Combinatorial Variations on Cantor's Diagonal

We discuss counting problems linked to finite versions of Cantor's diagonal of infinite tableaux. We extend previous results of [2] by refining an equivalence relation that reduces significantly the exhaustive generation. New enumerative results follow and allow to look at the sub-class of the so- called bi-Cantorian tableaux. We conclude with a correspondence between Cantorian-type tableaux and coloring of hypergraphs having a square number of vertices

Languages invariant under more symmetries: overlapping factors versus palindromic richness

Factor complexity $\mathcal{C}$ and palindromic complexity $\mathcal{P}$ of infinite words with language closed under reversal are known to be related by the inequality $\mathcal{P}(n) + \mathcal{P}(n+1) \leq 2 + \mathcal{C}(n+1)-\mathcal{C}(n)$ for any $n\in \mathbb{N}$\,. Word for which the equality is attained for any $n$ is usually called rich in palindromes. In this article we study words whose languages are invariant under a finite group $G$ of symmetries. For such words we prove a stronger version of the above inequality. We introduce notion of $G$-palindromic richness and give several examples of $G$-rich words, including the Thue-Morse sequence as well.Comment: 22 pages, 1 figur

Proof of Brlek-Reutenauer conjecture

Brlek and Reutenauer conjectured that any infinite word u with language closed under reversal satisfies the equality 2D(u) = \sum_{n=0}^{\infty}T_u(n) in which D(u) denotes the defect of u and T_u(n) denotes C_u(n+1)-C_u(n) +2 - P_U(n+1) - P_u(n), where C_u and P_u are the factor and palindromic complexity of u, respectively. This conjecture was verified for periodic words by Brlek and Reutenauer themselves. Using their results for periodic words, we have recently proved the conjecture for uniformly recurrent words. In the present article we prove the conjecture in its general version by a new method without exploiting the result for periodic words.Comment: 9 page

Words and polynomial invariants of finite groups in non-commutative variables

Let V be a complex vector space with basis {x_1,x_2,...,x_n} and G be a finite subgroup of GL(V). The tensor algebra T(V) over the complex is isomorphic to the polynomials in the non-commutative variables x_1, x_2,..., x_n with complex coefficients. We want to give a combinatorial interpretation for the decomposition of T(V) into simple G-modules. In particular, we want to study the graded space of invariants in T(V) with respect to the action of G. We give a general method for decomposing the space T(V) into simple modules in terms of words in a Cayley graph of the group G. To apply the method to a particular group, we require a homomorphism from a subalgebra of the group algebra into the character algebra. In the case of G as the symmetric group, we give an example of this homomorphism from the descent algebra. When G is the dihedral group, we have a realization of the character algebra as a subalgebra of the group algebra. In those two cases, we have an interpretation for the graded dimensions of the invariant space in term of those words

Benjamin Franklin: Religion and Freedom

The paper considers Benjamin Franklin’s writings on religious matters, as well as his interaction with religious personae and institutions, on a culturological level. In this, his Autobiography (1791) is the primary source, as are three principal essays he published on the matter during his lifetime: “A Dissertation on Liberty and Necessity” (1725), “Articles of Belief and Acts of Religion” (1728), and “On the Providence of God in the Government of the World” (1732). From these sources, an attempt to reconstruct Franklin’s curious approach to religion, cosmology and the concept of God is made, and the trajectory along which his opinions seem to have shifted is traced. Most importantly, it is argued that, for all the different approaches to religion Franklin exhibited throughout his lifetime, his stance on religion is in a metonymic relation with his political orientation as a Founding Father of the United States. That is, religious freedom he advocated is ostensibly a manifestation of his grander approach to freedom of any kind, which American cultural identity is based on. This freedom is also considered in relation to Franklin’s stance towards slave owning and towards Native Americans

Religion, Imagination and Revolution in William Blake’s “The Tyger” and Percy Bysshe Shelley’s “Mont Blanc”

The Romantic era represented a considerable artistic and philosophical paradigm shift towards subjectivity in perceiving and portraying the world around us, and thus heavily relied on the fusion of several distinct topics: religion, politics, social matters, nature, art, etc. In the context of British Romanticism, or more precisely British Romantic poetry, one might point out the importance of the “great six” poets: William Blake, William Wordsworth, Samuel Taylor Coleridge, Lord Byron, Percy Bysshe Shelley, and John Keats. This paper comparatively takes into consideration Blake and Shelley, and singles out two of their poems which are generally considered to represent some of their finest work – Blake’s “The Tyger” and Shelley’s “Mont Blanc”. The comparison of the poems yields, in spite of their rarely being considered as companion pieces, great philosophical and poetic resemblances. Both authors saw nature, albeit in unique ways, as a logical extension of the poet’s imagination and of religious considerations, which becomes evident upon reading the poems. The similitudes demonstrated here serve as evidence of the complex entanglement of Romantic ideologies, both on the level of a single author or a single poem and on the level of time and place (turn-of-the-nineteenth-century England). Furthermore, the paper situates the said poems in a broader European political context and delineates the possible influence of that context on their creation

On the critical exponent of generalized Thue-Morse words

For certain generalized Thue-Morse words t, we compute the "critical exponent", i.e., the supremum of the set of rational numbers that are exponents of powers in t, and determine exactly the occurrences of powers realizing it.Comment: 13 pages; to appear in Discrete Mathematics and Theoretical Computer Science (accepted October 15, 2007

Benjamin Franklin: Religion and Freedom

The paper considers Benjamin Franklin’s writings on religious matters, as well as his interaction with religious personae and institutions, on a culturological level. In this, his Autobiography (1791) is the primary source, as are three principal essays he published on the matter during his lifetime: “A Dissertation on Liberty and Necessity” (1725), “Articles of Belief and Acts of Religion” (1728), and “On the Providence of God in the Government of the World” (1732). From these sources, an attempt to reconstruct Franklin’s curious approach to religion, cosmology and the concept of God is made, and the trajectory along which his opinions seem to have shifted is traced. Most importantly, it is argued that, for all the different approaches to religion Franklin exhibited throughout his lifetime, his stance on religion is in a metonymic relation with his political orientation as a Founding Father of the United States. That is, religious freedom he advocated is ostensibly a manifestation of his grander approach to freedom of any kind, which American cultural identity is based on. This freedom is also considered in relation to Franklin’s stance towards slave owning and towards Native Americans