x / log x provided that c (1,12 /11)


 Logan Mills
 1 years ago
 Views:
Transcription
1 Piatetski Shapiro Meets Chebotarev Yıldırım AKBAL Bilkent University This is a joint work with Ahmet Muhtar Güloğlu. In 1953 Ilya PiatetskiShapiro proved in [?] an analogo of the prime number theorem for primes of the form n c, where x max n IN : n x, n runs through positive integers and c 0 is fixed. He showed that the number ( x) of these primes not exceeding a given number x is asymptotic to 1/ c x / log x provided that c (1,12 /11) c. Since then, the admissible range of c has been extended by many authors and the result is currently knownn for c (1,2817 / 2426) (cf. [?] ). In this talk, we give an asymptotic formula for Shapiro primes lying in a specified Chebotarev class. We also apply our theorem to show that there are infinitely Shapiro primes of the form x ny. 2 2 [1] Y. Akbal, A. M. Güloğlu. Piatetski Shapiro meets Chebotarev (Submitted). [2] I.I. PiatetskiShapiro. On the distribution of prime numbers in sequences of the form [ f( n )], (Russian) Mat. Sbornik N.S. 33 (75), (1953) c [3] J. Rivat, P. Sargos. Nombres premiers de la forme [ n ], Canad. J. Math, 53 (2001), no. 2,
2 Some Results on Universal Modules of Differential Operators Melis Tekin AKÇĠN Department Of Mathematics, Hacettepe University Let R be a commutative kalgebra where k is a field of characteristic zero. We have the following exact sequence where is defined as for and is the kernel of. Note that is generated by the set { }. Let and Here is called the universal derivation of order. The left  module is called the universal module of order derivations and is denoted by Let be a hypersurface represented by Then it is known that Besides it, let be a coordinate ring of the product of a reduced hypersurface and an afine tspace, then In this talk, we will give some recent results on Universal Modules. 15
3 Flicker s correction Arguement For The Trace Formula Of GL(2) On A Number Field Ali AYDOĞDU 1, Rukiye ÖZTÜRK 1 and Engin ÖZKAN 2 Department of Mathematics, Science Faculty, Atatürk University, Erzurum, Turkey Department of Mathemaics, Science and Art Faculty, Erzincan University, Erzincan, Turkey. The correction arguement for the weighted orbital integrals that appears in the trace formula was introduced by Flicker in [4]. We give the correction arguement for the trace formula of GL(2) over a number field. The correction arguement is to introduce a global summand which does not change the global formula. However it changes the local weighted orbital integrals at the hyberbolic terms Mathematics Subject Classification. 11F72, 22E35 Key words and phrases. Trace Formula, number field, orbital integrals, weighted orbital integrals. [1] Arthur, J. "The local behaviour of weighted orbital integrals." Duke Math. J 56.2 (1988): [2] Arthur, J. "An introduction to the trace formula." In : Arthur J, Ellwood D, Kottwitz R, editor. Harmonic analysis, the trace formula, and Shimura varieties Clay Math Proc 4, Amer Math. Soc. Providence RI 2005; [3] Borel A, and Jacquet H. "Automorphic forms and automorphic representations." Proc. Sympos. Pure Math. Vol. 33. No. part I. Amer Math. Soc. Providence RI 1979; [4] Flicker, YZ. The trace formula and base change for GL (3). Lecture Notes in Math. Vol. 114 Berlin and New York: SpringerVerlag, [5] Flicker YZ. Eisenstein series and the trace Formula for GL(2) over a function field. Documenta Mathematica 2014: 19, [6] Flicker YZ., and Kazhdan D.A. "A simple trace formula." Journal d Analyse Mathématique 50.1 (1988):
4 [7] Gelbart S, and Jacquet H. "Forms of GL (2) from the analytic point of view." Proc. Sympos. Pure Math. Vol. 33. No. part I. Amer. Math. Soc. Providence RI 1979; [8] Godement R. The spectral decomposition of cusp forms. Proc. Sympos. Pure Math., 9, Amer.Math. Soc., Providence, R.L., 1966, [9] Jacquet, H., and Langlands P.R. "Automorphic forms on GL (2)." Lect. Notes Math. Vvol. 114, Berlin and New York: SpringerVerlag, (1970). [10] Platonov V, Rapinchuk A. "Algebraic groups and number theory." Academic Press, Inc., San Diego,
5 Soft Matrix Product and Soft Cyrptosystem Emin AYGÜN, Akın Osman ATAGÜN, BüĢra KILIÇ Department of Mathematics, Erciyes University, Kayseri, Turkey Department of Mathematics, Bozok University, Yozgat, Turkey Soft set theory, proposed by Molodstov, has been regarded as an effective mathematical tool to dela with uncertainties. In this work, we define two new operations on the set of soft matrices, called inverse production and charactersitic production and give their properties. We introduce soft cryptosystem as a new cryptosystem method by using inverse production and characteristic production of soft matrices. We finally define soft encryption and soft decryption. Some applications are given G25, 15A23, 94A60, 11T71 Soft sets, Soft matrix, soft matrix products, cyrptosystem, soft cyrptosystem [1] H. Aktaş,, and N. Çağman, Soft sets and soft groups, Information Sciences (2007): [2] H. Aktaş,, and N. Çağman, Erraturn to "Soft sets and soft groups", Information Sciences 3 (2009), 338. [Inform. Sci. 177 (2007), ] [3] M. I.Ali, F.Feng, X. Liu, W. K Min and M. Shabir, On some new operations in soft set theory, Computers & Mathematics with Applications, 57 (9) (2009), [4] A.O. Atagün, and A. Sezgin. Soft substructures of rings, fields and modules, Computers and Mathematics with Applications 61 (3 ) (2011): [5] K. Atanassov, Operators over interval valued intuitionistic fuzzy set, Fuzzy sets and systems 64 (1994), [6] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986),
6 [7] D. Chen, E. C. C. Tsang, D. S Yeung, and X. Wang, The parameterization reduction of soft sets and its applications. Computers and Mathematics with Applications, 44(2002), [8] D. Chen, E. C. C. Tsang, D. S Yeung, and X. Wang, Some notes on the parameterization reduction of soft sets, in: International Conference on Machine Leraning and Cybernetics. Vol. 3, 2003, pp [9] N. Çağman, and S. Enginoğlu. Soft matrix theory and its decision making. Computers & Mathematics with Applications 59 (2010): [10] F. Feng, Y. B. Jun, and X. Zhao. Soft semirings, Computers & Mathematics with Applications 56 (2008), [11] Y. B. Jun, Soft BCK/BCIalgebras. Computers & Mathematics with Applications 56 (12) (2008), [12] Z. Kong, L. Gao, L. Wang and S. Li, The parameter reduction of soft sets and algorithm, Computers & Mathematics with Applications 56 (12) (2008), [13] D. V. Kovkov, V. M. Kolbanov, D. A. Molodtsov. Soft sets theorybased optimization. Journal of Computer and Systems Sciences International 46 (6) (2007) [14] P. K. Maji, R. Biswas, A. R. Roy. Soft set theory, Computers & Mathematics with Applications 45 (2003): [15] P. K. Maji, R. Biswas, A. R. Roy. Fuzzy soft sets, Journal of Fuzzy Mathematics 9(3) (2001) [16] P. K. Maji, R. Biswas, A. R. Roy.An application of soft sets in a decision making problem, Computers and Mathematics with Applications 44.8 (2002), [17] P. Majumdar, S. K. Samanta. Similarity measure of soft sets, New Mathematics and Natural Computation 4.01 (2008): [18] D. Molodstov, Soft set theoryfirst results, Computers and Mathematics with Applications 37 (1999), [19] D. Molodstov, The description of a dependence with the help of soft sets, Journal of Computer Science and Systems Sciences International 40 (6) (2001), [20] ] D. Molodstov, The theory of Soft Sets, URSS publishets, Moscow, 2004, (in Russian) [21] M. M. Mushrif,S. Sengupta, A. K. Ray. Texture classification using a novel, softset theory based classification algorithm, Lecture Notes in Computer Sciences 3851 (2006)
7 [22] Z. Pawlok and A. Skowron, Rudiments of seft sets, Information Sciences 177, (2007), [23] D. Pei, and D Miao. From soft sets to information systems.in: X. Hu. Q. Liu, A. Skowron, T.Y. Lin, R.R. Yager, B. Zhang (Eds.), Proceedings of Granular Computing Computing, vol. 2 IEEE, 2005, pp [24] R. Rivest, L. Adleman, and M. Dertouzos, On data banks and privacy homomorphisms, In Foundations of Secure Computation, 1978, [25] A. R. Roy, and P. K. Maji. A fuzzy soft set theoretic approach to decision making problems, Journal of Computational and Applied Mathematics 203 (2007), [26] A. Sezgin and A. O. Atagün. On operations of soft sets, Computers and Mathematics with Applications 61 (2011), [27] A. Sezgin, A. O. Atagün, Emin Aygün, A note on soft nearrings and idealistic soft nearrings, Filomat (2011) Vol. 25.(1), [28] D. Stinson, Cryptography: theory and practice, CRC Press New Jersey 1995, 573pp. [29] Q.M. Sun, ZL. Zhang, and J. Liu. Soft sets and soft modules, in Gouyin Wang, Tianrui Li, Jarzy W. GryzymalaBusse, Duoqian Miao, Andrzej Skowron, Yiyu Yao (Eds.) Rough Sets and Knowledge Technology. RST2008, Proceedings, Springer, 2008, pp [30] Z. Xiao, L. Chen, B. Zhong, S. Ye, Recognition for soft information based on the theory of soft sets in: J. Chen (Ed.), Proceedings of ICSSSM05.vol.2, IEEE, 2005, PP [31] C. F. Yang, A note on soft set theory, Computers and Mathematics with Applications 56 (2008), [Comput. &Math. Appl. 45 (45) (203), ]. [32] X. Yang, D. Yu, J. Yang, C. Wu, Generalization of soft set theory: from crisp to fuzzy case, in: BingYuan Cao (Ed.), Fuzzy Information and Engineering: Proceedings of ICFIE 2007, in: Advances in Soft Computing, vol. 40. Springer, pp [33] L. A. Zadeh,. Fuzzy sets. Information and Control 8 (1965), [34] L. Zhou, and W. Z. Wu. On generalized intuitionistic fuzzy rough approximation operators, Information Sciences 178.(11) (2008),
8 On the dot Product graph of a commutative ring Ayman BADAWI Department of Mathematical & Statistics, American University of Sharjah, P.O. Box 26666, Sharjah, United Arab Emirates Let A be a commutative ring with nonzero identity, be an integer, and (n times). The total dot product graph of R is the (undirected) graph with vertices { }, and two distinct vertices and are adjacent if and only if (where denote the normal dot product of and Let denote the set of all zerodivisors of R. Then the zero divisor dot product graph of R is the induced subgraph of with vertices { }. It follows that each edge (path) of the classical zero divisor graph is an edge (path) of We observe that if then is a disconnected graph and is identical to the wellknown zero divisor graph of R in the sense of BeckAndersonLivingston, and hence it is connected. In this paper, we study both graphs and. For a commutative ring A and we show that is connected with diameter two (at most three) and with girth three. Among other things, for we show that is identical to the zerodivisor graph of R if and only if either and A is an integral domain or R is ringisomorphic to [1] Badawi, A., On the dot product graph of a commutative ring, to appear in Comm. Algebra (2015). [2] Badawi, A., On the annihilator graph of a commutative ring, Comm. Algebra, Vol.(42)(1), (2014), DOI: / [3] Anderson, D. F., Badawi, A., On the zerodivisor graph of a ring, Comm. Algebra (36)(2008), [4] Beck, I., Coloring of commutative rings, J. Algebra 116(1988), [5] Anderson, D. F., Livingston, P. S., The zerodivisor graph of a commutative ring, J. Algebra 217(1999), [6] Anderson, D. F., Mulay, S. B., On the diameter and girth of a zerodivisor graph, J. Pure Appl. Algebra 210(2007), [7] Lucas, T. G., The diameter of a zerodivisor graph, J. Algebra 301(2006),
9 The Lattice of Generalized LSubgroups Dilek BAYRAK (1), Sultan YAMAK (2) Karadeniz Technical University, Trabzon, TURKEY, Karadeniz Technical University, Trabzon, TURKEY, Many studies have invastigated the lattice structure of fuzzy substructures of algebraic sets such as group, ring and module. Some important results about modularity and distributivity have been obtained in these studies. We discuss some properties of the lattices of normal Lsubgroups and obtain that the lattice of all normal Lsubgroups of a group is modular. As consequence, we obtain that the lattices of all normal fuzzy subgroups and all normal Lsubgroups of a group are modular. We characterize abelian groups by the latice of Lsubgroups of these groups. We show that a finity group G is cyclic if and only if the lattice of Lsubgroups is distributive. As consequence, we obtain that the lattices of all fuzzy subgroups and all fuzzy subgroups of a finite cyclic group are distributive. The lattice of fuzzy subgroups is obtained pseudocomplemented. We also show that the lattice of subgroups of G is a chain if and only if the lattice of fuzzy subgroups of G is pseudocomplemented. Key Words: modular lattice; distributive lattice; Lsubgroups; Lsubgroups fuzzy subgroups; [1] Ajmal, Naseem. "The lattice of fuzzy normal subgroups is modular." Information sciences 83.3 (1995): [2] Ajmal, Naseem, and K. V. Thomas. "The lattices of fuzzy subgroups and fuzzy normal subgroups." Information sciences 76.1 (1994): [3] Bayrak, Dilek, and Sultan Yamak. "The lattice of generalized normal Lsubgroups." Journal of Intelligent and Fuzzy Systems. (DOI: /IFS ). [4] Bhakat, S. K., and P. Das. "On the definition of a fuzzy subgroup." Fuzzy sets and systems 51.2 (1992): [5] Head, Tom. "A metatheorem for deriving fuzzy theorems from crisp versions." Fuzzy Sets and Systems 73.3 (1995): [6] Jahan, Iffat. "The lattice of Lideals of a ring is modular." Fuzzy Sets and Systems 199 (2012):
10 [7] Jahan, Iffat. "Modularity of Ajmal for the lattices of fuzzy ideals of a ring." Iranian Journal of fuzzy systems 5.2 (2008): [8] Jun, YoungBae, MinSu Kang, and ChulHwan Park. "Fuzzy subgroups based on fuzzy points." Communications of the Korean Mathematical Society 26.3 (2011): [9] Ore, Oystein. "Structures and group theory. II." Duke Mathematical Journal 4.2 (1938): [10] Tarnauceanu, Marius. "Distributivity in lattices of fuzzy subgroups." Information Sciences (2009): [11] Yao, Bingxue. fuzzy normal subgroups and fuzzy quotient subgroups." Journal of Fuzzy Mathematics 13.3 (2005): [12] Yuan, Xuehai, Cheng Zhang, and Yonghong Ren. "Generalized fuzzy groups and manyvalued implications." Fuzzy sets and Systems (2003):
11 RAD SUPPLEMENTED LATTICES Çiğdem BĠÇER 1 and Celil NEBĠYEV 2 1,2 Ondokuz Mayis University, Department of Mathematics KurupelitAtakumSamsun/Turkey In this work, Radsupplemented lattices are defined and investigate some properties of these lattices. Let be a complete modular lattice and 1=. If is Rad supplemented for every then is Radsupplemented. Key words: Small elemets in lattices, Supplemented Lattices, Radical Supplemented Lattices, Radsupplemented lattices. [1]Calugareanu, G.,2000, Lattice Concepts of Module Theory, Kluwer Academic Publisher, Dordrecht, Boston, London. [2] Alizade, R., Toksoy E., Cofinitely Supplemented Moduler Lattices Arabian Journal for Science and Engineering, Volume 36, Issue 6, Page , [3] Çalışıcı, H., Türkmen, E., Generalized Supplemented Modules, Algebra and Discrete Mathematics Volume 10, Number 2. pp ,
12 Coprimely Structured Modules Zehra BĠLGĠN, KürĢat Hakan ORAL and Ünsal TEKĠR Department of Mathematics, Yildiz Technical University, Esenler, İstanbul, Turkey Department of Mathematics, Marmara University, Kadıköy, İstanbul, Turkey Let R be acommutative ring with identity. In [5], the authors define strongly prime submodules of multiplication Rmodules and call a multiplication Rmodule M strongly 0 dimensional provided each prime submodule of M is strongly prime. Then theu give some properties of such modules. A prime submodule P of a multiplication Rmodule M is called coprimely structured if, whenever P is coprime to each element of an arbitrary family of submodules of M, the intersection of the family is not contained in P. A multiplication R module M is called coprimely structured provided each prime submodule of M is coprimely structured. We show that every strongly 0 dimensional module is coprimely structured and we conclude that every artinian multiplication module is coprimely structured. Then we give some characterizations of coprimely structured modules. [1] Abd E Bast, Z., Smith, P. P., (1988). Multiplication Modules. Communications in Algebra, 16(4): [2] Ali, M. M., (2008). Idempotent and Nilpotent Submodules of Multiplication Modules. Communications in Algebra, 6(412): [3] Ameri, R., (2003). On the Prime Submodules Of Multiplication Modules. International Journal of Mathematics and Mathematical Sciences, 2003(27): [4]Brewer, J., Richman, F.. (2006). Subrings of Zerodimensional Rings. Multiplicative Ideal Theory in Commutative Algebra, Springer US, [5] Oral, K. H.. Özkirişçi, N. A.. Tekir, Ü., (2014). Strongly 0dimensional Modules. Canadian Mathematical Bulletin, 57(1):
13 On Reciprocity Formula of Character Dedekind Sums Mümün CAN and M. Cihat DAĞLI Departments of Mathematics, Akdeniz University, Antalya, Turkey Applying character analogue of the EulerMaclaurian summation formulato generalized Bernoulli function we obtain a relation involving an integral and character Dedekind sum. With the help of this relation we give an alternative prof for the reciprocity formula of character Dedekind sum. [1] B.C. Berndt, Character transformation formulae similar to those fort he Dedekind Etafunction, in Analytic Number Theory,Proc. Sym. Pure Math. XXIV, Amer. Math. Soc., Prodivence. R. I., (1973) [2] B.C. Berndt, Character analogues of Poisson and EulerMaclaurin summation formulas with applications, J. Number Theory 7 (1975) [3] M. Cenkci, M. Can and V. Kurt, Degenerate and character Dedekind sums, J. Number Theory 124 (2007) [4] M. Can and V. Kurt, Character analogues of certain HardyBerndt sums, Int. J. Number Theory, 10 (2014),
14 p adic Character Dedekind Sums Mehmet CENKCĠ Akdeniz University, Department of Mathematics, Antalya, 07058, Turkey Using padic measure theory we give an explicit representation of padic character Dedekind sums and their reciprocity laws. Keywords: Dedekind sums, character Dedekind sums, padic measure theory. MSC: 11F20 27
15 Permuting nfderivations on Lattices Sahin CERAN, Mustafa AġÇI, and Utku PEHLĠVAN Pamukkale University Science and Arts Faculty Depatrment of Mathematics KINIKLI Denizli /TURKEY In this paper as a generalization of permuting triderivation on a lattice we introduced the notion of permuting nfderivation of a lattice. We defined the isotone permuting nfderivation and got some interesting results about isotoneness. We characterized the distributive and isotone lattices by permuting nfderivation Mathematics Subject Classification. 06B35, 06B99, 16B70. Key words and phrases. Lattice, Derivation, Permuting nfderivation. 28
16 Some Power Series with Rational Coefficients for Lioville Number Arguments Fatma ÇALIġKAN Istanbul University, Faculty of Science, Department of Mathematics, Vezneciler/Istanbul, Turkey, In the present talk, we discuss that some power series with rational coefficients belong to either the rational number field or the transcendental number set for Liouville number arguments under certain conditions. [1] Bugeand, Y. Approxiamation by Algebraic Numbers, Cambridge University Press, [2] Koksma, J. F. Über die Mahlersce Klasseneinteilung der transzentean Zahlen und die Approximation komplexer Zahlen durch algebraische Zahlen, Monatsh Math Phys 48, , [3] Lang, S. Integral Points on Curves, Institut des Hautes Etudes Scientifiques Publication Mahematiques 6, 2743, [4] Leveque, W. J. Topics in Number Theory Vol. II (AddisonWesley Publishing Company, 1961). [5] Long, X. X. On Mahler s Classification of padic Numbers, Pure Apply Math 5, 7380, [6] Mahler, K. Zur Approximation der Exponantialfunktion und des Logarithmus I, Journal fr reine und angewandte Mathematik 166), , [7] Mahler, K. Über eine KLassenEinteilung der padischen Zahlen Mathematica Leiden 3, , [8] Oryan, M.H. Über gewisse Potenzreihen deren Funktionwerte für Argümente aus der Menge der Liouvilleschen zahlen transzendent sind, İstanbul Üniv. Fen Fak. Mat Derg 49, 3738, [9] Wirsing, E. Approximation mit Algebraischen Zahlen beschrankten Grades, J reine angew Math 206, 6777,
17 Yusuf DANIġMAN Mevlana University, Konya, Turkey, Let be a nonarchimedean field. The local Langlands conjecture describes a correspondence between the irreducible admissible representations of a reductive group and the representations of the WeilDeligne group. In this conjecture, local factors of irreducible representations of are defined as the local factors of the corresponding representation of the WeilDeligne group[?],[?]. Local factors of can be also defined by attaching an integral representation, which would compute the local factor and provides more information about the correspondence. For the group ) of symplectic similitudes of rank, PiatetskiShapirodefined the local factors ( and ) by using the theory of integral representations depends on the Bessel model [?]. For the irreducible representations of induced from Borel subgroup we compute the factors by using the Jacquet module structure. Key words: Lfunction, Lfactor, [1] Deligne P., Les constantes des quations fonctionnelles des fonctions L, Lecture Notes in Math. Vol. 349, Springer Verlag, 1973, [2] Langlands R.P, On the functional Equation of Artin's L function, unpublished manuscript. [3] PiatetskiShapiro I. Lfunctions for GSp4. Pacific J Math Olga Taussky Todd Memorial Issue 1997;
18 On Generalızed Semiperfect Rings Yılmaz Mehmet DEMĠRCĠ Sinop University, Department of Mathematics, Osmaniye Köyü Nasuhbaşoğlu Mevkii, 57000, Sinop, Turkey, Throughout all rings are associative with identity element unless otherwise stated and all modules are unitary right modules. Let R be a ring and M be an R module, A flat cover of M is an epimorphism where F is a flat Rmodule and Amini et. al. called a ring R right generalized perfect (right Gperfect for short) if every right Rmodule has a flat cover. We introduce right generalized semi perfect rings (right Gsemiperfect for short) as the rings pver which every simple right Rmodule has a flat cover. Since every projective module is flat, every semiperfect ring is Gsemiperfect and therefore Gsemiperfect rings are generalizations of semiperfect rings. We give some examples along with the following results concerning Gsemiperfect rings. Theoem 0.1. Let R and S be Morita equivalent rings. If R is right Gsemiperfect, then so is S. Proposition 0.1. Let R and S be Gsemiperfect rings. Then, a) Every factor ring of R is Gsemiperfect, b) RxS is Gsemiperfect ring. Theoem 0.2. ([5], Theorem 3.8). Let R be a semilocal ring. Then R is Gsemiperfect if and only if R is semiperfect. Proposition 0.2. Let R be a commutative Gsemiperfect ringi Then is semiperfect for every finite number of maximal ideals,, and. Proposition 0.3. Let R be a right Gsemiperfect ring and J(R) be nil. Then R is right Noetherian if and only if R is right Artinian Mathematical Subject Classification. Primary: 16L30; Secondary: 16D40. Key words and phrases. Generalized perfect ring, generalized semiperfect ring. [1] A. Amini. B. Amini M. Ershad, and H. Sharif, On Generalized Semiperfect Rings, Comm. Algebra 35 (2007),
19 [2] F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer, New York, [3] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Westview Press [4]T, Y. Lam. A First Course in Noncommutative Rings, Graduate Texts in Matheatics, Springer, New York, [5] C. Lomp, On Semilocal Rings and Modules, Comm. Algebra 27(4) (1999),
20 On Skew Cyclic and QuasiCylic Codes Over Abdullah DERTLĠ a, Yasemin ÇENGELLENMĠġ b, ġenol EREN a (a) Ondokuz Mayıs University, Faculty of Arts and Sciences, Mathematics Department Samsun, Turkey (b) Trakya University, Faculty of Arts and Sciences, Mathematics Department Edirne, Turkey By defining the Lee distance and the Lee weight of linear codes over the ring,, we construct a Gray map which is both an isometry and a weight preserving map from to. It was shown that the Gray image of cyclic code over is quasicyclic codes of index and the Gray image of quasicyclic code over is quasicyclic code of index. Moreover, the skew cyclic and skew quasicyclic codes over introduced and the Gray images of them are determined. D. Boucher, W. Geiselmann, F. Ulmer, Skew cyclic codes, Appl. Algebra, Eng. Commun Comput., Vol, No D. Boucher, P. Sole, F. Ulmer, Skew constacyclic codes over Galois rings, Advance of Mathematics of Communication, Vol., Number. D. Boucher, F. Ulmer, Coding with skew polynomial rings, Journal of Symbolic Computation, I. Siap, T. Abualrub, N. Aydın, P. Seneviratne, Skew cyclic codes of arbitrary length, Int. Journal of Information and Coding Theory,. J. F. Qian, L. N. Zhang, S. X. Zhu, constacyclic and cyclic codes over, Applied Mathematics Letters,. J. Gao, L. Shen, F. W. Fu, Skew generalized quasicyclic codes over finite fields, arxiv: M. Bhaintwal, Skew quasicyclic codes over Galois rings, Des. Codes Cryptogr., DOI. M. Wu, Skew cyclic and quasicyclic codes of arbitrary length over Galois rings, International Journal of Algebra, vol,, no,. 33
21 S. Jitman, S. Ling, P. Udomkovanich, Skew constacyclic codes over finite chain rings, AIMS Journal. T. Abualrub, A. Ghrayeb, N. Aydın, I. Siap, On the construction of skew quasicyclic codes, IEEE Transsactions on Information Theory, Vol, No,,. T. Abualrub, P. Seneviratne, Skew codes over ring, Proceeding of the interntional Multi Conference of Eng. And Comp. Sci., IMECS, March,, Hong Kong. Y. Cengellenmis, On cyclic codes over, Int.J. Contemp. Math. Sciences, vol,, no,. 34
22 TLRing Homomorphisms Ümit DENĠZ 1 1 Recep Tayyip Erdoğan University Department of Matematics, Rize, Turkey We study TLring homomorphism. TLring homomorphism is a TLsubset. In this paper we use fuzzy function definition and with this definition and with this definition we introduce TLring homomorphism and prove some theorems of ring homomorphisms in classical algebra. 1. Klement E. P., Mesiar R. And Pap E., Triangular Norms. Position Paper I: Basic Analytical and Algebraic Properties, Fuzzy Sets and Systems 143 (2004) Klement E. P., Mesiar R. And Pap E., Triangular Norms. Position Paper II: General Constructions and Parameterized Families, Fuzzy Sets and Systems 145 (2004) Klement E. P., Mesiar R. And Pap E., Triangular Norms. Position Paper III: Continuous t Norms, Fuzzy Sets and Systems 145 (2004) Demirci M. and Recasens J., Fuzzy Groups, Fuzzy Functions and Fuzzy Equivalence Relations, Fuzzy Sets and Systems 144 (2004) Demirci M., Fuzzy Functions and Their Applications, Journal of Mathematical Analysis and Applications 252 (2000) Demirci M., Fundamentals of Mvague Algebra and Mvague Arithmetic Operations, Int. J. Uncertainly, Fuzziness KnowledgeBased Systems 10, 1 (2002) Sostak A. P., Fuzzy Functions and an Extension of the Category LTop of Chang Goguen LTopological Spaces, Proceedings of the Ninth Prague Symposium, pp , Topology Atlas, Toronto, Wang Z.D. and Yu Y. D., TLsubrings and TLideals, Part 2: Generated TLideals, Fuzzy Sets and Systems 87 (1997) Wang Z.D. and Yu Y. D., TLsubrings and TLideals, Part 1: Basic concepts, Fuzzy Sets and Systems 68 (1994) Zadeh L. A., Fuzzy Sets, Information and Control, 8 (1965)
23 On the Character Degrees of Solvable Groups Temha ERKOÇ ¹ ¹ Department of Mathematics, University of Istanbul, Istanbul, Turkey A difficult problem in the character theory of finite solvable groups, known as the IsaacsSeitz conjecture, asserts that the derived length of a finite solvable group is bounded above by the number of distinct irreducible character degrees of that group. The first result for this conjecture appeared in the paper [1]. Also in 1976, T. Berger proved that the conjecture is true for solvable groups of odd order. In this talk we will provide some affirmative answers to the conjecture. Keywords and phrases: Character degrees, derived lenght, solvable groups Mathematic Subject Classification: 20C15 [1] I.M., Isaacs, Character degrees and derived lenght of a solvable group, Canad. J. Math, 27 (1975), [2] T.R. Berger, Characters and derived length in groups of odd order, J. Algebra 1976, 39: [3] S. Garrison, On Groups with a Small Number of Character Degree,Ph.D. Thesis, University of Wisconsin, Madison, [4] I.M., Isaacs, Groups having at most three irreducible character degrees, Proc. Amer. Math. Soc., 1969, 21: [5] I.M. Isaacs, G. Knutson, Irreducible character degrees and normal subgroups, J. Algebra, 1998, 199: [6] I.M. Isaacs, Character Theory of Finite Groups, AMS Chelsea Publishing, Providence, RI, [7] M. Lewis, Derived lengths of solvable groups having five irreducible character degrees I, Algebras Represent. Theory :
Amply Fws Modules (Aflk n, Sonlu Zay f Eklenmifl Modüller)
Çankaya Üniversitesi FenEdebiyat Fakültesi, Journal of Arts and Sciences Say : 4 / Aral k 2005 Amply Fws Modules (Aflk n, Sonlu Zay f Eklenmifl Modüller) Gökhan B LHA * Abstract In this work amply finitely
More informationOn the Algebraic Structures of Soft Sets in Logic
Applied Mathematical Sciences, Vol. 8, 2014, no. 38, 18731881 HIKARI Ltd, www.mhikari.com http://dx.doi.org/10.12988/ams.2014.43127 On the Algebraic Structures of Soft Sets in Logic Burak Kurt Department
More informationADDITIVE GROUPS OF RINGS WITH IDENTITY
ADDITIVE GROUPS OF RINGS WITH IDENTITY SIMION BREAZ AND GRIGORE CĂLUGĂREANU Abstract. A ring with identity exists on a torsion Abelian group exactly when the group is bounded. The additive groups of torsionfree
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationF. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)
Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationU.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory
More informationSign changes of Hecke eigenvalues of Siegel cusp forms of degree 2
Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2 Ameya Pitale, Ralf Schmidt 2 Abstract Let µ(n), n > 0, be the sequence of Hecke eigenvalues of a cuspidal Siegel eigenform F of degree
More informationIdeal Class Group and Units
Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals
More informationGroup Theory. Contents
Group Theory Contents Chapter 1: Review... 2 Chapter 2: Permutation Groups and Group Actions... 3 Orbits and Transitivity... 6 Specific Actions The Right regular and coset actions... 8 The Conjugation
More informationFACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z
FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization
More informationOSTROWSKI FOR NUMBER FIELDS
OSTROWSKI FOR NUMBER FIELDS KEITH CONRAD Ostrowski classified the nontrivial absolute values on Q: up to equivalence, they are the usual (archimedean) absolute value and the padic absolute values for
More informationThe Open University s repository of research publications and other research outputs
Open Research Online The Open University s repository of research publications and other research outputs The degreediameter problem for circulant graphs of degree 8 and 9 Journal Article How to cite:
More informationIRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible
More informationEXERCISES FOR THE COURSE MATH 570, FALL 2010
EXERCISES FOR THE COURSE MATH 570, FALL 2010 EYAL Z. GOREN (1) Let G be a group and H Z(G) a subgroup such that G/H is cyclic. Prove that G is abelian. Conclude that every group of order p 2 (p a prime
More informationSECRET sharing schemes were introduced by Blakley [5]
206 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 52, NO. 1, JANUARY 2006 Secret Sharing Schemes From Three Classes of Linear Codes Jin Yuan Cunsheng Ding, Senior Member, IEEE Abstract Secret sharing has
More informationLOWDEGREE PLANAR MONOMIALS IN CHARACTERISTIC TWO
LOWDEGREE PLANAR MONOMIALS IN CHARACTERISTIC TWO PETER MÜLLER AND MICHAEL E. ZIEVE Abstract. Planar functions over finite fields give rise to finite projective planes and other combinatorial objects.
More informationNonunique factorization of polynomials over residue class rings of the integers
Comm. Algebra 39(4) 2011, pp 1482 1490 Nonunique factorization of polynomials over residue class rings of the integers Christopher Frei and Sophie Frisch Abstract. We investigate nonunique factorization
More informationON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP. A. K. Das and R. K. Nath
International Electronic Journal of Algebra Volume 7 (2010) 140151 ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP A. K. Das and R. K. Nath Received: 12 October 2009; Revised: 15 December
More informationCURRICULUM VITAE of Giuseppe Molteni
CURRICULUM VITAE of Giuseppe Molteni Personal information Address: Dipartimento di Matematica, Università di Milano, Via Saldini 50, I20133 Milano (Italy) Phone number: (+39) 02 503 16144 email: giuseppe.molteni1@unimi.it
More informationRINGS WITH A POLYNOMIAL IDENTITY
RINGS WITH A POLYNOMIAL IDENTITY IRVING KAPLANSKY 1. Introduction. In connection with his investigation of projective planes, M. Hall [2, Theorem 6.2]* proved the following theorem: a division ring D in
More informationON GALOIS REALIZATIONS OF THE 2COVERABLE SYMMETRIC AND ALTERNATING GROUPS
ON GALOIS REALIZATIONS OF THE 2COVERABLE SYMMETRIC AND ALTERNATING GROUPS DANIEL RABAYEV AND JACK SONN Abstract. Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Q p for
More informationSOLVING POLYNOMIAL EQUATIONS BY RADICALS
SOLVING POLYNOMIAL EQUATIONS BY RADICALS Lee Si Ying 1 and Zhang DeQi 2 1 Raffles Girls School (Secondary), 20 Anderson Road, Singapore 259978 2 Department of Mathematics, National University of Singapore,
More informationTotal colorings of planar graphs with small maximum degree
Total colorings of planar graphs with small maximum degree Bing Wang 1,, JianLiang Wu, SiFeng Tian 1 Department of Mathematics, Zaozhuang University, Shandong, 77160, China School of Mathematics, Shandong
More informationChapter 13: Basic ring theory
Chapter 3: Basic ring theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 3: Basic ring
More informationCURRICULUM VITAE. : Department of Mathematics, Hacettepe University, Beytepe Campus, Ankara, Türkiye.
CURRICULUM VITAE Personel Information Name and Surname : Adnan TERCAN Place of birth and year : Ankara, 1963. Nationality Marital status Address : Turkish : Married : Department of Mathematics, Hacettepe
More informationEMBEDDING DEGREE OF HYPERELLIPTIC CURVES WITH COMPLEX MULTIPLICATION
EMBEDDING DEGREE OF HYPERELLIPTIC CURVES WITH COMPLEX MULTIPLICATION CHRISTIAN ROBENHAGEN RAVNSHØJ Abstract. Consider the Jacobian of a genus two curve defined over a finite field and with complex multiplication.
More informationInternational Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1, No.3,August 2013
FACTORING CRYPTOSYSTEM MODULI WHEN THE COFACTORS DIFFERENCE IS BOUNDED Omar Akchiche 1 and Omar Khadir 2 1,2 Laboratory of Mathematics, Cryptography and Mechanics, Fstm, University of Hassan II MohammediaCasablanca,
More informationOn Bhargava s representations and Vinberg s invariant theory
On Bhargava s representations and Vinberg s invariant theory Benedict H. Gross Department of Mathematics, Harvard University Cambridge, MA 02138 gross@math.harvard.edu January, 2011 1 Introduction Manjul
More informationMaster of Arts in Mathematics
Master of Arts in Mathematics Administrative Unit The program is administered by the Office of Graduate Studies and Research through the Faculty of Mathematics and Mathematics Education, Department of
More informationGalois representations with open image
Galois representations with open image Ralph Greenberg University of Washington Seattle, Washington, USA May 7th, 2011 Introduction This talk will be about representations of the absolute Galois group
More informationA REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries
Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do
More informationThe fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33 37) Bart de Smit
The fundamental group of the Hawaiian earring is not free Bart de Smit The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992),
More informationCOUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS
COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS Alexander Burstein Department of Mathematics Howard University Washington, DC 259, USA aburstein@howard.edu Sergey Kitaev Mathematics
More informationSequence of Mathematics Courses
Sequence of ematics Courses Where do I begin? Associates Degree and Nontransferable Courses (For math course below prealgebra, see the Learning Skills section of the catalog) MATH M09 PREALGEBRA 3 UNITS
More informationIrreducible Representations of Wreath Products of Association Schemes
Journal of Algebraic Combinatorics, 18, 47 52, 2003 c 2003 Kluwer Academic Publishers. Manufactured in The Netherlands. Irreducible Representations of Wreath Products of Association Schemes AKIHIDE HANAKI
More informationRobert Langlands. Bibliography
Robert Langlands Bibliography (1) Some holomorphic semigroups, Proc. Nat. Acad. Sci. 46 (1960), 361363. (2) On Lie semigroups, Can. J. Math. 12 (1960), 686693. (3) Dirichlet series associated with
More informationFactoring of Prime Ideals in Extensions
Chapter 4 Factoring of Prime Ideals in Extensions 4. Lifting of Prime Ideals Recall the basic AKLB setup: A is a Dedekind domain with fraction field K, L is a finite, separable extension of K of degree
More informationMathematics Review for MS Finance Students
Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,
More informationIntroduction to Modern Algebra
Introduction to Modern Algebra David Joyce Clark University Version 0.0.6, 3 Oct 2008 1 1 Copyright (C) 2008. ii I dedicate this book to my friend and colleague Arthur Chou. Arthur encouraged me to write
More informationShort Programs for functions on Curves
Short Programs for functions on Curves Victor S. Miller Exploratory Computer Science IBM, Thomas J. Watson Research Center Yorktown Heights, NY 10598 May 6, 1986 Abstract The problem of deducing a function
More informationUNIVERSITY GRADUATE STUDIES PROGRAM SILLIMAN UNIVERSITY DUMAGUETE CITY. Master of Science in Mathematics
1 UNIVERSITY GRADUATE STUDIES PROGRAM SILLIMAN UNIVERSITY DUMAGUETE CITY Master of Science in Mathematics Introduction The Master of Science in Mathematics (MS Math) program is intended for students who
More informationComplexity problems over algebraic structures
Complexity problems over algebraic structures Theses of PhD dissertation Created by: Horváth Gábor Mathematical Doctoral School Theor. Mathematical Doctoral Program Dir. of School: Dr. Laczkovich Miklós
More informationCHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY
January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.
More informationA number field is a field of finite degree over Q. By the Primitive Element Theorem, any number
Number Fields Introduction A number field is a field of finite degree over Q. By the Primitive Element Theorem, any number field K = Q(α) for some α K. The minimal polynomial Let K be a number field and
More informationON TORI TRIANGULATIONS ASSOCIATED WITH TWODIMENSIONAL CONTINUED FRACTIONS OF CUBIC IRRATIONALITIES.
ON TORI TRIANGULATIONS ASSOCIATED WITH TWODIMENSIONAL CONTINUED FRACTIONS OF CUBIC IRRATIONALITIES. O. N. KARPENKOV Introduction. A series of properties for ordinary continued fractions possesses multidimensional
More informationChapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize
More informationON HENSEL S ROOTS AND A FACTORIZATION FORMULA IN Z[[X]]
#A47 INTEGERS 4 (204) ON HENSEL S ROOTS AND A FACTORIZATION FORMULA IN Z[[X]] Daniel Birmaer Department of Mathematics, Nazareth College, Rochester, New Yor abirma6@naz.edu Juan B. Gil Penn State Altoona,
More informationRow Ideals and Fibers of Morphisms
Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion
More informationDegree Hypergroupoids Associated with Hypergraphs
Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated
More informationGRAPHS AND ZERODIVISORS. In an algebra class, one uses the zerofactor property to solve polynomial equations.
GRAPHS AND ZERODIVISORS M. AXTELL AND J. STICKLES In an algebra class, one uses the zerofactor property to solve polynomial equations. For example, consider the equation x 2 = x. Rewriting it as x (x
More informationTwo classes of ternary codes and their weight distributions
Two classes of ternary codes and their weight distributions Cunsheng Ding, Torleiv Kløve, and Francesco Sica Abstract In this paper we describe two classes of ternary codes, determine their minimum weight
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22 RAVI VAKIL CONTENTS 1. Discrete valuation rings: Dimension 1 Noetherian regular local rings 1 Last day, we discussed the Zariski tangent space, and saw that it
More informationOn the largest prime factor of x 2 1
On the largest prime factor of x 2 1 Florian Luca and Filip Najman Abstract In this paper, we find all integers x such that x 2 1 has only prime factors smaller than 100. This gives some interesting numerical
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More informationModule MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013
Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 D. R. Wilkins Copyright c David R. Wilkins 1997 2013 Contents A Cyclotomic Polynomials 79 A.1 Minimum Polynomials of Roots of
More informationSMALL SKEW FIELDS CÉDRIC MILLIET
SMALL SKEW FIELDS CÉDRIC MILLIET Abstract A division ring of positive characteristic with countably many pure types is a field Wedderburn showed in 1905 that finite fields are commutative As for infinite
More informationThe sum of digits of polynomial values in arithmetic progressions
The sum of digits of polynomial values in arithmetic progressions Thomas Stoll Institut de Mathématiques de Luminy, Université de la Méditerranée, 13288 Marseille Cedex 9, France Email: stoll@iml.univmrs.fr
More informationOn the representability of the biuniform matroid
On the representability of the biuniform matroid Simeon Ball, Carles Padró, Zsuzsa Weiner and Chaoping Xing August 3, 2012 Abstract Every biuniform matroid is representable over all sufficiently large
More informationFINITE DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROSSHEN S UNIMODULARITY CONJECTURE AARON TIKUISIS
FINITE DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROSSHEN S UNIMODULARITY CONJECTURE AARON TIKUISIS Abstract. It is shown that, for any field F R, any ordered vector space structure
More informationSOME PROPERTIES OF SYMBOL ALGEBRAS OF DEGREE THREE
SOME PROPERTIES OF SYMBOL ALGEBRAS OF DEGREE THREE CRISTINA FLAUT and DIANA SAVIN Communicated by the former editorial board In this paper, we study some properties of the matrix representations of the
More informationCURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXXI, 1 (2012), pp. 71 77 71 CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC E. BALLICO Abstract. Here we study (in positive characteristic) integral curves
More informationON ROUGH (m, n) BIΓHYPERIDEALS IN ΓSEMIHYPERGROUPS
U.P.B. Sci. Bull., Series A, Vol. 75, Iss. 1, 2013 ISSN 12237027 ON ROUGH m, n) BIΓHYPERIDEALS IN ΓSEMIHYPERGROUPS Naveed Yaqoob 1, Muhammad Aslam 1, Bijan Davvaz 2, Arsham Borumand Saeid 3 In this
More informationZERODIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS
ZERODIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS M. AXTELL, J. COYKENDALL, AND J. STICKLES Abstract. We recall several results of zero divisor graphs of commutative rings. We
More informationCOMMUTATIVITY DEGREE, ITS GENERALIZATIONS, AND CLASSIFICATION OF FINITE GROUPS
COMMUTATIVITY DEGREE, ITS GENERALIZATIONS, AND CLASSIFICATION OF FINITE GROUPS ABSTRACT RAJAT KANTI NATH DEPARTMENT OF MATHEMATICS NORTHEASTERN HILL UNIVERSITY SHILLONG 793022, INDIA COMMUTATIVITY DEGREE,
More informationA DETERMINATION OF ALL NORMAL DIVISION ALGEBRAS OVER AN ALGEBRAIC NUMBER FIELD*
A DETERMINATION OF ALL NORMAL DIVISION ALGEBRAS OVER AN ALGEBRAIC NUMBER FIELD* BY A. ADRIAN ALBERT AND HELMUT HASSE t 1. Introduction. The principal problem in the theory of linear algebras is that of
More informationCycles in a Graph Whose Lengths Differ by One or Two
Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDEBERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS
More informationCOMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS
Bull Austral Math Soc 77 (2008), 31 36 doi: 101017/S0004972708000038 COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS IGOR V EROVENKO and B SURY (Received 12 April 2007) Abstract We compute
More informationMOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu
Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing
More informationMonogenic Fields and Power Bases Michael Decker 12/07/07
Monogenic Fields and Power Bases Michael Decker 12/07/07 1 Introduction Let K be a number field of degree k and O K its ring of integers Then considering O K as a Zmodule, the nicest possible case is
More informationPrime Numbers and Irreducible Polynomials
Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.
More informationGroups with the same orders and large character degrees as PGL(2, 9) 1. Introduction and preliminary results
Quasigroups and Related Systems 21 (2013), 239 243 Groups with the same orders and large character degrees as PGL(2, 9) Behrooz Khosravi Abstract. An interesting class of problems in character theory arises
More information11 Ideals. 11.1 Revisiting Z
11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(
More informationMATHEMATICS (MATH) 3. Provides experiences that enable graduates to find employment in sciencerelated
194 / Department of Natural Sciences and Mathematics MATHEMATICS (MATH) The Mathematics Program: 1. Provides challenging experiences in Mathematics, Physics, and Physical Science, which prepare graduates
More information7. Some irreducible polynomials
7. Some irreducible polynomials 7.1 Irreducibles over a finite field 7.2 Worked examples Linear factors x α of a polynomial P (x) with coefficients in a field k correspond precisely to roots α k [1] of
More informationQUADRATIC RECIPROCITY IN CHARACTERISTIC 2
QUADRATIC RECIPROCITY IN CHARACTERISTIC 2 KEITH CONRAD 1. Introduction Let F be a finite field. When F has odd characteristic, the quadratic reciprocity law in F[T ] (see [4, Section 3.2.2] or [5]) lets
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationNOTES ON CATEGORIES AND FUNCTORS
NOTES ON CATEGORIES AND FUNCTORS These notes collect basic definitions and facts about categories and functors that have been mentioned in the Homological Algebra course. For further reading about category
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationResearch Article On the Rank of Elliptic Curves in Elementary Cubic Extensions
Numbers Volume 2015, Article ID 501629, 4 pages http://dx.doi.org/10.1155/2015/501629 Research Article On the Rank of Elliptic Curves in Elementary Cubic Extensions Rintaro Kozuma College of International
More informationInteger roots of quadratic and cubic polynomials with integer coefficients
Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street
More informationIntroduction to finite fields
Introduction to finite fields Topics in Finite Fields (Fall 2013) Rutgers University Swastik Kopparty Last modified: Monday 16 th September, 2013 Welcome to the course on finite fields! This is aimed at
More informationTHE BIRCH AND SWINNERTONDYER CONJECTURE
THE BIRCH AND SWINNERTONDYER CONJECTURE ANDREW WILES A polynomial relation f(x, y) = 0 in two variables defines a curve C 0. If the coefficients of the polynomial are rational numbers, then one can ask
More informationNonzero degree tangential maps between dual symmetric spaces
ISSN 14722739 (online) 14722747 (printed) 709 Algebraic & Geometric Topology Volume 1 (2001) 709 718 Published: 30 November 2001 ATG Nonzero degree tangential maps between dual symmetric spaces Boris
More informationPOSTECH SUMMER SCHOOL 2013 LECTURE 4 INTRODUCTION TO THE TRACE FORMULA
POSTECH SUMMER SCHOOL 2013 LECTURE 4 INTRODUCTION TO THE TRACE FORMULA 1. Kernel and the trace formula Beginning from this lecture, we will discuss the approach to Langlands functoriality conjecture based
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationIrreducible Divisor Graphs
Irreducible Divisor Graphs Jim Coykendall Department of Mathematics North Dakota State University Fargo, ND 581055075 Jack Maney Department of Mathematical Sciences The University of South Dakota 414
More informationFACTORING IN QUADRATIC FIELDS. 1. Introduction. This is called a quadratic field and it has degree 2 over Q. Similarly, set
FACTORING IN QUADRATIC FIELDS KEITH CONRAD For a squarefree integer d other than 1, let 1. Introduction K = Q[ d] = {x + y d : x, y Q}. This is called a quadratic field and it has degree 2 over Q. Similarly,
More informationBasic Algorithms In Computer Algebra
Basic Algorithms In Computer Algebra Kaiserslautern SS 2011 Prof. Dr. Wolfram Decker 2. Mai 2011 References Cohen, H.: A Course in Computational Algebraic Number Theory. Springer, 1993. Cox, D.; Little,
More informationSéminaire Dubreil. Algèbre
Séminaire Dubreil. Algèbre DAVID EISENBUD Notes on an extension of Krull s principal ideal theorem Séminaire Dubreil. Algèbre, tome 28, n o 1 (19741975), exp. n o 20, p. 14.
More informationTHE AVERAGE DEGREE OF AN IRREDUCIBLE CHARACTER OF A FINITE GROUP
THE AVERAGE DEGREE OF AN IRREDUCIBLE CHARACTER OF A FINITE GROUP by I. M. Isaacs Mathematics Department University of Wisconsin 480 Lincoln Dr. Madison, WI 53706 USA EMail: isaacs@math.wisc.edu Maria
More informationAn example of a computable
An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept
More informationAssociativity condition for some alternative algebras of degree three
Associativity condition for some alternative algebras of degree three Mirela Stefanescu and Cristina Flaut Abstract In this paper we find an associativity condition for a class of alternative algebras
More informationPrime numbers and prime polynomials. Paul Pollack Dartmouth College
Prime numbers and prime polynomials Paul Pollack Dartmouth College May 1, 2008 Analogies everywhere! Analogies in elementary number theory (continued fractions, quadratic reciprocity, Fermat s last theorem)
More informationFour Unsolved Problems in Congruence Permutable Varieties
Four Unsolved Problems in Congruence Permutable Varieties Ross Willard University of Waterloo, Canada Nashville, June 2007 Ross Willard (Waterloo) Four Unsolved Problems Nashville, June 2007 1 / 27 Congruence
More informationMA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES
MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2016 47 4. Diophantine Equations A Diophantine Equation is simply an equation in one or more variables for which integer (or sometimes rational) solutions
More informationGROUP ALGEBRAS. ANDREI YAFAEV
GROUP ALGEBRAS. ANDREI YAFAEV We will associate a certain algebra to a finite group and prove that it is semisimple. Then we will apply Wedderburn s theory to its study. Definition 0.1. Let G be a finite
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More information3 1. Note that all cubes solve it; therefore, there are no more
Math 13 Problem set 5 Artin 11.4.7 Factor the following polynomials into irreducible factors in Q[x]: (a) x 3 3x (b) x 3 3x + (c) x 9 6x 6 + 9x 3 3 Solution: The first two polynomials are cubics, so if
More informationCOMBINATORIAL PROPERTIES OF THE HIGMANSIMS GRAPH. 1. Introduction
COMBINATORIAL PROPERTIES OF THE HIGMANSIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the HigmanSims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact
More information