Classification of graded Hecke algebras for complex reflection groups

The graded Hecke algebra for a finite Weyl group is intimately related to the geometry of the Springer correspondence. A construction of Drinfeld produces an analogue of a graded Hecke algebra for any finite subgroup of GL(V). This paper classifies all the algebras obtained by applying Drinfeld's construction to complex reflection groups. By giving explicit (though nontrivial) isomorphisms, we show that the graded Hecke algebras for finite real reflection groups constructed by Lusztig are all isomorphic to algebras obtained by Drinfeld's construction. The classification shows that there exist algebras obtained from Drinfeld's construction which are not graded Hecke algebras as defined by Lusztig for real as well as complex reflection groups. Received: July 25, 2001     

On Linear Representations of Analytic Moufang Loops

Mathematical Notes -     

A Construction of Difference Sets in High Exponent 2-Groups Using Representation Theory

Nontrivial difference sets in groups of order a power of 2 are part of the family of difference sets called Menon difference sets (or Hadamard), and they have parameters (2^2d+2, 2^2d+1±2 ^d , 2^2d ±2 ^d ). In the abelian case, the group has a difference set if and only if the exponent of the group is less than or equal to 2 ^d+2. In [14], the authors construct a difference set in a nonabelian group of order 64 and exponent 32. This paper generalizes that result to show that there is a difference set in a nonabelian group of order 2^2d+2 with exponent 2 ^d+3. We use representation theory to prove that the group has a difference set, and this shows that representation theory can be used to verify a construction similar to the use of character theory in the abelian case.     

Cyclic reduction and FACR methods for piecewise hermite bicubic orthogonal spline collocation

Cyclic reduction and Fourier analysis-cyclic reduction (FACR) methods are presented for the solution of the linear systems which arise when orthogonal spline collocation with piecewise Hermite bicubics is applied to boundary value problems for certain separable partial differential equations on a rectangle. On anN×N uniform partition, the cyclic reduction and Fourier analysis-cyclic reduction methods requireO(N ²log₂ N) andO(N ²log₂log₂ N) arithmetic operations, respectively.     

Lax表示的变形与Hamilton方程族的Lax表示

本文首先给出了构造演化方程族的Ｌａｘ表示的马文秀方法的一种变形，后对这一方法作了改进，使之适用于Ｈａｍｉｌｔｏｎ形式的方程族．作为应用，得到了具有非等谱Ｌａｘ表示的杨方程族．     

二元样条的积分表示及分层三角剖分下二次样条空间的维数

本文通过引入一个积分协调条件，首次给出了二元样条的一个积分表示．文中还定义了平面单连通多边形区域的所谓分层三角剖分，并确定了此剖分下二次样条空间的维数．     

Neural network representations of quantum many-body states

Machine learning is currently the most active interdisciplinary field having numerous applications;additionally,machine-learning techniques are used to research quantum many-body problems.In this study,we first propose neural network quantum states(NNQSs)with general input observables and explore a few related properties,such as the tensor product and local unitary operation.Second,we determine the necessary and sufficient conditions for the representability of a general graph state using normalized NNQS.Finally,to quantify the approximation degree of a given pure state,we define the best approximation degree using normalized NNQSs.Furthermore,we observe that some 7V-qubit states can be represented by a normalized NNQS,such as separable pure states,Bell states and GHZ states.     

New Interfaces and Approaches to Machine Learning When Classifying Gestures within Music

Interactive music uses wearable sensors (i.e., gestural interfaces—GIs) and biometric datasets to reinvent traditional human–computer interaction and enhance music composition. In recent years, machine learning (ML) has been important for the artform. This is because ML helps process complex biometric datasets from GIs when predicting musical actions (termed performance gestures). ML allows musicians to create novel interactions with digital media. Wekinator is a popular ML software amongst artists, allowing users to train models through demonstration. It is built on the Waikato Environment for Knowledge Analysis (WEKA) framework, which is used to build supervised predictive models. Previous research has used biometric data from GIs to train specific ML models. However, previous research does not inform optimum ML model choice, within music, or compare model performance. Wekinator offers several ML models. Thus, we used Wekinator and the Myo armband GI and study three performance gestures for piano practice to solve this problem. Using these, we trained all models in Wekinator and investigated their accuracy, how gesture representation affects model accuracy and if optimisation can arise. Results show that neural networks are the strongest continuous classifiers, mapping behaviour differs amongst continuous models, optimisation can occur and gesture representation disparately affects model mapping behaviour; impacting music practice.     

The vertical spectrum of H2CO revisited: (SC)2-CI and CC calculations

The vertical electronic spectrum of formaldehyde has been studied by means of (SC)²-MR-SDCI and CCLR methods. Two basis sets of atomic natural orbitals (ANOs) complemented with a one-centre series of Rydberg orbitals were used. The first was taken from the CASPT2 study by Merchán, M., and Roos, B. O., 1995, Theoret. Chim. Acta, 92, 221, and may be described as C,O[4s3pld]/H[2slp] with a lslpld Rydberg series centred in the charge centroid of the ²B₂ state of the cation. The second was a larger basis set that may be described as C,O[6s5p3d2f]/H[4s3p2d] + 3s3p3d in the same centre. The (SC)² dressing may be applied efficiently to an MR-SDCI method and comparison with the dressed CAS-SDCI is satisfactory, in spite of the remarkable reduction in the CI space dimension. The consistency of the (SC)²-MR-SDCI results was tested also against the CCLR and CASPT2 results using the same basis sets and against the CCLR results using Dunning's aug- and daug-cc-pVQZ basis sets. The ³A₁(π → π *) state is correctly placed as the second excited triplet while the highly multi-configurational nature of the ¹A₁(π → π *) state is confirmed as well as its greatly mixed valence-Rydberg nature. This state is predicted as lying under the 10 eV level, on top of the (n_y → 3d) Rydberg states that are predicted in the 8.9–9.5eV region. The 5 ¹B₂(n^y → 4s) Rydberg state and the ¹B₂(σ_y → π*) also are predicted in this region. The triplet states also were calculated with the (SC)²-MR-SDCI method. The vertical ordering of the 2 ¹A₁(n_y → 3p_y) and 2 ¹B₂(n_y → 3p_z) states is discussed, as well as that of the ¹B₁(σ → π*) and the Rydberg ¹B₁(n_y → 3d_xy) states. This work shows the highly reliable values that may be reached applying the dressing method along with a large basis set. Such a procedure is made possible using an MR-SDCI selection of spaces instead of the CAS-SDCI that was used up to now in most (SC)² dressing applications.     

Application of three-mode Einstein-Podolsky-Rosen entangled state with continuous variables to teleportation

We consider how to teleport two- and three-mode Einstein--Podolsky--Rosen entangled states (|\eta> and | p_t,\chi₂,\chi₃>) via a | p_t,\chi₂,\chi₃> quantum channel for continuous variables. Using the complete and orthogonal representation of the entangled states, we can not only find the a complete basis set for the joint measurement but also propose the specific scheme of teleportation. Our calculation can be greatly simplified by using their Schmidt decompositions.