# Your search: "author:"Rodriguez, Jose A""

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## Scholarly Works (36 results)

Numerical algebraic geometry provides numerical descriptions of solution sets of polynomial systems of equations in several unknown. Such sets are called algebraic varieties.

In algebraic statistics, a statistical model is associated to an algebraic variety to study its geometric structure.

This thesis contains my work at UC Berkeley that uses numerical algebraic geometry for the algebraic statistics problem of maximum likelihood estimation.

In Chapter 2 we study the maximum likelihood estimation problem on manifolds of matrices with bounded rank. These represent mixtures of distributions of two independent discrete random variables. We determine the maximum likelihood degree for a range of determinantal varieties, and we apply numerical algebraic geometry to compute all critical points of their likelihood functions.

In Chapter 3 we prove a bijection between critical points of the likelihood function on the complex variety of matrices of rank r and critical points on the complex variety of matrices of corank r-1.

From the perspective of statistics, we show that maximum likelihood estimation for matrices of rank r is the same problem as minimum likelihood estimation for matrices of corank r-1,

and vice versa.

In Chapter 4, a description of the maximum likelihood estimation problem in terms of dual varieties and conormal varieties is given. With this description, we define the dual likelihood equations. We show how solving these dual likelihood equations give solutions to the maximum likelihood estimation problem without having the defining equations of the model itself.

In Chapter 5,

discrete algebraic statistical models are considered and solutions to the likelihood equations when the data contain zeros are studied.

Focusing on sampling and model zeros, we show that the solutions of the likelihood equations in these cases are contained in a previously studied variety, the likelihood correspondence. The number of solutions give a lower bound on the ML degree, and the problem of finding critical points to the likelihood function can be partitioned into computationally easier problems involving sampling and model zeros.

In Chapter 6

the Macaulay2 package Bertini.m2 is introduced.

Macaulay2 is a software system designed to support research in algebraic geometry, and Bertini is a popular software system for numerical algebraic geometry.

The package Bertini.m2 provides an interface to Bertini via Macaulay2.

The receptor for transferrin (Tf), otherwise known as transferrin receptor 1 (TfR1 or CD71), is a single pass type two transmembrane protein ubiquitously expressed in most vertebrate cells. TfR1 is responsible for facilitating the import of iron into cells from transferrin bound iron in circulation. In cells, release of iron from Tf occurs near pH 5.5 and requires clathrin-mediated endocytosis of the Tf-TfR1 complex to acidified endosomes, followed by its recycling to the cell surface without dissociation.

Since its discovery, high expression of TfR1 is typically associated with rapidly proliferating cells and a variety of tumor cell types. Because of its association with cancer, its cell surface accessibility, its facile entry into cells, and its critical role in the pathophysiology of cancer, TfR1 is an active target for anti-cancer therapies. Antibodies were the first vehicles designed to specifically target TfR1 with the aim of blocking Tf uptake into malignant cells and induce cytotoxicity.

Based on a panel of thirty-two mouse monoclonal antibodies developed, in the late 1980s, to recognize the extracellular domain of human TfR1, several were identified that did not appear to block binding of the receptor to Tf. One of these antibodies, otherwise referred to as 128.1, inspired the generation of a chimeric antibody-avidin fusion protein (ch128.1Av). This molecule was originally designed to act as a delivery vehicle for biotinylated agents into cancer cells. However, this molecule possesses a significant intrinsic anti-proliferative/pro-apoptotic activity against malignant B cells expressing TfR1, as did its parental chimeric antibody (ch128.1) although to a lesser extent. The detailed nature of the interactions of these antibodies (based on the murine monoclonal 128.1) with TfR1 is addressed in part by this thesis.

The experiments presented in this thesis are those from two published articles addressing the nature of the interactions between ch128.1 or ch128.1Av and TfR1, and the effects of these interactions on malignant B cells expressing the receptor. Special emphasis is placed on how these antibodies affect the recycling of TfR1 in order to reduce Tf uptake and induce lethal iron deprivation in malignant hematopoietic cells, as well as the transcriptional response of affected cells to insult by these antibodies.

While investigating the effect of a low pH (late endosome or lysosome like) environment on the stability of TfR1 in the presence of these antibodies, the unexpected observation was made of an intrinsic instability of TfR1 under these conditions. Thus, data that provide insights into the inducible proteolysis of the extracellular domain of TfR1 in a low pH environment (pH 3-5) are presented. These results show a distinct and consistent pattern of degradation of the TfR1 ectodomain selectively within this pH range, which is consistent with the enzymatic activity of a pH-dependent cysteine/serine protease.

Atmospheric circulation leads to an accumulation of debris from meteors in the Antarctic stratosphere at the beginning of austral spring. The major component of meteoric material is alkaline, comprised predominantly of the oxides of magnesium and iron. These metals may neutralize the natural acidity of stratospheric aerosols, remove nitric acid from the gas phase, and bond it as metal nitrates in the aerosol phase. Removal of nitric acid vapor has been previously shown to be a critical link in the photochemical depletion of ozone in the Antarctic spring, by allowing for increased catalytic loss from chlorine and bromine.

Phase slip and beat phenomena are ubiquitous in many forms of wave superposition. Likewise, it is trivially true that cumulative summation of white noise gives rise to “brown noise” (1/f2 power spectrum). We assume that the background electrocorticogram (ECoG) to a first approximation results from integration of innumerable action potentials by dendrites. This paper presents a simulation that relates these phenomena with cortical function. We report that cumulative summation of random numbers simulates resting ECoG, and that phase interference in the distributed frequencies of oscillation in bandpass-filtered brown noise gives null power spikes like those in the ECoG from test subjects. The null spikes coincide with the onsets of frames in which the spatial amplitude patterns are classifiable with respect to conditioned stimuli. We report similarity in the waveforms and amplitude distributions of null spikes upon filtering brown noise in bands corresponding to the theta, alpha, beta and gamma ranges in experimental and simulated ECoG. We estimate a threshold in null spike minimal amplitudes below which perceptual frames having gamma oscillations may recur at theta rates.

The contribution from the chlorine and bromine species in the formation of the Antarctic ozone hole is evaluated. Since chlorine and bromine compounds are of different industrial origin, it is desirable, from a policy point of view, to be able to attribute chlorine-catalyzed loss of ozone with those reactions directly involving chlorine species, and likewise for bromine-catalyzed loss. In the stratosphere, however, most of the chemical families are highly coupled, and, for example, changes in the chlorine abundance will alter the partitioning in other families and thus the rate of ozone loss. This modeling study examines formation of the Antarctic ozone hole for a wide range of bromine concentrations (5–25 pptv) and for chlorine concentrations typical of the last two decades (1.5, 2.5 and 3.5 ppbv). We follow the photochemical evolution of a single parcel of air, typical of the inner Antarctic vortex (50 mbar, 70°S, NO_{y}=2 ppbv, with polar stratospheric clouds(PSC)) from August 1 to November 1. For all of these ranges of chlorine and bromine loading, we would predict a substantial ozone hole (local depletion greater than 90%) within the de-nitrified, PSC-perturbed vortex. The contributions of the different catalytic cycles responsible for ozone loss are tabulated. The deep minimum in ozone is driven primarily by the chlorine abundance. As bromine levels decrease, the magnitude of the chlorine-catalyzed ozone loss increases to take up the slack. This is because bromine suppresses ClO by accelerating the conversion of ClO and Cl_{2}O_{2} back to HCl. For this range of conditions, the local relative efficiency of ozone destruction per bromine atom to that per chlorine atom (α-factor) ranges from 33 to 55, decreasing with increase of bromine.