The Integral on Quantum Super Groups of Type Ar∣sA_{r|s}

We compute the integral on matrix quantum (super) groups of type $A_{r|s}$ and derive from it the quantum analogue of (super) HCIZ integral.Comment: 11 pages, latex 2.09 using amsart style file

Poincar\'e Series of Quantum Spaces Associated to Hecke Operators

We study the Poincar\'e series of the quantum spaces associated to a Hecke operator, i.e., a Yang-Baxter operator satisfying the equation $(x+1)(x-q)=0$. The Poincar\'e series of the corresponding matrix bialgebra is also considered. Using an old result on Poly\'a frequency sequence, we show that the Poincar\'e series of quantum spaces are always rational functions having negative roots and positive poles. In particular, we show that the rank of an even Hecke operator should be rational functions having negative roots and positive poles. In particular, we show that the rank of an even Hecke operator should be greater than the dimension of the vector space it is acting on.Comment: latex 2.09, amsart style, 8 page

On the representation categories of matrix quantum groups of type AA

A quantum groups of type $A$ is defined in terms of a Hecke symmetry. We show in this paper that the representation category of such a quantum group is uniquely determined as an abelian braided monoidal category by the bi-rank of the Hecke symmetry.Comment: 9 page

The homological determinant of quatum groups of type A

A quantum group of type A is defined as a Hopf algebra associated to a Hecke symmetry. We show the homology of a Koszul complex associated to the Hecke symmetry is one dimensional and determines a group-like element in the Hopf algebra. This group-like element can be interpreted as a homological determinant as suggested by Yu. Manin.Comment: 6 page

Batch Arrival Multiserver Queue with Setup Time

Queues with setup time are extensively studied because they have application in performance evaluation of power-saving data centers. In a data center, there are a huge number of servers which consume a large amount of energy. In the current technology, an idle server still consumes about 60\% of its peak processing a job. Thus, the only way to save energy is to turn off servers which are not processing a job. However, when there are some waiting jobs, we have to turn on the OFF servers. A server needs some setup time to be active during which it consumes energy but cannot process a job. Therefore, there exists a trade-off between power consumption and delay performance. Gandhi et al. \cite{Gandhi10a,Gandhi10} analyze this trade-off using an M/M/$c$ queue with staggered setup (one server in setup at a time). In this paper, using an alternative approach, we obtain generating functions for the joint stationary distribution of the number of active servers and that of jobs in the system for a more general model with batch arrivals and state-dependent setup time. We further obtain moments for the queue size. Numerical results reveal that keeping the same traffic intensity, the mean power consumption decreases with the mean batch size for the case of fixed batch size. One of the main theoretical contribution is a new conditional decomposition formula showing that the number of waiting customers under the condition that all servers are busy can be decomposed to the sum of two independent random variables where the first is the same quantity in the corresponding model without setup time while the second is the number of waiting customers before an arbitrary customer

Realizations of quantum hom-spaces, invariant theory and quantum determinantal ideals

For a Hecke operator $R$, one defines the matrix bialgebra \E_R, which is considered as the function algebra on the quantum space of endomorphisms of the quantum space associated to $R$. One generalizes this notion, defining the function algebra \M_{RS} on the quantum space of homomorphisms of two quantum spaces associated to two Hecke operators $R$ and $S$ respectively. \M_{RS} can be considered as a quantum analogue (or a deformation) of the function algebra on the variety of matrices of a certain degree. We provide two realiztions of \M_{RS} as a quotient algebra and as a subalgebra of a tensor algebra, whence derive interesting informations about \M_{RS}, for instance the Koszul property, a formula for computing the Poincar\'e series. On \M_{RS} coact the bialgebras \E_R and \E_S. We study the two-sided ideals in \M_{RS}, invariant with respect to these actions, in particular, the determinantal ideals. We prove analogies of the fundamental theorems on invariant theory for these quantum groups and quantum hom-spaces.Comment: latex 2.09, amsart style, 28 page

Koszul Property and Poincare' Series of Matrix-Bialgebras of Type ANA_N

Bialgebras associated to Yang-Baxter operators satisfying the Hecke equation, are considered. It is shown that they are Koszul algebras. Their Poincare' series are calculated via the Poincare' series of the corresponding quantum spaces.Comment: 15 pages, standard latex fil

On Matrix Quantum Groups of type AnA_n

Given a Hecke symmetry $R$, one can define a matrix bialgebra $E_R$ and a matrix Hopf algebra $H_R$, which are called function rings on the matrix quantum semi-group and matrix quantum groups associated to $R$. We show that for an even Hecke symmetry, the rational representations of the corresponding quantum group are absolutely reducible and that the fusion coefficients of simple representations depend only on the rank of the Hecke symmetry. Further we compute the quantum rank of simple representations. We also show that the quantum semi-group is ``Zariski'' dense in the quantum group. Finally we give a formula for the integral.Comment: Ams-Latex file, 26 pages, bezier style, use style file grcalc.st

The cost of approximate controllability for semilinear heat equations in one space dimension

This paper deals with the approximate controllability for the semilinear heat equation in one space dimension. Our aim is to provide an estimate of the cost of the control

Integrals on Hopf algebras and Application to Representation Theory of Quantum Groups of Type A0∣0A_{0|0}

In this work we study some properties of comldules over (non-cosemisimple) Hopf algebras possessing integrals, which are also called co-Frobenius Hopf algebras. We apply the result obtained to the classification of representations of quantum groups of type $A_{0|0}$.Comment: 20 pages, latex 2.09, amsart styl