1,721,005 research outputs found
On the Number of Rich Lines in Truly High Dimensional Sets
Get PDFWe prove a new upper bound on the number of r-rich lines (lines with at least r points) in a 'truly' d-dimensional configuration of points v_1,...,v_n over the complex numbers. More formally, we show that, if the number of r-rich lines is significantly larger than n^2/r^d then there must exist a large subset of the points contained in a hyperplane. We conjecture that the factor r^d can be replaced with a tight r^{d+1}. If true, this would generalize the classic Szemeredi-Trotter theorem which gives a bound of n^2/r^3 on the number of r-rich lines in a planar configuration. This conjecture was shown to hold in R^3 in the seminal work of Guth and Katz and was also recently proved over R^4 (under some additional restrictions) by Solomon and Sharir. For the special case of arithmetic progressions (r collinear points that are evenly distanced) we give a bound that is tight up to lower order terms, showing that a d-dimensional grid achieves the largest number of r-term progressions.
The main ingredient in the proof is a new method to find a low degree polynomial that vanishes on many of the rich lines. Unlike previous applications of the polynomial method, we do not find this polynomial by interpolation. The starting observation is that the degree r-2 Veronese embedding takes r-collinear points to r linearly dependent images. Hence, each collinear r-tuple of points, gives us a dependent r-tuple of images. We then use the design-matrix method of Barak et al. to convert these 'local' linear dependencies into a global one, showing that all the images lie in a hyperplane. This then translates into a low degree polynomial vanishing on the original set
Sylvester-Gallai for Arrangements of Subspaces
Get PDFIn this work we study arrangements of k-dimensional subspaces V_1,...,V_n over the complex numbers. Our main result shows that, if every pair V_a, V_b of subspaces is contained in a dependent triple (a triple V_a, V_b, V_c contained in a 2k-dimensional space), then the entire arrangement must be contained in a subspace whose dimension depends only on k (and not on n). The theorem holds under the assumption that the subspaces are pairwise non-intersecting (otherwise it is false). This generalizes the Sylvester-Gallai theorem (or Kelly's theorem for complex numbers), which proves the k=1 case. Our proof also handles arrangements in which we have many pairs (instead of all) appearing in dependent triples, generalizing the quantitative results of Barak et. al. One of the main ingredients in the proof is a strengthening of a theorem of Barthe (from the k=1 to k>1 case) proving the existence of a linear map that makes the angles between pairs of subspaces large on average. Such a mapping can be found, unless there is an obstruction in the form of a low dimensional subspace intersecting many of the spaces in the arrangement (in which case one can use a different argument to prove the main theorem)
Recommended from our members
Fourier and Circulant Matrices Are Not Rigid
Get PDFThe concept of matrix rigidity was first introduced by Valiant in [Friedman, 1993]. Roughly speaking, a matrix is rigid if its rank cannot be reduced significantly by changing a small number of entries. There has been extensive interest in rigid matrices as Valiant showed in [Friedman, 1993] that rigidity can be used to prove arithmetic circuit lower bounds.
In a surprising result, Alman and Williams showed that the (real valued) Hadamard matrix, which was conjectured to be rigid, is actually not very rigid. This line of work was extended by [Dvir and Edelman, 2017] to a family of matrices related to the Hadamard matrix, but over finite fields. In our work, we take another step in this direction and show that for any abelian group G and function f:G - > {C}, the matrix given by M_{xy} = f(x - y) for x,y in G is not rigid. In particular, we get that complex valued Fourier matrices, circulant matrices, and Toeplitz matrices are all not rigid and cannot be used to carry out Valiant’s approach to proving circuit lower bounds. This complements a recent result of Goldreich and Tal [Goldreich and Tal, 2016] who showed that Toeplitz matrices are nontrivially rigid (but not enough for Valiant’s method). Our work differs from previous non-rigidity results in that those works considered matrices whose underlying group of symmetries was of the form {F}_p^n with p fixed and n tending to infinity, while in the families of matrices we study, the underlying group of symmetries can be any abelian group and, in particular, the cyclic group {Z}_N, which has very different structure. Our results also suggest natural new candidates for rigidity in the form of matrices whose symmetry groups are highly non-abelian.
Our proof has four parts. The first extends the results of [Josh Alman and Ryan Williams, 2016; Dvir and Edelman, 2017] to generalized Hadamard matrices over the complex numbers via a new proof technique. The second part handles the N x N Fourier matrix when N has a particularly nice factorization that allows us to embed smaller copies of (generalized) Hadamard matrices inside of it. The third part uses results from number theory to bootstrap the non-rigidity for these special values of N and extend to all sufficiently large N. The fourth and final part involves using the non-rigidity of the Fourier matrix to show that the group algebra matrix, given by M_{xy} = f(x - y) for x,y in G, is not rigid for any function f and abelian group G
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe 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
Deterministic Extractors for Algebraic Sources
No full textAn algebraic source is a random variable distributed uniformly over the set of common zeros of one or more multivariate polynomials defined over a finite field F. Our main result is the construction of an explicit deterministic extractor for algebraic sources over exponentially large prime fields. More precisely, we give an explicit (and arguably simple) function E: F n ↦ → {0, 1} m such that the output of E on any algebraic source in F n is close to the uniform distribution, provided that the degrees of the defining polynomials are not too high and that the algebraic source contains ‘enough ’ points. This extends previous works on extraction from affine sources (sources distributed over subspaces) and from polynomial sources (sources defined as the image of a low degree polynomial mapping). We also give an additional construction of a deterministic extractor for algebraic sources with support larger than |F | n/2. This construction works over fields as small as d O(1) , where d is the maximal degree of a polynomial used to define the source
- …
