The Extended Bigraded Toda hierarchy

We generalize the Toda lattice hierarchy by considering N+M dependent variables. We construct roots and logarithms of the Lax operator which are uniquely defined operators with coefficients that are $\epsilon$-series of differential polynomials in the dependent variables, and we use them to provide a Lax pair definition of the extended bigraded Toda hierarchy. Using R-matrix theory we give the bihamiltonian formulation of this hierarchy and we prove the existence of a tau function for its solutions. Finally we study the dispersionless limit and its connection with a class of Frobenius manifolds on the orbit space of the extended affine Weyl groups of the $A$ series.Comment: 32 pages, corrected typo

Integrable discrete systems on R and related dispersionless systems

The general framework for integrable discrete systems on R in particular containing lattice soliton systems and their q-deformed analogues is presented. The concept of regular grain structures on R, generated by discrete one-parameter groups of diffeomorphisms, through which one can define algebras of shift operators is introduced. Two integrable hierarchies of discrete chains together with bi-Hamiltonian structures are constructed. Their continuous limit and the inverse problem based on the deformation quantization scheme are considered.Comment: 19 page

Tau function and Hirota bilinear equations for the Extended bigraded Toda Hierarchy

In this paper we generalize the Sato theory to the extended bigraded Toda hierarchy (EBTH). We revise the definition of the Lax equations,give the Sato equations, wave operators, Hirota bilinear identities (HBI) and show the existence of $tau$ function $\tau(t)$. Meanwhile we prove the validity of its Fay-like identities and Hirota bilinear equations (HBEs) in terms of vertex operators whose coefficients take values in the algebra of differential operators. In contrast with HBEs of the usual integrable system, the current HBEs are equations of product of operators involving $e^{\partial_x}$ and $\tau(t)$.Comment: 29 pages, to appear Journal of Mathematical Physics(2010

Motor Parameter-Free Predictive Current Control of Synchronous Motors by Recursive Least-Square Self-Commissioning Model

This article deals with a finite-set model predictive current control in synchronous motor drives. The peculiarity is that it does not require the knowledge of any motor parameter. The inherent advantage of this method is that the control is self-adapting to any synchronous motor, thus easing the matching between motor and inverter coming from different manufacturers. Overcoming the flaws of the existing lookup table based parameter-free techniques, the article elaborates the past current measurements by a recursive least-square algorithm to estimate the future behavior of the current in response to a finite set of voltage vectors. The article goes through the mathematical basis of the algorithm till a complete set of experiments that prove the feasibility and the advantages of the proposed technique

An Effective Model-Free Predictive Current Control for Synchronous Reluctance Motor Drives

The performances of a model predictive control algorithm largely depend on the knowledge of the system model. A model-free predictive control approach skips all the effects of parameters variations or mismatches, as well as of model nonlinearity and uncertainties. A finite-set model-free current predictive control is proposed in this paper. The current variations predictions induced by the eight base inverter voltage vectors are estimated by means of the previous measurements stored into lookup tables. To keep the current variations information up to date, the three current measurements due to the three most recent feeding voltages are combined together to reconstruct all the others. The reconstruction is performed by taking advantage of the relationships between the three different base voltage vectors involved in the process. In particular, 210 possible combinations of three-state voltage vectors can be found, but they can be gathered together in six different groups. A light and computationally fast algorithm for the group identification is proposed in this paper. Finally, the current reconstruction for the prediction of future steps is thoroughly analyzed. A compensation of the motor rotation effect on the input voltages is proposed, too. The control scheme is evaluated by means of both simulation and experimental evidences on two different synchronous reluctance motors

The multicomponent 2D Toda hierarchy: Discrete flows and string equations

The multicomponent 2D Toda hierarchy is analyzed through a factorization problem associated to an infinite-dimensional group. A new set of discrete flows is considered and the corresponding Lax and Zakharov--Shabat equations are characterized. Reductions of block Toeplitz and Hankel bi-infinite matrix types are proposed and studied. Orlov--Schulman operators, string equations and additional symmetries (discrete and continuous) are considered. The continuous-discrete Lax equations are shown to be equivalent to a factorization problem as well as to a set of string equations. A congruence method to derive site independent equations is presented and used to derive equations in the discrete multicomponent KP sector (and also for its modification) of the theory as well as dispersive Whitham equations.Comment: 27 pages. In the revised paper we improved the presentatio

Doubly Perfect Nonlinear Boolean Permutations

Due to implementation constraints the XOR operation is widely used in order to combine plaintext and key bit-strings in secret-key block ciphers. This choice directly induces the classical version of the differential attack by the use of XOR-kind differences. While very natural, there are many alternatives to the XOR. Each of them inducing a new form for its corresponding differential attack (using the appropriate notion of difference) and therefore block-ciphers need to use S-boxes that are resistant against these nonstandard differential cryptanalysis. In this contribution we study the functions that offer the best resistance against a differential attack based on a finite field multiplication. We also show that in some particular cases, there are robust permutations which offers the best resistant against both multiplication and exponentiation base differential attacks. We call them doubly perfect nonlinear permutations

On the Complexity of Computing Two Nonlinearity Measures

We study the computational complexity of two Boolean nonlinearity measures: the nonlinearity and the multiplicative complexity. We show that if one-way functions exist, no algorithm can compute the multiplicative complexity in time $2^{O(n)}$ given the truth table of length $2^n$, in fact under the same assumption it is impossible to approximate the multiplicative complexity within a factor of $(2-\epsilon)^{n/2}$. When given a circuit, the problem of determining the multiplicative complexity is in the second level of the polynomial hierarchy. For nonlinearity, we show that it is #P hard to compute given a function represented by a circuit