1$ but no other conjugates in the open unit disc. Our applications concern meromorphic functions defined in $|z|<|u|^{a\ell}$ by power series $\sum_{n\ge1}z^n/(\prod_{0\le\lambda<\ell}R_{a(n+\lambda)+b})$, where $R_m:=gu^m+hv^m$ with non-zero $u,v,g,h$ satisfying $|u|>|v|, R_m\ne0$ for any $m\ge1$, and $a,b+1,\ell$ are positive rational integers. Clearly, the case where $R_m$ are the Fibonacci or Lucas numbers is of particular interest. It should be noted that power series of the above type were first studied by Wynn from the analytical point of view.

scholarly journals A factorization formula for power series

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Daniel Birmajer ◽  
Juan B. Gil ◽  
Michael D. Weiner
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Inversion Formula ◽  
Constant Term ◽  
Bell Polynomials ◽  
Integer Coefficients ◽  
Factorization Formula ◽  
International Audience ◽  
Formal Power

International audience Given an odd prime p, we give an explicit factorization over the ring of formal power series with integer coefficients for certain reducible polynomials whose constant term is of the form $p^w$ with $w>1$. Our formulas are given in terms of partial Bell polynomials and rely on the inversion formula of Lagrange.

Global CLF Stabilization of Nonlinear Systems: A Geometric Point of View

2011 ◽  
Vol 44 (1) ◽  
pp. 11139-11144
Author(s):  
Julio Solĺs-Daun
Keyword(s):  
Nonlinear Systems ◽  
Point Of View ◽  
Geometric Point

Formal power series and nearly analytic functions

1991 ◽  
Vol 57 (1) ◽  
pp. 61-70 ◽  
Author(s):  
H�kan Hedenmalm
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Analytic Functions ◽  
Formal Power

scholarly journals On transformations of formal power series

2003 ◽  
Vol 184 (2) ◽  
pp. 369-383 ◽  
Author(s):  
Manfred Droste ◽  
Guo-Qiang Zhang
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Formal Power

scholarly journals On the Square Root of a Bell Matrix

2021 ◽  
Vol 76 (1) ◽  
Author(s):  
Donatella Merlini
Keyword(s):  
Power Series ◽  
Closed Form ◽  
Formal Power Series ◽  
Computational Method ◽  
Square Root ◽  
Riordan Arrays ◽  
Formal Power

AbstractIn the context of Riordan arrays, the problem of determining the square root of a Bell matrix $$R={\mathcal {R}}(f(t)/t,\ f(t))$$ R = R ( f ( t ) / t , f ( t ) ) defined by a formal power series $$f(t)=\sum _{k \ge 0}f_kt^k$$ f ( t ) = ∑ k ≥ 0 f k t k with $$f(0)=f_0=0$$ f ( 0 ) = f 0 = 0 is presented. It is proved that if $$f^\prime (0)=1$$ f ′ ( 0 ) = 1 and $$f^{\prime \prime }(0)\ne 0$$ f ″ ( 0 ) ≠ 0 then there exists another Bell matrix $$H={\mathcal {R}}(h(t)/t,\ h(t))$$ H = R ( h ( t ) / t , h ( t ) ) such that $$H*H=R;$$ H ∗ H = R ; in particular, function h(t) is univocally determined by a symbolic computational method which in many situations allows to find the function in closed form. Moreover, it is shown that function h(t) is related to the solution of Schröder’s equation. We also compute a Riordan involution related to this kind of matrices.

On the intersection of null spaces for matrix substitutions in a non-commutative rational formal power series

2004 ◽  
Vol 339 (8) ◽  
pp. 533-538 ◽  
Author(s):  
Daniel Alpay ◽  
Dmitry S. Kalyuzhnyı̆-Verbovetzkiı̆
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Formal Power
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