Dept. of Mathematics
http://hdl.handle.net/2142/16339
Tue, 20 Feb 2018 00:06:49 GMT2018-02-20T00:06:49ZCohomology of associative algebras and spectral sequences
http://hdl.handle.net/2142/98983
Cohomology of associative algebras and spectral sequences
Shih, Kung-Sing
Algebras, Abstract; Mathematics
Mon, 11 May 1953 00:00:00 GMThttp://hdl.handle.net/2142/989831953-05-11T00:00:00ZShih, Kung-SingStructures and dynamics
http://hdl.handle.net/2142/98378
Structures and dynamics
Panagiotopoulos, Aristotelis
Our results are divided in three independent chapters.
In Chapter 2, we show that if g is a generic isometry of a generic subspace X of the Urysohn metric space U then g does not extend to a full isometry of U. The same applies to the Urysohn sphere S. Let M be a Fraisse L-structure, where L is a relational countable language and M has no algebraicity. We provide necessary and sufficient conditions for the following to hold: "For a generic substructure A of M, every automorphism f in Aut(A) extends to a full automorphism f' in Aut(M)." From our analysis, a dichotomy arises and some structural results are derived that, in particular, apply to omega-stable Fraisse structures without algebraicity. Results in Chapter 2 are separately published in [Pan15].
In Chapter 3, we develop a game-theoretic approach to anti-classi cation results for orbit equivalence relations and use this development to reorganize conceptually the proof of Hjorth's turbulence theorem. We also introduce a new dynamical criterion providing an obstruction to classi cation by orbits of Polish groups which admit a complete left invariant metric (CLI groups). We apply this criterion to the relation of equality of countable sets of reals and we show that the relations of unitary conjugacy of unitary and selfadjoint operators on the separable in nite-dimensional Hilbert space are not classi able by CLI-group actions. Finally we show how one can adapt this approach to the context of Polish groupoids. Chapter 3 is joint work with Martino Lupini and can also be found in [LP16].
In Chapter 4, we develop a theory of projective Fraisse limits in the spirit of Irwin-Solecki. The structures here will additionally support dual semantics as in [Sol10, Sol12]. Let Y be a compact metrizable space and let G be a closed subgroup of Homeo(Y ). We show that there is always a projective Fraisse limit K and a closed equivalence relation r on its domain K that is de finable in K, so that the quotient of K under r is homeomorphic to Y and the projection of K to Y induces a continuous group embedding of Aut(K) in G with dense image. The main results of Chapter 4 can also be found in [Pan16].
Polish groups; Fraisse; Turbulence; Hjorth; Left invariant; Becker; Projective Fraisse; Infinite games; Borel complexity
Thu, 13 Jul 2017 00:00:00 GMThttp://hdl.handle.net/2142/983782017-07-13T00:00:00ZPanagiotopoulos, AristotelisAsymptotically optimal shapes for counting lattice points and eigenvalues
http://hdl.handle.net/2142/98364
Asymptotically optimal shapes for counting lattice points and eigenvalues
Liu, Shiya
In Part I, we aim to maximize the number of first-quadrant lattice points under a concave (or convex) curve with respect to reciprocal stretching in the coordinate directions. The optimal domain is shown to be asymptotically balanced, meaning that the optimal stretch factor approaches 1 as the "radius" approaches infinity. In particular, the result implies when 1 < p < ∞ that among all p-ellipses (or Lamé curves), the p-circle x^p+y^p=r^p is asymptotically optimal for enclosing the most first-quadrant lattice points as the radius approaches infinity.
The case p = 2 corresponds to minimization of high eigenvalues of the Dirichlet Laplacian on rectangles, and so our work generalizes a result of Antunes and Freitas. Similarly, we generalize a Neumann eigenvalue maximization result of van den Berg, Bucur and Gittins. Further, Ariturk and Laugesen recently handled 0 < p < 1 by building on our results here.
The case p = 1 remains open: which right triangles in the first quadrant (with two sides along the axes) will enclose the most lattice points for given area, and what are the limiting shapes of those triangles as the area tends to infinity?
In Part II, we translate the positive-integer lattice points in the first quadrant by some amount in the horizontal and vertical directions. We seek to identify the limiting shape of the curve that encloses the greatest number of shifted lattice points in the same family of reciprocal stretching curves as in Part I.
The limiting shape is shown to depend explicitly on the lattice shift. The result holds for all positive shifts, and for negative shifts satisfying a certain condition. When the shift becomes too negative, the optimal curve no longer converges to a limiting shape, and instead it degenerates.
Our results handle the p-circle when p > 1 (concave) and also when 0 < p < 1 (convex). The straight line case (p = 1) generates an open problem about minimizing high eigenvalues of quantum harmonic oscillators with normalized parabolic potentials.
Lattice points; Planar convex domain; P-ellipse; Spectral optimization; Laplacian; Dirichlet eigenvalues; Neumann eigenvalues; Translated lattice; Schrödinger eigenvalues; Harmonic oscillator
Tue, 11 Jul 2017 00:00:00 GMThttp://hdl.handle.net/2142/983642017-07-11T00:00:00ZLiu, ShiyaColoring and covering problems on graphs
http://hdl.handle.net/2142/98358
Coloring and covering problems on graphs
Loeb, Sarah Jane
The \emph{separation dimension} of a graph $G$, written $\pi(G)$, is the minimum number of linear orderings of $V(G)$ such that every two nonincident edges are ``separated'' in some ordering, meaning that both endpoints of one edge appear before both endpoints of the other. We introduce the \emph{fractional separation dimension} $\pi_f(G)$, which is the minimum of $a/b$ such that some $a$ linear orderings (repetition allowed) separate every two nonincident edges at least $b$ times.
In contrast to separation dimension, we show fractional separation dimension is bounded: always $\pi_f(G)\le 3$, with equality if and only if $G$ contains $K_4$. There is no stronger bound even for bipartite graphs, since $\pi_f(K_{m,m})=\pi_f(K_{m+1,m})=\frac{3m}{m+1}$. We also compute $\pi_f(G)$ for cycles and some complete tripartite graphs. We show that $\pi_f(G)<\sqrt{2}$ when $G$ is a tree and present a sequence of trees on which the value tends to $4/3$. We conjecture that when $n=3m$ the $K_4$-free $n$-vertex graph maximizing $\pi_f(G)$ is $K_{m,m,m}$.
We also consider analogous problems for circular orderings, where pairs of nonincident edges are separated unless their endpoints alternate. Let $\pi^\circ(G)$ be the number of circular orderings needed to separate all pairs, and let $\pi_f^\circ(G)$ be the fractional version. Among our results: (1) $\pi^\circ(G)=1$ if and only $G$ is outerplanar. (2) $\pi^\circ(G)\le2$ when $G$ is bipartite. (3) $\pi^\circ(K_n)\ge\log_2\log_3(n-1)$. (4) $\pi_f^\circ(G)\le\frac{3}{2}$, with equality if and only if $K_4\subseteq G$. (5) $\pi_f^\circ(K_{m,m})=\frac{3m-3}{2m-1}$.
A \emph{star $k$-coloring} is a proper $k$-coloring where the union of any two color classes induces a star forest. While every planar graph is 4-colorable, not every planar graph is star 4-colorable. One method to produce a star 4-coloring is to partition the vertex set into a 2-independent set and a forest; such a partition is called an \emph{\Ifp}. We use discharging to prove that every graph with maximum average degree less than $\frac{5}{2}$ has an \Ifp, which is sharp and improves the result of Bu, Cranston, Montassier, Raspaud, and Wang (2009). As a corollary, we gain that every planar graph with girth at least 10 has a star 4-coloring.
A proper vertex coloring of a graph $G$ is \emph{$r$-dynamic} if for each $v\in V(G)$, at least $\min\{r,d(v)\}$ colors appear in $N_G(v)$. We investigate $3$-dynamic versions of coloring and list coloring. We prove that planar and toroidal graphs are 3-dynamically 10-choosable, and this bound is sharp for toroidal graphs.
Given a proper total $k$-coloring $c$ of a graph $G$, we define the \emph{sum value} of a vertex $v$ to be $c(v) + \sum_{uv \in E(G)} c(uv)$. The smallest integer $k$ such that $G$ has a proper total $k$-coloring whose sum values form a proper coloring is the \emph{neighbor sum distinguishing total chromatic number} $\chi''_{\Sigma}(G)$. Pil{\'s}niak and Wo{\'z}niak~(2013) conjectured that $\chi''_{\Sigma}(G)\leq \Delta(G)+3$ for any simple graph with maximum degree $\Delta(G)$. We prove this bound to be asymptotically correct by showing that $\chi''_{\Sigma}(G)\leq \Delta(G)(1+o(1))$. The main idea of our argument relies on Przyby{\l}o's proof (2014) for neighbor sum distinguishing edge-coloring.
Graph coloring; Graph covering
Mon, 10 Jul 2017 00:00:00 GMThttp://hdl.handle.net/2142/983582017-07-10T00:00:00ZLoeb, Sarah JaneTowards a model theory of logarithmic transseries
http://hdl.handle.net/2142/98343
Towards a model theory of logarithmic transseries
Gehret, Allen R
The ordered valued differential field $\mathbb{T}_{\log}$ of logarithmic transseries is conjectured to have good model theoretic properties. This thesis records our progress in this direction and describes a strategy moving forward. As a first step, we turn our attention to the value group of $\mathbb{T}_{\log}$. The derivation on $\mathbb{T}_{\log}$ induces on its value group $\Gamma_{\log}$ a certain map $\psi$; together forming the pair $(\Gamma_{\log},\psi)$, the \emph{asymptotic couple of $\mathbb{T}_{\log}$}. We study the asymptotic couple $(\Gamma_{\log},\psi)$ and show that it has a nice model theory. Among other things, we prove that $\Th(\Gamma_{\log},\psi)$ has elimination of quantifiers in a natural language, is model complete, and has the non-independence property (NIP). As a byproduct of our work, we also give a complete characterization of when an $H$-field has exactly one or exactly two Liouville closures. Finally, we present an outline for proving a model completeness result for $\mathbb{T}_{\log}$ in a reasonable language. In particular, we introduce and study the notion of \emph{$\LD$-fields} and also the property of a differentially-valued field being \emph{$\Psi$-closed}.
Logarithmic transseries; Model theory
Sun, 09 Jul 2017 00:00:00 GMThttp://hdl.handle.net/2142/983432017-07-09T00:00:00ZGehret, Allen RThe distribution of k-free numbers and integers with fixed number of prime factors
http://hdl.handle.net/2142/98338
The distribution of k-free numbers and integers with fixed number of prime factors
Meng, Xianchang
This thesis includes four chapters. In Chapter 1, we briefly introduce the history and the main results of the topics of this thesis: the distribution of $k$-free numbers and the derivative of the Riemann zeta-function, the generalization of Chebyshev's bias to products of any $k\geq 1$ primes, and the distribution of integers with prime factors from specific arithmetic progressions.
In Chapter 2, for any $k\geq 2$, we study the distribution of $k$-free numbers. It is known that the number of $k$-free numbers up to $x$ is $\widetilde{M}_k(x)\sim \frac{x}{\zeta(k)}$, where $\zeta(s)$ is the Riemann zeta-function. In this chapter, we focus on the distribution of the error term $M_k(x):=\widetilde{M}_k(x)-\frac{x}{\zeta(k)}$. Under the Riemann Hypothesis, we prove an equivalent relation between a mean square of the error term $M_k(x)$ and the negative moments of $|\zeta'(\rho)|$ as $\rho$ runs over the zeros of $\zeta(s)$. Under some reasonable conjectures, we show that $M_k(x)\ll x^{\frac{1}{2k}}(\log x)^{\frac{1}{2}-\frac{1}{2k}+\epsilon}$ for all $\epsilon>0$ except on a set of finite logarithmic measure, and that $e^{-\frac{y}{2k}}M_k(e^y)$ has a limiting distribution. Finally, based on the analysis of the tail of the limiting distribution, we make a precise conjecture on the maximal order of the error term.
In Chapter 3, we generalize the Chebyshev's bias and the so-called prime race problems to the distribution of products of any $k\geq 1$ primes in different arithmetic progressions. For any $k\geq 1$, we derive a formula for the difference between the number of integers $n\leq x$ with $\omega(n)=k$ or $\Omega(n)=k$ in two different arithmetic progressions, where $\omega(n)$ is the number of distinct prime factors of $n$ and $\Omega(n)$ is the number of prime factors of $n$ counted with multiplicity. Under the extended Riemann Hypothesis (ERH) and the Linear Independence Conjecture (LI) for Dirichlet $L$-functions, we show that, if $k$ is odd, the integers with $\Omega(n)=k$ have preference for quadratic non-residue classes; and if $k$ is even, such integers have preference for quadratic residue classes. This result confirms a conjecture of Richard Hudson. However, the integers with $\omega(n)=k$ always have preference for quadratic residue classes. Moreover, as $k$ increases, the biases decrease for both cases. For large $k$, we also give asymptotic formulas for the logarithmic densities of the sets on which the corresponding difference functions have a given sign.
In Chapter 4, we prove an asymptotic formula for the number of integers $\leq x$ which can be written as the product of $k ~(\geq 2)$ distinct primes $p_1\cdots p_k$ with each prime factor from a fixed arithmetic progression $p_j\equiv a_j \bmod q$, $(a_j, q)=1$ $(q \geq 3, 1\leq j\leq k)$. For any $A> 0$, our result is uniform for $2\leq k\leq A\log\log x$. Moreover, we show that, there are large biases toward certain arithmetic progressions $\boldsymbol{a}=(a_1, \cdots, a_k)$, and that such biases have connections with Mertens' theorem and the least prime in arithmetic progressions. Unlike the previous two topics, all results in this chapter are unconditional.
Riemann zeta-function; K-free numbers; Primes in arithmetic progressions; Dirichlet L-function
Fri, 07 Jul 2017 00:00:00 GMThttp://hdl.handle.net/2142/983382017-07-07T00:00:00ZMeng, XianchangRelative waring rank of binary forms
http://hdl.handle.net/2142/98327
Relative waring rank of binary forms
Tokcan, Neriman
Suppose $f(x,y)$ is a binary form of degree $d$ with coefficients in a field $K \subseteq \cc$. The {\it $K$-rank of $f$} is the smallest number of $d$-th powers of linear forms over $K$ of which $f$ is a $K$-linear combination. We prove that for $d \ge 5$, there always exists a form of degree $d$ with at least three different ranks over various fields. We also study the relation between the relative rank and the algebraic properties of the underlying field. In particular, we show that $K$-rank of a form $f$ (such as $x^3y^2$) may depend on whether $-1$ is a sum of two squares in $K.$ We provide lower bounds for the $\mathbb{C}$-rank (Waring rank) and for the $\mathbb{R}$-rank (real Waring rank) of binary forms depending on their factorization. We also give the rank of quartic and quintic binary forms based on their factorization over $\cc.$ We investigate the structure of binary forms with unique $\mathbb{C}$-minimal representation.
Waring rank; Real rank; Binary forms; Sums of powers; Sylvester; Tensor decompositions
Wed, 05 Jul 2017 00:00:00 GMThttp://hdl.handle.net/2142/983272017-07-05T00:00:00ZTokcan, NerimanStabilizing spectral functors of exact categories
http://hdl.handle.net/2142/98290
Stabilizing spectral functors of exact categories
Villeta-Garcia, Juan S
We define and study the $K$-theory of exact categories with coefficients in endofunctors of spectra in analogy with Mitchell's homology of categories. Generalizing computations of McCarthy, we determine, for a discrete ring $R$, the $K$-theory of the exact category of finitely-generated projective $R$-modules with coefficients in the $n$-fold smash product functor. This computation allows us to analyze the effects of applying this functorial construction to the Goodwillie Taylor tower of a homotopy endofunctor of spectra. In the case of $\Sigma^\infty\Omega^\infty$, the associated tower recovers the Taylor tower of relative $K$-theory as computed by Lindenstrauss and McCarthy.
Algebraic K-theory; Goodwillie calculus; Homotopy theory; Algebraic topology
Thu, 13 Jul 2017 00:00:00 GMThttp://hdl.handle.net/2142/982902017-07-13T00:00:00ZVilleta-Garcia, Juan SPartition asymptotics; zeros of zeta functions; and Apéry-like numbers
http://hdl.handle.net/2142/98291
Partition asymptotics; zeros of zeta functions; and Apéry-like numbers
Malik, Amita
PART I
G. H. Hardy and S. Ramanujan established an asymptotic formula for the number of
unrestricted partitions of a positive integer, and claimed a similar asymptotic formula for
the number of partitions into perfect kth powers, which was later proved by E. M. Wright.
Recently, R. C. Vaughan provided a simpler asymptotic formula in the case k = 2. In the
first part of the thesis, we study the number of partitions into parts from a specific set
Ak(a0; b0) :={mk : m 2 N;m _ a0 (mod b0)}, for fixed positive integers k, a0; and b0. Using
the Hardy-Littlewood circle method, we give an asymptotic formula for the number of such partitions, thus generalizing the aforementioned results of Wright and Vaughan. We also consider the parity problem for such partitions and prove that the number of such partitions is even (odd) infinitely often, which generalizes O. Kolberg's theorem for the ordinary partition function. This material builds on the joint work with B. C. Berndt and A. Zaharescu.
PART II
The Riemann Hypothesis implies that the zeros of all the derivatives of the Riemann-_
function lie on the critical line. Results on the proportion of zeros on the critical line of derivatives of _(s) have been investigated before by B. Conrey, and I. Rezvyakova. The percentage of zeros of _(k)(s) on the critical line approaches 100% percent as k increases. The second part of this thesis builds on the joint work with S. Chaubey, N. Robles, and A. Zaharescu. We study the zeros of combinations of derivatives of _(s). Although such combinations do not always have all their zeros on the critical line, we show that the proportion of zeros on the critical line still tends to 1.
PART III
The third part of this thesis focuses on the work on Apéry-like numbers joint with Armin
Straub. In 1982, Gessel showed that the Apéry numbers associated to the irrationality of
_(3) satisfy Lucas congruences. Our main result is to prove corresponding congruences for all known sporadic Apéry-like sequences. In several cases, we are able to employ approaches due to McIntosh, Samol-van Straten and Rowland-Yassawi to establish these congruences. However, for the sequences labeled s18 and (_) we require a finer analysis. As an application, we investigate modulo which numbers these sequences are periodic. In particular, we show that the Almkvist-Zudilin numbers are periodic modulo 8, a special property which they share with the Apéry numbers. We also investigate primes which do not divide any term of a given Apéry -like sequence.
Partitions; Arithmetic progressions; Parity; Asymptotics; Zeros; Riemann zeta function; Proportion; Apéry numbers
Thu, 13 Jul 2017 00:00:00 GMThttp://hdl.handle.net/2142/982912017-07-13T00:00:00ZMalik, AmitaCorrelations of sequences modulo one and statistics of geometrical objects associated to visible points
http://hdl.handle.net/2142/98203
Correlations of sequences modulo one and statistics of geometrical objects associated to visible points
Chaubey, Sneha
This thesis is divided into two major topics. In the first, we study the topic of distribution of sequences modulo one. In particular, we look at the spacing distributions between members of rational valued sequences modulo one. We come up with examples of many such sequences which behave as randomly chosen numbers from the unit interval. These include examples from the class of exponentially as well as sub-exponentially growing sequences. In the second part, we examine distribution questions for certain geometrical objects, for example, Farey-Ford and generalized Farey-Ford polygons and Farey-Ford parabolas associated to visible lattice points. As the names suggest, these objects are constructed based on the relation between visible points/Farey fractions and their geometrical interpretation in the form of Ford circles. We study the distribution of moments of various geometrical parameters associated to these objects by giving asymptotic formulas employing tools from analytic number theory.
Pair correlation; Riemann zeta; Visible lattice points
Mon, 10 Jul 2017 00:00:00 GMThttp://hdl.handle.net/2142/982032017-07-10T00:00:00ZChaubey, Sneha