Scalar-flat K?hler orbifolds via quaternionic-complex reduction

We prove that any asymptotically locally Euclidean scalar-flat K?hler 4-orbifold whose isometry group contains a 2-torus is isometric, up to an orbifold covering, to a quaternionic-complex quotient of a k-dimensional quaternionic vector space by a (k−1)-torus. In order to do so, we first prove that any compact anti-self-dual 4-orbifold with positive Euler characteristic whose isometry group contains a 2-torus is conformally equivalent, up to an orbifold covering, to a quaternionic quotient of k-dimensional quaternionic projective space by a (k − 1)-torus.     

The rank reduction procedure of Egerváry

Here we give a survey of results concerning the rank reduction algorithm developed by Egerváry between 1953 and 1958 in a sequence of papers.     

Boundary Nevanlinna�CPick interpolation via reduction and augmentation

We give an elementary proof of Sarason??s solvability criterion for the Nevanlinna?CPick problem with boundary interpolation nodes and boundary target values. We also give a concrete parametrization of all solutions of such a problem. The proofs are based on a reduction method due to Julia and Nevanlinna. Reduction of functions corresponds to Schur complementation of the corresponding Pick matrices.     

Minimal reduction type and the Kazhdan–Lusztig map

We introduce the notion of minimal reduction type of an affine Springer fiber, and use it to define a map from the set of conjugacy classes in the Weyl group to the set of nilpotent orbits. We show that this map is the same as the one defined by Lusztig in Lfromto, (2011) and that the Kazhdan–Lusztig map in Kazhdan and Lusztig, (1998) is a section of our map. This settles several conjectures in the literature. For classical groups, we prove more refined results by introducing and studying the “skeleta” of affine Springer fibers.     

On ℋ︁2-model reduction of linear parameter-varying systems

In this paper, we will discuss an advantageous relation between a special class of linear parameter-varying systems and bilinear control systems. This will automatically lead to parameter-preserving model reduction techniques. Furthermore, we review a recently introduced interpolation-based ℋ₂-framework for bilinear control systems. By means of a numerical example, the efficiency of the new method will be underlined. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Invariance analysis and reduction of discrete Painlevé equations

The construction and role of symmetries for difference equations have been established, relatively, recently. In this paper, a symmetry analysis and reductions of the discrete Painlevé equations are considered. We assume that the characteristics of the ‘vector fields’ have a particular dependence since the general form lead to cumbersome calculations. Where possible, these symmetries are used to construct exact solutions in some cases.     

Deligne’s congruence and supersingular reduction of Drinfeld modules

Most rank two Drinfeld modules are known to have infinitely many supersingular primes. But how many supersingular primes of a given degree can a fixed Drinfeld module have? In this paper, a congruence between the Hasse invariant and a certain Eisenstein series is used for obtaining a bound on the number of such supersingular primes. Certain exceptional cases correspond to zeros of certain Eisenstein series with rational j-invariants.     

Reorthogonalization for the Golub–Kahan–Lanczos bidiagonal reduction

The Golub–Kahan–Lanczos (GKL) bidiagonal reduction generates, by recurrence, the matrix factorization of $X \in \mathbb{R }^{m \times n}, m \ge n$ , given by $$\begin{aligned} X = UBV^T \end{aligned}$$ where $U \in \mathbb{R }^{m \times n}$ is left orthogonal, $V \in \mathbb{R }^{n \times n}$ is orthogonal, and $B \in \mathbb{R }^{n \times n}$ is bidiagonal. When the GKL recurrence is implemented in finite precision arithmetic, the columns of $U$ and $V$ tend to lose orthogonality, making a reorthogonalization strategy necessary to preserve convergence of the singular values. The use of an approach started by Simon and Zha (SIAM J Sci Stat Comput, 21:2257–2274, 2000) that reorthogonalizes only one of the two left orthogonal matrices $U$ and $V$ is shown to be very effective by the results presented here. Supposing that $V$ is the matrix reorthogonalized, the reorthogonalized GKL algorithm proposed here is modeled as the Householder Q–R factorization of $\left( \begin{array}{c} 0_{n \times k} \\ X V_k \end{array}\right) $ where $V_k = V(:,1:k)$ . That model is used to show that if $\varepsilon _M $ is the machine unit and $$\begin{aligned} \bar{\eta }= \Vert \mathbf{tril }(I-V^T\!~V)\Vert _F, \end{aligned}$$ where $\mathbf{tril }(\cdot )$ is the strictly lower triangular part of the contents, then: (1) the GKL recurrence produces Krylov spaces generated by a nearby matrix $X + \delta X$ , $\Vert \delta X\Vert _F = \mathcal O (\varepsilon _M + \bar{\eta }) \Vert X\Vert _F$ ; (2) singular values converge in the Lanczos process at the rate expected from the GKL algorithm in exact arithmetic on a nearby matrix; (3) a new proposed algorithm for recovering leading left singular vectors produces better bounds on loss of orthogonality and residual errors.     

Nonnegative rank factorization—a heuristic approach via rank reduction

Given any nonnegative matrix $A \in \mathbb{R}^{m \times n}$ , it is always possible to express A as the sum of a series of nonnegative rank-one matrices. Among the many possible representations of A, the number of terms that contributes the shortest nonnegative rank-one series representation is called the nonnegative rank of A. Computing the exact nonnegative rank and the corresponding factorization are known to be NP-hard. Even if the nonnegative rank is known a priori, no simple procedure exists presently that is able to perform the nonnegative factorization. Based on the Wedderburn rank reduction formula, this paper proposes a heuristic approach to tackle this difficult problem numerically. Starting with A, the idea is to recurrently extrat, whenever possible, a rank-one nonnegative portion from the previous matrix while keeping the residual nonnegative and lowering its rank by one. With a slight modification for symmetry, the method can equally be applied to another important class of completely positive matrices. No convergence can be guaranteed, but repeated restart might help alleviate the difficulty. Extensive numerical testing seems to suggest that the proposed algorithm might serve as a first-step numerical means for exploring the intriguing problem of nonnegative rank factorization.     

Grauert’s line bundle convexity,reduction and Riemann domains

We consider a convexity notion for complex spaces X with respect to a holomorphic line bundle L over X. This definition has been introduced by Grauert and, when L is analytically trivial, we recover the standard holomorphic convexity. In this circle of ideas, we prove the counterpart of the classical Remmert’s reduction result for holomorphically convex spaces. In the same vein, we show that if H⁰(X,L) separates each point of X, then X can be realized as a Riemann domain over the complex projective space Pⁿ, where n is the complex dimension of X and L is the pull-back of O(1).     

Explaining the Gentzen–Takeuti reduction steps: a second-order system

Using the concept of notations for infinitary derivations we give an explanation of Takeuti's reduction steps on finite derivations (used in his consistency proof for Π¹ ₁-CA) in terms of the more perspicious infinitary approach from [BS88]. Received: 27 April 1999 / Published online: 21 March 2001     

Higher asymptotics of unitarity in ��quantization commutes with reduction��

Let M be a compact K?hler manifold equipped with a Hamiltonian action of a compact Lie group G. Guillemin and Sternberg (Invent Math 67:515?C538, 1982, no. 3), showed that there is a geometrically natural isomorphism between the G-invariant quantum Hilbert space over M and the quantum Hilbert space over the symplectic quotient M //G. This map, though, is not in general unitary, even to leading order in ${\hslash}$ . Hall and Kirwin (Commun Math Phys 275:401?C422, 2007, no. 2), showed that when the metaplectic correction is included, one does obtain a map which, while not in general unitary for any fixed ${\hslash}$ , becomes unitary in the semiclassical limit ${\hslash\rightarrow0}$ (cf. the work of Ma and Zhang (C R Math Acad Sci Paris 341:297?C302, 2005, no. 5), and (Astérisque No. 318:viii+154, 2008). The unitarity of the classical Guillemin?CSternberg map and the metaplectically corrected analogue is measured by certain functions on the symplectic quotient M //G. In this paper, we give precise expressions for these functions, and compute complete asymptotic expansions for them as ${\hslash\rightarrow0}$ .     

Degree reduction of SG-Bézier surfaces based on grey wolf optimizer

Aiming at the problem of approximate degree reduction of SG-Bézier surfaces, a method is proposed to achieve the degree reduction from (n × n) to (m × m) (m < n). Starting from the idea of grey wolf optimizer (GWO) algorithm and combining the geometric properties of SG-Bézier surfaces, this method transforms the degree reduction problem of SG-Bézier surfaces into an optimization problem. By choosing the fitness function, the degree reduction approximation of shape-adjustable SG-Bézier surfaces under unconstrained and angular interpolation constraints is realized. At the same time, some concrete examples of degree reduction and its errors are given. The results show that this method not only achieves good degree reduction effect but also is easy to implement and has high precision.     

Energy preserving model order reduction of the nonlinear Schrödinger equation

An energy preserving reduced order model is developed for two dimensional nonlinear Schrödinger equation (NLSE) with plane wave solutions and with an external potential. The NLSE is discretized in space by the symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting system of Hamiltonian ordinary differential equations are integrated in time by the energy preserving average vector field (AVF) method. The mass and energy preserving reduced order model (ROM) is constructed by proper orthogonal decomposition (POD) Galerkin projection. The nonlinearities are computed for the ROM efficiently by discrete empirical interpolation method (DEIM) and dynamic mode decomposition (DMD). Preservation of the semi-discrete energy and mass are shown for the full order model (FOM) and for the ROM which ensures the long term stability of the solutions. Numerical simulations illustrate the preservation of the energy and mass in the reduced order model for the two dimensional NLSE with and without the external potential. The POD-DMD makes a remarkable improvement in computational speed-up over the POD-DEIM. Both methods approximate accurately the FOM, whereas POD-DEIM is more accurate than the POD-DMD.