Browsing by Author "Simos, T. E."
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Item Applications of hecke operator to generalized dedekind eta functions(Amer Inst Pyhsics, 2009) Açıkgöz, Mehmet; Kim, Daeyeoul; Şimşek, Yılmaz; Simos, T. E.; Psihoyios, G.; Tsitouras, C.; Cangül, İsmail Naci; Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Anabilim Dalı.; 0000-0002-0700-5774; J-3505-2017; 57189022403The aim of this paper is to give relations between generalized Dedekind eta functions, theta functions, Dedekind sums, Hardy-Berndt sums and Hecke operators.Item Calculation of the minimal polynomial of 2cos(π/n) over Q with Maple(American Inst Physics, 2012) Simos, T. E.; Psihoyios, G.; Tsitouras, C.; Anastassi, Z.; Yurttaş, Aysun; Özgür, Birsen; Cangül, İsmail Naci; Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü.; 0000-0002-0700-5774; 0000-0002-0700-5774; ABA-6206-2020; ABI-4127-2020; J-3505-2017; AAG-8470-2021; 37090056000; 54403501400; 57189022403The number lambda(q) = 2cos pi/q, q is an element of N, q >= 3, appears in the study of Hecke groups which are Fuchsian groups and in the study of regular polyhedra. There are many results about the minimal polynomial of this algebraic number. Here we obtain the minimal polynomial of these numbers over the field of rationals by means of the better known Chebycheff polynomials and the Maple language.Item Classification of normal subgroups of Hecke group H6 in terms of parabolic class number(AIP, 2011) Simos, T. E.; Yurttaş, Aysun; Demirci, Musa; Cangül, İsmail Naci; Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Anabilim Dalı.; 0000-0002-0700-5774; 0000-0002-0700-5774; AAG-8470-2021; ABA-6206-2020; J-3505-2017; 37090056000; 23566581100; 57189022403In [3], Greenberg showed that n <= 6t(3) so that mu - nt <= 6t(4) for a normal subgroup N of level n and index mu having t parabolic classes in the modular group Gamma. Accola, [1], improved these to n <= 6t(2) always and n <= t(2) if Gamma/N is not abelian. Newman, [5], obtained another generalisation of these results. Hecke groups are generalisations of the modular group. We particularly deal with one of the most important cases, q = 6.Item Deterrmining the minimal polynomial of cos(2π/n) over Q with Maple(Amer Inst Physics, 2012) Simos, T. E.; Psihoyios, G.; Tsitouras, C.; Anastassi, Z.; Özgür, Birsen; Yurttaş, Aysun; Cangül, İsmail Naci; Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü.; 0000-0002-0700-5774; 0000-0002-0700-5774; AAG-8470-2021; J-3505-2017; ABA-6206-2020; ABI-4127-2020; 54403501400; 37090056000; 57189022403The number lambda(q) = 2cos pi/q, q is an element of N, q >= 3, appears in the study of Hecke groups which are Fuchsian groups, and in the study of regular polyhedra. There are many results about the minimal polynomial of this algebraic number and in some of these methods, the minimal polynomials of several algebraic numbers are used. Here we obtain the minimal polynomial of one of those numbers, cos(2 pi/n), over the field of rationals by means of the better known Chebycheff polynomials for odd q and give some of their properties. We calculated this minimal polynomial for n is an element of N by using the Maple language and classifying the numbers n is an element of N into different classes.Item Hurwitz type multiple genocchi zeta function(Amer Inst Pyhsics, 2009) Şimşek, Yılmaz; Simos, T. E.; Maroulis, G.; Özden, Hacer; Cangül, İsmail Naci; Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Anabilim Dalı.; 0000-0002-0700-5774; 0000-0002-0700-5774; AAH-5090-2021; J-3505-2017; 23973633900; 57189022403Main purpose of this paper is to construct higher-order w-q-Genocchi numbers and polynomials by using p-adic q-deformed fermionic integral on Z(p). We derive some interesting identities related to higher-order w-q-Genocchi numbers and polynomials. We also construct Hurwitz type multiple w-Genocchi zeta function which interpolates these polynomials at negative integers.Item Integer solutions of a special Diophantine equation(Amer Inst Pyhsics, 2011) Simos, T. E.; Özkoç, Arzu; Tekcan, Ahmet; Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Anabilim Dalı.; AAH-8518-2021; 24485340700; 55883777900Let t not equal 1 be an integer. In this work, we determine the integer solutions of Diophantine equation D : x(2) + (2-t(2))y(2)+(-2t(2) - 2t + 2)x+(2t(5) - 6t(3) + 4t)y - t(8) + 4t(6) - 4t(4) + 2t(3) + t(2) - 2t - 0 over Z and also over finite fields F-p for primes p >= 2. Also we derive some recurrence relations on the integer solutions (x(n), y(n)) of D and formulate the the n-th solution (x(n), y(n)) by using the simple continued fraction expansion of x(n)/y(n).Item The minimal polynomials of 2cos(π/2k) over the rationals(Amer Inst Pyhsics, 2011) İkikardeş, Nazlı Y.; Simos, T. E.; Demirci, Musa; Özgür, Birsen; Cangül, İsmail Naci; Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Anabilim Dalı.; 0000-0002-0700-5774; 0000-0002-0700-5774; ABA-6206-2020; ABI-4127-2020; J-3505-2017; 23566581100; 54403501400; 57189022403The number lambda(q) = 2cos pi/q, q is an element of N, q >= 3, appears in the study of Hecke groups which are Fuchsian groups of the first kind, and in the study of regular polyhedra. Here we obtained the minimal polynomial of this number by means of the better known Chebycheff polynomials and the set of roots on the extension Q(lambda(q)). We follow some kind of inductive method on the number q. The minimal polynomial is obtained for even q.Item On some geometric structures and local rings(Amer Inst Physics, 2011) Simos, T. E.; Erdoǧan, Fatma Özen; Çelik, Basri; Çiftçi, Süleyman; Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü.; 0000-0001-7234-8063; AAE-2600-2019; AAG-8274-2021; 54402700700; 23026643900; 26635052900In this paper, we investigate some combinatoric properties of the Projective Klingenberg planes coordinatized with a finite local ring R when the cardinality of set I of the non-unit elements of R is k. As a result we arrive at the result that the order of the projective plane underlying projective Klingenberg plane must be n k, which is the index of I in R when |R| = n. Although some of the results given here can be found in the literature [1], [3] and [4] we approach to them in a direct way and give alternative proofs.Item A p-adic Look at the Diophantine equation x2 + 112k = yn(Amer Inst Physics, 2009) Şimşek, Yılmaz; Soydan, Gökhan; Simos, T. E.; Psihoyios, G.; Tsitouras, C.; Cangül, İsmail Naci; Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü.; 0000-0002-0700-5774; ABA-6206-2020; J-3505-2017; 57189022403We find all solutions of Diophantine equation x(2) + 11(2k) = y(n), x >= 1, y >= 1, k is an element of N, n >= 3. We give p-adic interpretation of this equation.Item Some formulae for the Zagreb indices of graphs(Amer Inst Physics, 2012) Çevik, Ahmet Sinan; Simos, T. E.; Psihoyios, G.; Tsitouras, C.; Anastassi, Z.; Cangül, İsmail Naci; Yurttaş, Aysun; Togan, Müge; Uludaǧ Üniversitesi/Fen Edebiyat Fakültesi/Matematik Bölümü.; 0000-0002-0700-5774; 0000-0002-0700-5774; AAG-8470-2021; J-3505-2017; ABA-6206-2020; 57189022403; 37090056000; 54403978300In this study, we first find formulae for the first and second Zagreb indices and coindices of certain classical graph types including path, cycle, star and complete graphs. Secondly we give similar formulae for the first and second Zagreb coindices.Item Some properties of the minimal polynomials of 2cos(pi/q) for odd q(Amer Inst Pyhsics, 2011) Simos, T. E.; Özgür, Birsen; Demirci, Musa; Yurttaş, Aysun; Cangül, İsmail Naci; Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Anabilim Dalı.; 0000-0002-0700-5774; 0000-0002-0700-5774; ABA-6206-2020; ABI-4127-2020; J-3505-2017; AAG-8470-2021; 54403501400; 23566581100; 37090056000; 57189022403The number lambda(q) = 2cos pi/q, q is an element of N, q >= 3, appears in the study of Hecke groups which are Fuchsian groups, and in the study of regular polyhedra. There are many results about the minimal polynomial of this algebraic number. Here we obtain the minimal polynomial of this number by means of the better known Chebycheff polynomials for odd q and give some of their properties.Item Some special cases of the minimal polynomial of 2cos(pi/q) over q(Amer Inst Pyhsics, 2011) Simos, T. E.; Togan, Müge; Özgür, Birsen; Cangül, İsmail Naci; Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Anabilim Dalı.; 0000-0002-0700-5774; 0000-0002-0700-5774; ABA-6206-2020; J-3505-2017; ABI-4127-2020; 54403978300; 54403501400; 57189022403The number lambda(q) - 2cos pi/q, q is an element of N, q >= 3, appears in the study of Hecke groups which are Fuchsian groups of the first kind, and in the study of regular polyhedra. Here we obtained some results on the values of the minimal polynomial of this number in modulo prime p. This results help in the calculation of the congruence subgroups of the Hecke groups which is an important problem in discrete group theory.Item Upper bounds for the level of normal subgroups of Hecke groups(Amer Inst Pyhsics, 2011) Simos, T. E.; Demirci, Musa; Yurttaş, Aysun; Cangül, İsmail Naci; Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Anabilim Dalı.; 0000-0002-0700-5774; 0000-0002-0700-5774; ABA-6206-2020; J-3505-2017; AAG-8470-2021; 23566581100; 37090056000; 57189022403In [4], Greenberg showed that n <= 6t(3) so that mu - nt <= 6t(4) for a normal subgroup N of level n and index mu having t parabolic classes in the modular group Gamma. Accola, [1], improved these to n <= 6t(2) always and n <= t(2) if Gamma/N is not abelian. In this work we generalise these results to Hecke groups. We get results between three parameters of a normal subgroup, i.e. the index mu, the level n and the parabolic class number t. We deal with the case q = 4, and then obtain the generalisation to other q. Two main problems here are the calculation of the number of normal subgroups and the determination of the bounds on the level n for a given t.