Daniel L. Silver - University of South Alabama

Daniel L. Silver
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Name
Daniel L. Silver
Affiliation
University of South Alabama
City
Mobile
Country
United States

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Mathematics - Geometric Topology (37)
 
Mathematics - Dynamical Systems (8)
 
Mathematics - Group Theory (4)
 
Mathematics - Combinatorics (4)
 
Computer Science - Computer Vision and Pattern Recognition (2)
 
Computer Science - Learning (2)
 
Mathematics - Quantum Algebra (1)
 
Mathematics - Commutative Algebra (1)
 
Computer Science - Neural and Evolutionary Computing (1)
 
Computer Science - Artificial Intelligence (1)
 
Mathematics - History and Overview (1)

Publications Authored By Daniel L. Silver

Understanding and discovering knowledge from GPS (Global Positioning System) traces of human activities is an essential topic in mobility-based urban computing. We propose TrajectoryNet-a neural network architecture for point-based trajectory classification to infer real world human transportation modes from GPS traces. To overcome the challenge of capturing the underlying latent factors in the low-dimensional and heterogeneous feature space imposed by GPS data, we develop a novel representation that embeds the original feature space into another space that can be understood as a form of basis expansion. Read More

The (torsion) complexity of a finite edge-weighted graph is defined to be the order of the torsion subgroup of the abelian group presented by its Laplacian matrix. When G is d-periodic (i.e. Read More

The complexity of a finite connected graph is its number of spanning trees; for a non-connected graph it is the product of complexities of its connected components. If $G$ is an infinite graph with cofinite free ${\mathbb Z}^d$-symmetry, then the logarithmic Mahler measure $m(\Delta)$ of its Laplacian polynomial $\Delta$ is the exponential growth rate of the complexity of finite quotients of $G$. It is bounded below by $m(\Delta({\mathbb G}_d))$, where ${\mathbb G}_d$ is the grid graph of dimension $d$. Read More

In 1986, W. Thurston introduced a (possibly degenerate) norm on the first cohomology group of a 3-manifold. Inspired by this definition, Turaev introduced in 2002 a analogous norm on the first cohomology group of a finite 2-complex. Read More

A short, elementary proof is given of the result: The number of components of a link arising from a medial graph M(G) by resolving vertices is equal to the nullity of the mod-2 Laplacian matrix of the graph G. Read More

The space C of conservative vertex colorings (over a field F) of a countable, locally finite graph G is introduced. The subspace of based colorings is shown to be isomorphic to the bicycle space of the graph. For graphs G with a free Z^d-action by automorphisms, C is a finitely generated module over the polynomial ring F[Z^d], and for this a polynomial invariant, the Laplacian polynomial, is defined. Read More

This paper presents an unsupervised multi-modal learning system that learns associative representation from two input modalities, or channels, such that input on one channel will correctly generate the associated response at the other and vice versa. In this way, the system develops a kind of supervised classification model meant to simulate aspects of human associative memory. The system uses a deep learning architecture (DLA) composed of two input/output channels formed from stacked Restricted Boltzmann Machines (RBM) and an associative memory network that combines the two channels. Read More

Let K be a knot of genus g. If K is fibered, then it is well known that the knot group pi(K) splits only over a free group of rank 2g. We show that if K is not fibered, then pi(K) splits over non-free groups of arbitrarily large rank. Read More

2013Apr
Affiliations: 1University of South Alabama, 2University of South Alabama, 3University of South Alabama

A group invariant for links in thickened closed orientable surfaces is studied. Associated polynomial invariants are defined. The group detects nontriviality of a virtual link and determines its virtual genus. Read More

The virtual genus of a virtual satellite link is equal to that of its companion. Read More

2013Jan
Affiliations: 1University of South Alabama, 2University of South Alabama, 3University of South Alabama

This survey article discusses three aspects of knot colorings. Fox colorings are assignments of labels to arcs, Dehn colorings are assignments of labels to regions, and Alexander-Briggs colorings assign labels to vertices. The labels are found among the integers modulo n. Read More

Let \phi be an endomorphism of a finitely generated free group F, and let H be a finite-index subgroup of F that is invariant under \phi. The nonzero eigenvalues of \phi are contained in the eigenvalues of \phi restricted to H. Read More

Let L be an oriented (d+1)-component link in the 3-sphere, and let L(q) be the d-component link in a homology 3-sphere that results from performing 1/q-surgery on the last component. Results about the Alexander polynomial and twisted Alexander polynomials of L(q) corresponding to finite-image representations are obtained. The behavior of the invariants as q increases without bound is described. Read More

Twisted Alexander invariants have been defined for any knot and linear representation of its group. The invariants are generalized for any periodic representation of the commutator subgroup of the knot group. Properties of the new twisted invariants are given. Read More

A finitely generated module over the ring L=Z[t, t^{-1}] of integer Laurent polynomials that has no Z-torsion is determined by a pair of sub-lattices of L^d. Their indices are the absolute values of the leading and trailing coefficients of the order of the module. This description has applications in knot theory. Read More

The group of any nontrivial torus knot, hyperbolic 2-bridge knot, or hyperbolic knot with unknotting number one contains infinitely many elements, none the automorphic image of another, such that each normally generates the group. Read More

We generalize a theorem of Burde and de Rham characterizing the zeros of the Alexander polynomial. Given a representation of a knot group $\pi$, we define an extension of $\pi$, the Crowell group. For any GL(n,C) representation of $\pi$, the zeros of the associated twisted Alexander polynomial correspond to representations of the Crowell group into the group of dilations of C^n. Read More

X.S. Lin's original definition of twisted Alexander knot polynomial is generalized for arbitrary finitely presented groups. Read More

2009May
Affiliations: 1University of Sydney, 2University of South Alabama, 3University of South Alabama

Given a knot and an SL(n,C) representation of its group that is conjugate to its dual, the representation that replaces each matrix with its inverse-transpose, the associated twisted Reidemeister torsion is reciprocal. An example is given of a knot group and SL(3,Z) representation that is not conjugate to its dual for which the twisted Reidemeister torsion is not reciprocal. Read More

We consider the relations $\ge$ and $\ge_p$ on the collection of all knots, where $k \ge k'$ (respectively, $k \ge_p k'$) if there exists an epimorphism $\pi k \to \pi k'$ of knot groups (respectively, preserving peripheral systems). When $k$ is a torus knot, the relations coincide and $k'$ must also be a torus knot; we determine the knots $k'$ that can occur. If $k$ is a 2-bridge knot and $k \ge_p k'$, then $k'$ is a 2-bridge knot with determinant a proper divisor of the determinant of $k$; only finitely many knots $k'$ are possible. Read More

The Pontryagin dual of the twisted Alexander module for a d-component link and GL(N,Z) representation is an algebraic dynamical system with an elementary description in terms of colorings of a diagram. In the case of a knot, its associated topological entropy is the logarithmic growth rate of the number of torsion elements in the twisted first-homology group of r-fold cyclic covers of the knot complement, as r goes to infinity. Total twisted representations are introduced, and their properties are studied. Read More

For any knot, the following are equivalent. (1) The infinite cyclic cover has uncountably many finite covers; (2) there exists a finite-image representation of the knot group for which the twisted Alexander polynomial vanishes; (3) the knot group admits a finite-image representation such that the image of the fundamental group of an incompressible Seifert surface is a proper subgroup of the image of the commutator subgroup of the knot group. Read More

The authors conjectured previously that a knot is nonfibered if and only if its infinite cyclic cover has uncountably many finite covers. We prove the conjecture for a class of knots that includes all knots of genus 1, using techniques from symbolic dynamics. Read More

Necessary and sufficient conditions are given for a satellite knot to be fibered. Any knot $\tilde k$ embeds in an unknotted solid torus $\tilde V$ with arbitrary winding number in such a way that no satellite knot with pattern $(\tilde V, \tilde k)$ is fibered. In particular, there exist nonfibered satellite knots with fibered pattern and companion knots and nonzero winding number. Read More

2007Mar
Affiliations: 1Univ. of South Ala., 2Univ. of South Fla., 3Univ. of South Fla., 4Univ. of South Ala., 5Univ. of South Ala.

A group-theoretical method, via Wada's representations, is presented to distinguish Kishino's virtual knot from the unknot. Biquandles are constructed for any group using Wada's braid group representations. Cocycle invariants for these biquandles are studied. Read More

The group of a nontrivial knot admits a finite permutation representation such that the corresponding twisted Alexander polynomial is not a unit. Read More

Lehmer's question is equivalent to one about generalized growth rates of Lefschetz numbers of iterated pseudo-Anosov surface homeomorphisms. One need consider only homeomorphisms that arise as monodromies of fibered knots in lens spaces L(n,1), n>0. Lehmer's question for Perron polynomials is equivalent to one about generalized growth rates of words under injective free group endomorphisms. Read More

The derived group of a permutation representation, introduced by R.H. Crowell, unites many notions of knot theory. Read More

We describe a pair of invariants for actions of finite groups on shifts of finite type, the left-reduced and right-reduced shifts. The left-reduced shift was first constructed by U. Fiebig, who showed that its zeta function is an invariant, and in fact equal to the zeta function of the quotient dynamical system. Read More

Given any knot k, there exists a hyperbolic knot tilde k with arbitrarily large volume such that the knot group pi k is a quotient of pi tilde k by a map that sends meridian to meridian and longitude to longitude. The knot tilde k can be chosen to be ribbon concordant to k and also to have the same Alexander invariant as k. Read More

If the twist numbers of a collection of oriented alternating link diagrams are bounded, then the Alexander polynomials of the corresponding links have bounded euclidean Mahler measure (see Definition 1.2). The converse assertion does not hold. Read More

Extended Alexander groups are used to define an invariant for open virtual strings. Examples of non-commuting open strings and a ribbon-concordance obstruction are given. An example is given of a slice virtual open string that is not ribbon. Read More

2004May
Affiliations: 1University of South Alabama, 2University of South Alabama

Let K be the kernel of an epimorphism G -> Z, where G is a finitely presented group. If K has infinitely many subgroups of index 2, 3, or 4, then it has uncountably many. Moreover, if K is the commutator subgroup of a classical knot group G, then any homomorphism from K onto the symmetric group S_2 lifts to a homomorphism onto S_3, and any homomorphism from K onto Z_3 lifts to a homomorphism onto the alternating group A_4. Read More

2004May
Affiliations: 1University of South Alabama, 2University of South Alabama

Alexander group systems for virtual long knots are defined and used to show that any virtual knot is the closure of infinitely many long virtual knots. Manturov's result that there exists a pair of long virtual knots that do not commute is reproved. Read More

2004May
Affiliations: 1University of South Alabama

Any knot group is the image of the group of a prime knot by a homomorphism that preserves peripheral structure. In fact, there are infinitely many such prime knots. A related partial order on knots is defined, and its properties are discussed. Read More

2004May
Affiliations: 1George Washington University, 2University of South Alabama, 3University of South Alabama

New obstructions for embedding one compact oriented 3-manifold in another are given. A theorem of D. Krebes concerning 4-tangles embedded in links arises as a special case. Read More

Torsion and Betti numbers for knots are special cases of more general invariants associated to a finitely generated group G and epimorphism from G to the integers. The sequence of Betti numbers is always periodic; under mild hypotheses, the sequence of torsion numbers satisfies a linear homogeneous recurrence relation with constant coeffiencts. Generally, the torsion number sequence exhibits exponential growth rate. Read More

2001May
Affiliations: 1University of South Alabama, 2University of South Alabama

Let l be an oriented link of d components in a homology 3-sphere. For any nonnegative integer q, let l(q) be the link of d-1 components obtained from l by performing 1/q surgery on the dth component. Then the Mahler measure of the Alexander polynomial of l(q) converges to the Mahler measure of the Alexander polynomial of l as q goes to infinity, provided that some other component of l has nonzero linking number with the dth. Read More

2000Mar
Affiliations: 1University of South Alabama, 2University of South Alabama

Let l be a link of d components. For every finite-index lattice in Z^d there is an associated finite abelian cover of S^3 branched over l. We show that the order of the torsion subgroup of the first homology of these covers has exponential growth rate equal to the logarithmic Mahler measure of the Alexander polynomial of l, provided this polynomial is nonzero. Read More