Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of weakly continuous convolution semigroups of functionals on a C∗-bialgebra, the noncommutative counterpart of a locally compact semigroup. On locally compact quantum groups we obtain a bijective cor...

We study decoupling limits and S-dualities for noncommutative open string/ Yang-Mills theory in a gravity setup by considering an SL(2, Z) invariant super-gravity solutions of the form ((F, D1), D3) bound state of type IIB string theory. This configuration can be regarded as D3-brane solution with both electric and magnetic fields turned on along one of the spatial directions of the brane and p...

For a smooth harmonic map flow u : M× [0, T ) → N with blow-up as t ↑ T , it has been asked ([6], [5], [7]) whether the weak limit u(T ) : M→ N is continuous. Recently, in [12], we showed that in general it need not be. Meanwhile, the energy function E(u(·)) : [0, T ) → R, being weakly positive, smooth and weakly decreasing, has a continuous extension to [0, T ]. Here we show that if this exten...

Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of weakly continuous convolution semigroups of functionals on a C∗bialgebra, the noncommutative counterpart of locally compact semigroup. On locally compact quantum groups we obtain a bijective corres...

We consider the DC (difference of two convex functions) optimization problem ðPÞ inf x∈Xfðf 1ðxÞ− f 2ðxÞÞ þ ðg1ðAxÞ− g2ðAxÞÞg, where f 1, f 2, g1, and g2 are proper convex functions defined on locally convex Hausdorff topological vector spaces X and Y , and A is a linear continuous operator from X to Y . Adopting different tactics, two types of the Fenchel dual problems of ðPÞ are given. By usi...