Diagonal Projections Research Articles

AdS3/RMT2 duality

We introduce a framework for quantifying random matrix behavior of 2d CFTs and AdS3 quantum gravity. We present a 2d CFT trace formula, precisely analogous to the Gutzwiller trace formula for chaotic quantum systems, which originates from the SL(2, ℤ) spectral decomposition of the Virasoro primary density of states. An analogy to Berry’s diagonal approximation allows us to extract spectral statistics of individual 2d CFTs by coarse-graining, and to identify signatures of chaos and random matrix universality. This leads to a necessary and sufficient condition for a 2d CFT to display a linear ramp in its coarse-grained spectral form factor.Turning to gravity, AdS3 torus wormholes are cleanly interpreted as diagonal projections of squared partition functions of microscopic 2d CFTs. The projection makes use of Hecke operators. The Cotler-Jensen wormhole of AdS3 pure gravity is shown to be extremal among wormhole amplitudes: it is the minimal completion of the random matrix theory correlator compatible with Virasoro symmetry and SL(2, ℤ)-invariance. We call this MaxRMT: the maximal realization of random matrix universality consistent with the necessary symmetries. Completeness of the SL(2, ℤ) spectral decomposition as a trace formula allows us to factorize the Cotler-Jensen wormhole, extracting the microscopic object ZRMT(τ) from the coarse-grained product. This captures details of the spectrum of BTZ black hole microstates. ZRMT(τ) may be interpreted as an AdS3 half-wormhole. We discuss its implications for the dual CFT and modular bootstrap at large central charge.

Geodesic fiber tracking in white matter using activation function

Background and objective:The geodesic ray-tracing method has shown its effectiveness for the reconstruction of fibers in white matter structure. Based on reasonable metrics on the spaces of the diffusion tensors, it can provide multiple solutions and get robust to noise and curvatures of fibers. The choice of the metric on the spaces of diffusion tensors has a significant impact on the outcome of this method. Our objective is to suggest metrics and modifications of the algorithms leading to more satisfactory results in the construction of white matter tracts as geodesics.Methods:Starting with the DTI modality, we propose to rescale the initially chosen metric on the space of diffusion tensors to increase the geodetic cost in the isotropic regions. This change should be conformal in order to preserve the angles between crossing fibers. We also suggest to enhance the methods to be more robust to noise and to employ the fourth order tensor data in order to handle the fiber crossings properly.Results: We propose a way to choose the appropriate conformal class of metrics where the metric gets scaled according to tensor anisotropy. We use the logistic functions, which are commonly used in statistics as cumulative distribution functions. To prevent deviation of geodesics from the actual paths, we propose a hybrid ray-tracing approach. Furthermore, we suggest how to employ diagonal projections of 4th order tensors to perform fiber tracking in crossing regions.Conclusions: The algorithms based on the newly suggested methods were succesfuly implemented, their performance was tested on both synthetic and real data, and compared to some of the previously known approaches.

Reconstruction of hv-Convex Binary Images with Diagonal and Anti-Diagonal Projections using Genetic Algorithm

This paper is related to reconstruct the binary images from small number of projections. Objectives: To enhance the unambiguousness of image reconstructed. Other goal isto reduce the time of reconstruction of image. Method: In this paper, we put forward a new approach for reconstruction. Firstly, image is reconstructed using Chang’s Algorithm and we utilize two projections first diagonal projections and second anti-diagonal projections for reconstruction. Then Genetic Algorithm is applied on image reconstructed after Chang’s algorithm. Findings: The images reconstructed after Chang’s algorithm and Genetic Algorithm are compared with original images and they are contrasted with each other also. We also calculate the percentage change of images reconstructed after Chang’s Algorithm and Genetic Algorithm with original images. It is concluded that images reconstructed after Genetic Algorithm have less percentage difference than Chang’s Algorithm. Improvements: The accuracy of images reconstructed after Genetic Algorithm is much better than reconstructed by Chang’s Algorithm alone.

Triangularizability of operators with increasing spectrum

We establish finite- and infinite-dimensional versions of the following assertion. If M is a matrix with the property that whenever P and Q are diagonal projections with P ⩽ Q , the spectrum of PMP (considered as an operator on the range of P) is contained in that of QMQ (considered as an operator on the range of Q), then there is a permutation matrix U such that U − 1 M U is triangular.

$P(\omega)/{\rm fin}$ and projections in the Calkin algebra

We investigate the set-theoretic properties of the lattice of projections in the Calkin algebra of a separable infinite-dimensional Hilbert space in relation to those of the Boolean algebra $P(\omega)/{\rm fin}$, which is isomorphic to the sublattice of diagonal projections. In particular, we prove some basic consistency results about the possible cofinalities of well-ordered sequences of projections and the possible cardinalities of sets of mutually orthogonal projections that are analogous to well-known results about $P(\omega)/{\rm fin}$.

Reconstruction of 8-connected but not 4-connected [formula omitted]-convex discrete sets

The reconstruction of 8-connected but not 4-connected hv -convex discrete sets from few projections is considered. An algorithm is given with worst case complexity of O ( mn min { m , n } ) to reconstruct all sets with given horizontal and vertical projections. Experimental results are also presented. It is shown, that using also the diagonal projections the algorithm can be speeded up having complexity of O ( mn ) and in this case the solution is uniquely determined. Finally, we consider the possible generalizations of our results to solve the problem in more general classes.

Factorization of triangular operators and ideals through the diagonal

We give a necessary and sufficient condition to determine when an operator in the nest algebra of doubly infinite block upper triangular operators factors through a diagonal projection. An example shows that this condition does not extend to more general nest algebras, but a similar criterion yields a description of the ideals of nest algebras generated by diagonal projections.

Matrix norm inequalities and the relative Dixmier property

If x is a selfadjoint matrix with zero diagonal and non-negative entries, then there exists a decomposition of the identity into k diagonal orthogonal projections {pm} for which $$\parallel \sum p_m xp_m \parallel \leqslant (1/k)\parallel x\parallel $$ From this follows that all bounded matrices with non-negative entries satisfy the relative Dixmier property or, equivalently, the Kadison Singer extension property. This inequality fails for large Hadamard matrices. However a similar inequality holds for all matrices with respect to the Hilbert-Schmidt norm with constant k−1/2 and for Hadamard matrices with respect to the Schatten 4-norm with constant 21/4k−1/2.

3-D reconstruction of molecular shape from microcrystal thin sections

Three-dimensional reconstruction techniques using electron microscopy have been principally developed for application to 2-D arrays (i.e. monolayers) of biological molecules and symmetrical single particles (e.g. helical viruses). However many biological molecules that crystallise form multilayered microcrystals which are unsuitable for study by either the standard methods of 3-D reconstruction or, because of their size, by X-ray crystallography. The grid sectioning technique enables a number of different projections of such microcrystals to be obtained in well defined directions (e.g. parallel to crystal axes) and poses the problem of how best these projections can be used to reconstruct the packing and shape of the molecules forming the microcrystal.Given sufficient projections there may be enough information to do a crystallographic reconstruction in Fourier space. We however have considered the situation where only a limited number of projections are available, as for example in the case of catalase platelets where three orthogonal and two diagonal projections have been obtained (Fig. 1).