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Papers by Elinor L . Velasquez

Describe the periodic non-abelian Toda lattice generalized to a finite analog of a symmetric spac... more Describe the periodic non-abelian Toda lattice generalized to a finite analog of a symmetric space. Describe the algebro-geometric method for this setting. Discuss the inverse scattering approach for this setting (Optional). Describe how the Korteweg-de Vries equation can be realized on a hyperbolic space (Optional).

Geometric analogues of the Toda lattice are studied on a type of connected graph. Graph-theoretic... more Geometric analogues of the Toda lattice are studied on a type of connected graph. Graph-theoretic Hamiltonian systems are constructed. Toda lattices and its generalization to this setting are given as examples.

A prototype for a web application was designed and implemented as a guide to be used by clinician... more A prototype for a web application was designed and implemented as a guide to be used by clinicians when designing the best drug therapy for a specific cancer patient, given biological data derived from the patients tumor tissue biopsy. A representation of the patients metabolic pathways is displayed as a graph in the application, with nodes as substrates and products and edges as enzymes. The top metabolically active subpaths in the pathway, ranked using an algorithm based on both the patients biological data and the graph topology, are also displayed and can be individually highlighted to examine potential enzymatic sites to be disrupted by a drug these sites serve as a guide for designing the patients specific drug therapy. Displayed next to each sub-path is the sub-path score used to decide its rank, as well as the predicted patient survival time to indicate how effective that specific drug will be in alleviating that patient's cancer. Future work includes an animation component to track the patients progress and redesign the drug therapy as needed.

Abstract. The Radon transform on Zkn averages a function over its values on a translate of a fixe... more Abstract. The Radon transform on Zkn averages a function over its values on a translate of a fixed subset S in Zkn. We discuss invertibility conditions and computer inverse formulas based on the Moore–Penrose inverse and on linear algorithms. We expect the results to be of use in directional and toroidal time series.

The Radon transform on Zn averages a function over its values on a translate of a fixed subset S ... more The Radon transform on Zn averages a function over its values on a translate of a fixed subset S in Zn. We discuss invertibility conditions and computer inverse formulas based on the Moore-Penrose inverse and on linear algorithms. We expect the results to be of use in directional and toroidal time series.

19th IEEE Symposium on Computer-Based Medical Systems (CBMS'06), 2006

The state of the art in modern drug discovery involves investigating a large number of drug-like ... more The state of the art in modern drug discovery involves investigating a large number of drug-like molecules using medium or high-throughput assays, often being conducted against multiple targets. Managing the information generated in such processes requires the ability to deal with complex, multifarious data as well as the development of new user-data interaction paradigms that help glean patterns hidden in the multitude of data by emphasizing exploration and information assimilation. This paper describes our research in developing FreeFlowDB, a drug discovery information database system that is geared towards storing both structural as well as high-throughput assay information generated as part of a typical drug discovery process. FreeFlowDB supports powerful structural querying facilities that subsume within a common algorithmic framework exact structural matching, sub-structure querying, and inexact matching. Furthermore, the system supports unified visualization-query facilities that allow interacting with assay as well as structure-activity information. This allows efficacious and intuitive query-analysis of large amounts of data for knowledge discovery. Case studies and experimental results demonstrate the capabilities of the system.

Sixth IEEE Symposium on BioInformatics and BioEngineering (BIBE'06), 2006

Discerning the similarity between two molecules is a challenging problem in drug discovery as wel... more Discerning the similarity between two molecules is a challenging problem in drug discovery as well as in molecular biology. The importance of this problem is due to the fact that the biochemical characteristics of a molecule are closely related to its structure. Therefore molecular similarity is a key notion in investigations targeted at understanding existing molecules as well as in guiding the synthesis of new molecules. Additionally, the notion of molecular similarity plays a central role in structure query-retrieval. This paper presents a Wavelet-based Riemannian metric for determining molecular similarity. The proposed metric extends traditional molecular similarity measures in terms of its ability to capture and compare nonlinear molecular descriptors, thus allowing more accurate characterization of the true nature of the factors involved. Furthermore, owing to its metric properties and wavelet nature, this similarity measure supports highly efficient query-retrieval strategies. To compare graph-based molecular representations using the wavelet-based Riemannian metric, the paper uses a two-phase molecular graph matching strategy. In the first step, an efficient nonlinear graph-matching technique based on the graduated assignment algorithm is used to obtain a preliminary correspondence between molecular graphs in terms of their topological characteristics. Starting from this correspondence, the second stage directly optimizes the proposed metric on arbitrary molecular descriptors using a branch-and-bound search strategy. Various experiments, many in comparative settings, study the retrieval performance of this similarity formulation and underline its efficacy and efficiency.

Siam Journal on Discrete Mathematics, 2004

Congressus numerantium, 1970

The integer N may be fixed as finite or infinite, or N could be chosen to be an element from a fi... more The integer N may be fixed as finite or infinite, or N could be chosen to be an element from a finite cyclic group to invoke periodicity in the lattice. Then , for a given lattice particle, mj, we have that: j + N ≡ j for j ∈ Z+ ∪ {0}. The system we describe is called the Toda lattice [5] where ...

Contemporary Mathematics, 1994

This paper follows [49], [15], and [2], which were written in that order. As such, we will be ske... more This paper follows [49], [15], and [2], which were written in that order. As such, we will be sketchy in places covered by those 3 papers. In particular, we will let the reader look in these papers for the detailed theory o f the graphs associated to finite upper half planes. Here we will concentrate only on a certain kind o f eigenfunction o f the Laplacian on these graphs, namely the spherical functions. This is the content o f Section 2. In the previous papers, we considered other kinds of eigenfunctio11s of the Laplacian; namely K-Bessel function analogs and principal series spherical functions. See also Evans [67] and [68].
In Section 3 o f this paper we consider two aspects o fanalysis o f the spherical Fourier transform on finite upper half planes Hq - the uncertainty principle and the Selberg trace formula. Most of our results depend only on the hypothesis that L2(K\G/K) is a commutative algebra under convolution - where G is a finite group
3 versions o f the uncertainty results are parallel to those o f Donoho and Stark [20] in the finite and real abelian cases. Finally
with subgroup K. We work out principle in Theorems 2-4. The
we find a general version of the Selberg trace formula in Theorem 5. We end with a computation of some of the terms in this trace formula when the subgroup is H=GL(2,fp) in the group G=GL(2,fq), q=pr. There is a surprising similarity to the terms in the calculation over the real numbers as found, for example in Terras [50, Chapter 3].

Pacific Journal of Mathematics, 1997

The Radon transform belongs to the area of Inverse Problems. The reconstruction of a function fro... more The Radon transform belongs to the area of Inverse Problems. The reconstruction of a function from its projection or averages is a central point of study. Besides having direct applications in medical tomography, geophysics, there are also applications in signal processing, statistics and probability. Hence, it is useful to consider discretized versions of the Radon transform.

Hamiltonian Systems with Three or More Degrees of Freedom, 1999

Abstract. In these informal notes, we describe one way to think about expanding the idea behind s... more Abstract. In these informal notes, we describe one way to think about expanding the idea behind support vector machines.

In this thesis prospectus, Bayesian networks are to be constructed from microarray data obtained ... more In this thesis prospectus, Bayesian networks are to be constructed from microarray data obtained from the human placenta in order to determine gene regulatory pathways in the human placenta.

Abstract: In these informal notes, we describe a novel approach to genomic evolutionary theory.

We employed machine learning to identify plausible molecular biomarkers for Parkinson's disea... more We employed machine learning to identify plausible molecular biomarkers for Parkinson's disease (PD) using an approach previously employed in cancer biomarker discovery. The top five biomarker gene candidates were SLC39A5, RAB42, BTNL9, INPP5A and a hypothetical gene. We compared a traditional gene/feature selection method with the recursive feature elimination using the support vector machines algorithm (RFE-SVM) and found that RFE-SVM out-performs the traditional ranking method in both classification of PD and accuracy, using the NIPS 2003 Feature Selection Challenge criterion. We used the Naive Bayes as the baseline classifier as well as SVMs. These results indicate that it may be possible to computationally identify novel biomarkers for PD using our approach.

Contemporary Mathematics, 1993

IS THERE LIFE ON FINITE UPPER HALF PLANES? 67 (3) Xp (d, a) is a regular graph of degree p+ 1 pro... more IS THERE LIFE ON FINITE UPPER HALF PLANES? 67 (3) Xp (d, a) is a regular graph of degree p+ 1 provided that d is a non-square in Fp and a^ 0 or 4d. Proof. (1) This is clear from (3). (2) Note that the map Tc (x+ yVd)= cx+ y^/d is an isomorphism when viewed as a map of the ...

Pacific Journal of Mathematics, 1998

Heisenberg's uncertainty principle is extended to certain finite graphs. The fundamental theorem ... more Heisenberg's uncertainty principle is extended to certain finite graphs. The fundamental theorem of calculus, integration by parts, and vanishing boundary terms for graphs are defined as well as functions of random variables, expectation values, and moments on graphs. Section 3 gives three versions of Heisenberg's uncertainty principle for graphs. For the 2nd version, we assume that our graph is the Cayley graph of a finite abelian group. We work out the example of a finite cycle graph in detail and compare it to the uncertainty principle on the continuous circle obtained by Grünbaum around 1990.

19th IEEE Symposium on Computer-Based Medical Systems (CBMS'06), 2006

Sixth IEEE Symposium on BioInformatics and BioEngineering (BIBE'06), 2006

Siam Journal on Discrete Mathematics, 2004

Congressus numerantium, 1970

Contemporary Mathematics, 1994

Pacific Journal of Mathematics, 1997

Hamiltonian Systems with Three or More Degrees of Freedom, 1999

Abstract: In these informal notes, we describe a novel approach to genomic evolutionary theory.

Contemporary Mathematics, 1993

Pacific Journal of Mathematics, 1998

These notes expand the area of predictive analytics by defining the viewpoint arising from a "cel... more These notes expand the area of predictive analytics by defining the viewpoint arising from a "cellular" predictive theme (the definition for 'cellular' lives in the topic area of mathematical biomimicry). Prediction is reinterpreted as acting like that of a biological cell, so, in terms of 'form and function.' The result is a new methodology for prediction theory, with a special emphasis on predictions involving Big Data. As an application, the concept of a "ghost-child" is introduced in order to describe a universe in which the Second Law of Thermodynamics is never violated.

The Central Limit Theorem explains that the equilibrium probability distribution of a sequence of... more The Central Limit Theorem explains that the equilibrium probability distribution of a sequence of independent, identically distributed random variables is Gaussian. But, why is this so? The distribution is Gaussian since the distribution is dependent on the actual data under consideration. It is always the data that "drives" the long-term behavior-the limiting probability distribution (hence the theorem outcome), rather than the other way around. Historically, we have imagined data as living in a flat space, i.e., a space of zero curvature. As a result, the Central Limit Theorem only features a Gaussian for long-term behavior: Recall that equations describing "basic" physics in such flat spaces-heat or diffusion-have Gaussians as fundamental solutions, given non-perverse initial or boundary conditions. As we achieve a better understanding of the spaces that data inhabit (our data landscapes), the story that the Central Limit Theorem tells us naturally evolves. The equations and other mathematical tools that we will want to construct for an improved comprehension will naturally result in much more sophisticated versions of the Central Limit Theorem. To summarize, a well posed Central Limit Theorem, one that is based on a well studied "data landscape," is essential for effective problem solving in data science and its myriad applications, such as predictive analytics modeling for climate change model solutions, biomedical applications, finance models based on complex dynamical systems, and so forth.

This unpublished final manuscript succinctly discusses how poorly conceived mathematical models m... more This unpublished final manuscript succinctly discusses how poorly conceived mathematical models may create global crises, and proposes novel questions to help in constructing valuable mathematical models used to alleviate pressing locally, regionally, and globally situated social, economic, and ecodemic problems.

Preliminary Report, 2010

An algorithm for predicting minimal paths between genomes. Preliminary report. Genomic sequencing... more An algorithm for predicting minimal paths between genomes. Preliminary report. Genomic sequencing has permitted us the opportunity to use comparative genomics to reconstruct species evolution. However, genomic diversity has made the connection between an ancestral genome and present day species computationally challenging. A genome undergoes rearrangements, translocation and speciation of genes. Just modeling the rearrangements of genes that a genome undertakes is difficult. If we try to model the rearrangements of a single chromosome to another chromosome and attempt to compute the minimum number of rearrangements that a chromosome must undergo to become another chromosome, then we have an NP-hard problem. To address this issue, we will instead construct the geodesic path between two genomes. As an example of this technique, we tackle the geodesic path between the tobacco genome and the Lobelia fervens genome using the calculus of variations.

Dissertation Abstract: The Radon transform and its inverse transform belong to the area of mathem... more Dissertation Abstract: The Radon transform and its inverse transform belong to the area of mathematics known as inverse problems. The continuous Radon transform and its inverse transform are applied to reconstruction problems in tomography, radio astronomy, seismology, and other areas. Kung (1979) and then Bolker (1984) consider finite analogs of the Radon transform. Diaconis and Graham (1986) construct the Radon transform based on a translate of a subset S within a finite group. Fill (1989) examines the Radon transform based on translates of a subset S contained in the group of integers modulo n. We continue the study of the Radon transform group by studying the Radon transform associated with the group of k-tuples whose components belong to the group of integers modulo n, and the Radon transform associated to the group of invertible matrices of rank 2 over a finite field, modulo a stabilizer subgroup of the matrices. We find conditions of invertibility for both groups and construct specific inversion formulas. Our methods rely chiefly on the Fourier transform, and in the case of the second group, on construction of both finite analogs of continuous eigenfunctions and matrix analogs of spherical functions. With respect to the first group, the invertibility condition is found to reduce to a condition on a Krawtchouk polynomial, if the subset S is a shell of radius r. With respect to the second group, the invertibility condition partially reduces to conditions on eigenvalues associated with Kloosterman sums, the finite analogs of k-Bessel eigenfunctions. It is anticipated that continued study of the finite Radon transform will help with reconstruction problems in both mathematics and science, as well as encouraging further insights toward Radon transforms.

Abstract In these notes, we describe a novel approach to genomic evolutionary theory.

A Bayesian network is a technique from computer science that visualizes a machine learning method... more A Bayesian network is a technique from computer science that visualizes a machine learning methodology. In this study Bayesian networks will be constructed from microarray data obtained from the human placenta in order to determine gene regulatory pathways in the human placenta. 1. Relevant Background: 1. Microarray data for human placentas Preeclampsia is a pregnancy complication associated with faulty placentation, affecting at least 5-8 percent of all pregnancies, often characterized by high blood pressure and the presence of protein in urine. Identifying transcriptional changes associated with preeclampsia and other diseases is the focal point of this research. By identifying transcriptional changes between normal placentas and placentas with preeclampsia, specifically identified gene products may be targeted for drug therapy. This research begins by studying, using in-silico tools, transcriptional changes within normal placentas as a reference point. By using the same in-silico tools, transcriptional changes within placentas from preeclamptic pregnancies will be evaluated. The human placenta is a ideal environment for considering gene expression and cell differentiation because of the unique cell-cell interactions and this fact provides another motivation for studying the human placenta. Microarray data is derived by members of the Fisher laboratory at U.C. San Francisco. Experiments are comprised of hybridization onto Affymetrix high-density HG-U133A and HG-U133B GeneChips with 45,000-oligomer probe sets representing 39,000 transcripts.

A differential-difference operator is used to model the heat equation on a finite graph analogue ... more A differential-difference operator is used to model the heat equation on a finite graph analogue of Poincaré's upper half-plane. Finite analogues of the classical theta functions are shown to be solutions to the heat equation in this setting.

We employed machine learning to identify plausible molecular biomark-ers for Parkinson's disease ... more We employed machine learning to identify plausible molecular biomark-ers for Parkinson's disease (PD) using an approach previously employed in cancer biomarker discovery. The top five biomarker gene candidates were SLC39A5, RAB42, BTNL9, INPP5A and a hypothetical gene. We compared a traditional gene/feature selection method with the recursive feature elimination using the support vector machines algorithm (RFE-SVM) and found that RFE-SVM out-performs the traditional ranking method in both classification of PD and accuracy, using the NIPS 2003 Feature Selection Challenge criterion. We used the Naive Bayes as the baseline classifier as well as SVMs. These results indicate that it may be possible to computationally identify novel biomarkers for PD using our approach.

A prototype for a web application was designed and implemented as a guide to be used by clinician... more A prototype for a web application was designed and implemented as a guide to be used by clinicians when designing the best drug therapy for a specific cancer patient, given biological data derived from the patients tumor tissue biopsy. A representation of the patients metabolic pathways is displayed as a graph in the application, with nodes as substrates and products and edges as enzymes. The top metabolically active sub-paths in the pathway, ranked using an algorithm based on both the patients biological data and the graph topology, are also displayed and can be individually highlighted to examine potential enzymatic sites to be disrupted by a drug these sites serve as a guide for designing the patients specific drug therapy. Displayed next to each sub-path is the sub-path score used to decide its rank, as well as the predicted patient survival time to indicate how effective that specific drug will be in alleviating that patient's cancer. Future work includes an animation component to track the patients progress and redesign the drug therapy as needed. Corresponding author's academic email: elinor@soe.ucsc.edu.

The axioms of probability are listed.