Related papers: An alternative to Vaughan's identity
In this note we present a method for obtaining a wide class of combinatorial identities. We give several examples, in particular, based on the Gamma and Beta functions. Some of them have already been considered by previously, and other are…
We prove that non-trivial representations of the alternating group $A_n$ are reducible over a primitive proper subgroup which is isomorphic to some alternating group $A_m$.
Starting from the characteristic polynomial for ordinary matrices we give a combinatorial deduction of the Mandelstam identities and viceversa, thus showing that the two sets of relations are equivalent. We are able to extend this…
We give a pen and paper and (comparatively) much simpler proof to verify of the Four Colour Theorem.
Many assumptions that underpin human concepts of identity do not hold for machine minds that can be copied, edited, or simulated. We argue that there exist many different coherent identity boundaries (e.g.\ instance, model, persona), and…
We consider certain functional identities on the matrix algebra $M_n$ that are defined similarly as the trace identities, except that the "coefficients" are arbitrary polynomials, not necessarily those expressible by the traces. The main…
Several combinatorial identities are presented, involving Stirling functions of the second kind with a complex variable. The identities involve also Stirling numbers of the first kind, binomial coefficients and harmonic numbers.
This purpose of this paper is to note an interesting identity derived from an integral in Gradshteyn and Ryzhik using techniques from George Boros'(deceased) Ph.D thesis. The idenity equates a sum to a product by evaluating an integral in…
We give an alternative proof for the equivalence of two definitions of the totally positive grassmannian.
A very short, bijective proof, of Touchard's Catalan identity is given, using Dyck paths.
In the context of chit-chat dialogues it has been shown that endowing systems with a persona profile is important to produce more coherent and meaningful conversations. Still, the representation of such personas has thus far been limited to…
We claim to resolve the P=?NP problem via a formal argument for P=NP.
In this paper we introduce two general identities relating Eisenstein series on split classical groups, as well as double covers of symplectic groups. The first identity can be viewed as an extension of the doubling construction introduced…
An agent may strategically employ a vague message to mislead an audience's belief about the state of the world, but this may cause the agent to feel guilt or negatively impact how the audience perceives the agent. Using a novel experimental…
An technically interesting proof of a known theorem.
We show that the Bailey lattice can be extended to a bilateral version in just a few lines from the bilateral Bailey lemma, using a very simple lemma transforming bilateral Bailey pairs relative to $a$ into bilateral Bailey pairs relative…
The normal distribution is well-known for several results that it is the only to fulfil. The aim of the present paper is to show that many of these characterizations actually follow from the fact that the derivative of the log-density of…
A. Lacasse conjectured a combinatorial identity in his study of learning theory. Various people found independent proofs. Here is another one that is based on the study of the tree function, with links to Lamberts $W$-function and…
Motivated by a polynomial identity of certain iterated integrals, first observed in [CGM20] in the setting of lattice paths, we prove an intriguing combinatorial identity in the shuffle algebra. It has a close connection to de Bruijn's…
A straightforward argument shows that, by allowing counterfactual elements of physical reality, any arbitrary discrete finite-dimensional operator corresponds to an observable.