arXiv subjects

M. Teicher

Publications and source records attributed to M. Teicher.

At least 19 recordsLinked to original sources

arXiv2010

Length-based conjugacy search in the Braid group

Several key agreement protocols are based on the following "Generalized Conjugacy Search Problem": Find, given elements b_1,...,b_n and xb_1x^{-1},...,xb_nx^{-1} in a nonabelian group G, the conjugator x. In the case of subgroups of the braid group B_N, Hughes and Tannenbaum suggested a length-based approach to finding x. Since the introduction of this approach, its effectiveness and successfulness were debated. We introduce several effective realizations of this approach. In particular, a new length function is defined on B_N which possesses significantly better properties than the natural length associated to the Garside normal form. We give experimental results concerning the success probability of this approach, which suggest that very large computational power is required for this method to successfully solve the Generalized Conjugacy Search Problem when its parameters are as in existing protocols.

math.GR↗

arXiv2008

Fundamental group for the complement of the Cayley's singularities

Given a singular surface X, one can extract information on it by investigating the fundamental group $π_1(X - Sing_X)$. However, calculation of this group is non-trivial, but it can be simplified if a certain invariant of the branch curve of X - called the braid monodromy factorization - is known. This paper shows, taking the Cayley cubic as an example, how this fundamental group can be computed by using braid monodromy techniques and their liftings. This is one of the first examples that uses these techniques to calculate this sort of fundamental group.

math.AG↗

arXiv2008

Braid monodromy factorization for a non-prime $K3$ surface branch curve

This paper is the second in a series. The first one describes pillow degenerations of a $K3$ surface with genus $g$. In this paper we study the $(2,2)$-pillow degeneration of a non-prime $K3$ surface and the braid monodromy of the branch curve of the surface with respect to a generic projection onto $\C¶^2$. In future papers we study the fundamental group of the complement of the branch curve and the fundamental group of the Galois cover of the surface with respect to this generic projection.

math.AG↗

arXiv2008

Coxeter covers of the classical Coxeter groups

Let $C(T)$ be a generalized Coxeter group, which has a natural map onto one of the classical Coxeter groups, either $B_n$ or $D_n$. Let $C_Y(T)$ be a natural quotient of $C(T)$, and if $C(T)$ is simply-laced (which means all the relations between the generators has order 2 or 3), $C_Y(T)$ is a generalized Coxeter group, too . Let $A_{t,n}$ be a group which contains $t$ Abelian groups generated by $n$ elements. The main result in this paper is that $C_Y(T)$ is isomorphic to $A_{t,n} \semidirect B_n$ or $A_{t,n} \semidirect D_n$, depends on whether the signed graph $T$ contains loops or not, or in other words C(T) is simply-laced or not, and $t$ is the number of the cycles in $T$. This result extends the results of Rowen, Teicher and Vishne to generalized Coxeter groups which have a natural map onto one of the classical Coxeter groups.

math.GR↗

arXiv2008

On the Well-possedness of the Problem of Reconstruction of Non-separate Boundary Conditions

We consider an inverse spectral problem with the third-order differential equation and the non-separated boundary conditions. Two theorems on the uniqueness of the solution of this problem are proved, and a method for establishing the unknown conditions is obtained, using 19 eigenvalues. The method of approximate calculation of unknown boundary conditions is explained, with the help of an example.

math.SP↗

arXiv2008

Identification of Boundary Conditions Using Natural Frequencies in Case of a Ring Membrane

The problem of finding boundary conditions for fastening of a ring membrane, which are inaccessible for direct observation from the natural frequencies of its flexural oscillations, is considered. Two theorems on the uniqueness of this problem are proved, and a method for establishing the unknown conditions for fastening of the membrane to the walls is indicated. An approximate formula for determining the unknown conditions is obtained, using first three natural frequencies. The method of approximate calculation of unknown boundary conditions, is explained with the help of an example. Keywords: Boundary conditions, inverse spectral problem, membrane, natural frequencies, Plucker coordinates, Plucker relation.

math.SP↗

arXiv2008

Can One Hear Fastening of a Rod?

Rods are parts of various devices. If it is impossible to observe the rod directly, the only source of information about possible defects of its fastening can be the natural frequencies of its flexural vibrations. The question arises whether one would be able to detect damage in rod fastening by the natural frequencies of its flexural vibrations. This paper gives and substantiates a positive answer to this question.

math.SP↗

arXiv2007

Trisecant Lemma for Non Equidimensional Varieties

The classic trisecant lemma states that if $X$ is an integral curve of $\PP^3$ then the variety of trisecants has dimension one, unless the curve is planar and has degree at least 3, in which case the variety of trisecants has dimension 2. In this paper, our purpose is first to present another derivation of this result and then to introduce a generalization to non-equidimensional varities. For the sake of clarity, we shall reformulate our first problem as follows. Let $Z$ be an equidimensional variety (maybe singular and/or reducible) of dimension $n$, other than a linear space, embedded into $\PP^r$, $r \geq n+1$. The variety of trisecant lines of $Z$, say $V_{1,3}(Z)$, has dimension strictly less than $2n$, unless $Z$ is included in a $(n+1)-$dimensional linear space and has degree at least 3, in which case $\dim(V_{1,3}(Z)) = 2n$. Then we inquire the more general case, where $Z$ is not required to be equidimensional. In that case, let $Z$ be a possibly singular variety of dimension $n$, that may be neither irreducible nor equidimensional, embedded into $\PP^r$, where $r \geq n+1$, and $Y$ a proper subvariety of dimension $k \geq 1$. Consider now $S$ being a component of maximal dimension of the closure of $\{l \in \G(1,r) \vtl \exists p \in Y, q_1, q_2 \in Z \backslash Y, q_1,q_2,p \in l\}$. We show that $S$ has dimension strictly less than $n+k$, unless the union of lines in $S$ has dimension $n+1$, in which case $dim(S) = n+k$. In the latter case, if the dimension of the space is stricly greater then $n+1$, the union of lines in $S$ cannot cover the whole space. This is the main result of our work. We also introduce some examples showing than our bound is strict.

math.AG↗

arXiv2007

Fundamental groups of some special quadric arrangements

In this work we obtain presentations of fundamental groups of the complements of three families of quadric arrangements in $\mathbb{P}^2$. The first arrangement is a union of $n$ quadrics, which are tangent to each other at two common points. The second arrangement is composed of $n$ quadrics which are tangent to each other at one common point. The third arrangement is composed of $n$ quadrics, $n-1$ of them are tangent to the $n$'th one and each one of the $n-1$ quadrics is transversal to the other $n-2$ ones.

math.AG↗

arXiv2007

Probabilistic Solutions of Equations in the Braid Group

Given a system of equations in a "random" finitely generated subgroup of the braid group, we show how to find a small ordered list of elements in the subgroup, which contains a solution to the equations with a significant probability. Moreover, with a significant probability, the solution will be the first in the list. This gives a probabilistic solution to: The conjugacy problem, the group membership problem, the shortest representation of an element, and other combinatorial group-theoretic problems in random subgroups of the braid group. We use a memory-based extension of the standard length-based approach, which in principle can be applied to any group admitting an efficient, reasonably behaving length function.

math.GR↗

arXiv2006

Several Applications of Bezout Matrices

The notion of Bezout matrix is an essential tool in studying broad variety of subjects: zeroes of polynomials, stability of differential equations, rational transformations of algebraic curves, systems of commuting nonselfadjoint operators, boundaries of quadrature domains etc. We present a survey of several properties of Bezout matrices and their applications in all mentioned topics. We use the framework of Vandermonde vectors because such approach allows us to give new proofs of both classical and modern results and in many cases to obtain new explicit formulas. These explicit formulas can significantly simplify various computational problems and, in particular, make the research of algebraic curves and their applications easier. In addition we wrote a Maple software package, which computes all the formulas. For instance, as Bezout matrices are used in order to compute the image of a rational transformation of an algebraic curve, we used these results to study some connections between small degree rational transformation of an algebraic curve and the braid monodromy of its image.

math.AG↗

arXiv2005

The Hurwitz Equivalence Problem is Undecidable

In this paper, we prove that the Hurwitz equivalence problem for 1-factorizations in $F_2 \oplus F_2$ is undecidable, and as a consequence, the Hurwitz equivalence problem for $Δ^2$-factorizations in the braid groups $B_n, n\geq 5$ is also undecidable.

math.LO↗

arXiv2005

Palindromic Braids

The braid group $B_{n}$, endowed with Artin's presentation, admits an antiautomorphism $B_{n} \to B_{n}$, such that $v \mapsto \bar{v}$ is defined by reading braids in reverse order (from right to left instead of left to right). We prove that the map $B_{n} \to B_{n}$, $v \mapsto v \bar{v}$ is injective. We also give some consequences arising due to this injectivity.

math.GT↗

arXiv2004

Braid Monodromy Computation of Real Singular Curves

We generalize the Moishezon Teicher algorithm that was suggested for the computation of the braid monodromy of an almost real curve. The new algorithm suits a larger family of curves, and enables the computation of braid monodromy not only of caspidal curves, but of general algebraic curves, with some non simple singularities. Moreover, it works also when in the fiber the curve admits any number of imaginary points. We also provide two examples of how to use the generalized algorithm.

math.AG↗

arXiv2004

Braid Monodromy Type and Rational Transformations of Plane Algebraic Curves

We combine the newly discovered technique, which computes explicit formulas for the image of an algebraic curve under rational transformation, with techniques that enable to compute braid monodromies of such curves. We use this combination in order to study properties of the braid monodromy of the image of curves under a given rational transformation. A description of the general method is given along with full classification of the images of two intersecting lines under degree 2 rational transformation. We also establish a connection between degree 2 rational transformations and the local braid monodromy of the image at the intersecting point of two lines. Moreover, we present an example of two birationally isomorphic curves with the same braid monodromy type and non diffeomorphic real parts.

math.AG↗

arXiv2004

Three-Dimensional Face Orientation and Gaze Detection from a Single Image

Gaze detection and head orientation are an important part of many advanced human-machine interaction applications. Many systems have been proposed for gaze detection. Typically, they require some form of user cooperation and calibration. Additionally, they may require multiple cameras and/or restricted head positions. We present a new approach for inference of both face orientation and gaze direction from a single image with no restrictions on the head position. Our algorithm is based on a face and eye model, deduced from anthropometric data. This approach allows us to use a single camera and requires no cooperation from the user. Using a single image avoids the complexities associated with of a multi-camera system. Evaluation tests show that our system is accurate, fast and can be used in a variety of applications, including ones where the user is unaware of the system.

cs.CV↗

arXiv2002

Identifying Powers of Half-Twists and Computing its Root

In this paper we give an algorithm for solving a main case of the conjugacy problem in the braid groups. We also prove that half-twists satisfy a special root property which allows us to reduce the solution for the conjugacy problem in half-twists into the free group. Using this algorithm one is able to check conjugacy of a given braid to one of E. Artin's generators in any power, and compute its root. Moreover, the braid element which conjugates a given half-twist to one of E. Artin's generators in any power can be restored. The result is applicable to calculations of braid monodromy of branch curves and verification of Hurwitz equivalence of braid monodromy factorizations, which are essential in order to determine braid monodromy type of algebraic surfaces and symplectic 4-manifolds.

math.AG↗

arXiv2002

Identifying Half-Twists Using Randomized Algorithm Methods

Since the braid group was discovered by E. Artin, the question of its conjugacy problem has been solved by Garside and Birman, Ko and Lee. However, the solutions given thus far are difficult to compute with a computer, since the number of operations needed is extremely large. Meanwhile, random algorithms used to solve difficult problems such as primality of a number were developed, and the random practical methods have become an important tool. We give a random algorithm, along with a conjecture of how to improve its convergence speed, in order to identify elements in the braid group, which are conjugated to its generators for a given power. These elements of the braid group, the half-twists, are important in themselves, as they are the key players in some geometrical and algebraical methods, the building blocks of quasipositive braids and they construct endless sets of generators for the group.

math.AG↗