270 research outputs found
Michael Polyak
No full text. The classical Whitney formula relates the algebraic number of selfintersections of a generic plane curve to its winding number. We generalize it to an infinite family of identities, expressing the winding number in terms of the internal geometry of a plane curve. This enables us to split the Whitney formula by some characteristic of double points. It turns out, that only crossings of a very specific type contribute to the computation of the winding number. We also provide a "difference integration" of these formulae, establishing a new family of simple formulae with the base point pushed off the curve. Similar new identities are obtained for Arnold's invariant Strangeness of plane curves. 1. Whitney formula and its generalizations 1.1. Introduction. The classical Whitney formula [4] relates the algebraic number of times that a generic immersed plane curve intersects itself to the Whitney index, or winding number, of this curve. Since it was discovered in 1937, this formula remained m..
Michael Polyak
No full text: Bennequin invariant l() of a Legendrian curve in the space of cooriented contact elements of the plane counts its self-linking number. We present a new formula for computation of this invariant via a state summation over crossings of the corresponding Legendrian front. Using this state sum we obtain a quantization l q () 2 Z[q;q \Gamma1 ] of l(). Other generalizations of l() are discussed. SUR L'INVARIANT DE BENNEQUIN DES COURBES LEGENDRIENNES ET SA QUANTIFICATION R' esum' e: L'invariant de Bennequin l() d'une courbe de Legendre dans l'espace des 'el'ements de contact coorient'es du plan est un nombre d'auto-enlacement de . Nous pr'esentons une nouvelle formule de calcul de cet invariant utilisant une somme statistique sur les croisements du front legendrien correspondant. Une quantification l q () 2 Z[q;q \Gamma1 ] de l() est d'eduite de cette somme statistique. D'autres g'en'eralisations de l() sont consid'er'ees. Version francaise abr'eg'ee. 1. Consid'erons l'espace M =..
Review of Student Voice: From Invisible to Invaluable
No full textReview of the book Student Voice: From Invisible to Invaluable by Michael Lubefeld, Nick Polyak, and P.J. Capose
Iterate averaging methods for solving non-linear programming problems: Applied to a transportation network equilibrium problem
No full textTraffic congestion is an unresolved problem and it has effects not only to the transportation system, but also on other aspects of life (economic, spatial and social). The idea is that 'Road Pricing' can be used to solve this problem. There is a need for an appropriate tool for predicting the effects of 'Road Pricing'. Such a tool could be a traffic assignment model. Traffic is by nature dynamic and hence only dynamic models can describe traffic process adequately. It appears, however, that iterate averaging methods have not yet been applied to transportation network problems. In this research iterate averaging methods are investigated and also the possibility of applying these methods in transportation network problems. Recently the Polyak method was introduced, which is supposed to have better convergence qualities than the method that is normally used, the Method of Successive Averages (MSA). The three topics of this research of iterate averaging methods are: \u95 To find out how the Polyak method works, after which the Polyak method is implemented. \u95 To find out if the Polyak method indeed converges faster than MSA. \u95 To find out if there exist alternative methods that are faster in convergence than the Polyak method and MSA. To find answers on these three topics the literature was studied. By reading the nature of the Polyak method is found out. When the Polyak method is understood, the method is implemented (and also MSA is implemented). That was needed for analysing the convergence of the methods. After the Polyak method is implemented research for alternative methods is done. Finally, all the described methods are compared and illustrations are given. In order to satisfy the increasing demand for more accurate model outcomes and to be able to compute the effects of different traffic policies, new and improved traffic assignment models are needed. While Static Traffic Assignment models may provide basic insights, only dynamic assignment models are able capture the true dynamic nature of traffic and therefore provide the analyst with more accurate forecast. An iterative process is needed to solve the Dynamic Traffic Assignment (DTA) model. This is because network conditions may change after performing network loading. At all the iterations, the path flows are updated by combining the results from the current iteration with the previous iteration. The 'classical' (e.g., derivative-based) fixed-point solution methods are often inappropriate for some problems. In such cases, the fixed-points are usually computed using one of the iterate averaging methods introduced by Robbins and Monro [3.1]. MSA, introduced by Sheffi and Powell [3.2], is probably the best known and most widely-used instance of iterate averaging methods. In iterate averaging methods estimates for the fixed-point are found. These estimates are called design points. MSA computes each new design point by adding a part of the observation evaluated in the previous design point with a part of the previous design point. MSA has the advantages of avoiding (potentially expensive) step size calculations, working directly with map outputs without requiring derivative calculations or other transformations, and being able to handle 'noisy' map evaluations (where the evaluation returns a value affected by a zero-mean disturbance). Other advantages of MSA are that it is simple to understand and that it is simple to implement. In many cases, however, the method's empirically observed convergence properties are disappointing: while it exhibits generally effective performance in the initial iterations, this is followed by a pronounced 'tail' effect, resulting in overall slow convergence. Approximately ten years ago, B.T. Polyak and J.A. Bather proposed two relatively minor modifications of iterate averaging methods which were rigorously shown to produce fixed-point estimates with asymptotically optimal properties. The Polyak method is a two-pass method. The first pass resembles MSA except that the step sizes are larger; this allows the algorithm to explore the solution space more aggressively but leads to greater variability in the outputs. The second pass is carried out offline (i.e., without influencing the first pass); it calculates an average of iterates that are generated by the first pass. The average calculated by the second pass at termination is the fixed-point solution estimate. A somewhat different approach was proposed by J.A. Bather. Here, the design point is derived from a combination of the average of previous design points with the average of previous evaluation results. Apart from the Polyak method and the Bather method, alternative methods are proposed. In total eight methods are applied and presented in this report. They were all compared with different stop criterions. MSA and the Polyak method were compared. To compare these methods and to compare also other methods a traffic problem with three cities and two routes is considered. This traffic problem is solved by using a DTA algorithm. For stopping this DTA algorithm there are different stop criteria. The stop criterion that is a combination of the route costs and flows is the best stop criterion for stopping the DTA algorithm. This stop criterion is reached after 190 iterations of MSA and after 226 iterations of the Polyak method. The conclusion is that MSA is faster in convergence than the Polyak method for this stop criterion. The Bather method, the Bliemer method and the Bliemer Moving method were compared. After 118 iterations of the Bather method, after 112 iterations of the Bliemer method and after 40 iterations of the Bliemer Moving method the stop criterion is reached. It can be concluded that the Bather method and the Bliemer method solve the problem in almost the same number of iterations. For different stop criterions the Bliemer Moving method much faster than is the four other methods. The Bliemer Moving method is the fastest method, but by combining two methods it's possible to get a method that is even faster in convergence than the methods shown before. Therefore, the MSA-Biiemer method, the MSA-Bather method and the Bliemer-Bather method are compared. The fastest method in convergence, for the stop criterion we chose, is the Bliemer-Bather method. The conclusion is that there are alternative methods that are much faster in convergence than the Method of Successive Averages and the Polyak method. The best alternative methods are the MSA-Bather method and the Bliemer-Bather method. The recommendation is to use the Bliemer-Bather method for solving Non Linear Programming (NLP) problems in transportation networks and to do further research how the values of the variables used in the Bliemer-Bather method have to be chosen.Transport & PlanningCivil Engineering and Geoscience
Invariants of curves and fronts via Gauss diagrams
No full textAbstractWe use a notion of chord diagrams to define their representations in Gauss diagrams of plane curves. This enables us to obtain invariants of generic plane and spherical curves in a systematic way via Gauss diagrams. We define a notion of invariants are of finite degree and prove that any Gauss diagram invariants are of finite degree. In this way we obtain elementary combinatorial formulas for the degree 1 invariants J± and St of generic plane curves introduced by Arnold [1] and for the similar invariants J±SandStS of spherical curves. These formulas allow a systematic study and an easy computation of the invariants and enable one to answer several questions stated by Arnold. By a minor modification of this technique we obtain similar expressions for the generalization of the invariants J± and St to the case of Legendrian fronts. Different generalizations of the invariants and their relations to Vassiliev knot invariants are discussed
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Identifying Combinatorial Drug Targets for Ras Pathway-Driven Melanomas
No full textCutaneous melanoma is a highly metastatic skin cancer, with ~100,000 new cases estimated to occur in 2020 in the US. Melanomas are defined by oncogenic “driver” mutations that constitutively activate the Ras/Raf/MEK/ERK signaling (henceforth called Ras/ERK) pathway. Clinical agents that target various components of this pathway have been developed, although they are not typically curative. For BRAF-mutant melanomas, combined BRAF and MEK inhibitors prolong survival, although patients still relapse within ~11 months. Unfortunately, Ras/ERK pathway inhibitors are even less effective in NRAS-mutant tumors, with MEK inhibitors stimulating only partial responses in 15-20% of patients. Thus, NRAS mutations are associated with poor overall survival, and there is a great need for improved therapies. The goal of my thesis has been to identify agents that potentiate the effects of Ras/ERK pathway inhibitors as a means of developing more effective therapies for NRAS and other Ras pathway-driven melanomas.
This dissertation describes three promising therapeutic combinations. In one approach, we discovered a new combinatorial therapy for NRAS-mutant melanomas by performing an unbiased negative selection CRISPR-Cas9 screen. Specifically, we identified the de-ubiquitinase USP7 as a target that when suppressed, sensitizes NRAS-mutant melanomas to MEK inhibitors. Further genetic and mechanistic analysis revealed USP7 interacts with two other hits from our screen and together form a complex that regulates H2B ubiquitination, a marker of transcriptional elongation. Through epigenetic and transcriptional studies, we found that this complex regulates the expression of CABLES1, which is required for triggering cell death and tumor regression.
Using candidate-based approaches, we also identified two additional combination therapies that target BRAF-, NRAS-, and NF1-mutant melanomas. Specifically, we discovered that histone de-acetylase (HDAC) inhibitors potentiate responses to Ras/ERK signaling pathway inhibitors in BRAF-, NRAS-, and NF1-mutant melanomas by suppressing DNA repair pathways (Maertens et al. 2019). Additionally, we demonstrated that the combination of Ras/ERK signaling pathway and bromodomain and extraterminal domain (BET) inhibitors represents a promising new therapeutic approach to reduce acquired and intrinsic resistance in BRAF-mutant melanomas (Katherine R. Singleton 2017). In summary, this dissertation details three promising combination therapies for Ras/ERK pathway-driven melanomas and describes the distinct mechanisms by which they function.Medical SciencesMedical Science
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Identification and Characterization of Novel Cancer Genetic Dependencies
No full textCancer cells have unique genetic dependencies that can be exploited therapeutically, leading to successful targeted therapeutic drugs that benefit patients. However, many cancers remain undruggable, and many drug development efforts targeting cancer genetic dependencies have failed. To address these challenges, my thesis work focuses on two specific aspects.
First, the paradigm of cancer dependencies and targeted therapies has been largely focused on inhibition of critical pathways in cancer. Conversely, conditional activation of signaling pathways as a new source of selective cancer vulnerabilities has not been systematically characterized. In Chapter 2, I developed a method for enacting gain-of-function genetic perturbations simultaneously across ~500 barcoded cancer cell lines to query the pan-cancer vulnerability landscape upon activating 10 key pathway nodes, revealing selective activation dependencies of MAPK, PI3K, p53, and cell cycle pathways associated with specific biomarkers. Notably, we discovered novel pathway hyperactivation dependencies in subsets of APC-mutant colorectal cancers where further activation of the WNT pathway by APC knockdown or direct β-catenin overexpression led to robust anti-tumor effects in xenograft and patient-derived organoid models. Together, this chapter reveals a new class of conditional gene activation dependencies in cancer.
Second, despite remarkable successes in the clinic, cancer targeted therapy development remains challenging and the failure rate is disappointingly high. In chapter 3, I identified that the problem is partly due to the misapplication of the targeted therapy paradigm to therapeutics targeting pan-essential genes, which can result in therapeutics where efficacy is attenuated by dose-limiting toxicity. I summarized the key features of successful chemotherapy and targeted therapy agents, and used case studies to outline recurrent challenges to drug development efforts targeting pan-essential genes. Finally, I suggested strategies to avoid previous pitfalls for ongoing and future development of pan-essential therapeutics.Medical SciencesMedical Science
Shadows Of Legendrian Links And J + -Theory Of Curves
No full text. We introduce invariants of 2-component fronts similar to Arnold's [1] invariants J \Sigma following approach of Viro [22] and generalize Viro's formulas to invariants of 1 and 2-component 0-homologous fronts on surfaces of non-zero Euler characteristic. We modify Turaev's construction [19] of link shadows and define shadows of Legendrian links in ST S 2 . This enables us to relate integral formulas for J + -type invariants of fronts to Turaev's [19] shadow formulas for linking and self-linking numbers applied to Legendrian shadows. Other applications of Legendrian shadows, e.g. quantum J \Sigma -type invariants of fronts are discussed. 1. Introduction Generic circle immersions into the plane have only transversal double points. Immersions with points of self-tangency create a singular discriminant in the space of all immersions. The consideration of this discriminant enabled Arnold [1] to introduce recently new basic invariants J + and J \Gamma of generic immersions a..
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