On the cotangent cohomology of rational surface singularities with almost reduced fundamental cycle

We prove dimension formulas for the cotangent spaces T ¹ and T ² for a class of rational surface singularities by calculating a correction term in the general dimension formulas. We get that it is zero if the dual graph of the rational surface singularity X does not contain a particular type of configurations, and this generalizes a result of Theo de Jong stating that the correction term c (X ) is zero for rational determinantal surface singularities. In particular our result implies that c (X ) is zero for Riemenschneiders quasi‐determinantal rational surface singularities, and this also generalizes results for quotient singularities. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Metric homology

Metric homology is a homology theory constructed on semialgebraic (or compact subanalytic) sets with singularities. Metric homology is an invariant under semialgebraic (subanalytic) bi‐Lipschitz homeomorphisms and not a topological invariant. As in intersection homology theory, classes of admissible chains are defined using a semialgebraic stratification and a perversity function. In contrast to intersection homology, the perversity is a rational‐valued function. Instead of the topological dimension of the intersection of a chain with a stratum, we consider the so‐called volume‐growth number. This number is a sort of generalization of Hausdorff dimension. In the second part of the paper we describe one‐dimensional metric homology for spaces with isolated singularities and calculate some concrete examples. © 2000 John Wiley & Sons, Inc.     

Simply connected Voisin manifolds of dimension four

Voisin constructed a series of examples of simply connected compact Kähler manifolds of even dimension, which do not have the rational homotopy type of a complex projective manifold starting from dimension six. In this note, we prove that Voisin's examples of dimension four also do not have the rational homotopy type of a complex projective manifold. Oguiso constructed simply connected compact Kähler manifolds starting from dimension four, which cannot deform to a complex projective manifold under a small deformation. We also prove that Oguiso's examples do not have the rational homotopy type of a complex projective manifold.     

Exact solutions for fractional DDEs via auxiliary equation method coupled with the fractional complex transform

Dynamical behavior of many nonlinear systems can be described by fractional‐order equations. This study is devoted to fractional differential–difference equations of rational type. Our focus is on the construction of exact solutions by means of the (G'/G)‐expansion method coupled with the so‐called fractional complex transform. The solution procedure is elucidated through two generalized time‐fractional differential–difference equations of rational type. As a result, three types of discrete solutions emerged: hyperbolic, trigonometric, and rational. Copyright © 2016 John Wiley & Sons, Ltd.     

The projective translation equation and rational plane flows. II. Corrections and additions

In this second part of the work, we correct the flaw which was left in the proof of the main Theorem in the first part. This affects only a small part of the text in this first part and two consecutive papers. Yet, some additional arguments are needed to claim the validity of the classification results. With these new results, algebraic and rational flows can be much more easily and transparently classified. It also turns out that the notion of an algebraic projective flow is a very natural one. For example, we give an inductive (on dimension) method to build algebraic projective flows with rational vector fields, and ask whether these account for all such flows. Further, we expand on results concerning rational flows in dimension 2. Previously we found all such flows symmetric with respect to a linear involution \(i_{0}(x,y)=(y,x)\). Here we find all rational flows symmetric with respect to a non-linear 1-homogeneous involution \(i(x,y)=(\frac{y^2}{x},y)\). We also find all solenoidal rational flows. Up to linear conjugation, there appears to be exactly two non-trivial examples.     

Rational curves of degree 16 on a general heptic fourfold

According to a conjecture of H. Clemens, the dimension of the space of rational curves on a general projective hypersurface should equal the number predicted by a naïve dimension count. In the case of a general hypersurface of degree 7 in P⁵${\mathbb{P}}^5$, the conjecture predicts that the only rational curves should be lines. This has been verified by Hana and Johnsen for rational curves of degree at most 15. Here we extend their results to show that no rational curves of degree 16 lie on a general heptic fourfold.     

CM–fields and skew–symmetric matrices

Cohen and Odoni prove that every CM–field can be generated by an eigenvalue of some skew–symmetric matrix with rational coefficients. It is natural to ask for the minimal dimension of such a matrix. They show that every CM–field of degree 2n is generated by an eigenvalue of a skew–symmetric matrix over Q of dimension at most 4n+2. The aim of the present paper is to improve this bound.     

On Finite Dimension Exchange Algorithms

Load balancing on parallel computers aims at equilibrating some initial load which is different from one processor to another. We consider only nearest neighbour algorithms: in each step a processor communicates only with its direct neighbours. Load balancing algorithms can be divided into two classes: diffusion and dimension exchange. Whereas the first is appropriate for the so‐called all‐port‐model where a processor can send tokens to all its neighbours at a time, the latter is useful for the one‐port‐model. Both kinds of algorithms can be viewed as methods for solving certain singular linear systems. Since a few years there exist finite diffusion algorithms which have the property that they compute l₂‐minimal flows. In this paper a new finite dimension exchange method will be presented that is based on edge‐colourings of the underlying graph. It is usually faster and more stable than its diffusion counterpart. The flows computed by the new method are not minimal but it can be shown that they are bounded. Our analysis is based on techniques from numerical linear algebra.     

Coverings over Tori and topological approach to Klein’s resolvent problem

This paper answers the question: what coverings over a topological torus can be induced from a covering over a space of dimension k? The answer to this question is then applied in algebro-geometric context to present obstructions to transforming an algebraic equation depending on several parameters to an equation depending on fewer parameters by means of a rational transformation.     

Non-formal homogeneous spaces

Several large classes of homogeneous spaces are known to be formal—in the sense of rational homotopy theory. However, it seems that far fewer examples of non-formal homogeneous spaces are known. In this article we provide several construction principles and characterisations for non-formal homogeneous spaces, which will yield a lot of examples. This will enable us to prove that, from dimension 72 on, such a space can be found in each dimension.     

On the minimal sum of Betti numbers of an almost complex manifold

We show that the only rational homology spheres which can admit almost complex structures occur in dimensions two and six. Moreover, we provide infinitely many examples of six-dimensional rational homology spheres which admit almost complex structures, and infinitely many which do not. We then show that if a closed almost complex manifold has sum of Betti numbers three, then its dimension must be a power of two.     

Endomorphism rings and automorphism groups of almost completely decomposable groups

To every non-cuspidal K-rational point on the modular curve X_l(n) a non-commutative Noetherian domain of global dimension 3 can be associated : the Sklyanin algebra. In this paper we give the defining equations of the Sklyanin algebras and their centers when X_l(n) is rational, i.e. n <= 10 or n = 12.     

On Countable-Dimensional Spaces with the Menger Property, Rational Dimension and a Question of S. D. Iliadis

 This paper deals with the class of spaces which are countable unions of zero-dimensional sets and with the larger class of Haver’s C-spaces. All spaces are assumed to be separable and metrizable. We are concerned with various aspects of universality of these classes when they are combined with the covering analogue for σ-compactness defined by Menger and the rational dimension introduced by Menger and N?beling. A solution of a problem of S. D. Iliadis [16] concerning universal spaces for rational dimension will result. Received 11 September 1998; in revised form 12 January 1999     

On a class of extremal solutions of a moment problem for rational matrix‐valued functions in the nondegenerate case I

The main theme of this paper is the discussion of a parametrized family of solutions of a finite moment problem for rational matrix‐valued functions in the nondegenerate case. We will show that each member of this family is extremal in several directions concerning some point of the open unit disk. These investigations are inspired by the authors' paper [23], where a similar topic is studied in the context of the matricial Carathéodory problem. We will see that larger parts of the results presented there can be extended to the rational case studied here.The main theme of this paper is the discussion of a parametrized family of solutions of a finite moment problem for rational matrix‐valued functions in the nondegenerate case. We will show that each member of this family is extremal in several directions concerning some point of the open unit disk. These investigations are inspired by the authors' paper [23], where a similar topic is studied in the context of the matricial Carathéodory problem. We will see that larger parts of the results presented there can be extended to the rational case studied here (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generatrices of rational curves

We investigate the one-parametric set of projective subspaces that is generated by a set of rational curves in projective relation. The main theorem connects the algebraic degree of , the number of degenerate subspaces in and the dimension of the variety of all rational curves that can be used to generate . It generalizes classical results and is related to recent investigations on projective motions with trajectories in proper subspaces of the fixed space. Received 9 May 2001.     

An extended TODIM method for multiple attribute group decision‐making based on 2‐dimension uncertain linguistic Variable

The significant characteristic of the TODIM (an acronym in Portuguese of Interactive and Multiple Attribute Decision Making) method is that it can consider the bounded rationality of the decision makers. However, in the classical TODIM method, the rating of the attributes only can be used in the form of crisp numbers. Because 2‐dimension uncertain linguistic variables can easily express the fuzzy information, in this article, we extend the TODIM method to 2‐dimension uncertain linguistic information. First of all, the definition, characteristics, expectation, comparative method and distance of 2‐dimension uncertain linguistic information are introduced, and the steps of the classical TODIM method for Multiple attribute decision making (MADM) problems are presented. Second, on the basis of the classical TODIM method, the extended TODIM method is proposed to deal with MADM problems in which the attribute values are in the form of 2‐dimension uncertain linguistic variables, and detailed decision steps are given. Its significant characteristic is that it can fully consider the bounded rationality of the decision makers, which is a real action in real decision making. Finally, a numerical example is provided to verify the developed approach and its practicality and effectiveness. © 2014 Wiley Periodicals, Inc. Complexity 21: 20–30, 2016     

Projecting the one-dimensional Sierpinski gasket

LetS⊂ℝ² be the Cantor set consisting of points (x,y) which have an expansion in negative powers of 3 using digits {(0,0), (1,0), (0,1)}. We show that the projection ofS in any irrational direction has Lebesgue measure 0. The projection in a rational directionp/q has Hausdorff dimension less than 1 unlessp+q ≡ 0 mod 3, in which case the projection has nonempty interior and measure 1/q. We compute bounds on the dimension of the projection for certain sequences of rational directions, and exhibit a residual set of directions for which the projection has dimension 1. This work was partially completed while the author was at the Institut Fourier, Grenoble, France.     

Optimal finite difference grids and rational approximations of the square root I. Elliptic problems

The main objective of this paper is optimization of second‐order finite difference schemes for elliptic equations, in particular, for equations with singular solutions and exterior problems. A model problem corresponding to the Laplace equation on a semi‐infinite strip is considered. The boundary impedance (Neumann‐to‐Dirichlet map) is computed as the square root of an operator using the standard three‐point finite difference scheme with optimally chosen variable steps. The finite difference approximation of the boundary impedance for data of given smoothness is the problem of rational approximation of the square root on the operator's spectrum. We have implemented Zolotarev's optimal rational approx‐imant obtained in terms of elliptic functions. We have also found that a geometrical progression of the grid steps with optimally chosen parameters is almost as good as the optimal approximant. For bounded operators it increases from second to exponential the convergence order of the finite difference impedance with the convergence rate proportional to the inverse of the logarithm of the condition number. For the case of unbounded operators in Sobolev spaces associated with elliptic equations, the error decays as the exponential of the square root of the mesh dimension. As an example, we numerically compute the Green function on the boundary for the Laplace equation. Some features of the optimal grid obtained for the Laplace equation remain valid for more general elliptic problems with variable coefficients. © 2000 John Wiley & Sons, Inc.     

Tilting on non-commutative rational projective curves

In this article we introduce a new class of non-commutative projective curves and show that in certain cases the derived category of coherent sheaves on them has a tilting complex. In particular, we prove that the right bounded derived category of coherent sheaves on a reduced rational projective curve with only nodes and cusps as singularities, can be fully faithfully embedded into the right bounded derived category of the finite dimensional representations of a certain finite dimensional algebra of global dimension two. As an application of our approach we show that the dimension of the bounded derived category of coherent sheaves on a rational projective curve with only nodal or cuspidal singularities is at most two. In the case of the Kodaira cycles of projective lines, the corresponding tilted algebras belong to a well-known class of gentle algebras. We work out in details the tilting equivalence in the case of the Weierstrass nodal curve zy ² = x ³ + x ² z.