scholarly journals A remark on a theorem of A. E. Ingham.

2006 ◽  
Vol Volume 29 ◽  
Author(s):  
K G Bhat ◽  
K Ramachandra
Keyword(s):  
Functional Equation ◽  
Absolute Constant ◽  
International Audience

International audience Referring to a theorem of A. E. Ingham, that for all $N\geq N_0$ (an absolute constant), the inequality $N^3\leq p\leq(N+1)^3$ is solvable in a prime $p$, we point out in this paper that it is implicit that he has actually proved that $\pi(x+h)-\pi(x) \sim h(\log x)^{-1}$ where $h=x^c$ and $c (>\frac{5}{8})$ is any constant. Further, we point out that even this stronger form can be proved without using the functional equation of $\zeta(s)$.

scholarly journals A generalization of Bochner's formula

2004 ◽  
Vol Volume 27 ◽  
Author(s):  
S Kanemitsu ◽  
Y Tanigawa ◽  
H Tsukada
Keyword(s):  
Functional Equation ◽  
Dirichlet Series ◽  
Hierarchy Theorems ◽  
International Audience ◽  
Natural Way

International audience In this note we expound our general hierarchy theorems by the example of a Ramified-Type Functional Equarion H, which gives all possbile forms, in terms of se-ries with H-function coefficients, of the functional equation of higher hierarchy arising from the original ramified one satisfied by the Dirichlet series. Then by sepcifying the parameters, we shall deduce a few concrete examples scattered in the literature in the most natural way.

scholarly journals Balanced binary trees in the Tamari lattice

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Samuele Giraudo
Keyword(s):  
Functional Equation ◽  
Binary Trees ◽  
Tamari Lattice ◽  
Tree Patterns ◽  
Generating Series ◽  
Nous Montrons ◽  
International Audience ◽  
Balanced Tree

International audience We show that the set of balanced binary trees is closed by interval in the Tamari lattice. We establish that the intervals $[T_0, T_1]$ where $T_0$ and $T_1$ are balanced trees are isomorphic as posets to a hypercube. We introduce tree patterns and synchronous grammars to get a functional equation of the generating series enumerating balanced tree intervals. Nous montrons que l'ensemble des arbres équilibrés est clos par intervalle dans le treillis de Tamari. Nous caractérisons la forme des intervalles du type $[T_0, T_1]$ où $T_0$ et $T_1$ sont équilibrés en montrant qu'en tant qu'ensembles partiellement ordonnés, ils sont isomorphes à un hypercube. Nous introduisons la notion de motif d'arbre et de grammaire synchrone dans le but d'établir une équation fonctionnelle de la série génératrice qui dénombre les intervalles d'arbres équilibrés.

scholarly journals Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Matthieu Josuat-Vergès ◽  
Jang-Soo Kim
Keyword(s):  
Functional Equation ◽  
Triple Product ◽  
Combinatorial Proof ◽  
Jacobi’S Triple Product Identity ◽  
Product Identity ◽  
Nous Montrons ◽  
Schröder Paths ◽  
International Audience ◽  
Finite Sum ◽  
Jacobi's Triple Product Identity

International audience We give a combinatorial proof of a Touchard-Riordan-like formula discovered by the first author. As a consequence we find a connection between his formula and Jacobi's triple product identity. We then give a combinatorial analog of Jacobi's triple product identity by showing that a finite sum can be interpreted as a generating function of weighted Schröder paths, so that the triple product identity is recovered by taking the limit. This can be stated in terms of some continued fractions called T-fractions, whose important property is the fact that they satisfy some functional equation. We show that this result permits to explain and generalize some Touchard-Riordan-like formulas appearing in enumerative problems. Nous donnons une preuve combinatoire d'une formule à la Touchard-Riordan due au premier auteur. En conséquence, nous faisons appara\^ıtre un lien entre cette formule et l'identité du produit triple de Jacobi. Nous donnons un analogue combinatoire à l'identité du produit triple en montrant qu'une somme finie peut être interprétée comme fonction génératrice de chemins de Schröder pondérés, de sorte que l'identité du produit triple s'obtient en passant à la limite. Ceci peut être énoncé en termes de fractions continues appelées T-fractions, dont la propriété importante est le fait qu'elle satisfont certaines équations fonctionnelles. Nous montrons que ce résultat permet d'expliquer et généraliser certaines formules à la Touchard-Riordan apparaissant dans des problèmes d'énumération.

scholarly journals Counting quadrant walks via Tutte's invariant method (extended abstract)

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Olivier Bernardi ◽  
Mireille Bousquet-Mélou ◽  
Kilian Raschel
Keyword(s):  
Differential Equation ◽  
Functional Equation ◽  
Linear Differential Equation ◽  
Generating Function ◽  
Algebraic Approach ◽  
Rational Functions ◽  
The Past ◽  
Lattice Walks ◽  
International Audience ◽  
Linear Differential

Extended abstract presented at the conference FPSAC 2016, Vancouver. International audience In the 1970s, Tutte developed a clever algebraic approach, based on certain " invariants " , to solve a functional equation that arises in the enumeration of properly colored triangulations. The enumeration of plane lattice walks confined to the first quadrant is governed by similar equations, and has led in the past decade to a rich collection of attractive results dealing with the nature (algebraic, D-finite or not) of the associated generating function, depending on the set of allowed steps. We first adapt Tutte's approach to prove (or reprove) the algebraicity of all quadrant models known or conjectured to be algebraic (with one small exception). This includes Gessel's famous model, and the first proof ever found for one model with weighted steps. To be applicable, the method requires the existence of two rational functions called invariant and decoupling function respectively. When they exist, algebraicity comes out (almost) automatically. Then, we move to an analytic viewpoint which has already proved very powerful, leading in particular to integral expressions of the generating function in the non-D-finite cases, as well as to proofs of non-D-finiteness. We develop in this context a weaker notion of invariant. Now all quadrant models have invariants, and for those that have in addition a decoupling function, we obtain integral-free expressions of the generating function, and a proof that this series is differentially algebraic (that is, satisfies a non-linear differential equation).

scholarly journals On the enumeration of column-convex permutominoes

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Nicholas R. Beaton ◽  
Filippo Disanto ◽  
Anthony J. Guttmann ◽  
Simone Rinaldi
Keyword(s):  
Functional Equation ◽  
Asymptotic Behavior ◽  
Numerical Analysis ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Recursive Construction ◽  
Nous Obtenons ◽  
Comportement Asymptotique ◽  
Generating Trees ◽  
International Audience

International audience We study the enumeration of \emphcolumn-convex permutominoes, i.e. column-convex polyominoes defined by a pair of permutations. We provide a direct recursive construction for the column-convex permutominoes of a given size, based on the application of the ECO method and generating trees, which leads to a functional equation. Then we obtain some upper and lower bounds for the number of column-convex permutominoes, and conjecture its asymptotic behavior using numerical analysis. Nous étudions l'énumeration des \emphpermutominos verticalement convexes, c.à.d. les polyominos verticalement convexes définis par un couple de permutations. Nous donnons une construction recursive directe pour ces permutominos de taille fixée, basée sur une application de la méthode ECO et les arbres de génération, qui nous amène à une équat ion fonctionelle. Ensuite nous obtenons des bornes superieures et inférieures pour le nombre de ces permutominos convexes et nous conjecturons leur comportement asymptotique à l'aide d'analyses numériques.

scholarly journals On the number of decomposable trees

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Stephan G. Wagner
Keyword(s):  
Functional Equation ◽  
Generating Functions ◽  
Limit Case ◽  
Asymptotic Results ◽  
Spanning Forest ◽  
International Audience ◽  
Generated Family ◽  
Fixed Tree

International audience A tree is called $k$-decomposable if it has a spanning forest whose components are all of size $k$. Analogously, a tree is called $T$-decomposable for a fixed tree $T$ if it has a spanning forest whose components are all isomorphic to $T$. In this paper, we use a generating functions approach to derive exact and asymptotic results on the number of $k$-decomposable and $T$-decomposable trees from a so-called simply generated family of trees - we find that there is a surprisingly simple functional equation for the counting series of $k$-decomposable trees. In particular, we will study the limit case when $k$ goes to $\infty$. It turns out that the ratio of $k$-decomposable trees increases when $k$ becomes large.

scholarly journals Mean square of the Hurwitz zeta-function and other remarks

2004 ◽  
Vol Volume 27 ◽  
Author(s):  
R Balasubramanian ◽  
K Ramachandra
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Analytic Continuation ◽  
Hurwitz Zeta Function ◽  
Complex Numbers ◽  
Approximate Functional Equation ◽  
The Riemann Zeta Function ◽  
International Audience ◽  
Approximate Function ◽  
Riemann Zeta

International audience The Hurwitz zeta-function associated with the parameter $a\,(0< a\leq1)$ is a generalisation of the Riemann zeta-function namely the case $a=1$. It is defined by $$\zeta(s,a)=\sum_{n=0}^{\infty}(n+a)^{-s},\,(s=\sigma+it,\,\sigma>1)$$ and its analytic continuation. %In fact $$\zeta(s,a)=\sum_{n=0}^{\infty}\left((n+a)^{-s}-\int_{n}^{n+1}\frac{du}{(u+a)^s} \right)+\frac{a^{1-s}}{s-1}$$ gives the analytic continuation to $(\sigma>0)$. A repetition of this several times shows that $$\zeta-\frac{a^{1-s}}{s-1}$$ can be continued as an entire function to the whole plane. In $Re(s)\geq-1,\,t\geq2,\,\zeta(s,a)-a^{-s}=O(t^3)$ and by the functional equation (see \S2) it is $$O\left(\left(\frac{\vert s\vert}{2\pi}\right)^{\frac{1}{2}-Re(s)}\right)$$ in $Re(s)\leq-1,\,t\geq2$. From these facts In this paper, we deduce an `Approximate function equation' (see \S3), which is a generalisation of the approximate functional equation for $\zeta(s)$. Combining this with an important theorem due to van-der-Corput, we prove $$T^{-\frac{1}{3}}\int_{T}^{T+T^{\frac{1}{3}}} \vert\zeta(\frac{1}{2}+it)-a^{-\frac{1}{2}-it}\vert^2 dt <\!\!\!< (\log T)^3$$ uniformly in $a(0< a\leq1)$. From this we deduce similar results for quasi $L$-functions and more general functions. %Let $a_1, a_2,\ldots$, be any periodic sequence of complex numbers for which the sum over a period is zero. Let $b_1, b_2,\ldots$ be any sequence of complex numbers for which $\sum_{j=2}^{n}\vert b_j-b_{j-1}\vert+\vert b_n\vert\leq n^{\varepsilon}$ for every $\varepsilon>0$ and every $n\geq n_0(\varepsilon)$. Then we prove $$T^{-\frac{1}{3}}\int_{T}^{T+T^{\frac{1}{3}}} \vert\sum_{n=1}^{\infty}\frac{a_nb_n}{(n+a)^{\frac{1}{2}+it}}\vert^2\,dt\leq T^{\varepsilon}$$ for every $\varepsilon>0$ and every $T\geq T_0(\varepsilon)$. Here, as usual, $0<a\leq1$ and $T_0(\varepsilon)$ is independent of $a$.

scholarly journals Bounded-degree graphs have arbitrarily large queue-number

2008 ◽  
Vol Vol. 10 no. 1 (Graph and Algorithms) ◽  
Author(s):  
David R. Wood
Keyword(s):  
Absolute Constant ◽  
Bounded Degree ◽  
Large N ◽  
Vertex Graph ◽  
International Audience ◽  
Bounded Degree Graphs

Graphs and Algorithms International audience It is proved that there exist graphs of bounded degree with arbitrarily large queue-number. In particular, for all \Delta ≥ 3 and for all sufficiently large n, there is a simple \Delta-regular n-vertex graph with queue-number at least c√\Delta_n^{1/2-1/\Delta} for some absolute constant c.

The one-level functional equation of multi-rate loss systems

2003 ◽  
Vol 14 (2) ◽  
pp. 107-118 ◽  
Author(s):  
Harro L. Hartmann ◽  
Martin Knoke
Keyword(s):  
Functional Equation ◽  
Loss Systems ◽  
The One ◽  
Rate Loss
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