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Nicolau C. Saldanha

Publications and source records attributed to Nicolau C. Saldanha.

At least 37 records · Page 2Linked to original sources
arXiv2016

Spaces of completions of elementary theories and convergence laws for random hypergraphs

Consider the binomial model $G^{d+1}(n,p)$ of the random $(d+1)$-uniform hypergraph on $n$ vertices, where each edge is present, independently of one another, with probability $p:\mathbb{N}\to[0,1]$. We prove that, for all logarithmo-exponential $p\ll n^{-d+ε}$, the probabilities of all elementary properties of hypergraphs converge, with particular emphasis in the ranges $p(n)\sim C/n^d$ and $p(n) \sim C\log(n)/n^d$. The exposition is unified by constructing, for each such function $p$, the topological space of all completions of its almost sure theory. This space turns out to be compact, metrizable and totally disconnected, but further properties depend on the range of $p$. The convergence of the probabilities of elementary properties is associated with a borelian probability measure on the space.

math.CO↗
arXiv2016

Scaling limits of discrete copulas are bridged Brownian sheets

For large $n$, take a random $n \times n$ permutation matrix and its associated discrete copula $X_n$. For $a, b = 0, 1, \ldots, n$, let $y_n(\frac{a}{n},\frac{b}{n}) = \frac{1}{n} ( X_{a,b} - \frac{ab}{n} )$; define $y_n: [0,1]^2 \to R$ by interpolating quadratically on squares of side $\frac{1}{n}$. We prove a Donsker type central limit theorem: $\sqrt{n} y_n$ approaches a bridged Brownian sheet on the unit square.

math.PR↗
arXiv2015

Domino tilings of three-dimensional regions: flips, trits and twists

In this paper, we consider domino tilings of regions of the form $\mathcal{D} \times [0,n]$, where $\mathcal{D}$ is a simply connected planar region and $n \in \mathbb{N}$. It turns out that, in nontrivial examples, the set of such tilings is not connected by flips, i.e., the local move performed by removing two adjacent dominoes and placing them back in another position. We define an algebraic invariant, the twist, which partially characterizes the connected components by flips of the space of tilings of such a region. Another local move, the trit, consists of removing three adjacent dominoes, no two of them parallel, and placing them back in the only other possible position: performing a trit alters the twist by $\pm 1$. We give a simple combinatorial formula for the twist, as well as an interpretation via knot theory. We prove several results about the twist, such as the fact that it is an integer and that it has additive properties for suitable decompositions of a region.

math.CO↗
arXiv2015

Components of spaces of curves with constrained curvature on flat surfaces

Let $S$ be a complete flat surface, such as the Euclidean plane. We obtain direct characterizations of the connected components of the space of all curves on $S$ which start and end at given points in given directions, and whose curvatures are constrained to lie in a given interval, in terms of all parameters involved. Many topological properties of these spaces are investigated. Some conjectures of L. E. Dubins are proved.

math.GT↗
arXiv2015

Some examples of asymptotic combinatorial behavior, zero-one and convergence results on random hypergraphs

This is an extended version of the thesis presented to the Programa de Pós-Graduação em Matemática of the Departamento de Matemática, PUC-Rio, in September 2013, incorporating some suggestions from the examining commission. Random graphs (and more generally hypergraphs) have been extensively studied, including their first order logic. In this work we focus on certain specific aspects of this vast theory. We consider the binomial model $G^{d+1}(n,p)$ of the random $(d+1)$-uniform hypergraph on $n$ vertices, where each edge is present, independently of one another, with probability $p=p(n)$. We are particularly interested in the range $p(n) \sim C\log(n)/n^d$, after the double jump and near connectivity. We prove several zero-one, and, more generally, convergence results and obtain combinatorial applications of some

math.CO↗
arXiv2014

Twists for duplex regions

This note relies heavily on arXiv:1404.6509 and arXiv:1410.7693. Both articles discuss domino tilings of three-dimensional regions, and both are concerned with flips, the local move performed by removing two parallel dominoes and placing them back in the only other possible position. In the second article, an integer $\operatorname{Tw}(t)$ is defined for any tiling $t$ of a large class of regions $\mathcal{R}$: it turns out that $\operatorname{Tw}(t)$ is invariant by flips. In the first article, a more complicated polynomial invariant $P_t(q)$ is introduced for tilings of two-story regions. It turns out that $\operatorname{Tw}(t) = P_t'(1)$ whenever $t$ is a tiling of a duplex region, a special kind of two-story region for which both invariants are defined. This identity is proved in arXiv:1410.7693 in an indirect and nonconstructive manner. In the present note, we provide an alternative, more direct proof.

math.CO↗
arXiv2014

Flip invariance for domino tilings of three-dimensional regions with two floors

We investigate tilings of cubiculated regions with two simply connected floors by 2 x 1 x 1 bricks. More precisely, we study the flip connected component for such tilings, and provide an algebraic invariant that "almost" characterizes the flip connected components of such regions, in a sense that we discuss in the paper. We also introduce a new local move, the trit, which, together with the flip, connects the space of domino tilings when the two floors are identical.

math.CO↗
arXiv2013

The homotopy type of spaces of locally convex curves in the sphere

A smooth curve $γ: [0,1] \to \Ss^2$ is locally convex if its geodesic curvature is positive at every point. J. A. Little showed that the space of all locally convex curves $γ$ with $γ(0) = γ(1) = e_1$ and $γ'(0) = γ'(1) = e_2$ has three connected components $L_{-1,c}$, $L_{+1}$, $L_{-1,n}$. The space $\cL_{-1,c}$ is known to be contractible. We prove that $\cL_{+1}$ and $\cL_{-1,n}$ are homotopy equivalent to $(Ω\Ss^3) \vee \Ss^2 \vee \Ss^6 \vee \Ss^{10} \vee \cdots$ and $(Ω\Ss^3) \vee \Ss^4 \vee \Ss^8 \vee \Ss^{12} \vee \cdots$, respectively. As a corollary, we deduce the homotopy type of the components of the space $\Free(\Ss^1,\Ss^2)$ of free curves $γ: \Ss^1 \to \Ss^2$ (i.e., curves with nonzero geodesic curvature). We also determine the homotopy type of the spaces $\Free([0,1], \Ss^2)$ with fixed initial and final frames.

math.GT↗
arXiv2012

The homotopy and cohomology of spaces of locally convex curves in the sphere -- I

A smooth curve $γ: [0,1] \to S^2$ is locally convex if its geodesic curvature is positive at every point. J. A. Little showed that the space of all locally positive curves $γ$ with $γ(0) = γ(1) = e_1$ and $γ'(0) = γ'(1) = e_2$ has three connected components $L_{-1,c}$, $L_{+1}$, $L_{-1,n}$. The space $L_{-1,c}$ is known to be contractible but the topology of the other two connected components is not well understood. We study the homotopy and cohomology of these spaces. In particular, for $L_{-1} = L_{-1,c} \sqcup L_{-1,n}$, we show that $\dim H^{2k}(L_{(-1)^{k}}, \RR) \ge 1$, that $\dim H^{2k}(L_{(-1)^{(k+1)}}, \RR) \ge 2$, that $π_2(L_{+1})$ contains a copy of $Z^2$ and that $π_{2k}(L_{(-1)^{(k+1)}})$ contains a copy of $Z$.

math.GT↗
arXiv2012

The homotopy and cohomology of spaces of locally convex curves in the sphere -- II

A smooth curve $γ: [0,1] \to S^2$ is locally convex if its geodesic curvature is positive at every point. J. A. Little showed that the space of all locally positive curves $γ$ with $γ(0) = γ(1) = e_1$ and $γ'(0) = γ'(1) = e_2$ has three connected components $L_{-1,c}$, $L_{+1}$, $L_{-1,n}$. The space $L_{-1,c}$ is known to be contractible but the topology of the other two connected components is not well understood. We prove that all connected components of $L_I$ are simply connected, that $H^2(L_{+1};Z) = Z^2$ and $H^2(L_{-1,n};Z) = Z$.

math.GT↗
arXiv2012

Spaces of locally convex curves in S^n and combinatorics of the group B_{n+1}

In the 1920's Marston Morse developed what is now known as Morse theory trying to study the topology of the space of closed curves on S^2. We propose to attack a very similar problem, which 80 years later remains open, about the topology of the space of closed curves on S^2 which are locally convex (i.e., without inflection points). One of the main difficulties is the absence of the covering homotopy principle for the map sending a non-closed locally convex curve to the Frenet frame at its endpoint. In the present paper we study the spaces of locally convex curves in S^n with a given initial and final Frenet frames. Using combinatorics of B^{+}_{n+1} = B_{n+1} \cap SO_{n+1}, where B_{n+1} \subset O_{n+1} is the usual Coxeter-Weyl group, we show that for any n \ge 2 these spaces fall in at most $\lceil\frac{n}{2}\rceil+1$ equivalence classes up to homeomorphism. We also study this classification in the double cover Spin(n+1). For $n = 2$ our results complete the classification of the corresponding spaces into two topologically distinct classes, or three classes in the spin case.

math.GT↗
arXiv2011

Dynamics of the symmetric eigenvalue problem with shift strategies

A common algorithm for the computation of eigenvalues of real symmetric tridiagonal matrices is the iteration of certain special maps $F_σ$ called shifted $QR$ steps. Such maps preserve spectrum and a natural common domain is ${\cal T}_Λ$, the manifold of real symmetric tridiagonal matrices conjugate to the diagonal matrix $Λ$. More precisely, a (generic) shift $s \in \RR$ defines a map $F_s: {\cal T}_Λ\to {\cal T}_Λ$. A strategy $σ: {\cal T}_Λ\to \RR$ specifies the shift to be applied at $T$ so that $F_σ(T) = F_{σ(T)}(T)$. Good shift strategies should lead to fast deflation: some off-diagonal coordinate tends to zero, allowing for reducing of the problem to submatrices. For topological reasons, continuous shift strategies do not obtain fast deflation; many standard strategies are indeed discontinuous. Practical implementation only gives rise systematically to bottom deflation, convergence to zero of the lowest off-diagonal entry $b(T)$. For most shift strategies, convergence to zero of $b(T)$ is cubic, $|b(F_σ(T))| = Θ(|b(T)|^k)$ for $k = 3$. The existence of arithmetic progressions in the spectrum of $T$ sometimes implies instead quadratic convergence, $k = 2$. The complete integrability of the Toda lattice and the dynamics at non-smooth points are central to our discussion. The text does not assume knowledge of numerical linear algebra.

math.DS↗
arXiv2010

Convergence rates to deflation of simple shift strategies

The computation of eigenvalues of real symmetric tridiagonal matrices frequently proceeds by a sequence of QR steps with shifts. We introduce simple shift strategies, functions sigma satisfying natural conditions, taking each n x n matrix T to a real number sigma(T). The strategy specifies the shift to be applied by the QR step at T. Rayleigh and Wilkinson's are examples of simple shift strategies. We show that if sigma is continuous then there exist initial conditions for which deflation does not occur, i.e., subdiagonal entries do not tend to zero. In case of deflation, we consider the rate of convergence to zero of the (n, n-1) entry: for simple shift strategies this is always at least quadratic. If the function sigma is smooth in a suitable region and the spectrum of T does not include three consecutive eigenvalues in arithmetic progression then convergence is cubic. This implies cubic convergence to deflation of Wilkinson's shift for generic spectra. The study of the algorithm near deflation uses tubular coordinates, under which QR steps with shifts are given by a simple formula.

math.NA↗
arXiv2009

Cut-and-paste of quadriculated disks and arithmetic properties of the adjacency matrix

We define cut-and-paste, a construction which, given a quadriculated disk obtains a disjoint union of quadriculated disks of smaller total area. We provide two examples of the use of this procedure as a recursive step. Tilings of a disk $Δ$ receive a parity: we construct a perfect or near-perfect matching of tilings of opposite parities. Let $B_Δ$ be the black-to-white adjacency matrix: we factor $B_Δ= L\tilde DU$, where $L$ and $U$ are lower and upper triangular matrices, $\tilde D$ is obtained from a larger identity matrix by removing rows and columns and all entries of $L$, $\tilde D$ and $U$ are equal to 0, 1 or -1.

math.CO↗
arXiv2008

The Asymptotics of Wilkinson's Iteration: Loss of Cubic Convergence

One of the most widely used methods for eigenvalue computation is the $QR$ iteration with Wilkinson's shift: here the shift $s$ is the eigenvalue of the bottom $2\times 2$ principal minor closest to the corner entry. It has been a long-standing conjecture that the rate of convergence of the algorithm is cubic. In contrast, we show that there exist matrices for which the rate of convergence is strictly quadratic. More precisely, let $T_X$ be the $3 \times 3$ matrix having only two nonzero entries $(T_X)_{12} = (T_X)_{21} = 1$ and let $T_L$ be the set of real, symmetric tridiagonal matrices with the same spectrum as $T_X$. There exists a neighborhood $U \subset T_L$ of $T_X$ which is invariant under Wilkinson's shift strategy with the following properties. For $T_0 \in U$, the sequence of iterates $(T_k)$ exhibits either strictly quadratic or strictly cubic convergence to zero of the entry $(T_k)_{23}$. In fact, quadratic convergence occurs exactly when $\lim T_k = T_X$. Let $X$ be the union of such quadratically convergent sequences $(T_k)$: the set $X$ has Hausdorff dimension 1 and is a union of disjoint arcs $X^σ$ meeting at $T_X$, where $σ$ ranges over a Cantor set.

math.SP↗
arXiv2008

The geometry of the critical set of nonlinear periodic Sturm-Liouville operators

We study the critical set C of the nonlinear differential operator F(u) = -u" + f(u) defined on a Sobolev space of periodic functions H^p(S^1), p >= 1. Let R^2_{xy} \subset R^3 be the plane z = 0 and, for n > 0, let cone_n be the cone x^2 + y^2 = tan^2 z, |z - 2 pi n| < pi/2; also set Sigma = R^2_{xy} U U_{n > 0} cone_n. For a generic smooth nonlinearity f: R -> R with surjective derivative, we show that there is a diffeomorphism between the pairs (H^p(S^1), C) and (R^3, Sigma) x H where H is a real separable infinite dimensional Hilbert space.

math.FA↗
arXiv2008

An atlas for tridiagonal isospectral manifolds

Let ${\cal T}_Λ$ be the compact manifold of real symmetric tridiagonal matrices conjugate to a given diagonal matrix $Λ$ with simple spectrum. We introduce {\it bidiagonal coordinates}, charts defined on open dense domains forming an explicit atlas for ${\cal T}_Λ$. In contrast to the standard inverse variables, consisting of eigenvalues and norming constants, every matrix in ${\cal T}_Λ$ now lies in the interior of some chart domain. We provide examples of the convenience of these new coordinates for the study of asymptotics of isospectral dynamics, both for continuous and discrete time.

math.SP↗