In this paper the ideas of Rota and Ramanujan are shown to be central to understanding problems in additive number theory. The circle and sieve methods are two different facets of the same theme of interplay between probability and Fourier series used to great advantage by Wiener in engineering. Norbert Wiener forged a powerful tool for electrical engineers by combining two distinct branches of...

The Bogoliubov recursion is a particular procedure appearing in the process of renormalization in perturbative quantum field theory. It provides convergent expressions for otherwise divergent integrals. We develop here a theory of functional identities for noncommutative Rota–Baxter algebras which is shown to encode, among others, this process in the context of Connes–Kreimer’s Hopf algebra of ...

A Hilbert space operator is called universal (in the sense of Rota) if every on similar to a multiple restriction universal one its invariant subspaces. We exhibit an analytic Toeplitz whose adjoint in Rota and which commutes with quasinilpotent, injective, compact dense range, but unlike other examples, it acts Bergman instead Hardy this associated “hyperbolic” composition ope...

We formulate the Hopf algebraic approach of Connes and Kreimer to renormalization in perturbative quantum field theory using triangular matrix representation. We give a Rota– Baxter anti-homomorphism from general regularized functionals on the Feynman graph Hopf algebra to triangular matrices with entries in a Rota–Baxter algebra. For characters mapping to the group of unipotent triangular matr...

A rectangular rod is placed in a flow field flowing parallel to a flat plate. Effect of chord-thickness ratio of rectangular rod on developing vortex shedding downstream to the rod is studied. Then, for each one of the aspect ratios, the distance of the rod from the neighboring flat plate is reduced until the rod sticks to the flat plate. In each case, the effect of the flat plate boundary laye...

In this paper, first we introduce the notion of quadratic Rota–Baxter Lie algebras arbitrary weight, and show that there is a one-to-one correspondence between factorizable bialgebras nonzero weight. Then notions matched pairs, Manin triples bialgebras, certain pairs are equivalent. The coadjoint representations play important roles in whole study. Finally generalize some results to group conte...

A LieYRep pair consists of a Lie-Yamaguti algebra and its representation. In this paper, we establish the cohomology theory pairs characterize their linear deformations by second group. Then introduce notion relative Rota-Baxter-Nijenhuis structures on pairs, investigate properties, prove that structure gives rise to compatible Rota-Baxter operators under certain condition. Finally, show equiva...

One presents here the * ψ-Bernoulli-Taylor* formula of a new sort with the rest term of the Cauchy type recently derived by the author in the case of the so called ψ-difference calculus which constitutes the representative for the purpose case of extended umbral calculus. The central importance of such a type formulas is beyond any doubt-and recent publications do confirm this historically esta...

Mitragyna speciosa Korth. leaves have been traditionally used by Malay and Thai natives for its opiumlike effect and coca-like stimulant ability to enhance tolerance for hard work. Mitragynine is the major indole alkaloid in this plant. In the present study, we investigated the effects of the alkaloid extract of M. speciosa leaves and its major alkaloid mitragynine on exploratory behaviour, sho...