The standard OpenMath is an enabling technology for creating an integrated computer environment in which software packages for computer algebra and for proof checking can be combined. Here we demonstrate how OpenMath can be employed for generating interactive mathematical documents containing primality proofs. Our case study takes place within a browser; once a prime number is specified, a docu...

Let M be a compact manifold and D a Dirac type differential operator on M . Let A be a C∗-algebra. Given a bundle W (with connection) of A-modules over M , the operator D can be twisted with this bundle. One can then use a trace on A to define numerical indices of this twisted operator. We prove an explicit formula for these indices. Our result does complement the Mishchenko–Fomenko index theor...

We present recent results about Capelli polynomials with involution or graded and their asymptotics. In the associative case, asymptotic equality between codimensions of $T$-ideal generated by polynomial rank $k^2+1$ matrix algebra $M_k(F)$ was proved. This result extended to superalgebras. Recently, similar have been determined authors in case algebras superalgebras involution.

In this paper, we study the correlation functions of quantum toroidal $\mathfrak{gl}_1$ algebra. The first key properties establish are similar to those affine algebras $U_q\mathfrak{n}_+$ as established by Enriquez in (Eneiquez, 2000), while proof remaining ``vanishing property" relies on a certain ``Master Equality'' formal series, which constitutes main technical res...

The boundary map in K-theory arising from the Wiener-Hopf extension of a crossed product algebra with R is the Connes-Thom isomorphism. In this article the Wiener Hopf extension is combined with the Heisenberg group algebra to provide an elementary construction of a corresponding map on higher traces (and cyclic cohomology). It then follows directly from a non-commutative Stokes theorem that th...

It was proposed recently that the near-horizon states of an extremal four-dimensional Kerr black hole could be identified with a certain chiral conformal field theory whose Virasoro algebra arises as an asymptotic symmetry algebra of the near-horizon Kerr geometry. Supportive evidence for the proposed duality came from the equality of the microscopic entropy of the CFT, calculated by means of t...

For a twisted partial action Θ of a group G on an (associative nonnecessarily unital) algebra A over a commutative unital ring k, the crossed product A⋊ΘG is proved to be associative. Given a G-graded k-algebra B = ⊕g∈GBg with the mild restriction of homogeneous non-degeneracy, a criteria is established for B to be isomorphic to the crossed product B1⋊ΘG for some twisted partial action of G on ...

This paper addresses the dialectical relation between theoretical thinking and technique, as they coemerge in a combined computer algebra and paper-and-pencil environment. The theoretical framework in this ongoing study consists of the instrumental approach to tool use and an adaptation of Chevallard’s anthropological theory. The main aim is to unravel the subtle intertwining of students’ theor...

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