A nilsoliton is a nilpotent Lie algebra $$\mathfrak {g}$$ with metric such that $${{\,\mathrm{Ric}\,}}=\lambda \mathrm {Id}+D$$ , D derivation. For indefinite metrics, this determines four different geometries, according to whether $$\lambda $$ and are zero or not. We illustrate examples the greater flexibility of case compared Riemannian setting. determine algebraic properties must satisfy whe...

A Standard Quadratic Optimization Problem (StQP) consists of maximizing a (possibly indefinite) quadratic form over the standard simplex. Likewise, in a multi-StQP we have to maximize a (possibly indefinite) quadratic form over the cartesian product of several standard simplices (of possibly different dimensions). Two converging monotone interior point methods are established. Further, we prove...

In this work we propose an outcome space approach for globally solving generalized concave multiplicative problems, a special class of nonconvex problems which involves the maximization of a finite sum of products of concave functions over a nonempty compact convex set. It is shown that this nonconvex maximization problem can be reformulated as an indefinite quadratic problem with infinitely ma...

We detail the implementation of basic operations on multivariate Chebyshev approximations. In most cases, they can be derived directly from well known properties of univariate Chebyshev polynomials. Besides addition, subtraction and multiplication, we discuss integration, indefinite di↵erentiation, indefinite integration and interpolation. The latter three, can be written as matrix-vector produ...

We apply the homological mirror symmetry for elliptic curves to the study of indefinite theta series. We prove that every such series corresponding to a quadratic form of signature (1,1) can be expressed in terms of theta series associated with split quadratic forms and the usual theta series. We also show that indefinite theta series corresponding to univalued Massey products between line bund...

We consider minimization of a quadratic objective function subject to a sign-indefinite quadratic equality constraint. We derive necessary and sufficient conditions for the existence of solutions to the constrained minimization problem. These conditions involve a generalized eigenvalue of the matrix pencil consisting of a symmetric positivesemidefinite matrix and a symmetric indefinite matrix. ...

After a brief introduction to the spectral analysis of saddle point matrices, in this talk we present new estimates for the eigenvalue intervals for symmetric saddle-point and regularised saddle-point matrices in the case where the (1,1) block may be indefinite. These generalise known results for the definite (1,1) case. We also discuss spectral properties of the equivalent augmented formulatio...

We consider nonsingular saddle-point matrices whose leading block is maximally rank deficient, and show that the inverse in this case has unique mathematical properties. We then develop a class of indefinite block preconditioners that rely on approximating the null space of the leading block. The preconditioned matrix is a product of two indefinite matrices, but under certain conditions the con...

We consider the application of the preconditioned conjugate gradient (PCG) method to the solution of indefinite linear systems arising in nonlinear optimization. Our approach is based on the choice of quasidefinite preconditioners and of a suitable factorization routine. Some theoretical and numerical results about these preconditioners are obtained. Furthermore, we show the behaviour of the PC...

This paper proposes a new analysis of the use of bare nouns vs. indefinite NPs in predicative position in French. We distinguish between predicational sentences (with the bare noun version) and equative sentences (with the indefinite version). We argue that bare nouns ascribe permanent properties to aspects of entities. As for the indefinites, we claim that they exhibit their specific reading a...