Unitary Matrices and Hermitian Matrices Recall that the conjugate of a complex number a + bi is a − bi. The conjugate of a + bi is denoted a+ bi or (a+ bi)∗. In this section, I’ll use ( ) for complex conjugation of numbers of matrices. I want to use ( )∗ to denote an operation on matrices, the conjugate transpose. Thus, 3 + 4i = 3− 4i, 5− 6i = 5 + 6i, 7i = −7i, 10 = 10. Complex conjugation sati...

The main theorem of this paper states that ifM is a closed orientable hyperbolic 3-manifold of volume at most 3.08, then the dimension of H1(M ;Z2) is at most 7, and that it is at most 6 unless M is “strange.” To say that a closed, orientable 3-manifold M , for which H1(M ;Z2) has dimension 7, is strange means that the Z2-vector space H1(M ;Z2) has a 2-dimensional subspace X such that for every...

F3 and TAG-1 are two closely related adhesion glycoproteins of the Ig superfamily that are both expressed by the axons of cerebellar granule cells. In an in vitro system in which cerebellar granule cells were cultured on monolayers of transfected Chinese hamster ovary (CHO) cells, we show that F3 and TAG-1 interact functionally. F3 transfectants have been shown to inhibit outgrowth and induce f...

We study D1-branes on the fourfold C/(Z2 × Z2 × Z2), in the presence of discrete torsion. Discrete torsion is incorporated in the gauge theory of the D1-branes by considering a projective representation of the finite group Z2 × Z2 × Z2. The corresponding orbifold is then deformed by perturbing the F-flatness condition of the gauge theory. The moduli space of the resulting gauge theory retains a...

Given a group-based Markov model on a tree, one can compute the vertex representation of a polytope which describes the associated toric variety. The half-space representation, however, is not easily computable. In the case of Z2 or Z2 × Z2, these polytopes have applications in the field of phylogenetics. We provide a half-space representation for the m-claw tree where G = Z2 × Z2, which corres...

In this paper we introduce self-dual cyclic and quantum codes over Zα2 × (Z2 + uZ2) . We determine the conditions for any Z2Z2[u]-cyclic code to be self-dual, that is, C = C. Since the binary image of a selforthogonal Z2Z2[u]-linear code is also a self-orthogonal binary linear code, we introduce quantum codes over Zα2 × (Z2 + uZ2) β . Finally, we present some examples of self-dual cyclic and qu...

Inspired by the Z2Z4-additive codes, linear codes over Z r 2×(Z2 + uZ2) s have been introduced by Aydogdu et al. more recently. Although these family of codes are similar to each other, linear codes over Zr2×(Z2 + uZ2) s have some advantages compared to Z2Z4-additive codes. A code is called constant weight(one weight) if all the codewords have the same weight. It is well known that constant wei...

This paper reports on two listening experiments which explore the relative contributions of F2 and F3 in the perception of rhoticity. Experiment 1 tests the hypothesis that a low-frequency F3 is a crucial spectral component for a signal to be perceived as rhotic. Its results suggest that the removal of F3 may in fact strengthen the rhoticity percept. Experiment 2 manipulates the relative amplit...

The main theorem of this paper states that ifM is a closed orientable hyperbolic 3-manifold of volume at most 3.08, then the dimension of H1(M ;Z2) is at most 7, and that it is at most 6 unless M is “strange.” To say that a closed, orientable 3-manifold M , for which H1(M ;Z2) has dimension 7, is strange means that the Z2-vector space H1(M ;Z2) has a 2-dimensional subspace X such that for every...