Related papers: Experimenting with the Garsia-Milne Involution Pri…
In 1917, G.H.Hardy and S.Ramanujan proved that the `typical' number of prime factors of a positive integer $n$ is approximately $\ln\ln n$. In this technical paper we proffer a complete exposition of this proof, and further provide novel…
In his 1984 proof of the Bieberbach and Milin conjectures de Branges used a positivity result of special functions which follows from an identity about Jacobi polynomial sums that was found by Askey and Gasper in 1973, published in 1976. In…
We study linearizability properties of finite subgroups of the Cremona group ${\mathrm{Cr}}_n(k)$ in the case where $k$ is a global field, with the focus on the local-global principle. For every global field $k$ of characteristic different…
In this paper we prove a refined major-balance identity on the $321$-avoiding involutions of length $n$, respecting the leading element of permutations. The proof is based on a sign-reversing involution on the lattice paths within a…
An identity by Ramanujan is expressed using polar coordinates, so that its proof reduces to the verification of an elementary trigonometric identity. This approach produces a few variations on Ramanujan's original identity.
Inductive inference is a recursion-theoretic theory of learning, first developed by E. M. Gold (1967). This paper surveys developments in probabilistic inductive inference. We mainly focus on finite inference of recursive functions, since…
Let $G$ be a connected reductive group defined over a finite field $\mathbb{F}_q$ with corresponding Frobenius $F$. Let $\iota_G$ denote the duality involution defined by D. Prasad under the hypothesis $2\mathrm{H}^1(F,Z(G))=0$, where…
We defend a new theory of statistical evidence, which we call Robust Bayesianism (RB). We prove that, under widely accepted assumptions, RB entails the law of likelihood [Royall, 1997], the likelihood principle [Berger and Wolpert, 1988],…
Ramanujan stated an identity to the effect that if three sequences $\{a_n\}$, $\{b_n\}$ and $\{c_n\}$ are defined by $r_1(x)=:\sum_{n=0}^{\infty}a_nx^n$, $r_2(x)=:\sum_{n=0}^{\infty}b_nx^n$ and $r_3(x)=:\sum_{n=0}^{\infty}c_nx^n$ (here each…
In the note, the authors give a unified proof of Identities~67, 84, and~85 in the monograph "M. Z. Spivey, The Art of Proving Binomial Identities, Discrete Mathematics and its Applications, CRC Press, Boca Raton, FL, 2019; available online…
This paper verifies Abhyankar's Inertia Conjecture for certain sporadic groups $G$ in particular characteristics by showing that all possible inertia groups occur for $G$-Galois covers of the affine line. For a larger set of sporadic…
Baxter numbers are known to count several families of combinatorial objects, all of which come equipped with natural involutions. In this paper, we add a combinatorial family to the list, and show that the known bijections between these…
Laguerre histories (restricted or not) are certain weighted Motzkin paths with two types of level steps. They are, on one hand, in natural bijection with the set of permutations, and on the other hand, yield combinatorial interpretations…
A long-standing conjecture asserts that the polynomial \[p(t) = \text{Tr}[(A+tB)^m]\] has nonnegative coefficients whenever $m$ is a positive integer and $A$ and $B$ are any two $n \times n$ positive semidefinite Hermitian matrices. The…
In this survey article, we present an expanded version of Lucy Slater's famous list of identities of the Rogers-Ramanujan type, including identities of similar type, which were discovered after the publication of Slater's papers, and older…
We show that there is a bijection between real-linear automorphisms of the multicomplex numbers of order $n$ and signed permutations of length $2^{n-1}$. This allows us to deduce a number of results on the multicomplex numbers, including a…
The phenomenon of gene conservation is an interesting evolutionary problem related to speciation and adaptation. Conserved genes are acted upon in evolution in a way that preserves their function despite other structural and functional…
We give direct bijective proofs of the symmetry of the distributions of the number of ascents and descents over standard Young tableaux of shape $\lambda$, where $\lambda$ is a rectangle $(n,n,\dots,n)$ or a truncated staircase…
We give another proof of the second Rogers-Ramanujan identity by Kashiwara crystals.
Several new transformations for q-binomial coefficients are found, which have the special feature that the kernel is a polynomial with nonnegative coefficients. By studying the group-like properties of these positivity preserving…