Mini-system: un software di calcolo numerico

Questo elaborato descrive il lavoro di tesi che ha portato all' inserimento del MiniSystem come strumento di supporto e di esercitazione per gli studenti nell' apprendimento del "Calcolo Numerico".

Neurofuzzy network model construction using Bézier-Bernstein polynomial functions

Neurofuzzy modelling systems combine fuzzy logic with quantitative artificial neural networks via a concept of fuzzification by using a fuzzy membership function usually based on B-splines and algebraic operators for inference, etc. The paper introduces a neurofuzzy model construction algorithm using Bezier-Bernstein polynomial functions as basis functions. The new network maintains most of the properties of the B-spline expansion based neurofuzzy system, such as the non-negativity of the basis functions, and unity of support but with the additional advantages of structural parsimony and Delaunay input space partitioning, avoiding the inherent computational problems of lattice networks. This new modelling network is based on the idea that an input vector can be mapped into barycentric co-ordinates with respect to a set of predetermined knots as vertices of a polygon (a set of tiled Delaunay triangles) over the input space. The network is expressed as the Bezier-Bernstein polynomial function of barycentric co-ordinates of the input vector. An inverse de Casteljau procedure using backpropagation is developed to obtain the input vector's barycentric co-ordinates that form the basis functions. Extension of the Bezier-Bernstein neurofuzzy algorithm to n-dimensional inputs is discussed followed by numerical examples to demonstrate the effectiveness of this new data based modelling approach.

Bézier曲线间最近距离的计算方法

针对Bzier曲线间最近距离计算问题,提出一种简捷、可靠的计算方法.该方法以Bernstein多项式算术运算为工具,建立Bzier曲线间最近距离的计算模型;然后充分利用Bzier曲面的凸包性质和de Casteljau分割算法进行求解.该方法几何意义明确,能有效地避免迭代初始值的选择和非线性方程组的求解,并可进一步推广应用于计算Bzier曲线/曲面间的最近距离.实验结果表明,该方法简捷、可靠且容易实现,与Newton-Raphson方法的融合可进一步提高该方法的运行速度.

Data based constructive identification - overcoming the curse of dimensionality

The modelling of a nonlinear stochastic dynamical processes from data involves solving the problems of data gathering, preprocessing, model architecture selection, learning or adaptation, parametric evaluation and model validation. For a given model architecture such as associative memory networks, a common problem in non-linear modelling is the problem of "the curse of dimensionality". A series of complementary data based constructive identification schemes, mainly based on but not limited to an operating point dependent fuzzy models, are introduced in this paper with the aim to overcome the curse of dimensionality. These include (i) a mixture of experts algorithm based on a forward constrained regression algorithm; (ii) an inherent parsimonious delaunay input space partition based piecewise local lineal modelling concept; (iii) a neurofuzzy model constructive approach based on forward orthogonal least squares and optimal experimental design and finally (iv) the neurofuzzy model construction algorithm based on basis functions that are Bézier Bernstein polynomial functions and the additive decomposition. Illustrative examples demonstrate their applicability, showing that the final major hurdle in data based modelling has almost been removed.

La base di Bernstein in spazi polinomiali generalizzati

Nella tesi si illustra il passaggio dagli spazi polinomiali agli spazi polinomiali generalizzati, gli spazi di Chebyshev estesi (spazi EC), e viene dato un metodo per costruirli a partire da opportuni sistemi di funzioni dette funzioni peso. Successivamente si tratta il problema dell'esistenza di un analogo della base di Bernstein negli spazi EC: si presenta, in analogia ad una particolare costruzione nel caso polinomiale, una dimostrazione costruttiva dell'esistenza di tale base. Infine viene studiato il problema delle lunghezze critiche di uno spazio EC: si tratta di determinare l'ampiezza dell'intervallo oltre la quale lo spazio considerato perde le proprietà di uno spazio EC, o non possiede più una base di Bernstein generalizzata; l'approccio adottato è di tipo sperimentale: nella tesi sono presentati i risultati ottenuti attraverso algoritmi di ricerca che analizzano le proprietà delle funzioni di transizione e ne traggono informazioni sullo spazio di studio.

La base di Bernstein in spazi polinomiali generalizzati a tratti

Le funzioni polinomiali possono essere utilizzate per approssimare le funzioni continue. Il vantaggio è che i polinomi, le loro derivate e primitive, possono essere rappresentati in maniera semplice attraverso i loro coefficienti ed esistono algoritmi stabili e veloci per valutarli. Inoltre gli spazi polinomiali godono di numerose proprietà importanti. In questo lavoro ci occuperemo di altri spazi funzionali, noti in letteratura come spazi di Chebyshev o polinomi generalizzati, per ragioni di riproducibilità. Infatti ciò che si ottiene attraverso i polinomi è soltanto una approssimazione che spesso risulta essere insufficiente. E' importante, quindi, considerare degli spazi in cui sia possibile avere una rappresentazione esatta di curve. Lo studio di questi spazi è possibile grazie alla potenza di elaborazione degli attuali calcolatori e al buon condizionamento di opportune basi di rappresentazione di questi spazi. Negli spazi polinomiali è la base di Bernstein a garantire quanto detto. Negli spazi di Chebyshev si definisce una nuova base equivalente. In questo lavoro andremo oltre gli spazi di Chebyshev ed approfondiremo gli spazi di Chebyshev a tratti, ovvero gli spazi formati dall'unione di più spazi del tipo precedente. Si dimostrerà inoltre l'esistenza di una base a tratti con le stesse proprietà della base di Bernstein per gli spazi polinomiali.

PAC-Bayes-empirical-Bernstein inequality

We present PAC-Bayes-Empirical-Bernstein inequality. The inequality is based on combination of PAC-Bayesian bounding technique with Empirical Bernstein bound. It allows to take advantage of small empirical variance and is especially useful in regression. We show that when the empirical variance is significantly smaller than the empirical loss PAC-Bayes-Empirical-Bernstein inequality is significantly tighter than PAC-Bayes-kl inequality of Seeger (2002) and otherwise it is comparable. PAC-Bayes-Empirical-Bernstein inequality is an interesting example of application of PAC-Bayesian bounding technique to self-bounding functions. We provide empirical comparison of PAC-Bayes-Empirical-Bernstein inequality with PAC-Bayes-kl inequality on a synthetic example and several UCI datasets.

Judge Max Frankenbach of Berlin hiking with his wife Grete Frankenbach nee Bernstein; Sassal-Massone Portraits Couples

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Bernstein-Kohlberg Portraits Family

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Bernard Bernstein Archive, 1941-2008 (bulk bulk 1960-1984).

Bernard Bernstein collection documents professional activities of Bernard Bernstein, a jeweler, metal smith, writer, and teacher. The collection includes artifacts, correspondence, documents, manuscripts, printed materials, photographs, other visual materials, and sketches.The larger part of the collection includes materials dealing with the artistic side of Bernard Bernstein. These materials are found throughout the collection and consist of artifacts produced during his schooling at City College (Series I: Artifacts), various jewelry designs produced by Bernard Bernstein for commercial use (Series III: Designs), certificates and awards (Series V: General), and materials pertaining to a number of shows and exhibits that Bernard Bernstein was a part of (Series IV: Exhibitions and Art Catalogues).Other materials include documents pertaining to Bernard Bernstein education, professional carrier as a teacher ( Series II: City College of the City University of New York, Series V: General), and his articles in professional journals (Series VI: Printed Materials).In some cases materials are accompanied by Bernard Bernstein’s notes explaining the significance and provenance of the documents.

Herman Bernstein 1897-1935

Correspondence, clippings, manuscripts, notes, reports, relating to Bernstein's journalistic, literary and diplomatic careers. Correspondence with well-known literary, political and communal, society personalities, 1908-1935. Includes Cyrus Adler, Viscount Allenby, Joseph Barondess, Bernard Baruch, Henri Bergson, Hayyim Nahman Bialik, Jacob Billikopf, Vladimir Bourtzeff, Louis Brandeis, Robert Cecil, Fyodor Chaliapin, Jacob de Haas, Albert Einstein, Henry Ford, Felix Frankfurter, Herbert Hoover, Vladimir Jabotinsky, Horace M. Kallen, Peretz Hirschbein, Peter Kropotkin, Herbert Lehman, Louis Lipsky, Judah L. Magnes, Louis Marshall, Henry Morgenthau, Max Nordau, Adolph Simon Ochs, David de Sola Pool, Bernard G. Richards, Theodore Roosevelt, Julius Rosenwald, Jacob Schiff, Harry Schneiderman, Maurice Schwartz, George Bernard Shaw, Sholem Aleichem, Nathan Straus, Henrietta Szold, Chaim Tchernowitz, Leo Tolstoy, Samuel Untermyer, Henry Van Dyke, Lillian Wald, Felix Warburg, Chaim Weizman n, Jefferson Williams, Stephen Wise, Israel Zangwill. Correspondence and other materials relating to Bernstein's post as U.S. ambassador to Albania. Materials pertaining to Bernstein's editorial work at *The Day*, *Jewish Tribune*, *New York Herald*, *Jewish Daily Bulletin*. Materials pertaining to Bernstein's involvement with the American Jewish Committee. Correspondence with organizations including American Jewish Congress, *American Hebrew*, HIAS, *Jewish Chronicle* (London), Jewish Community of New York, *Menorah Journal*, *New York American*, *New York Times*, ORT, U.S. Dept. of State, Yiddish Art Theater, Zionist Organization of America. Articles, clippings, correspondence and court materials relating to the Ford libel suit. Miscellaneous documents and reports relating to the Paris Peace Conference, the Jewish situation in Russia, 1917-1920, Russian revolutionary events of 1917. News dispatches from Russia, 1917-1920s. Translations by Bernstein of Russian wri Andre yev,

Uso da linguagem de especificação SDL como alternativa ao diagrama de estados proposto pela linguagem UML.

Linguagem UML. Fase do projeto na UML: o diagrama de estados. Linguagem SDL. Comparação entre as duas técnicas apresentadas.

Algunos aspectos de polinomios de Bernstein, Bezier y trazadores

Este trabajo consta de dos partes: la primera presenta, de manera elemental, la teoría de los polinomios de Bernstein en una variable; la segunda esta dedicada a curvas de Bezier y q-trazadores ("q-splines"). Nos parece importante el uso que se puede dar del software Mathematica.