On digital sequences associated with Pascal's triangle

We consider the sequence of integers whose nth term has base-p expansion given by the nth row of Pascal's triangle modulo p (where p is a prime number). We first present and generalize well-known relations concerning this sequence. Then, with the great help of Sloane's On-Line Encyclopedia of Integer Sequences, we show that it appears naturally as a subsequence of a 2-regular sequence. Its study provides interesting relations and surprisingly involves odious and evil numbers, Nim-sum and even Gray codes. Furthermore, we examine similar sequences emerging from prime numbers involving alternating sum-of-digits modulo p. This note ends with a discussion about Pascal's pyramid involving trinomial coefficients

A generalization of Bohr-Mollerup's theorem for higher order convex functions

In 1922, Harald Bohr and Johannes Mollerup established a remarkable characterization of the Euler gamma function using its log-convexity property. A decade later, Emil Artin investigated this result and used it to derive the basic properties of the gamma function using elementary methods of the calculus. Bohr-Mollerup's theorem was then adopted by Nicolas Bourbaki as the starting point for his exposition of the gamma function. This open access book develops a far-reaching generalization of Bohr-Mollerup's theorem to higher order convex functions, along lines initiated by Wolfgang Krull, Roger Webster, and some others but going considerably further than past work. In particular, this generalization shows using elementary techniques that a very rich spectrum of functions satisfy analogues of several classical properties of the gamma function, including Bohr-Mollerup's theorem itself, Euler's reflection formula, Gauss' multiplication theorem, Stirling's formula, and Weierstrass' canonical factorization. The scope of the theory developed in this work is illustrated through various examples, ranging from the gamma function itself and its variants and generalizations (q-gamma, polygamma, multiple gamma functions) to important special functions such as the Hurwitz zeta function and the generalized Stieltjes constants. This volume is also an opportunity to honor the 100th anniversary of Bohr-Mollerup's theorem and to spark the interest of a large number of researchers in this beautiful theory

A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions

In 1922, Harald Bohr and Johannes Mollerup established a remarkable characterization of the Euler gamma function using its log-convexity property. A decade later, Emil Artin investigated this result and used it to derive the basic properties of the gamma function using elementary methods of the calculus. Bohr-Mollerup's theorem was then adopted by Nicolas Bourbaki as the starting point for his exposition of the gamma function. This open access book develops a far-reaching generalization of Bohr-Mollerup's theorem to higher order convex functions, along lines initiated by Wolfgang Krull, Roger Webster, and some others but going considerably further than past work. In particular, this generalization shows using elementary techniques that a very rich spectrum of functions satisfy analogues of several classical properties of the gamma function, including Bohr-Mollerup's theorem itself, Euler's reflection formula, Gauss' multiplication theorem, Stirling's formula, and Weierstrass' canonical factorization. The scope of the theory developed in this work is illustrated through various examples, ranging from the gamma function itself and its variants and generalizations (q-gamma, polygamma, multiple gamma functions) to important special functions such as the Hurwitz zeta function and the generalized Stieltjes constants. This volume is also an opportunity to honor the 100th anniversary of Bohr-Mollerup's theorem and to spark the interest of a large number of researchers in this beautiful theory

A generalization of Bohr-Mollerup's theorem for higher order convex functions: A tutorial

In its additive version, Bohr-Mollerup's remarkable theorem states that the unique (up to an additive constant) convex solution $f(x)$ to the equation $\Delta f(x)=\ln x$ on the open half-line $(0,\infty)$ is the log-gamma function $f(x)=\ln\Gamma(x)$, where $\Delta$ denotes the classical difference operator and $\Gamma(x)$ denotes the Euler gamma function. In a recently published open access book, the authors provided and illustrated a far-reaching generalization of Bohr-Mollerup's theorem by considering the functional equation $\Delta f(x)=g(x)$, where $g$ can be chosen in a wide and rich class of functions that have convexity or concavity properties of any order. They also showed that the solutions $f(x)$ arising from this generalization satisfy counterparts of many properties of the log-gamma function (or equivalently, the gamma function), including analogues of Bohr-Mollerup's theorem itself, Burnside's formula, Euler's infinite product, Euler's reflection formula, Gauss' limit, Gauss' multiplication formula, Gautschi's inequality, Legendre's duplication formula, Raabe's formula, Stirling's formula, Wallis's product formula, Weierstrass' infinite product, and Wendel's inequality for the gamma function. In this paper, we review the main results of this new and intriguing theory and provide an illustrative application.1.

On Minimal and Minimum Cylindrical Algebraic Decompositions

peer reviewedWe consider cylindrical algebraic decompositions (CADs) as a tool for representing semi-algebraic subsets of $\mathbb{R}^n$. In this framework, a CAD $\mathscr{C}$ is adapted to a given set $S$ if $S$ is a union of cells of $\mathscr{C}$. Different algorithms computing an adapted CAD may produce different outputs, usually with redundant cell divisions. In this paper we analyse the possibility to remove the superfluous data. More precisely we consider the set CAD$(S)$ of CADs that are adapted to $S$, endowed with the refinement partial order and we study the existence of minimal and minimum elements in this poset. We show that for every semi-algebraic set $S$ of $\mathbb{R}^n$ and every CAD $\mathscr{C}$ adapted to $S$, there is a minimal CAD adapted to $S$ and smaller (i.e. coarser) than or equal to $\mathscr{C}$. Moreover, when $n=1$ or $n=2$, we strengthen this result by proving the existence of a minimum element in CAD$(S)$. Astonishingly for $n \geq 3$, there exist semi-algebraic sets whose associated poset of adapted CADs does not admit a minimum. We prove this result by providing explicit examples. We finally use a reduction relation on CAD$(S)$ to define an algorithm for the computation of minimal CADs. We conclude with a characterization of those semi-algebraic sets $S$ for which CAD$(S)$ has a minimum by means of confluence of the associated reduction system

Entanglement classification schemes : comparison between Majorana representation and algebraic geometry approaches

editorial reviewedQuantum entanglement can be of different kinds [1] and classifying the quantum states in this respect may represent a difficult challenge in general multipartite systems. In particular, entanglement classes that are inequivalent under stochastic local operations and classical communication (SLOCC) are of fundamental importance. For -qubit systems with > 3, there is an infinity of such SLOCC entanglement classes [1] and it makes sense to gather them into a finite number of families, as was done for symmetric states in Refs. [2,3] using two distinct approaches (Majorana representation and algebraic geometry tools, respectively). Here, we compare these two structures and identify whether they can be embedded into one another or not. To do so, we formulate the structure of Ref. [2] in terms of -secants and - tangents ( a positive integer) of the Veronese variety [3] and we prove that only the -tangent structuration provides a coherent structure compatible with that of Ref. [3]. [1] W. Dür et al., Phys. Rev. A 62, 062314 (2000). [2] T. Bastin et al., Phys. Rev. Lett. 103, 070503 (2009). [3] M. Sanz et al., J. Phys. A: Math. Theor. 50, 195303 (2017)Entanglement classification with algebraic geometr

Entanglement classification schemes : comparison between Majorana representation and algebraic geometry approaches

editorial reviewedQuantum entanglement can be of different kinds [1] and classifying the quantum states in this respect may represent a difficult challenge in general multipartite systems. In particular, entanglement classes that are inequivalent under stochastic local operations and classical communication (SLOCC) are of fundamental importance. For -qubit systems with > 3, there is an infinity of such SLOCC entanglement classes [1] and it makes sense to gather them into a finite number of families, as was done for symmetric states in Refs. [2,3] using two distinct approaches (Majorana representation and algebraic geometry tools, respectively). Here, we compare these two structures and identify whether they can be embedded into one another or not. To do so, we formulate the structure of Ref. [2] in terms of -secants and - tangents ( a positive integer) of the Veronese variety [3] and we prove that only the -tangent structuration provides a coherent structure compatible with that of Ref. [3]. [1] W. Dür et al., Phys. Rev. A 62, 062314 (2000). [2] T. Bastin et al., Phys. Rev. Lett. 103, 070503 (2009). [3] M. Sanz et al., J. Phys. A: Math. Theor. 50, 195303 (2017)Entanglement classification with algebraic geometr