Geometric Approach to Pontryagin's Maximum Principle

Since the second half of the 20th century, Pontryagin's Maximum Principle has been widely discussed and used as a method to solve optimal control problems in medicine, robotics, finance, engineering, astronomy. Here, we focus on the proof and on the understanding of this Principle, using as much geometric ideas and geometric tools as possible. This approach provides a better and clearer understanding of the Principle and, in particular, of the role of the abnormal extremals. These extremals are interesting because they do not depend on the cost function, but only on the control system. Moreover, they were discarded as solutions until the nineties, when examples of strict abnormal optimal curves were found. In order to give a detailed exposition of the proof, the paper is mostly self\textendash{}contained, which forces us to consider different areas in mathematics such as algebra, analysis, geometry.Comment: Final version. Minors changes have been made. 56 page

Montgomery Multiplication on the Cell

A technique to speed up Montgomery multiplication targeted at the Synergistic Processor Elements (SPE) of the Cell Broadband Engine is proposed. The technique consists of splitting a number into four consecutive parts. These parts are placed one by one in each of the four element positions of a vector, representing columns in a 4-SIMD organization. This representation enables arithmetic to be performed in a 4-SIMD fashion. An implementation of the Montgomery multiplication using this technique is up to 2.47 times faster compared to an unrolled implementation of Montgomery multiplication, which is part of the IBM multi-precision math library, for odd moduli of length 160 to 2048 bits. The presented technique can also be applied to speed up Montgomery multiplication on other SIMD-architectures

On the Use of the Negation Map in the Pollard Rho Method

The negation map can be used to speed up the Pollard rho method to compute discrete logarithms in groups of elliptic curves over finite fields. It is well known that the random walks used by Pollard rho when combined with the negation map get trapped in fruitless cycles. We show that previously published approaches to deal with this problem are plagued by recurring cycles, and we propose effective alternative countermeasures. As a result, fruitless cycles can be resolved, but the best speedup we managed to achieve is by a factor of only 1.29. Although this is less than the speedup factor of root 2 generally reported in the literature, it is supported by practical evidence

Mechanistic target of rapamycin (MTOR) protein expression in the tumor and its microenvironment correlates with more aggressive pathology at cystectomy

Background: The mechanistic target of rapamycin (mTOR) has been implicated in driving tumor biology in multiple malignancies, including urothelial carcinoma (UC). We investigate how mTOR and phosphorylated mTOR (pmTOR) protein expression correlate with chemoresponsiveness in the tumor and its microenvironment at final pathologic staging after neoadjuvant chemotherapy (NAC). Methods: A single-institution retrospective analysis was performed on 62 patients with cT2–4Nany UC undergoing NAC followed by radical cystectomy. Diagnostic (transurethral resection specimens, TURBT) and postchemotherapy radical cystectomy specimens were evaluated for mTOR and pmTOR protein expression using immunohistochemistry of the tumor, peritumoral stroma, and normal surrounding stroma. Protein expression levels were compared between clinical and pathologic stage. Whole transcriptome analysis was performed to evaluate mRNA expression relative to mTOR pathway activation. Results: Baseline levels of mTOR and pmTOR within TURBT specimens were not associated with clinical stage and response to chemotherapy overall. Nonresponders with advanced pathologic stage at cystectomy (ypT2–4/ypTanyN+) had significantly elevated mTOR tumor staining (P = 0.006) and a sustained mTOR and pmTOR staining in the peritumoral and surrounding normal stroma (NS). Several genes relevant to mTOR activity were found to be up-regulated in the tumors of nonresponders. Remarkably, complete responders at cystectomy (ypT0) had significant decreases in both mTOR and pmTOR protein expression in the peritumoral and normal stroma (P = 0.01–0.03). Conclusions: Our results suggest that mTOR pathway activity is increased in tumor and sustained in its microenvironment in patients with adverse pathologic findings at cystectomy. These findings suggest the relevance of targeting this pathway in bladder cancer

Evidence of Color Coherence Effects in W+jets Events from ppbar Collisions at sqrt(s) = 1.8 TeV

We report the results of a study of color coherence effects in ppbar collisions based on data collected by the D0 detector during the 1994-1995 run of the Fermilab Tevatron Collider, at a center of mass energy sqrt(s) = 1.8 TeV. Initial-to-final state color interference effects are studied by examining particle distribution patterns in events with a W boson and at least one jet. The data are compared to Monte Carlo simulations with different color coherence implementations and to an analytic modified-leading-logarithm perturbative calculation based on the local parton-hadron duality hypothesis.Comment: 13 pages, 6 figures. Submitted to Physics Letters