arXiv subjects

Tahl Nowik

Publications and source records attributed to Tahl Nowik.

At least 19 recordsLinked to original sources

arXiv2022

The homology of random simplicial complexes in the multi-parameter upper model

We study random simplicial complexes in the multi-parameter upper model. In this model simplices of various dimensions are taken randomly and independently, and our random simplicial complex $Y$ is then taken to be the minimal simplicial complex containing this collection of simplices. We study the asymptotic behavior of the homology of $Y$ as the number of vertices goes to $\infty$. We observe the following phenomenon asymptotically almost surely. The given probabilities with which the simplices are taken determine a range of dimensions $\ell \leq k \leq \ell'$ with $\ell' \leq 2\ell +1$, outside of which the homology of $Y$ vanishes. Within this range, the homologies diminish drastically from dimension to dimension. In particular, the homology in the critical dimension $\ell$ is significantly the largest.

math.AT↗

arXiv2022

Random Simplicial Complexes, Duality and The Critical Dimension

In this paper we discuss two general models of random simplicial complexes which we call the lower and the upper models. We show that these models are dual to each other with respect to combinatorial Alexander duality. The behaviour of the Betti numbers in the lower model is characterised by the notion of critical dimension, which was introduced by A. Costa and M. Farber: random simplicial complexes in the lower model are homologically approximated by a wedge of spheres of dimension equal the critical dimension. In this paper we study the Betti numbers in the upper model and introduce new notions of critical dimension and spread. We prove that (under certain conditions) an upper random simplicial complex is homologically approximated by a wedge of spheres of the critical dimension.

math.AT↗

arXiv2020

A systolic inequality with remainder in the real projective plane

The first paper in systolic geometry was published by Loewner's student P. M. Pu over half a century ago. Pu proved an inequality relating the systole and the area of an arbitrary metric on the real projective plane. We prove a stronger version of Pu's systolic inequality with a remainder term.

math.DG↗

arXiv2020

Metric completions, the Heine-Borel property, and approachability

We show that the metric universal cover of a plane with a puncture yields an example of a nonstandard hull properly containing the metric completion of a metric space. As mentioned by do Carmo, a nonextendible Riemannian manifold can be noncomplete, but in the broader category of metric spaces it becomes extendible. We give a short proof of a characterisation of the Heine-Borel property of the metric completion of a metric space M in terms of the absence of inapproachable finite points in *M.

math.DG↗

arXiv2020

Topological embeddings into random 2-complexes

We consider 2-dimensional random simplicial complexes $Y$ in the multi-parameter model. We establish the multi-parameter threshold for the property that every 2-dimensional simplicial complex $S$ admits a topological embedding into $Y$ asymptotically almost surely. Namely, if in the procedure of the multi-parameter model, each $i$-dimensional simplex is taken independently with probability $p_i=p_i(n)$, from a set of $n$ vertices, then the threshold is $p_0 p_1^3 p_2^2 = \frac{1}{n}$. This threshold happens to coincide with the previously established thresholds for uniform hyperbolicity and triviality of the fundamental group. Our claim in one direction is in fact slightly stronger, namely, we show that if $p_0 p_1^3 p_2^2$ is sufficiently larger than $\frac{1}{n}$ then every $S$ has a fixed subdivision $S'$ which admits a simplicial embedding into $Y$ asymptotically almost surely. The main geometric result we prove to this end is that given $ε>0$, there is a subdivision $S'$ of $S$ such that every subcomplex $T \subseteq S'$ has $\frac{f_0(T)}{f_1(T)}>\frac{1}{3}-ε$ and $\frac{f_0(T)}{f_2(T)}>\frac{1}{2}-ε$, where $f_i(T)$ denotes the number of simplices in $T$ of dimension $i$. In the other direction we show that if $p_0 p_1^3 p_2^2$ is sufficiently smaller than $\frac{1}{n}$, then asymptotically almost surely, the torus does not admit a topological embedding into $Y$. Here we use a result of Z. Gao which bounds the number of different triangulations of a surface.

math.GT↗

arXiv2018

Universal Knot Diagrams

We study collections of planar curves that yield diagrams for all knots. In particular, we show that a very special class called potholder curves carries all knots. This has implications for realizing all knots and links as special types of meanders and braids. We also introduce and apply a method to compare the efficiency of various classes of curves that represent all knots.

math.GT↗

arXiv2018

The Distribution of Knots in the Petaluma Model

The representation of knots by petal diagrams (Adams et al. 2012) naturally defines a sequence of distributions on the set of knots. In this article we establish some basic properties of this randomized knot model. We prove that in the random n-petal model the probability of obtaining every specific knot type decays to zero as n, the number of petals, grows. In addition we improve the bounds relating the crossing number and the petal number of a knot. This implies that the n-petal model represents at least exponentially many distinct knots. Past approaches to showing, in some random models, that individual knot types occur with vanishing probability, rely on the prevalence of localized connect summands as the complexity of the knot increases. However this phenomenon is not clear in other models, including petal diagrams, random grid diagrams, and uniform random polygons. Thus we provide a new approach to investigate this question.

math.GT↗

arXiv2018

Monotone subsequence via ultrapower

An ultraproduct can be a helpful organizing principle in presenting solutions of problems at many levels, as argued by Terence Tao. We apply it here to the solution of a calculus problem: every infinite sequence has a monotone infinite subsequence, and give other applications. Keywords: ordered structures; monotone subsequence; ultrapower; saturation; compactness

math.CA↗

arXiv2016

Gregory's sixth operation

In relation to a thesis put forward by Marx Wartofsky, we seek to show that a historiography of mathematics requires an analysis of the ontology of the part of mathematics under scrutiny. Following Ian Hacking, we point out that in the history of mathematics the amount of contingency is larger than is usually thought. As a case study, we analyze the historians' approach to interpreting James Gregory's expression ultimate terms in his paper attempting to prove the irrationality of pi. Here Gregory referred to the last or ultimate terms of a series. More broadly, we analyze the following questions: which modern framework is more appropriate for interpreting the procedures at work in texts from the early history of infinitesimal analysis? as well as the related question: what is a logical theory that is close to something early modern mathematicians could have used when studying infinite series and quadrature problems? We argue that what has been routinely viewed from the viewpoint of classical analysis as an example of an "unrigorous" practice, in fact finds close procedural proxies in modern infinitesimal theories. We analyze a mix of social and religious reasons that had led to the suppression of both the religious order of Gregory's teacher degli Angeli, and Gregory's books at Venice, in the late 1660s.

math.HO↗

arXiv2016

Invariants of Random Knots and Links

We study random knots and links in R^3 using the Petaluma model, which is based on the petal projections developed by Adams et al. (2012). In this model we obtain a formula for the distribution of the linking number of a random two-component link. We also obtain formulas for the expectations and the higher moments of the Casson invariant and the order-3 knot invariant v3. These are the first precise formulas given for the distributions of invariants in any model for random knots or links. We also use numerical computation to compare these to other random knot and link models, such as those based on grid diagrams.

math.GT↗

arXiv2016

Small oscillations of the pendulum, Euler's method, and adequality

Small oscillations evolved a great deal from Klein to Robinson. We propose a concept of solution of differential equation based on Euler's method with infinitesimal mesh, with well-posedness based on a relation of adequality following Fermat and Leibniz. The result is that the period of infinitesimal oscillations is independent of their amplitude. Keywords: harmonic motion; infinitesimal; pendulum; small oscillations

math.HO↗

arXiv2016

Blotto Games with Costly Winnings

We introduce a new variation of the m-player asymmetric Colonel Blotto game, where the n battles occur as sequential stages of the game, and the winner of each stage needs to spend resources for maintaining his win. The limited resources of the players are thus needed both for increasing the probability of winning and for the maintenance costs. We show that if the initial resources of the players are not too small, then the game has a unique Nash equilibrium, and the given equilibrium strategies guarantee the given expected payoff for each player.

math.OC↗

arXiv2015

Differential geometry via infinitesimal displacements

We present a new formulation of some basic differential geometric notions on a smooth manifold M, in the setting of nonstandard analysis. In place of classical vector fields, for which one needs to construct the tangent bundle of M, we define a prevector field, which is an internal map from *M to itself, implementing the intuitive notion of vectors as infinitesimal displacements. We introduce regularity conditions for prevector fields, defined by finite differences, thus purely combinatorial conditions involving no analysis. These conditions replace the more elaborate analytic regularity conditions appearing in previous similar approaches, e.g. by Stroyan and Luxemburg or Lutz and Goze. We define the flow of a prevector field by hyperfinite iteration of the given prevector field, in the spirit of Euler's method. We define the Lie bracket of two prevector fields by appropriate iteration of their commutator. We study the properties of flows and Lie brackets, particularly in relation with our proposed regularity conditions. We present several simple applications to the classical setting, such as bounds related to the flow of vector fields, analysis of small oscillations of a pendulum, and an instance of Frobenius' Theorem regarding the complete integrability of independent vector fields.

math.DG↗

arXiv2014

Resolution of the Surprise Exam Paradox

We present a resolution of the celebrated "Surprise Exam Paradox". We argue that if the surprise exam story is analyzed using the exact same meaning of the notion of "surprise" as is dictated by the story itself, then no paradox arises.

math.HO↗

arXiv2014

Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow

Fermat, Leibniz, Euler, and Cauchy all used one or another form of approximate equality, or the idea of discarding "negligible" terms, so as to obtain a correct analytic answer. Their inferential moves find suitable proxies in the context of modern theories of infinitesimals, and specifically the concept of shadow. We give an application to decreasing rearrangements of real functions.

math.HO↗

arXiv2009

Complexity of relations in the braid group

We show that for any given n, there exists a sequence of words a_k in the generators sigma_1, ... sigma_{n-1} of the braid group B_n, representing the identity element of B_n, such that the number of braid relations of the form sigma_i sigma_{i+1} sigma_i = sigma_{i+1} sigma_i sigma_{i+1} needed to pass from a_k to the empty word is quadratic with respect to the length of a_k.

math.GR↗

arXiv2009

The expected genus of a random chord diagram

To any generic curve in an oriented surface there corresponds an oriented chord diagram, and any oriented chord diagram may be realized by a curve in some oriented surface. The genus of an oriented chord diagram is the minimal genus of an oriented surface in which it may be realized. Let g_n denote the expected genus of a randomly chosen oriented chord diagram of order n. We show that g_n satisfies: g_n = n/2 - Theta(ln n).

math.GT↗

arXiv2008

New Permutation Representations of the Braid Group

We give a new infinite family of group homomorphisms from the braid group B_k to the symmetric group S_{mk} for all k and m \geq 2. Most known permutation representations of braids are included in this family. We prove that the homomorphisms in this family are non-cyclic and transitive. For any divisor l of m, 1\leq l < m, we prove in particular that if \frac{m}{l} is odd then there are 1 + \frac{m}{l} non-conjugate homomorphisms included in our family. We define a certain natural restriction on homomorphisms B_k to S_n, common to all homomorphisms in our family, which we term 'good', and of which there are two types. We prove that all good homomorphisms B_k to S_{mk} of type 1 are included in the infinite family of homomorphisms we gave. For m=3, we prove that all good homomorphisms B_k to S_{3k} of type 2 are also included in this family. Finally, we refute a conjecture made by Matei and Suciu regarding permutation representations of braids and give an updated conjecture.

math.GR↗