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Departman Tezleri
Sınır değer problemlerinin incelenmesi
Bu tez 6 bölümden oluşmaktadır.Birinci bölümde diferansiyel denklemlerin kullanım alanlarından bahsedilerek teze giriş yapılmıştır. Ayrıca Sınır Değer Problemleri ile ilgili yapılan ilk çalışmalardan bahsedilmiştir.İkinci bölümde tezde kullanılan temel tanım ve kavramlar verilmiştir.Üçüncü bölümde tek noktalı Sınır Değer Problemleri üzerinde durularak çözümlerinin varlığı ve tekliği incelenmiştir.Dördüncü bölümde iki noktalı Sınır Değer Problemleri incelenerek Green fonksiyonu ve özellikleri hakkında bilgiler verilmiştir.Beşinci bölümde ise özel bir Sınır Değer Problemi olan Sturm-Liouville Problemi üzerinde durulmuştur. Özellikleri verilerek, özdeğerleri ile özfonksiyonları incelenmiştir. Bu problemin özel halleri olan Özel Fonksiyonlardan birkaçı üzerinde durulmuştur. Son olarak da elde edilen verilerle Sturm-Liouville Teoremleri ispatlanmıştır.Altıncı bölümde ise tez çalışmasından elde edilen sonuçlar belirtilmiştir.
Mackey izleçlerinde green uyuşumu
The Green corespondence for modules of group algebras was introduced by Greenin 1964. A version for Mackey functors was introduced by Sasaki in 1982. Sasaki'scharacterization of Mackey functor correspondence was based on the theory ofGreen functors. In this thesis, we give Sasaki's characterization and an alternativecharacterization of the Mackey functor correspondence. Our characterization iscloser to Green's original module theoretic approach. We show that the twocharacterizations are equivalent. This yields a new way of determining vertices,sources and Green correspondents; we shall illustrate this with some examples.
Düğümlü klasik Zariski çiftleri
In this thesis we study complex projective sextic curves withsimple singularities. All curves constituting classical Zariskipairs, especially those with nodes, are enumerated and classifiedup to equisingular deformation. Every set of singularitiesconstituting a classical Zariski pair gives rise to at most twofamilies, called abundant and non-abundant except for one whichgives rise to three families, one abundant and two conjugatenon-abundant. This classification is done arithmetically with theaid of integral lattices and quadratic forms.
Pauli ilkesi, temsil kuramı ve bayrak çeşitlemlerinin geometrisi
According to the Pauli exclusion principle, discovered in 1925, no two identicalelectrons may occupy the same quantum state. In terms of electron density matrixthis amounts to an upper bound for its eigenvalues by 1. In 1926, it has beenreplaced by skew-symmetry of a multi-electron wave function. In this thesis wegive two different solutions to a problem about the impact of this replacement onthe electron density matrix, which goes far beyond the original Pauli principle.
Tek terimli tam kesişim eğrileri ve azalmayan hilbert fonksiyonları
In this thesis, we first study the problem of determining set theoretic completeintersection (s.t.c.i.) projective monomial curves. We are also interested in findingthe equations of the hypersurfaces on which the monomial curve lie as set theoreticcomplete intersection. We find these equations for symmetric ArithmeticallyCohen-Macaulay monomial curves.We describe a method to produce infinitely many s.t.c.i. monomial curves inP^{n+1} starting from one single s.t.c.i. monomial curve in P^{n}. Our approach hasthe side novelty of describing explicitly the equations of hypersurfaces on whichthese new monomial curves lie as s.t.c.i.. On the other hand, semigroup gluingbeing one of the most popular techniques of recent research, we develop numericalcriteria to determine when these new curves can or cannot be obtained via gluing.Finally, by using the technique of gluing semigroups, we give infinitely manynew families of affine monomial curves in arbitrary dimensions with Cohen-Macaulay tangent cones. This gives rise to large families of 1-dimensional localrings with arbitrary embedding dimensions and having non-decreasing Hilbertfunctions. We also construct infinitely many affine monomial curves in A^{n+1}whose tangent cone is not Cohen Macaulay and whose Hilbert function is nondecreasingfrom a single monomial curve in A^{n} with the same property.
Mackey funktorların genişletilmesi, kısıtlanması, hesaplanması, ve alt funktorları
In this thesis we try to relate the subfunctor structure of a given Mackey functorM for a finite group G to the submodule structure of the KNG(H)-module M(H)where H is a subgroup of G.We mainly study the socle and the radical of a Mackey functor M for a finitegroup G over a field K, (usually, of characteristic p > 0). For a subgroup Hof G, we construct bijections between some classes of the simple subfunctors ofM and some classes of the simple KNG(H)-submodules of M(H). We relate themultiplicity of a simple Mackey functor SGH,V in the socle of M to the multiplicityof V in the socle of a certain KNG(H)-submodule of M(H). We also obtainsimilar results for the maximal subfunctors of M. We specialize our results tosome specific kinds of Mackey functors for G that includes the functors obtainedby inducing or restricting a simple Mackey functor, Mackey functors for a p-group,the fixed point functor, and the Burnside functor BGK.Let M be the Mackey functor "GK SKH,W for G obtained by inducing a simpleMackey functor SKH,W for K. For example, we observe that the socle and theradical of M can be determined from the socle and the radical of the KNG(H)-module V ="NG(H)NK(H)W. We also find similar results for Mackey functors obtainedby restricting a simple Mackey functor. Moreover, we derive criterions for aMackey functor obtained by inducing or restricting a simple Mackey functor tobe simple, semisimple or indecomposable.Our results about induced or restricted Mackey functors include Mackey functorversions of two classical and frequently used results in the representationtheory finite groups, namely Clifford?s theorem and Green?s indecomposibilitytheorem.We also apply our general results to Mackey functors satisfying some specialconditions such as having a unique maximal or simple subfunctor, being uniserial,and being a functor for a p-group. We give some results about primordial andcoprimordial subgroups of G for such kind of functors, and we refine our generalresults and obtain, for instance, a criterion for a Mackey functor to be a quotientof a projective Mackey functor, and find some information about compositionseries.In later chapters the main Mackey functor to which we apply our generalresults is the Burnside functor BGK. We first find the maximal subfunctors of BGKfor any group G, and obtain some results about evaluations of the terms of theradical series of BGK. We also get some results about simple Mackey functors inradical layers of BGKwhose minimal subgroups are p-subgroup of G. Assumingthat G is a p-group we find the first four top factors of the radical series of BGK,and assuming further that G is an abelian p-group we find the radical series ofBGKcompletely, which means that in this case we find the evaluations of the termsof the radical series, and the simple Mackey functors appearing in radical layers,and the Loewy length of BGK. We also study the socle series of BGK. This seems tobe harder than the radical series. Nevertheless, we obtain similar results for thesocle series of BGKassuming mostly that G is an abelian p-group. To illustrateapplications of our general results we also study briefly the radical and the socleseries of fixed point functors FPGV where V is a one dimensional KG-module.We finish this thesis by trying to find possible relations between the socles andthe radicals of the Mackey functors of the form T and FT where T is a Mackeyfunctor and F is one of the functors restriction, inflation, evaluation, or adjointsof them, between Mackey functor categories.Keywords: Mackey functor, Mackey algebra, simple, indecomposable, restriction,induction, evaluation, socle, radical, Clifford?s theorem, Green?s indecomposibilitytheorem, maximal subfunctor, Brauer quotient, minimal subfunctor, restrictionkernel, primordial, coprimordial, uniserial, Burnside functor, socle series,radical series, composition series, composition factors, Loewy lenght, fixed pointfunctor, functors for p-groups.
Düzgün zaman skalasında integre edilebilir sistemler
We present two approaches to unify the integrable systems. Bothapproaches are based on the classical R-matrix formalism. Thefirst approach proceeds from the construction of(1+1)-dimensional integrable \Delta-differential systems onregular time scales together with bi-Hamiltonian structures andconserved quantities. The second approach is established upon thegeneral framework of integrable discrete systems on Real numbers andintegrable dispersionless systems. We discuss the deformationquantization scheme for the dispersionless systems. We also applythe theories presented in this dissertation, to several well-knownexamples.
Rastsal kodlar ve matrisler
Results of our study are two fold. From the code theoretical point of view ourstudy yields the expectations and the covariances of the coe±cients of the weightenumerator of a random code. Particularly interesting is that, the coe±cients ofthe weight enumerator of a code with random parity check matrix are uncorre-lated. We give conjectures for the triple correlations of the coe±cients of weightenumerator of random codes. From the random matrix theory point of view weobtain results in the rank distribution of column submatrices. We give the ex-pectations and the covariances between the ranks (q¡rank) of such submatricesover Fq. We conjecture the counterparts of these results for arbitrary subma-trices. The case of higher correlations gets drastically complicated even in thecase of three submatrices. We give a formula for the correlation of ranks of threesubmatrices and a conjecture for its closed form.
VH-uzaylarında genleşme teoremleri
In the Appendix of the book Leçons d'analyse fonctionnelle by F. Riesz andB. Sz.-Nagy, B. Sz.-Nagy [15] proved an important theorem on operator valuedpositive definite maps on *-semigroups, which today can be considered as one ofthe pioneering results of dilation theory. In the same year W.F. Stinespring [11]proved another celebrated theorem about dilation of operator valued completelypositive linear maps on C*-algebras. Then F.H. Szafraniec [14] showed that thesetheorems are actually equivalent.Due to reasons coming from multivariate stochastic processes R.M. Loynes [7],considered a generalization of B. Sz.-Nagy's Theorem for vector Hilbert spaces(that he called VH-spaces). These VH-spaces have "inner products" that arevector valued, into the so-called "admissible spaces".This work is aimed at providing a detailed proof of R.M. Loynes Theorem thatgeneralizes B. Sz.-Nagy, a detailed proof of the equivalence of Stinespring's Theoremin the Arveson formulation [2] for B*-algebras with B. Sz.-Nagy's Theoremfollowing the lines in [14] together with some ideas from [2], and to get VH-variantsof Stinespring's Theorem for C*-algebras and B*-algebras. Relationsbetween these theorems are also considered.
Tekil Burnside halkaları ve Tom Dieck dönüşümünün ayrışımı
This thesis is mainly concerned with a decomposition of the reduced tom Dieck map die from the real representation ring to the unit group of the Burnside ring into two maps positive tom Dieck map and negative tom Dieck map of the real monomial Burnside ring. The key idea is to introduce a real Lefschetz invariant as an element of the real monomial Burnside ring and to generalize the assertion that the image of an RG-module under the tom Dieck map coincides with the Lefschetz invariant of the sphere of the same module.