2,648 research outputs found
On scattered convex geometries
A convex geometry is a closure space satisfying the anti-exchange axiom. For
several types of algebraic convex geometries we describe when the collection of
closed sets is order scattered, in terms of obstructions to the semilattice of
compact elements. In particular, a semilattice , that does not
appear among minimal obstructions to order-scattered algebraic modular
lattices, plays a prominent role in convex geometries case. The connection to
topological scatteredness is established in convex geometries of relatively
convex sets.Comment: 25 pages, 1 figure, submitte
Complex Hadamard matrices contained in a Bose-Mesner algebra
A complex Hadamard matrix is a square matrix H with complex entries of
absolute value 1 satisfying , where stands for the Hermitian
transpose and I is the identity matrix of order . In this paper, we first
determine the image of a certain rational map from the -dimensional complex
projective space to . Applying this result with ,
we give constructions of complex Hadamard matrices, and more generally, type-II
matrices, in the Bose-Mesner algebra of a certain 3-class symmetric association
scheme. In particular, we recover the complex Hadamard matrices of order 15
found by Ada Chan. We compute the Haagerup sets to show inequivalence of
resulting type-II matrices, and determine the Nomura algebras to show that the
resulting matrices are not decomposable into generalized tensor products.Comment: 28 pages + Appendix A + Appendix
Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements
We prove a Gleason-type theorem for the quantum probability rule using frame
functions defined on positive-operator-valued measures (POVMs), as opposed to
the restricted class of orthogonal projection-valued measures used in the
original theorem. The advantage of this method is that it works for
two-dimensional quantum systems (qubits) and even for vector spaces over
rational fields--settings where the standard theorem fails. Furthermore, unlike
the method necessary for proving the original result, the present one is rather
elementary. In the case of a qubit, we investigate similar results for frame
functions defined upon various restricted classes of POVMs. For the so-called
trine measurements, the standard quantum probability rule is again recovered.Comment: 10 pages RevTeX, no figure
Multiscale Representations for Manifold-Valued Data
We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere , the special orthogonal group , the positive definite matrices , and the Grassmann manifolds . The representations are based on the deployment of Deslauriers--Dubuc and average-interpolating pyramids "in the tangent plane" of such manifolds, using the and maps of those manifolds. The representations provide "wavelet coefficients" which can be thresholded, quantized, and scaled in much the same way as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds but is particularly suited to the manifolds we consider, i.e., Riemannian symmetric spaces, such as , , , where the and maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper
Horizontal variation of Tate--Shafarevich groups
Let be an elliptic curve over . Let be an odd prime and
an embedding. Let
be an imaginary quadratic field and the corresponding Hilbert class
field. For a class group character over , let be
the field generated by the image of and the prime
of above determined via . Under mild
hypotheses, we show that the number of class group characters such that
the -isotypic Tate--Shafarevich group of over is finite with
trivial -part increases with the absolute value of the
discriminant of
Enhanced Symmetries in Multiparameter Flux Vacua
We give a construction of type IIB flux vacua with discrete R-symmetries and
vanishing superpotential for hypersurfaces in weighted projective space with
any number of moduli. We find that the existence of such vacua for a given
space depends on properties of the modular group, and for Fermat models can be
determined solely by the weights of the projective space. The periods of the
geometry do not in general have arithmetic properties, but live in a vector
space whose properties are vital to the construction.Comment: 32 pages, LaTeX. v2: references adde
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