Il polinomio strutturale di un'algebra monounaria

Questa tesi descrive alcune proprietà delle algebre monounarie finite e si propone di trovare un metodo per classificarle. Poiché infatti il numero di algebre di ordine n aumenta notevolmente con la crescita di quest’ultimo, si cerca un modo per suddividerle in classi d’isomorfismo. In particolare, dal momento che anche il numero di queste classi cresce esponenzialmente all’aumentare di n, utilizziamo una classificazione meno fine dell’isomorfismo basata sul polinomio strutturale. Grazie a questo strumento infatti è possibile risalire a famiglie di grafi orientati associati ad algebre monounarie, a due a due non isomorfi, ricavando perciò alcune specifiche caratteristiche di quest’ultime. Infine, calcolando l’ordine di gruppi particolari, detti automorfi, si può ottenere l’effettivo numero di algebre aventi un dato polinomio strutturale.

Quotients of Koszul Algebras and 2-D-Determined Algebras

Vatne [13] and Green and Marcos [9] have independently studied the Koszul-like homological properties of graded algebras that have defining relations in degree 2 and exactly one other degree. We contrast these two approaches, answer two questions posed by Green and Marcos, and find conditions that imply the corresponding Yoneda algebras are generated in the lowest possible degrees.

Computer algebra for x-ray spectral reconstruction between 6 and 25 MV

A previously presented algorithm for the reconstruction of bremsstrahlung spectra from transmission data has been implemented into MATHEMATICA. Spectra vectorial algebra has been used to solve the matrix system A * F = T. The new implementation has been tested by reconstructing photon spectra from transmission data acquired in narrow beam conditions, for nominal energies of 6, 15, and 25 MV. The results were in excellent agreement with the original calculations. Our implementation has the advantage to be based on a well-tested mathematical kernel. Furthermore it offers a comfortable user interface.

Accelerated partial breast irradiation with interstitial brachytherapy as second conservative treatment for ipsilateral breast tumour recurrence: multicentric study of the GEC-ESTRO Breast Cancer Working Group

PURPOSE To analyse the clinical outcome after salvage lumpectomy and multi-catheter brachytherapy (MCB) for ipsilateral breast tumour recurrence (IBTR). MATERIAL AND METHODS Between 09/00 and 09/10, 217 patients presenting an IBTR underwent lumpectomy and MCB (low, pulsed, or high-dose rate). Survival rates without second local recurrence (2nd LR), distant metastasis (DM), and overall survival (OS) were analysed as well as late effects and cosmetic results. Univariate and multivariate analyses (MVA) based on IBTR data were performed to find prognostic factors for 2nd LR, DM, and OS. RESULTS Median follow-up after the IBTR was 3.9 years [range: 1.1-10.3]. Five and 10-year actuarial 2nd LR rates were 5.6% [range: 1.5-9.5] and 7.2% [range: 2.1-12.1], respectively. Five and 10-year actuarial DM rates were 9.6% [range: 5.7-15.2] and 19.1% [range: 7.8-28.3], respectively. Five and 10-year actuarial OS rates were 88.7% [range: 83.1-94.8] and 76.4% [range: 66.9-87.3], respectively. In MVA, histological grade was prognostic factor for 2nd LR (p=0.008) and OS (p=0.02); while tumour size was prognostic factor for DM (p=0.03). G3-4 complication rate was 11%. Excellent/good cosmetic result was achieved in 85%. CONCLUSION This study suggests that in case of IBTR, lumpectomy plus MCB is feasible and effective in preventing 2nd LR with an OS rate at least equivalent to those achieved with salvage mastectomy.

A feasible theory of truth over combinatory algebra

We define an applicative theory of truth TPT which proves totality exactly for the polynomial time computable functions. TPT has natural and simple axioms since nearly all its truth axioms are standard for truth theories over an applicative framework. The only exception is the axiom dealing with the word predicate. The truth predicate can only reflect elementhood in the words for terms that have smaller length than a given word. This makes it possible to achieve the very low proof-theoretic strength. Truth induction can be allowed without any constraints. For these reasons the system TPT has the high expressive power one expects from truth theories. It allows embeddings of feasible systems of explicit mathematics and bounded arithmetic. The proof that the theory TPT is feasible is not easy. It is not possible to apply a standard realisation approach. For this reason we develop a new realisation approach whose realisation functions work on directed acyclic graphs. In this way, we can express and manipulate realisation information more efficiently.

Algebra of Classical Theoretical Term Reductions in the Sciences

An elementary algebra identifies conceptual and corresponding applicational limitations in John Kemeny and Paul Oppenheim’s (K-O) 1956 model of theoretical reduction in the sciences. The K-O model was once widely accepted, at least in spirit, but seems afterward to have been discredited, or in any event superceeded. Today, the K-O reduction model is seldom mentioned, except to clarify when a reduction in the Kemeny-Oppenheim sense is not intended. The present essay takes a fresh look at the basic mathematics of K-O comparative vocabulary theoretical term reductions, from historical and philosophical standpoints, as a contribution to the history of the philosophy of science. The K-O theoretical reduction model qualifies a theory replacement as a successful reduction when preconditions of explanatory adequacy and comparable systematicization are met, and there occur fewer numbers of theoretical terms identified as replicable syntax types in the most economical statement of a theory’s putative propositional truths, as compared with the theoretical term count for the theory it replaces. The challenge to the historical model developed here, to help explain its scope and limitations, involves the potential for equivocal theoretical meanings of multiple theoretical term tokens of the same syntactical type.

Recommendations from GEC ESTRO Breast Cancer Working Group (I): Target definition and target delineation for accelerated or boost Partial Breast Irradiation using multicatheter interstitial brachytherapy after breast conserving closed cavity surgery

OBJECTIVE The aim was to develop a delineation guideline for target definition for APBI or boost by consensus of the Breast Working Group of GEC-ESTRO. PROPOSED RECOMMENDATIONS Appropriate delineation of CTV (PTV) with low inter- and intra-observer variability in clinical practice is complex and needs various steps as: (1) Detailed knowledge of primary surgical procedure, of all details of pathology, as well as of preoperative imaging. (2) Definition of tumour localization before breast conserving surgery inside the breast and translation of this information in the postoperative CT imaging data set. (3) Calculation of the size of total safety margins. The size should be at least 2 cm. (4) Definition of the target. (5) Delineation of the target according to defined rules. CONCLUSION Providing guidelines based on the consensus of a group of experts should make it possible to achieve a reproducible and consistent definition of CTV (PTV) for Accelerated Partial Breast Irradiation (APBI) or boost irradiation after breast conserving closed cavity surgery, and helps to define it after selected cases of oncoplastic surgery.

Lehrbuch der Arithmetik und der niederen Algebra : für Gymnasien, höhere Bürgerschulen, Kriegsschulen und zum Selbstunterrichte

von Fr. Adrian Köcher

Grundlehren der Algebra : Zum Gebrauch in Schulen und zum Selbstunterricht

Von J. Ineichen, Professor der Physik am Lyceum zu Luzern

An example of Learning based on competences: Use of Maxima in Linear Algebra for Engineers

This paper analyzes the role of Computer Algebra Systems (CAS) in a model of learning based on competences. The proposal is an e-learning model Linear Algebra course for Engineering, which includes the use of a CAS (Maxima) and focuses on problem solving. A reference model has been taken from the Spanish Open University. The proper use of CAS is defined as an indicator of the generic ompetence: Use of Technology. Additionally, we show that using CAS could help to enhance the following generic competences: Self Learning, Planning and Organization, Communication and Writing, Mathematical and Technical Writing, Information Management and Critical Thinking.

Differential algebra techniques for space applications

This project investigates the utility of differential algebra (DA) techniques applied to the problem of orbital dynamics with initial uncertainties in the orbital determination of the involved bodies. The use of DA theory allows the splitting of a common Monte Carlo simulation in two parts: the generation of a Taylor map of the final states with regard to the perturbation in the initial coordinates, and the evaluation of the map for many points. A propagator is implemented exploiting DA techniques, and tested in the field of asteroid impact risk monitoring with the potentially hazardous 2011 AG5 and 2007 VK184 as test cases. Results show that the new method is able to simulate 2.5 million trajectories with a precision good enough for the impact probability to be accurately reproduced, while running much faster than a traditional Monte Carlo approach (in 1 and 2 days, respectively).

DERIVE and Linear Algebra

This work describes an experience with a methodology for learning based on competences in Linear Algebra for engineering students. The experience has been based in autonomous team work of students. DERIVE tutorials for Linear Algebra topics are provided to the students. They have to work with the tutorials as their homework. After, worksheets with exercises have been prepared to be solved by the students organized in teams, using DERIVE function previously defined in the tutorials. The students send to the instructor the solution of the proposed exercises and they fill a survey with their impressions about the following items: ease of use of the files, usefulness of the tutorials for understanding the mathematical topics and the time spent in the experience. As a final work, we have designed an activity directed to the interested students. They have to prepare a project, related with a real problem in Science and Engineering. The students are free to choose the topic and to develop it but they have to use DERIVE in the solution. Obviously they are guided by the instructor. Some examples of activities related with Orthogonal Transformations will be presented.

A toolbox with DERIVE: calculus on several variables, Applications of Computer Algebra

A toolbox is a set of procedures taking advantage of the computing power and graphical capacities of a CAS. With these procedures the students can solve math problems, apply mathematics to engineering or simply reinforce the learning of certain mathematical concepts. From the point of view of their construction, we can consider two types of toolboxes: (i) the closed box, built by the teacher, in which the utility files are provided to the students together with the respective tutorials and several worksheets with proposed exercises and problems,