The notion of singular reduction operators, i.e., of singular operators of nonclassical (conditional) symmetry, of partial differential equations in two independent variables is introduced. All possible reductions of these equations to first-order ODEs are are exhaustively described. As examples, properties of singular reduction operators of (1+ 1)-dimensional evolution and wave equations are s...

The construction of topological index maps for equivariant families of Dirac operators requires factoring a general smooth map through maps of a very simple type: zero sections of vector bundles, open embeddings, and vector bundle projections. Roughly speaking, a normally non-singular map is a map together with such a factorisation. These factorisations are models for the topological index map....

Gómez-Mont, Seade and Verjovsky introduced an index, now called GSV-index, generalizing the Poincaré-Hopf index to complex vector fields tangent to singular hypersurfaces. The GSV-index extends to the real case. This is a survey paper on the joint research with Gómez-Mont and Giraldo about calculating the GSV-index IndV±,0(X) of a real vector field X tangent to a singular hypersurface V = f(0)....

In this paper we discuss the notion of singular vector tuples of a complex valued d-mode tensor of dimension m1 × . . . × md. We show that a generic tensor has a finite number of singular vector tuples, viewed as points in the corresponding Segre product. We give the formula for the number of singular vector tuples. We show similar results for tensors with partial symmetry. We give analogous re...

We study the Pareto frontier for two competing norms ‖ · ‖X and ‖ · ‖Y on a vector space. For a given vector c, the Pareto frontier describes the possible values of (‖a‖X , ‖b‖Y ) for a decomposition c = a + b. The singular value decomposition of a matrix is closely related to the Pareto frontier for the spectral and nuclear norm. We will develop a general theory that extends the notion of sing...

We discuss compactifications of higher dimensional supergravities which are induced by scalars. In particular, we consider vector multiplets coupled to the supergravity multiplet in the case of D = 9,8 and D = 7 minimal supergravities. These vector multiplets contain scalars, which parametrize coset spaces of the general form SO(10−D, n)/SO(10−D)×SO(n), where n is the number of vector multiplet...

The normalized eight-point algorithm is broadly used for the computation of the fundamental matrix between two images given a set of correspondences. However, it performs poorly for low-size datasets due to the way in which the rank-two constraint is imposed on the fundamental matrix. We propose two new algorithms to enforce the rank-two constraint on the fundamental matrix in closed form. The ...

Traditionally, radiant enclosures have been designed using a forward " trial-and-error " methodology. Recently, however, inverse and optimization methodologies have been applied to design radiant enclosures. Both of these methodologies solve the design problem faster, and the quality of the solutions is usually better than those obtained through the forward design methodology. This paper presen...

In this article, nondegeneracy of singular vector tuples, Z-eigenvectors and eigenvectors tensors is studied, which have found many applications in diverse areas. The main results are: (i) each (Z-)eigenvector/singular tuple a generic tensor nondegenerate, (ii) nonzero Z-eigenvector/singular an orthogonally decomposable nondegenerate.

Many current watermarking algorithms insert data in the spatial or transform domains like the discrete cosine, the discrete Fourier, and the discrete wavelet transforms. In this paper, we present a data-hiding algorithm that exploits the singular value decomposition (SVD) representation of the data. We compute the SVD of the host image and the watermark and embed the watermark in the singular v...