arXiv subjects

Pedro Lopes

Publications and source records attributed to Pedro Lopes.

At least 37 records · Page 2Linked to original sources

arXiv2019

The Prevalence of Persistent Tangles

This article addresses persistent tangles. These are tangles whose presence in a knot diagram forces that diagram to be knotted. We provide new methods for constructing persistent tangles. Our techniques rely mainly on the existence of non-trivial colorings for the tangles in question. Our main result in this article is that any knot admitting a non-trivial coloring gives rise to persistent tangles. Furthermore, we discuss when these persistent tangles are non-trivial.

math.GT↗

arXiv2019

Understanding the Effectiveness of Ultrasonic Microphone Jammer

Recent works have explained the principle of using ultrasonic transmissions to jam nearby microphones. These signals are inaudible to nearby users, but leverage "hardware nonlinearity" to induce a jamming signal inside microphones that disrupts voice recordings. This has great implications on audio privacy protection. In this work, we gain a deeper understanding on the effectiveness of ultrasonic jammer under practical scenarios, with the goal of disabling both visible and hidden microphones in the surrounding area. We first experiment with existing jammer designs (both commercial products and that proposed by recent papers), and find that they all offer limited angular coverage, and can only target microphones in a particular direction. We overcome this limitation by building a circular transducer array as a wearable bracelet. It emits ultrasonic signals simultaneously from many directions, targeting surrounding microphones without needing to point at any. More importantly, as the bracelet moves with the wearer, its motion increases jamming coverage and diminishes blind spots (the fundamental problem facing any transducer array). We evaluate the jammer bracelet under practical scenarios, confirming that it can effectively disrupt visible and hidden microphones in the surrounding areas, preventing recognition of recorded speech. We also identify limitations and areas for improvement.

cs.CR↗

arXiv2019

Antidote SQL: Relaxed When Possible, Strict When Necessary

Geo-replication poses an inherent trade-off between low latency, high availability and strong consistency. While NoSQL databases favor low latency and high availability, relaxing consistency, more recent cloud databases favor strong consistency and ease of programming, while still providing high scalability. In this paper, we present Antidote SQL, a database system that allows application developers to relax SQL consistency when possible. Unlike NoSQL databases, our approach enforces primary key, foreign key and check SQL constraints even under relaxed consistency, which is sufficient for guaranteeing the correctness of many applications. To this end, we defined concurrency semantics for SQL constraints under relaxed consistency and show how to implement such semantics efficiently. For applications that require strict SQL consistency, Antidote SQL provides support for such semantics at the cost of requiring coordination among replicas.

cs.DB↗

arXiv2018

Quandles of cyclic type with several fixed points

A quandle of cyclic type of order $n$ with $f\geq 2$ fixed points is such that each of its permutations splits into $f$ cycles of length $1$ and one cycle of length $n-f$. In this article we prove that there is only one such connected quandle, up to isomorphism. This is a quandle of order $6$ and $2$ fixed points, known in the literature as octahedron quandle. We prove also that, for each $f\geq 2$, the non-connected versions of these quandles only occur for orders $n$ in the range $f+2 \leq n \leq 2f$ and that, for each $f>1$, there is only one such quandle of order $2f$ with $f$ fixed points, up to isomorphism. Still in the range $f+2 \leq n \leq 2f$, we present sufficient conditions for the existence of such quandles, writing down their permutations; we also show how to obtain new quandles form old ones, leaning on the notion of common fixed point.

math.GR↗

arXiv2018

Colorings beyond Fox: the other linear Alexander quandles

This article is about applications of linear algebra to knot theory. For example, for odd prime p, there is a rule (given in the article) for coloring the arcs of a knot or link diagram from the residues mod p. This is a knot invariant in the sense that if a diagram of the knot under study admits such a coloring, then so does any other diagram of the same knot. This is called p-colorability. It is also associated to systems of linear homogeneous equations over the residues mod p, by regarding the arcs of the diagram as variables and assigning the equation "twice the over-arc minus the sum of the under-arcs equals zero" to each crossing. The knot invariant is here the existence or non-existence of non-trivial solutions of these systems of equations, when working over the integers modulo p (a non-trivial solution is such that not all variables take up the same value). Another knot invariant is the minimum number of distinct colors (values) these non-trivial solutions require, should they exist. This corresponds to finding a basis, supported by a diagram, in which these solutions take up the least number of distinct values. The actual minimum is hard to calculate, in general. For the first few primes, less than 17, it depends only on the prime, p, and not on the specific knots that admit non-trivial solutions, modulo p. For primes larger than 13 this is an open problem. In this article, we begin the exploration of other generalizations of these colorings (which also involve systems of linear homogeneous equations mod p) and we give lower bounds for the number of colors.

math.GT↗

arXiv2017

Infinitely many prime knots with the same Alexander invariants

We revisit the issue of the existence of infinitely many distinct prime knots with the same Alexander invariant. We present infinitely many distinct families, each family made up of infinitely many distinct knots. Within each family, the Alexander invariant is the same. Unlike other examples in the literature, ours are elementary and based on a sub-collection of pretzel knots with three tassels.

math.GT↗

arXiv2016

Partial profiles of quasi-complete graphs

We enumerate graph homomorphisms to quasi-complete graphs, i.e., graphs obtained from complete graphs by removing one edge. The source graphs are complete graphs, quasi-complete graphs, cycles, paths, wheels and broken wheels. These enumerations give rise to sequences of integers with two indices; one of the indices is the number of vertices of the source graph, and the other index is the number of vertices of the target graph.

math.CO↗

arXiv2016

Delta Diagrams

We call a Delta Diagram any diagram of a knot or link whose regions (including the unbounded one) have 3, 4, or 5 sides. We prove that any knot or link admits a delta diagram. We define and estimate combinatorial link invariants stemming from this definition.

math.GT↗

arXiv2015

The minimum number of Fox colors modulo 13 is 5

In this article we show that if a knot diagram admits a non-trivial coloring modulo 13 then there is an equivalent diagram which can be colored with 5 colors. Leaning on known results, this implies that the minimum number of colors modulo 13 is 5.

math.GT↗

arXiv2015

The Minimization of the Number of Colors is Different at p=11

In this article we present the following new fact for prime p=11. For knots 6_2 and 7_2, mincol_{11} 6_2 = 5 = mincol_{11} 7_2, along with the following feature. There is a pair of diagrams, one for 6_2 and the other one for 7_2, each of them admitting only non-trivial 11-colorings using 5 colors, but neither of them admitting being colored with the sets of 5 colors that color the other one. This is in full contrast with the behavior exhibited by links admitting non-trivial p-colorings over the smaller primes, p=2, 3, 5 or 7. We also prove results concerning obstructions to the minimization of colors over generic odd moduli. We apply these to find the right colors to eliminate from non-trivial colorings. We thus prove that 5 is the minimum number of colors for each knot of prime determinant 11 or 13 from Rolfsen's table.

math.GT↗

arXiv2015

Minimal sufficient sets of colors and minimum number of colors

In this paper we first investigate minimal sufficient sets of colors for p=11 and 13. For odd prime p and any p-colorable link L with non-zero determinant, we give alternative proofs of mincol_p L \geq 5 for p \geq 11 and mincol_p L \geq 6 for p \geq 17. We elaborate on equivalence classes of sets of distinct colors (on a given modulus) and prove that there are two such classes of five colors modulo 11, and only one such class of five colors modulo 13. Finally, we give a positive answer to a question raised by Nakamura, Nakanishi, and Satoh concerning an inequality involving crossing numbers. We show it is an equality only for the trefoil and for the figure-eight knots.

math.GT↗

arXiv2014

The Delunification Process and Minimal Diagrams

A link diagram is said to be lune-free if, when viewed as a 4-regular plane graph it does not have multiple edges between any pair of nodes. We prove that any colored link diagram is equivalent to a colored lune-free diagram with the same number of colors. Thus any colored link diagram with a minimum number of colors (known as a minimal diagram) is equivalent to a colored lune-free diagram with that same number of colors. We call the passage from a link diagram to an equivalent lune-free diagram its delunification process. We then introduce a notion of grey sets in order to obtain higher lower bounds for minimum number of colors. We calculate these higher lower bounds for a number of prime moduli with the help of computer programs. For each number of crossings through 16, we list the lune-free diagrams and we color them. If the number of colors equals the corresponding higher lower bound we know we have a minimum number of colors. We also introduce and list the lune-free crossing number of a link i.e., the minimum number of crossings needed for a lune-free diagram of this link, and other related link invariants.

math.GT↗

arXiv2014

Equivalence Classes of Colorings

For any link and for any modulus $m$ we introduce an equivalence relation on the set of non-trivial m-colorings of the link (an m-coloring has values in Z/mZ). Given a diagram of the link, the equivalence class of a non-trivial m-coloring is formed by each assignment of colors to the arcs of the diagram that is obtained from the former coloring by a permutation of the colors in the arcs which preserves the coloring condition at each crossing. This requirement implies topological invariance of the equivalence classes. We show that for a prime modulus the number of equivalence classes depends on the modulus and on the rank of the coloring matrix (with respect to this modulus).

math.GT↗

arXiv2013

Removing Colors 2k, 2k-1, and k

We prove that if a link admits non-trivial (2k+1)-colorings, with prime 2k+1>7, it also admits non-trivial (2k+1)-colorings not involving colors 2k, 2k-1, nor k.

math.GT↗

arXiv2013

On the Maximum Number of Colors for Links

For each odd prime p, and for each non-split link admitting non-trivial p-colorings, we prove that the maximum number of Fox colors is p. We also prove that we can assemble a non-trivial p-coloring with any number of colors, from the minimum to the maximum number of colors. Furthermore, for any rational link, we prove that there exists a non-trivial coloring of any Schubert Normal Form of it, modulo its determinant, which uses all colors available. If this determinant is an odd prime, then any non-trivial coloring of this Schubert Normal Form, modulo the determinant, uses all available colors. We prove also that the number of crossings in the Schubert Normal Form equals twice the determinant of the link minus 2. Facts about torus links and their coloring abilities are also proved.

math.GT↗

arXiv2012

The regularity of the $η$ function for the Shubin calculus

We prove the regularity of the $η$ function for classical pseudodifferential operators with Shubin symbols. We recall the construction of complex powers and of the Wodzicki and Kontsevich-Vishik functionals for classical symbols on $\mathbb{R}^{n}$ with these symbols. We then define the $ζ$ and $η$ functions associated to suitable elliptic operators. We compute the $K_{0}$ group of the algebra of zero-order operators and use this knowledge to show that the Wodzicki trace of the idempotents in the algebra vanishes. From this, it follows that the $η$ function is regular at 0 for any self-adjoint elliptic operator of positive order.

math.OA↗

arXiv2012

The Teneva Game

For each prime p > 7 we obtain the expression for an upper bound on the minimum number of colors needed to non-trivially color T(2, p), the torus knots of type (2, p), modulo p. This expression is t + 2 l -1 where t and l are extracted from the prime p. It is obtained from iterating the so-called Teneva transformations which we introduced in a previous article. With the aid of our estimate we show that the ratio "number of colors needed vs. number of colors available" tends to decrease with increasing modulus p. For instance as of prime 331, the number of colors needed is already one tenth of the number of colors available. Furthermore, we prove that 5 is minimum number of colors needed to non-trivially color T(2, 11) modulo 11. Finally, as a preview of our future work, we prove that 5 is the minimum number of colors modulo 11 for two rational knots with determinant 11.

math.GT↗

arXiv2009

Determinants of rational knots

We study the Fox coloring invariants of rational knots. We express the propagation of the colors down the twists of these knots and ultimately the determinant of them with the help of finite increasing sequences whose terms of even order are even and whose terms of odd order are odd.

math.GT↗