Topoloji Çözümlü Sorular

Bir önceki yazıda belirttiğim gibi , TÜBA  ders notlarını paylaşıma sunmuş...Topoloji ile ilgili bir çok çözümlü soruyu buradan indirebilirsiniz...Yararlı olacaktır...

Akademik Ders Notları

Matematik,fizik,kimya.biyoloji ve sosyal bilimlerle ilgili tüba aracılığıyla acikders.org.tr de çok güzel kaynaklar bulunmakta, incelemekte fayda var...Matematik ile ilgili birkaç dosyayı buradan indirebilirsiniz.

Aday Öğretmen Hazırlayıcı Eğitim Soruları

İnternette yer alan bazı sınav soru örneklerini buradan indirebilirsiniz.

Analytic Combinatorics - Philippe Flajolet & Robert Sedgewick

"Analytic combinatorics aims to enable precise quantitative predictions of the properties
of large combinatorial structures. The theory has emerged over recent decades
as essential both for the analysis of algorithms and for the study of scientific models
in many disciplines, including probability theory, statistical physics, computational
biology and information theory. With a careful combination of symbolic enumeration
methods and complex analysis, drawing heavily on generating functions, results
of sweeping generality emerge that can be applied in particular to fundamental structures
such as permutations, sequences, strings, walks, paths, trees, graphs and maps.
This account is the definitive treatment of the topic. In order to make it selfcontained,
the authors give full coverage of the underlying mathematics and give a
thorough treatment of both classical and modern applications of the theory. The text is
complemented with exercises, examples, appendices and notes throughout the book to
aid understanding. The book can be used as a reference for researchers, as a textbook
for an advanced undergraduate or a graduate course on the subject, or for self-study.
PHILIPPE FLAJOLET is Research Director of the Algorithms Project at INRIA Rocquencourt.
ROBERT SEDGEWICK isWilliamO. Baker Professor of Computer Science at Princeton
University."

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David A. Santos - Number Theory for Mathematical Contests

"These notes started in the summer of 1993 when I was teaching Number Theory at the Center for Talented Youth Summer
Program at the Johns Hopkins University. The pupils were between 13 and 16 years of age.
The purpose of the course was to familiarise the pupils with contest-type problem solving. Thus the majority of the problems
are taken from well-known competitions:
AHSME American High School Mathematics Examination
AIME American Invitational Mathematics Examination
USAMO United States Mathematical Olympiad
IMO International Mathematical Olympiad
ITT International Tournament of Towns
MMPC Michigan Mathematics Prize Competition
(UM)2 University of Michigan Mathematics Competition
STANFORD Stanford Mathematics Competition
MANDELBROT Mandelbrot Competition
Firstly, I would like to thank the pioneers in that course: Samuel Chong, Nikhil Garg, Matthew Harris, Ryan Hoegg, Masha
Sapper, Andrew Trister, Nathaniel Wise and Andrew Wong. I would also like to thank the victims of the summer 1994: Karen
Acquista, Howard Bernstein, Geoffrey Cook, Hobart Lee, Nathan Lutchansky, David Ripley, Eduardo Rozo, and Victor Yang.
I would like to thank Eric Friedman for helping me with the typing, and Carlos Murillo for proofreading the notes.
Due to time constraints, these notes are rather sketchy. Most of the motivation was done in the classroom, in the notes
I presented a rather terse account of the solutions. I hope some day to be able to give more coherence to these notes. No
theme requires the knowledge of Calculus here, but some of the solutions given use it here and there. The reader not knowing
Calculus can skip these problems. Since the material is geared to High School students (talented ones, though) I assume very
little mathematical knowledge beyond Algebra and Trigonometry. Here and there some of the problems might use certain
properties of the complex numbers.
A note on the topic selection. I tried to cover most Number Theory that is useful in contests. I also wrote notes (which I
have not transcribed) dealing with primitive roots, quadratic reciprocity, diophantine equations, and the geometry of numbers.
I shall finish writing them when laziness leaves my weary soul.
I would be very glad to hear any comments, and please forward me any corrections or remarks on the material herein.
David A. SANTOS
dsantos@ccp.edu"

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All-Soviet Union math competition

"This file contains the problems, suggested for solving on the Russian national mathematical competitions
(final part).
I've posted this stuff in a number of articles in rec.puzzles. But I have got many requests for the missing
parts. So I have decided to put this material here, having provided it with the answers on the common
questions.
==================collected articles start======================
I'm going to send some problems from the book Vasil'ev N.B, Egorov A.A. "The problems of the All-
Soviet–Union mathematical competitions",–Moscow.:Nauka. 1988 ISBN 5–02–013730–8. (in Russian).
Those problems were submitted for the solving on the competition between the pupils of 8, 9, or 10 forms
for 4 hours. So they do not contain something of the advanced topics, –– all of them can be solved by a
schoolboy. They can not go out of the common school plan bounds.
Most of the problems are original and the book contains all the necessary references. I am not going to
translate all the book, so I shall not send the solutions. Please, accept those messages as they are, to say
more exactly – as I can. I have to do my job, and this is hobby only, but nevertheless, that should be
enjoyable to solve those problems.
"Nobody can embrace the unembraceable."
Kozma Prutkov.
(beginning of the XIX c.)
May be those postings are just a harassment, but I hope, that most of You will not only enjoy the problems
solving, but will be able to use them in Your work with the students.
"Zeal overcomes everything."
"Sometimes zeal overcomes even the common sense."
Kozma Prutkov.
There are some wonderful books in Russian, that have not been translated into English yet, for example,
"Problems of the Moscow mathematical competitions",–compiled by G.A.Galperin, A.K.Tolpygo.,–
Moscow, Prosveshchenie, 1986.
A.A.Leman "Collection of Moscow mathematical competitions problems", –Moscow, Prosveshchenie,
1965."


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Cahit Arf Matematik Günleri / Birinci Gün 17 Nisan 2002-4

Cahit Arf Matematik Günleri / Birinci Gün 17 Nisan 2002-3

2002²⁰⁰² sayısı 5’e, 7’ye ve 19’a bölündüğünde kalanlar kaçtır? (1, 2, 2 puan) 

Çözüm : 

2002²⁰⁰² = 2²⁰⁰² = 4¹⁰⁰¹ = (-1)¹⁰⁰¹ = -1 = 4 (mod 5).

2002²⁰⁰² = (7 x 286)²⁰⁰² = 0 (mod 7).

2002²⁰⁰² =7²⁰⁰² =7^{3x667 + 1} = (7³)⁶⁶⁷ x 7 = 7 (mod 19) çünkü 7³ = 49 x7 = 11 x 7 = 77 =1 (mod 19).

Cahit Arf Matematik Günleri / Birinci Gün 17 Nisan 2002-2

Problem:

2. a ve b sayıları 11x² - 3x - 5 = 0 denkleminin iki çözümü olsun.

(1 + a + a² + a³ + ...) (1 + b + b² + b³ + ...)

 sayısını hesaplayın. (10 puan)

Cahit Arf Matematik Günleri / Birinci Gün 17 Nisan 2002-1

Problem:

1. ARF sözcüğünün tüm harflerini kullanarak (anlamlı ya da anlamsız) şu sözcükleri yazabiliriz:

AFR, ARF, FAR, FRA, RAF, RFA.

YASA sözcüğünün tüm harflerini kullanarak 12 sözcük yazabiliriz:

AASY, AAYS, ASAY, ASYA, AYAS, AYSA,

SAAY, SAYA, SYAA, YAAS, YASA, YSAA

1a. ÜÇGEN sözcüğünün tüm harflerini kullanarak kaç sözcük yazabilirsiniz? (Sözcükleri teker teker bulup listelemeniz gerekmez; sadece kaç tane yazılabileceğini bulun.) (1 puan)

1b. CAHİTARF sözcüğünün tüm harflerini kullanarak kaç sözcük yazabilirsiniz? (A harfini iki kez kullanacaksınız, aşağıdaki sorularda da öyle.) (2 puan)

1c. MATEMATİK sözcüğünün tüm harflerini kullanarak kaç sözcük yazabilirsiniz? (2 puan)

1d. DERECE sözcüğünün tüm harflerini kullanarak kaç sözcük yazabilirsiniz? (2 puan)

1e. İKİBİNİKİ sözcüğünün tüm harflerini kullanarak kaç sözcük yazabilirsiniz? (3 puan)

Algebraic numbers and Fourier analysis (Heath mathematical monographs) by Raphaël Salem

"THIS SMALL BOOK contains, with but a few developments, the substance of the
lectures I gave in the fall of 1960 at Brandeis University at the invitation of its
Department of Mathematics.
Although some of the material contained in this book appears in the latest
edition of Zygmund's treatise, the subject matter covered here has never until
now been presented as a whole, and part of it has, in fact, appeared only in original
memoirs. This, together with the presentation of a number of problems which
remain unsolved, seems to justify a publication which, I hope, may be of some
value to research students. In order to facilitate the reading of the book, I have
included in an Appendix the definitions and the results (though elementary)
borrowed from algebra and from number theory.
I wish to express my thanks to Dr. Abram L. Sachar, President of Brandeis
University, and to the Department of Mathematics of the University for the invitation
which allowed me to present this subject before a learned audience, as
well as to Professor D. V. Widder, who has kindly suggested that I release my
manuscript for publication in the series of Heath Mathematical Monographs.
I am very grateful to Professor A. Zygmund and Professor J.-P. Kahane for
having read carefully the manuscript, and for having made very useful suggestions.
R. Salem
Paris, 1 November 1961"

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Özel Geometri Soru ve Çözümleri

Olimpiyatlarda çıkmış 8 adet geometri sorusunun değişik çözümlerini sunu şeklinde buradan indirebilirsiniz.Hazırlayanların ellerine sağlık:)

Algebraic Topology - Allen Hatcher

This book was written to be a readable introduction to Algebraic Topology with
rather broad coverage of the subject. Our viewpoint is quite classical in spirit, and
stays largely within the confines of pure Algebraic Topology. In a sense, the book
could have been written thirty years ago since virtually all its content is at least that
old. However, the passage of the intervening years has helped clarify what the most
important results and techniques are. For example, CW complexes have proved over
time to be the most natural class of spaces for Algebraic Topology, so they are emphasized
here much more than in the books of an earlier generation. This emphasis
also illustrates the book’s general slant towards geometric, rather than algebraic, aspects
of the subject. The geometry of Algebraic Topology is so pretty, it would seem
a pity to slight it and to miss all the intuition that it provides. At deeper levels, algebra
becomes increasingly important, so for the sake of balance it seems only fair to
emphasize geometry at the beginning.
Let us say something about the organization of the book. At the elementary level,
Algebraic Topology divides naturally into two channels, with the broad topic of Homotopy
on the one side and Homology on the other. We have divided this material
into four chapters, roughly according to increasing sophistication, with Homotopy
split between Chapters 1 and 4, and Homology and its mirror variant Cohomology
in Chapters 2 and 3. These four chapters do not have to be read in this order, however.
One could begin with Homology and perhaps continue on with Cohomology
before turning to Homotopy. In the other direction, one could postpone Homology
and Cohomology until after parts of Chapter 4. However, we have not pushed this
latter approach to its natural limit, in which Homology and Cohomology arise just as
branches of Homotopy Theory. Appealing as this approach is from a strictly logical
point of view, it places more demands on the reader, and since readability is one of
our first priorities, we have delayed introducing this unifying viewpoint until later in
the book.
There is also a preliminary Chapter 0 introducing some of the basic geometric
concepts and constructions that play a central role in both the homological and homotopical
sides of the subject.
Each of the four main chapters concludes with a selection of Additional Topics
that the reader can sample at will, independent of the basic core of the book contained
in the earlier parts of the chapters. Many of these extra topics are in fact rather
important in the overall scheme of Algebraic Topology, though they might not fit into
the time constraints of a first course. Altogether, these Additional Topics amount
to nearly half the book, and we have included them both to make the book more
comprehensive and to give the reader who takes the time to delve into them a more
substantial sample of the true richness and beauty of the subject.
Not included in this book is the important but somewhat more sophisticated
topic of spectral sequences. It was very tempting to include something about this
marvelous tool here, but spectral sequences are such a big topic that it seemed best
to start with them afresh in a new volume. This is tentatively titled ‘Spectral Sequences
in Algebraic Topology’ and referred to herein as [SSAT]. There is also a third book in
progress, on vector bundles, characteristic classes, and K–theory, which will be largely
independent of [SSAT] and also of much of the present book. This is referred to as
[VBKT], its provisional title being ‘Vector Bundles and K–Theory.’
In terms of prerequisites, the present book assumes the reader has some familiarity
with the content of the standard undergraduate courses in algebra and point-set
topology. One topic that is not always a part of a first point-set topology course but
which is quite important for Algebraic Topology is quotient spaces, or identification
spaces as they are sometimes called. Good sources for this are the textbooks by Armstrong
and J¨anich listed in the Bibliography.
A book such as this one, whose aim is to present classical material from a fairly
classical viewpoint, is not the place to indulge in wild innovation. Nevertheless there is
one new feature of the exposition that may be worth commenting upon, even though
in the book as a whole it plays a relatively minor role. This is a modest extension
of the classical notion of simplicial complexes, which we call Ñ complexes. These
have made brief appearances in the literature previously, without a standard name
emerging. The idea is to weaken the condition that each simplex be embedded, to
require only that the interiors of simplices are embedded. (In addition, an ordering
of the vertices of each simplex is also part of the structure of a Ñ complex.) For
example, if one takes the standard picture of the torus as a square with opposite
edges identified and divides the square into two triangles by cutting along a diagonal,
then the result is a Ñ complex structure on the torus having 2 triangles, 3 edges, and
1 vertex. By contrast, it is known that a simplicial complex structure on the torus
must have at least 14 triangles, 21 edges, and 7 vertices. So Ñ complexes provide
a significant improvement in efficiency, which is nice from a pedagogical viewpoint
since it cuts down on tedious calculations in examples. A more fundamental reason
for considering Ñ complexes is that they just seem to be very natural objects from
the viewpoint of Algebraic Topology. They are the natural domain of definition for
simplicial homology, and a number of standard constructions produce Ñ complexes
rather than simplicial complexes, for instance the singular complex of a space, or the
classifying space of a discrete group or category.

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Analysis and number theory - Sarnak

The goal is to prove Theorem 5 of Peter Sarnak’s Baltimore lecture notes.
These are preliminary notes, typed in real time during class; there will be errors
below (which should be attributed solely to the typist).

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LYS Hazırlık - Trigonometri testi

Dersanelerde öğretmenlik yaptığım vakitler, trigonometri konusunu anlattıktan sonra öğrencilerime trigonometri ile ilgili tüm soru tiplerini içeren bu testi dağıtırdım ...

Çeşitli kaynaklardan ve yaprak testlerden tarayıp derlediğim bu test içinde sınavda karşılaşılabilecek tüm soru tipleri mevcut...300 den fazla soru var, tavsiyem indirin, çıktısını alın , çözün ve sorularınız olursa matematiksorusu@hotmail.com adresine yollayın...İyi çalışmalar...

Buraya tıklayıp indirebilirsiniz...

Yalnız Pergel Kullanarak Yapılabilen Çizimler-7

Düzlemdeki A ve B noktalarından geçen doğrunun , O merkezli ve r yarıçaplı bir (O,r) çemberini kestiği noktaları bulalım...

Öncelikle problemi 2 şekilde düşünmeliyiz...

1)O merkez noktası AB doğrusu üzerinde olmasın:

O merkez noktasının AB doğrusuna göre simetriği olan O1 noktasını çizimle buluruz...Verilen çember ile aranan X ve Y noktalarında kesişen (O1,r) çemberini çizeriz...Bitti:)

2)O merkez noktası AB doğrusu üzerinde olsun:

A merkezli ve isteksel bir d yarıçaplı olan ve verilen çemberi C ve D noktalarında kesen bir çember çizeriz...(O,r) çemberinin CD yayını iki eşit parçaya böleriz...


Burada AX=AO+OX ve AY=AO-OX olduğunu görmekte de fayda var.;)

Yalnız Pergel Kullanarak Yapılabilen Çizimler-6

Düzlemdeki A ve B noktalarından yalnız bir doğrunun geçtiğini Öklid sayesinde biliyoruz:)
İyi, güzel de bu doğru ile ilgili ne yapalım...
Pergel yardımıyla, bu doğru üzerindeki diğer noktaları tespit edelim...Evet, sorumuz  açık:''Yalnız pergel kullanarak AB doğrusunun üzerindeki diğer noktaları bulalım...''

Düzlemde ve AB doğrusu dışında isteksel bir C noktası ele alalım.C noktasının AB doğrusuna göre simetriği olan C1 noktasını 2. problemde verilen yöntemle bulalım.Sonra yarıçapı yine keyfi bir r olan (C,r) ve (C1,r) çemberlerini çizelim...
Bu çemberlerin kesim noktalarında, AB doğrusu üzerinde bulunan ve aradığımız X ve X1 noktalarını elde ederiz...

r yarıçap uzunluğunu değiştirerek verilen doğruya ait istediğimiz kadar değişik noktayı elde etmemiz mümkün tabii ki...:)