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Articles 541 - 567 of 567

Full-Text Articles in Mathematics

Pigeon-Holing Monodromy Groups, Niles G. Johnson Dec 2002

Pigeon-Holing Monodromy Groups, Niles G. Johnson

Mathematical Sciences Technical Reports (MSTR)

A simple tiling on a sphere can be used to construct a tiling on a d-fold branched cover of the sphere. By lifting a so-called equatorial tiling on the sphere, the lifted tiling is locally kaleidoscopic, yielding an attractive tiling on the surface. This construction is via a correspondence between loops around vertices on the sphere and paths across tiles on the cover. The branched cover and lifted tiling give rise to an associated monodromy group in the symmetric group on d symbols. This monodromy group provides a beautiful connection between the cover and its base space. Our investigation …

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Fixed Point And Two-Cycles Of The Discrete Logarithm, Joshua Holden Oct 2002

Fixed Point And Two-Cycles Of The Discrete Logarithm, Joshua Holden

Mathematical Sciences Technical Reports (MSTR)

We explore some questions related to one of Brizolis: does every prime p have a pair (g, h) such that h is a fixed point for the discrete logarithm with base g? We extend this question to ask about not only fixed points but also two-cycles. Campbell and Pomerance have not only answered the fixed point question for sufficiently large p but have also rigorously estimated the number of such pairs given certain conditions on g and h. We attempt to give heuristics for similar estimates given other conditions on g and h and also in the case …

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Polynomial Continued Fractions, Douglas Bowman, James Mclaughlin Jan 2002

Polynomial Continued Fractions, Douglas Bowman, James Mclaughlin

Mathematics Faculty Publications

Continued fractions whose elements are polynomial sequences have been carefully studied mostly in the cases where the degree of the numerator polynomial is less than or equal to two and the degree of the denominator polynomial is less than or equal to one. Here we study cases of higher degree for both numerator and denominator polynomials, with particular attention given to cases in which the degrees are equal. We extend work of Ramanujan on continued fractions with rational limits and also consider cases where the limits are irrational.

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Proceedings Of The First International Conference On Neutrosophy, Neutrosophic Logic, Neutrosophic Set, Neutrosophic Probability And Statistics, Florentin Smarandache Jan 2002

Proceedings Of The First International Conference On Neutrosophy, Neutrosophic Logic, Neutrosophic Set, Neutrosophic Probability And Statistics, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In 1960s Abraham Robinson has developed the non-standard analysis, a formalization of analysis and a branch of mathematical logic, that rigorously defines the infinitesimals. Informally, an infinitesimal is an infinitely small number. Formally, x is said to be infinitesimal if and only if for all positive integers n one has xxx < 1/n. Let &>0 be a such infinitesimal number. The hyper-real number set is an extension of the real number set, which includes classes of infinite numbers and classes of infinitesimal numbers. Let’s consider the non-standard finite numbers 1+ = 1+&, where “1” is its standard part and “&” its non-standard part, …

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Isotropy And Factorization In Reduced Witt Rings, Robert W. Fitzgerald Mar 2001

Isotropy And Factorization In Reduced Witt Rings, Robert W. Fitzgerald

Articles and Preprints

We consider reduced Witt rings of finite chain length. We show there is a bound, in terms of the chain length and maximal signature, on the dimension of anisotropic, totally indefinite forms. From this we get the ascending chain condition on principal ideals and hence factorization of forms into products of irreducible forms.

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Norms Of Sums Of Squares, Robert W. Fitzgerald Mar 2001

Norms Of Sums Of Squares, Robert W. Fitzgerald

Articles and Preprints

For a finite separable extension K/F of fields of characteristic not 2, the norm of a sum of 2n squares in K is a sum of 2n squares in F. We find explicit identities.

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Ramanujan-Like Congreuences Of The Distinct Partition Function, Ian Blumenfeld, Christi Carlstead, Mimi Cukier, Wesley Terway Dec 2000

Ramanujan-Like Congreuences Of The Distinct Partition Function, Ian Blumenfeld, Christi Carlstead, Mimi Cukier, Wesley Terway

Mathematical Sciences Technical Reports (MSTR)

In his work with the partition function, Ramanujan observed several congruences of the form p(An + B) = 0 (mod m). We adapt this form to several congruences of the distinct partition function, p2(n). We show that one can determine all ordered pairs of integers (A;B) for which p2(An + B)=0 (mod 2) and show families of congruences modulo 4. Finally, we offer a proof of a congruence modulo 5 satisfied by the distinct partition function.

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Torsion-Free Modules Over Reduced Witt Rings, Robert W. Fitzgerald Sep 2000

Torsion-Free Modules Over Reduced Witt Rings, Robert W. Fitzgerald

Articles and Preprints

We compute the genus class group of a torsion-free module over a reduced Witt ring of finite stability index. This is applied to modules locally isomorphic to odd degree extensions of formally real fields.

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The Proof Of Fermat's Last Theorem, Mohamad Trad Jan 2000

The Proof Of Fermat's Last Theorem, Mohamad Trad

Theses Digitization Project

Fermat, Pierre de, is perhaps the most famous number theorist who ever lived. Fermat's Last Theorem states that the equation xn + yn = zn has no non-zero integer solutions for x, y and z when n>2.

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Algebra În Exercijii Şi Probleme Pentru Liceu, Florentin Smarandache, Ion Goian, Raisa Grigor, Vasila Marin Jan 2000

Algebra În Exercijii Şi Probleme Pentru Liceu, Florentin Smarandache, Ion Goian, Raisa Grigor, Vasila Marin

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

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Definitions, Solved And Unsolved Problems, Conjectures, And Theorems In Number Theory And Geometry, Florentin Smarandache Jan 2000

Definitions, Solved And Unsolved Problems, Conjectures, And Theorems In Number Theory And Geometry, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Florentin Smarandache, an American mathematician of Romanian descent has generated a vast variety of mathematical problems. Some problems are easy, others medium, but many are interesting or unsolved and this is the reason why the present book appears. Here, of course, there are problems from various types. Solving these problems is addictive like eating pumpkin seed: having once started, one cannot help doing it over and over again.

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Small Extensions Of Witt Rings, Robert W. Fitzgerald May 1999

Small Extensions Of Witt Rings, Robert W. Fitzgerald

Articles and Preprints

We consider certain Witt ring extensions S of a noetherian Witt ring R obtained by adding one new generator. The conditions on the new generator are those known to hold when R is the Witt ring of a Field F, S is the Witt ring of a Field K and K/F is an odd degree extension. We show that if R is of elementary type then so is S.

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Sextic Number Fields With Discriminant -J2A3B, John W. Jones, David P. Roberts Jan 1999

Sextic Number Fields With Discriminant -J2A3B, John W. Jones, David P. Roberts

Mathematics Publications

. Complete lists of number fields, of given degree n and unramified outside a given finite set S of primes, are both of intrinsic interest and useful in some applications. For degrees n ≤ 5 and S = {∞, 2, 3}, the complete lists have appeared previously; there are in total 85 such fields. Here we give the complete list for n = 6 and S = {∞, 2, 3}, finding in particular exactly 398 such fields. We use a three-pronged approach to obtain this classification: an exhaustive computer search, sextic twinning, and class field theory. Also we completely identify …

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Smarandache Type Functions Obtained By Duality, Florentin Smarandache, C Dumitrescu, N Varlan, St Zamfir, E Radescu, N Radescu Jan 1999

Smarandache Type Functions Obtained By Duality, Florentin Smarandache, C Dumitrescu, N Varlan, St Zamfir, E Radescu, N Radescu

Branch Mathematics and Statistics Faculty and Staff Publications

we extend the Smarandache function from the set N* of positive integers to the set Q of rationals

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Asupra Unor Noi Functii În Teoria Numerelor, Florentin Smarandache Jan 1999

Asupra Unor Noi Functii În Teoria Numerelor, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Performantele matematicii actuale,ca si descoperirile din viitor isi au,desigur, inceputul in cea mai veche si mai aproape de filozofie ramura a matematicii, in teoria numerelor. Matematicienii din toate timpurile au fost, sunt si vor fi atrasi de frumusetea si varietatea problemelor specifice acestei ramuri a matematicii. Regina a matematicii, care la randul ei este regina a stiintelor, dupa cum spunea Gauss, teoria numerelor straluceste cu lumina si atractiile ei, fascinandu-ne si usurandu-ne drumul cunoasterii legitatilor ce guverneaza macrocosmosul si microcosmosul. De la etapa antichitatii, cand teoria numerelor era cuprinsa in aritmetica, la etapa aritmeticii superioare din perioada Renasterii, cand teoria …

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Convergence Of Random Walks On The Circle Generated By An Irrational Rotation, Francis E. Su Jan 1998

Convergence Of Random Walks On The Circle Generated By An Irrational Rotation, Francis E. Su

All HMC Faculty Publications and Research

Fix . Consider the random walk on the circle which proceeds by repeatedly rotating points forward or backward, with probability , by an angle . This paper analyzes the rate of convergence of this walk to the uniform distribution under ``discrepancy'' distance. The rate depends on the continued fraction properties of the number . We obtain bounds for rates when is any irrational, and a sharp rate when is a quadratic irrational. In that case the discrepancy falls as (up to constant factors), where is the number of steps in the walk. This is the first example of a sharp …

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Subiecte Posibile Pentru Examenul De Admitere În Liceu Şi Examenul De Capacitate (In Romanian), Florentin Smarandache, Constantin Coanda, Gheorghe Duta Jan 1998

Subiecte Posibile Pentru Examenul De Admitere În Liceu Şi Examenul De Capacitate (In Romanian), Florentin Smarandache, Constantin Coanda, Gheorghe Duta

Branch Mathematics and Statistics Faculty and Staff Publications

Prezenta lucrare" încearcă să pună la dispoziţia elevilor şi profesorilor instrumente de evaluare a cunoştinţelor la toate capitolele din actuala programă de matematică. " În evoluţia adolescenţi/or din ţara noastră etapa admiterii în liceu mobilizează eforturi şi emoţii cu totul speciale. 'Actuala lucrare se doreşte a fi un sfljtuitor permanent În perioada fierbinte dintre e.;wmenul de capacitate şi admiterea În liceu. Testele cuprinse in această culegere au un caracter complementar faţă de multe materiale scrise În vederea sprijinirii celor ce se pregătesc pentru astfel de examene şi se referă la intreaga materie cuprinsă În programele analitice de aritmetică, algebră şi …

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Gorenstein Witt Rings Ii, Robert W. Fitzgerald Jan 1997

Gorenstein Witt Rings Ii, Robert W. Fitzgerald

Articles and Preprints

The abstract Witt rings which are Gorenstein have been classified when the dimension is one and the classification problem for those of dimension zero has been reduced to the case of socle degree three. Here we classify the Gorenstein Witt rings of fields with dimension zero and socle degree three. They are of elementary type.

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Sums Of Powers And The Bernoulli Numbers, Laura Elizabeth S. Coen Jan 1996

Sums Of Powers And The Bernoulli Numbers, Laura Elizabeth S. Coen

Masters Theses

This expository thesis examines the relationship between finite sums of powers and a sequence of numbers known as the Bernoulli numbers. It presents significant historical events tracing the discovery of formulas for finite sums of powers of integers, the discovery of a single formula by Jacob Bernoulli which gives the Bernoulli numbers, and important discoveries related to the Bernoulli numbers. A method of generating the sequence by means of a number theoretic recursive formula is given. Also given is an application of matrix theory to find a relation, first given by Johannes Faulhaber, between finite sums of odd powers and …

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Metode De Calcul În Analiza Matematică, Florentin Smarandache, C. Dumitrescu Jan 1995

Metode De Calcul În Analiza Matematică, Florentin Smarandache, C. Dumitrescu

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

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Some Conjectures Concerning Triangular Numbers, Bruce Brandt Jan 1994

Some Conjectures Concerning Triangular Numbers, Bruce Brandt

Journal of the Minnesota Academy of Science

Strong empirical evidence supports conjectures that certain number patterns always hold. These patterns concern the function cr, defined by the equation cr(n) = n - m2, m2 being the nearest square to n, on the domain of the triangular numbers. Triangular squares or triangular numbers of the form m2+m are also mentioned in most of the conjectures. One of the conjectures, for example, is that the sum of cr over the triangular numbers up to a triangular square is 0. Some of these patterns can be described by strings of symbols, such as "S" and "L," formed by first writing …

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Supplement To "Some Conjectures Concerning Triangular Numbers", Bruce Brandt Jan 1994

Supplement To "Some Conjectures Concerning Triangular Numbers", Bruce Brandt

Journal of the Minnesota Academy of Science

In a previous paper (1), I stated many conjectures about triangular numbers. Since submitting that paper I have discovered many more results, including generalizations, which are presented here.

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Only Problems, Not Solutions! (Fourth Edition), Florentin Smarandache Jan 1993

Only Problems, Not Solutions! (Fourth Edition), Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

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Simultaneous Rational Approximations And Related Diophantine Equations, John Rickert Sep 1992

Simultaneous Rational Approximations And Related Diophantine Equations, John Rickert

Mathematical Sciences Technical Reports (MSTR)

In this paper we consider simultaneous approximations to algebraic numbers a1,...,am .

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Problemes Avec Et, Sans Problemes!, Florentin Smarandache Jan 1983

Problemes Avec Et, Sans Problemes!, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

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A Development Of The Number System, Janet R. Olsen May 1964

A Development Of The Number System, Janet R. Olsen

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

This paper is based on Landau's book "Foundations of Analysis" which constitutes a development of the number system founded on the Peano axioms for natural numbers.

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Groups Of Order Eight, Ludwig Eugene Loos May 1934

Groups Of Order Eight, Ludwig Eugene Loos

Bachelors’ Theses

The problem of this thesis is to show that there can be only five groups of order eight. Its basis is Kronecker's Theorem, and it demonstrates that, regardless of the number of generators used in composing the groups, the result will always be limited to four groups of order eight. The fifth and last group is a cyclic group formed on eight letters.

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