We construct a deformation of the quantum algebra Fun(T G) associated with Lie group G to the case where G is replaced by a quantum group Gq which has a bicovariant calculus. The deformation easily allows for the inclusion of the current algebra of left and right invariant one forms. We use it to examine a possible generalization of the Gauss law commutation relations for gauge theories based o...

Let G be an exceptional Lie group with a maximal torus T ⊂ G. We present the integral cohomology ring H∗(G/T ) by a minimal set of Schubert classes on G/T . This completes the program of determining the integral cohomology of all complete flag manifolds G/T in the context of Schubert calculus. The results have been applied in [DZ3] to construct the integral cohomology of a simple Lie group G in...

Let G be an exceptional Lie group with a maximal torus T ⊂ G. We express the integral cohomology ring H(G/T ) by a minimal set of Schubert classes on G/T . This completes the program of determining the integral cohomology of all complete flag manifolds G/T in the context of Schubert calculus. The results are used in [DZ3] to construct the integral cohomology of a simple Lie group G in terms of ...

Let G be a multigraph. We study the space X (G) of all pictures of G in complex projective d-space. The main result is that the homology groups (with integer coefficients) of X (G) are completely determined by the Tutte polynomial of G. One application is a criterion in terms of the Tutte polynomial for independence in the d-parallel matroids studied in combinatorial rigidity theory. In the cas...

It was recently shown (by the second author and D\'{i}az Garc\'{i}a, Krutov, Somberg, Strung) that every relative line module over an irreducible quantum flag manifold $\mathcal{O}_q(G/L_S)$ admits a unique $\mathcal{O}_q(G)$-covariant connection with respect to Heckenberger-Kolb differential calculus $\Omega^1_q(G/L_S)$. In this paper we show these connections are bimodule invertible associate...

Motivated by Kurt’s work [Filomat 30 (4) 921-927, 2016], we …rst consider a class of a new generating function for (p; q)-analog of Apostol type polynomials of order including Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials of order . By making use of their generating function, we derive some useful identities. We also introduce (p; q)-analog of Stirling numbers of second kind...

In 1933 Godel introduced a modal logic of provability (S4) and left open the problem of a formal provability semantics for this logic. Since then numerous attempts have been made to give an adequate provability semantics to Godel's provability logic with only partial success. In this paper we give the complete solution to this problem in the Logic of Proofs (LP). LP implements Godel's sugges...

We study the low-energy effective theory of N = 2 supersymmetric Yang-Mills theory with the exceptional gauge group G 2. We obtain the Picard-Fuchs equations for the G 2 spectral curve and compute multi-instanton contribution to the prepotential. We find that the spectral curve is consistent with the microscopic su-persymmetric instanton calculus. It is also shown that G 2 hyperelliptic curve d...