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When we consider an error model in a quantum computing system, we assume a parametric model where a prepared qubit belongs. Keeping this in mind, we focus on the evaluation of the amount of information we obtain when we know the system belongs to the model within the parameter range. Excluding classical fluctuations, uncertainty still remains in the system. We propose an information quantity called purely quantum information to evaluate this and give it an operational meaning. For the qubit case, it is relevant to the facility location problem on the unit sphere, which is well known in operations research. For general cases, we extend this to the facility location problem in complex projective spaces. Purely quantum information reflects the uncertainty of a quantum system and is related to the minimum entropy rather than the von Neumann entropy.

Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra.

In this paper, we extend a result of Schwick concerning normality and sharing values in one complex variable for families of holomorphic curves taking values in ℙ[sup.n]. We consider wandering moving hyperplanes (i.e., depending on the respective holomorphic curve in the family under consideration) and establish a sufficient condition for normality concerning shared hyperplanes.

We provide a classification theorem for compact stable minimal immersions (CSMI) of codimension 1 or dimension 1 (codimension 1 and 2 or dimension 1 and 2) in the product of a complex (quaternionic) projective space with any other Riemannian manifold. We characterize the complex minimal immersions of codimension 2 or dimension 2 as the only CSMI in the product of two complex projective spaces. As an application, we characterize the CSMI of codimension 1 or dimension 1 (codimension 1 and 2 or dimension 1 and 2) in the product of a complex (quaternionic) projective space with any compact rank one symmetric space.

In this paper, we give a complete classification of Hopf pseudo-Ricci-Yamabe solitons on real hypersurfaces in the complex projective space ℂPⁿ. As its applications, first we give a complete classification of gradient pseudo-Ricci-Yamabe solitons on real hypersurfaces with isometric Reeb flow in the complex projective space ℂPⁿ. Next we prove that a contact real hypersurface in ℂPⁿ which admits the gradient pseudo-Ricci-Yamabe soliton is pseudo-Einstein.

In this paper we study isometric immersions f : M n → C ′ P n of an n -dimensional pseudo-Riemannian manifold M n into the n -dimensional para-complex projective space C ′ P n . We study the immersion f by means of a lift f of f into a quadric hypersurface in S n + 1 2 n + 1 . We find the frame equations and compatibility conditions. We specialize these results to dimension n = 2 and a definite metric on M 2 in isothermal coordinates and consider the special cases of Lagrangian surface immersions and minimal surface immersions. We characterize surface immersions with special properties in terms of primitive harmonicity of the Gauss maps.

Irreducible isoparametric foliations of arbitrary codimension qq on complex projective spaces CPn\\mathbb {C} P^n are classified, for (q,n)≠(1,15)(q,n)\\neq (1,15). Remarkably, there are noncongruent examples that pull back under the Hopf map to congruent foliations on the sphere. Moreover, there exist many inhomogeneous isoparametric foliations, even of higher codimension. In fact, every irreducible isoparametric foliation on CPn\\mathbb {C} P^n is homogeneous if and only if n+1n+1 is prime. The main tool developed in this work is a method to study singular Riemannian foliations with closed leaves on complex projective spaces. This method is based on a certain graph that generalizes extended Vogan diagrams of inner symmetric spaces.

Recently, Pipoli and Sinestrari [Pipoli, G. and Sinestrari, C., Mean curvature flow of pinched submanifolds of ℝℙ n , Comm. Anal. Geom. , 25 , 2017, 799–846] initiated the study of convergence problem for the mean curvature flow of small codimension in the complex projective space ℝℙ m . The purpose of this paper is to develop the work due to Pipoli and Sinestrari, and verify a new convergence theorem for the mean curvature flow of arbitrary codimension in the complex projective space. Namely, the authors prove that if the initial submanifold in ℝℙ m satisfies a suitable pinching condition, then the mean curvature flow converges to a round point in finite time, or converges to a totally geodesic submanifold as t → ∞. Consequently, they obtain a differentiable sphere theorem for submanifolds in the complex projective space.

In this paper, we study the isolation phenomena of Einstein manifolds from the viewpoint of submanifolds theory. First, for locally strongly convex Einstein affine hyperspheres we prove a rigidity theorem and as its direct consequence we establish a unified affine differential geometric characterization of the noncompact symmetric spaces E6(−26)/F4\\mathrm {E}_{6(-26)}/\\mathrm {F}_4 and SL(m,R)/SO(m)\\mathrm {SL}(m,\\mathbb {R})/\\mathrm {SO}(m), SL(m,C)/SU(m)\\mathrm {SL}(m,\\mathbb {C})/\\mathrm {SU}(m), SU∗(2m)/Sp(m)\\mathrm {SU}^*(2m)/\\mathrm {Sp}(m) for each m≥3m\\ge 3. Second and analogously, for Einstein Lagrangian minimal submanifolds of the complex projective space CPn(4)\\mathbb {C}P^n(4) with constant holomorphic sectional curvature 44, we prove a similar rigidity theorem and as its direct consequence we establish a unified differential geometric characterization of the compact symmetric spaces E6/F4\\mathrm {E}_{6}/\\mathrm {F}_4 and SU(m)/SO(m)\\mathrm {SU}(m)/\\mathrm {SO}(m), SU(m)\\mathrm {SU}(m), SU(2m)/Sp(m)\\mathrm {SU}(2m)/\\mathrm {Sp}(m) for each m≥3m\\ge 3.

We consider real hypersurfaces M in complex projective space equipped with both the Levi–Civita and generalized Tanaka–Webster connections. For any nonnull constant k and any symmetric tensor field of type (1, 1) L on M , we can define two tensor fields of type (1, 2) on M , L F ( k ) and L T ( k ) , related to both connections. We study the behaviour of the structure operator ϕ with respect to such tensor fields in the particular case of L = A , the shape operator of M , and obtain some new characterizations of ruled real hypersurfaces in complex projective space.