Department of Mathematical Sciences Articles

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Filling Jars to Measure Time
(Sciendo, 10-2021) Le, Tiffany-Chau; Sarkar, Jyotirmoy; Mathematical Sciences, School of Science
If water is flowing at the same constant rate through each of H>3 hoses, so that any one hose will fill any one of J>2 available jars in exactly one hour, then what are the fillable fractions of a jar, and what are the measurable fractions of an hour? Learning to systematically answer such questions will not only equip readers to fluently use fractions, but also introduce or reintroduce them gently to the Queen of Mathematics – Number Theory.
Rotation sets for unimodal maps of the interval
(American Institute of Mathematical Sciences, 2003-02) Cleveland, Christopher; Mathematical Sciences, School of Science
We relate the rotation interval ρ(f) of a unimodal map f of the interval with its kneading invariant K(f). In particular, we show that for any μ∈(0,12), there are kneading invariants νμ and νμ,hom such that ρ(f)=[μ,12] if and only if νμ⪯K(f)⪯νμ,hom.
Intermittency in local field potentials recorded from the thalamus of patients with essential tremor
(Springer (Biomed Central Ltd.), 2007-07-06) Sen, Asok; Kane, Abdoul; Hutchison, William D; Lozano, Andres M; Hodaie, Mojgan; Dostrovsky, Jonathan O; Department of Mathematical Sciences, School of Science
A semiparametric modeling framework for potential biomarker discovery and the development of metabonomic profiles
(BioMed Central, 2008-01-23) Ghosh, Samiran; Grant, David F.; Dey, Dipak K.; Hill, Dennis W.; Mathematical Sciences, School of Science
Background The discovery of biomarkers is an important step towards the development of criteria for early diagnosis of disease status. Recently electrospray ionization (ESI) and matrix assisted laser desorption (MALDI) time-of-flight (TOF) mass spectrometry have been used to identify biomarkers both in proteomics and metabonomics studies. Data sets generated from such studies are generally very large in size and thus require the use of sophisticated statistical techniques to glean useful information. Most recent attempts to process these types of data model each compound's intensity either discretely by positional (mass to charge ratio) clustering or through each compounds' own intensity distribution. Traditionally data processing steps such as noise removal, background elimination and m/z alignment, are generally carried out separately resulting in unsatisfactory propagation of signals in the final model. Results In the present study a novel semi-parametric approach has been developed to distinguish urinary metabolic profiles in a group of traumatic patients from those of a control group consisting of normal individuals. Data sets obtained from the replicates of a single subject were used to develop a functional profile through Dirichlet mixture of beta distribution. This functional profile is flexible enough to accommodate variability of the instrument and the inherent variability of each individual, thus simultaneously addressing different sources of systematic error. To address instrument variability, all data sets were analyzed in replicate, an important issue ignored by most studies in the past. Different model comparisons were performed to select the best model for each subject. The m/z values in the window of the irregular pattern are then further recommended for possible biomarker discovery. Conclusion To the best of our knowledge this is the very first attempt to model the physical process behind the time-of flight mass spectrometry. Most of the state of the art techniques does not take these physical principles in consideration while modeling such data. The proposed modeling process will apply as long as the basic physical principle presented in this paper is valid. Notably we have confined our present work mostly within the modeling aspect. Nevertheless clinical validation of our recommended list of potential biomarkers will be required. Hence, we have termed our modeling approach as a "framework" for further work.
Recurrence Plots for Diesel Engine Variability Tests
(De Gruyter, 2009-02-20) Longwic, Rafał; Litak, Grzegorz; Sen, Asok K.; Mathematical Sciences, School of Science
Cycle-to-cycle variations of maximum pressure in a diesel engine are studied by using the methods of recurrence plots and recurrence quantification analysis. The pressure variations are found to exhibit strong periodicities in low frequency bands and intermittent oscillations at higher frequencies. The results are confirmed by wavelet analysis.
A Note on Dirac Operators on the Quantum Punctured Disk
(National Academy of Science of Ukraine, 2010-07-16) Klimek, Slawomir; McBride, Matt; Mathematical Sciences, School of Science
We study quantum analogs of the Dirac type operator −2z¯¯¯∂∂z¯¯¯ on the punctured disk, subject to the Atiyah–Patodi–Singer boundary conditions. We construct a parametrix of the quantum operator and show that it is bounded outside of the zero mode.
Branching rules for quantum toroidal gl (n)
(2013) Feigin, B.; Jimbo, M.; Miwa, T.; Mukhin, Eugene; Department of Mathematical Sciences, School of Science
We construct an analog of the subalgebra Ugl(n) ⊗ Ugl(m) ⊂ Ugl(m + n) in the setting of quantum toroidal algebras and study the restrictions of various representations to this subalgebra.
Using a continuum model to predict closure time of gaps in intestinal epithelial cell layers
(Wiley Blackwell (Blackwell Publishing), 2013-03) Arciero, Julia C.; Mi, Qi; Branca, Maria; Hackam, David; Swigon, David; Department of Mathematical Sciences, School of Science
A two-dimensional continuum model of collective cell migration is used to predict the closure of gaps in intestinal epithelial cell layers. The model assumes that cell migration is governed by lamellipodia formation, cell-cell adhesion, and cell-substrate adhesion. Model predictions of the gap edge position and complete gap closure time are compared with experimental measures from cell layer scratch assays (also called scratch wound assays). The goal of the study is to combine experimental observations with mathematical descriptions of cell motion to identify effects of gap shape and area on closure time and to propose a method that uses a simple measure (e.g., area) to predict overall gap closure time early in the closure process. Gap closure time is shown to increase linearly with increasing gap area; however, gaps of equal areas but different aspect ratios differ greatly in healing time. Previous methods that calculate overall healing time according to the absolute or percent change in gap area assume that the gap area changes at a constant rate and typically underestimate gap closure time. In this study, data from scratch assays suggest that the rate of change of area is proportional to the first power or square root power of area.
Modeling the interactions of bacteria and Toll-like receptor-mediated inflammation in necrotizing enterocolitis
(Elsevier, 2013-03-21) Arciero, Julia; Bard Ermentrout, G.; Siggers, Richard; Afrazi, Amin; Hackam, David; Vodovotz, Yoram; Rubin, Jonathan; Department of Mathematical Sciences, School of Science
Necrotizing enterocolitis (NEC) is a severe disease of the gastrointestinal tract in premature infants, characterized by a disrupted intestinal epithelium and an exaggerated pro-inflammatory response. Since the activation of Toll-like receptor-4 (TLR4) blocks cell migration and proliferation and contributes to an uncontrolled inflammatory response within the intestine, this receptor has been identified as a key contributor to the development of NEC. Toll-like receptor-9 (TLR9) has been shown to sense bacterial genome components (CpG DNA) and to play an anti-inflammatory role in NEC. We present in vitro results demonstrating direct inhibition of TLR4 activation by CpG DNA, and we develop a mathematical model of bacteria-immune interactions within the intestine to investigate how such inhibition of TLR4 signaling might alter inflammation, associated bacterial invasion of tissue, and resulting outcomes. The model predicts that TLR9 can inhibit both the beneficial and detrimental effects of TLR4, and thus a proper balance of action by these two receptors is needed to promote intestinal health. The model results are also used to explore three interventions that could potentially prevent the development of NEC: reducing bacteria in the mucus layer, administering probiotic treatment, and blocking TLR4 activation. While the model shows that these interventions would be successful in most cases, the model is also used to identify situations in which the proposed treatments might be harmful.
Random Interval Homeomorphisms
(2014) Alsedà, Lluís; Misiurewicz, Michał; Department of Mathematical Sciences, School of Science
We investigate homeomorphisms of a compact interval, applied randomly. We consider this system as a skew product with the two-sided Bernoulli shift in the base. If on the open interval there is a metric in which almost all maps are contractions, then (with mild additional assumptions) there exists a global pullback attractor, which is a graph of a function from the base to the fiber. It is also a forward attractor. However, the value of this function depends only on the past, so when we take the one-sided shift in the base, it disappears. We illustrate those phenomena on an example, where there are two piecewise linear homeomorphisms, one moving points to the right and the other one to the left.

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