1,720,996 research outputs found
INdAM meeting: International meeting on numerical semigroups Cortona 2018
No full textIn this article we study the Hilbert function HR of one-dimensional semigroup rings R = k[[S]], with embedding dimension four over an infinite field k. Let S =< e, n2, n3, n4 > and let M = S ?{0}. Consider the Apéry set of S with respect to the multiplicity e and its subsets Ah = {s ? Apéry(S) | s ? hM ? (h + 1)M}, h ≥ 2. Further let D2 ?{n3, n4} be the set of generators with torsion order 1. We prove that HR is non-decreasing at level ≤ 3 and that HR is non decreasing in each of the following cases: if A2 has cardinality ≤ 4, if A3 has cardinality ≤ 3, if A4 = ?, if D2 has cardinality 2, if S has multiplicity ≤ 13
Monomials as sum of k-th powers of forms
No full textMotivated by recent results on the Waring problem for polynomial rings and representation of monomial as sum of powers of linear forms, we consider the problem of presenting monomials of degree kd as sums of kth-powers of forms of degree d. We produce a general bound on the number of summands for any number of variables which we refine in the two variables case. We completely solve the k = 3 case for monomials in two and three variables
Ranks of tensors: geometry and applications
Full text linkIn this article, we briefly survey some of the recent results in the geometry of tensors with a focus on bounds for various important notions of ranks. This is written in honor of Edoardo Ballico, who made important contributions to this field, on the occasion of his 70th birthday
Partially symmetric variants of Comon's problem via simultaneous rank
No full textA symmetric tensor may be regarded as a partially symmetric tensor in several different ways. These produce different notions of rank for the symmetric tensor which are related by chains of inequalities. By exploiting algebraic tools such as apolarity theory, we show how the study of the simultaneous symmetric rank of partial derivatives of the homogeneous polynomial associated to the symmetric tensor can be used to prove equalities among different partially symmetric ranks. This approach aims to understand to what extent the symmetries of a tensor affect its rank. We apply this to the special cases of binary forms, ternary and quaternary cubics, monomials, and elementary symmetric polynomials
On the quantum periods of del Pezzo surfaces with ⅓ (1, 1) singularities
No full textAbstractIn earlier joint work with collaborators we gave a conjectural classification of a broad class of orbifold del Pezzo surfaces, using Mirror Symmetry. We proposed that del Pezzo surfacesXwith isolated cyclic quotient singularities such thatXadmits a ℚ-Gorenstein toric degeneration correspond via Mirror Symmetry to maximally mutable Laurent polynomialsfin two variables, and that the quantum period of such a surfaceX, which is a generating function for Gromov–Witten invariants ofX, coincides with the classical period of its mirror partnerf.In this paper we give strong evidence for this conjecture. Contingent on conjectural generalisations of the Quantum Lefschetz theorem and the Abelian/non-Abelian correspondence, we compute many quantum periods for del Pezzo surfaces with13(1, 1) singularities. Our computations also give strong evidence for the extension of these two principles to the orbifold setting.</jats:p
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
The monic rank
No full textWe introduce the monic rank of a vector relative to an affinehyperplane section of an irreducible Zariski-closed affine cone X. We show that the monic rank is finite and greater than or equal to the usual X-rank. We describe an algorithmic technique based on classical invariant theory to determine, in concrete situations, the maximal monic rank. Using this technique, we establish three new instances of a conjecture due to B. Shapiro which states that a binary form of degree d · e is the sum of d dth powers of forms of degree e. Furthermore, in the case where X is the cone of highest weight vectors in an irreducible representation-this includes the well-known cases of tensor rank and symmetric rank-we raise the question whether the maximal rank equals the maximal monic rank. We answer this question affirmatively in several instances
A frequency-domain analysis of 3D reconstruction techniques based on defocusing
No full textA technique used for three-dimensional reconstruction of microscopic objects is based on linear processing of a collection of images taken at different focus positions, that is, acquired step by step while the objective moves along the microscope optical axis. As the relationship between the "object" I(x,y,z) and the collection of images (if these latter are enough) is known to be a convolution via the microscope point spread function, a deconvolution procedure yields I(x,y,z). Here I(x,y,z) is the local absorption of the object. Based on the fact that reasonable approximations for the point spread function are available in the literature, the above technique seems to work satisfactorily, and has been applied by various researchers. In the present work, the method is analyzed from a strictly mathematical point of view. It Is shown that being the optical transfer function (OTF), that is, the Fourier transform of the point spread function, identically vanishing in an infinite region of the frequency domain, a deconvolution procedure is not feasible. If the z-coordinate refers to the optical axis, the OTF is actually zero outside a conical region in the frequency domain. It is also shown that what can be actually recovered are projections of the given object. To this end, the so-called slice theorem, or projection theorem, is used. More exactly, we prove that one can obtain all projections of the object within an angle of projection - with respect to the optical axis - not greater than the aperture angle of the image formation system, that is, the microscope. Examples of simulations and of processing of experimental images are reporte
Geometric conditions for strict submultiplicativity of rank and border rank
No full textThe X-rank of a point p in projective space is the minimal number of points of an algebraic variety X whose linear span contains p. This notion is naturally submultiplicative under tensor product. We study geometric conditions that guarantee strict submultiplicativity. We prove that in the case of points of rank two, strict submultiplicativity is entirely characterized in terms of the trisecant lines to the variety. Moreover, we focus on the case of curves: we prove that for curves embedded in an even-dimensional projective space, there are always points for which strict submultiplicativity occurs, with the only exception of rational normal curves
- …
