On joint essential maximal numerical ranges

Paper presented at the 4th Strathmore International Mathematics Conference (SIMC 2017), 19 - 23 June 2017, Strathmore University, Nairobi, Kenya.The study of numerical range of an operator has been an area of intense research. The motivation for the development arose from the classical theory of quadratic forms. It forms a very important aspect in functional analysis, operator theory and its applications to economics, quantum chemistry and quantum computing amongst other fields. A lot of results have been obtained on numerical ranges particularly by Fong, Khan among others. The concept of maximal numerical ranges of a bounded operator T E B (H) was studied by Stampfli who used it to derive an identity for the norm of derivation. This concept was later generalized by Ghan to the joint maximal numerical ranges of m- tuples of operator. The joint essential maximal numerical range was studied by Khan and established that the joint essential maximal numerical range can be empty. In this paper we have showed that the joint essential maximal numerical range is nonempty, compact and convex. We also established that each element in the joint essential maximal numerical range is a star centre of the joint maximal numerical range. The result obtained show that star-shapedness is related to convexity in that a convex set is starshaped with all its points being star centres.Jaramogi Oginga Odinga University of Science and Technology, Box 210-40601, Bondo-Kenya

Continuity of the maximum-entropy inference: Convex geometry and numerical ranges approach

We study the continuity of an abstract generalization of the maximum-entropy inference - a maximizer. It is defined as a right-inverse of a linear map restricted to a convex body which uniquely maximizes on each fiber of the linear map a continuous function on the convex body. Using convex geometry we prove, amongst others, the existence of discontinuities of the maximizer at limits of extremal points not being extremal points themselves and apply the result to quantum correlations. Further, we use numerical range methods in the case of quantum inference which refers to two observables. One result is a complete characterization of points of discontinuity for $3\times 3$ matrices.Comment: 27 page

Efficient Bayesian inference for harmonic models via adaptive posterior factorization

NOTICE: this is the author’s version of a work that was accepted for publication in Neurocomputing. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in NEUROCOMPUTING, [VOL72, ISSUE 1-3, (2008)] DOI10.1016/j.neucom.2007.12.05

Exact extreme value statistics at mixed order transitions

We study extreme value statistics (EVS) for spatially extended models exhibiting mixed order phase transitions (MOT). These are phase transitions which exhibit features common to both first order (discontinuity of the order parameter) and second order (diverging correlation length) transitions. We consider here the truncated inverse distance squared Ising (TIDSI) model which is a prototypical model exhibiting MOT, and study analytically the extreme value statistics of the domain lengths. The lengths of the domains are identically distributed random variables except for the global constraint that their sum equals the total system size $L$. In addition, the number of such domains is also a fluctuating variable, and not fixed. In the paramagnetic phase, we show that the distribution of the largest domain length $l_{\max}$ converges, in the large $L$ limit, to a Gumbel distribution. However, at the critical point (for a certain range of parameters) and in the ferromagnetic phase, we show that the fluctuations of $l_{\max}$ are governed by novel distributions which we compute exactly. Our main analytical results are verified by numerical simulations.Comment: 25 pages, 6 figures, 1 tabl

Cloning of spin-coherent states

We consider optimal cloning of the spin coherent states in Hilbert spaces of different dimensionality d. We give explicit form of optimal cloning transformation for spin coherent states in the three-dimensional space, analytical results for the fidelity of the optimal cloning in d=3 and d=4 as well as numerical results for higher dimensions. In the low-dimensional case we construct the corresponding completely positive maps and exhibit their structure with the help of Jamiolkowski isomorphism. This allows us to formulate some conjectures about the form of optimal coherent cloning CP maps in arbitrary dimension.Comment: LateX, 9 pages, 1 figur

An inverse dynamics model for the analysis, reconstruction and prediction of bipedal walking

Walking is a constrained movement which may best be observed during the double stance phase when both feet contact the floor. When analyzing a measured movement with an inverse dynamics model, a violation of these constrains will always occur due to measuring errors and deviations of the segments model from reality, leading to inconsistent results. Consistency is obtained by implementing the constraints into the model. This makes it possible to combine the inverse dynamics model with optimization techniques in order to predict walking patterns or to reconstruct non-measured rotations when only a part of the three-dimensional joint rotations is measured. In this paper the outlines of the extended inverse dynamics method are presented, the constraints which define walking are defined and the optimization procedure is described. The model is applied to analyze a normal walking pattern of which only the hip, knee and ankle flexions/extensions are measured. This input movement is reconstructed to a kinematically and dynamically consistent three-dimensional movement, and the joint forces (including the ground reaction forces) and joint moments of force, needed to bring about this movement are estimated

Symmetry considerations and development of pinwheels in visual maps

Neurons in the visual cortex respond best to rod-like stimuli of given orientation. While the preferred orientation varies continuously across most of the cortex, there are prominent pinwheel centers around which all orientations a re present. Oriented segments abound in natural images, and tend to be collinear}; neurons are also more likely to be connected if their preferred orientations are aligned to their topographic separation. These are indications of a reduced symmetry requiring joint rotations of both orientation preference and the underl ying topography. We verify that this requirement extends to cortical maps of mo nkey and cat by direct statistical analysis. Furthermore, analytical arguments and numerical studies indicate that pinwheels are generically stable in evolving field models which couple orientation and topography