Browsing by Subject "Hilbert space."

Now showing 1 - 6 of 6

Open Access
Complete intersection monomial curves and non-decreasing Hilbert functions
(2008) Şahin, Mesut
In this thesis, we first study the problem of determining set theoretic complete intersection (s.t.c.i.) projective monomial curves. We are also interested in finding the equations of the hypersurfaces on which the monomial curve lie as set theoretic complete intersection. We find these equations for symmetric Arithmetically Cohen-Macaulay monomial curves. We describe a method to produce infinitely many s.t.c.i. monomial curves in P n+1 starting from one single s.t.c.i. monomial curve in P n . Our approach has the side novelty of describing explicitly the equations of hypersurfaces on which these new monomial curves lie as s.t.c.i.. On the other hand, semigroup gluing being one of the most popular techniques of recent research, we develop numerical criteria to determine when these new curves can or cannot be obtained via gluing. Finally, by using the technique of gluing semigroups, we give infinitely many new families of affine monomial curves in arbitrary dimensions with CohenMacaulay tangent cones. This gives rise to large families of 1-dimensional local rings with arbitrary embedding dimensions and having non-decreasing Hilbert functions. We also construct infinitely many affine monomial curves in A n+1 whose tangent cone is not Cohen Macaulay and whose Hilbert function is nondecreasing from a single monomial curve in A n with the same property.
Open Access
Extreme behavior of lex ideals on Betti numbers
(2013) Gürdoğan, Hubeyb Üsame
This paper mainly deals with the finitely generated graded modules of the polynomial ring k[x1, x2, ..., xn]. Free resolutions is an important tool to understand the structure of these modules. Betti numbers are an useful invariant that encodes the free resolutions. Our concentration accumulates on proving that the lex ideals provides an upper bound for Betti numbers of the graded ideals with the same Hilbert function in the polynomial ring k[x1, x2, ..., xn]. The material of this thesis is contemporary classical and includes the detailed study of the material that is scattered throughout the sources cited in the bibliography list.
Open Access
A measure disintegration approach to spectral multiplicity for normal operators
(2012) Ay, Serdar
In this thesis we studied the notion of direct integral Hilbert spaces, first introduced by J. von Neumann, and the closely related notion of decomposable operators, as defined in Kadison and Ringrose [1997] and Abrahamse and Kriete [1973]. Examples which show that some of the most familiar spaces in analysis are direct integral Hilbert spaces are presented in detail. Then we give a careful treatment of the notion of disintegration of a probability measure on a locally compact separable metric space, and using the machinery we obtain, a proof of the Spectral Multiplicity Theorem for Normal Operators employing the notion of disintegration of measures is given, based on Abrahamse and Kriete [1973], Arveson [1976], Arveson [2002]. In Chapter 5 the notion of essential preimage is presented in the sense of the article Abrahamse and Kriete [1973], and its relation with the spectral multiplicity function is discussed.
Open Access
Monomial Gotzmann sets
(2011) Pir, Ata Fırat
A homogeneous set of monomials in a quotient of the polynomial ring S := F[x1, . . . , xn] is called Gotzmann if the size of this set grows minimally when multiplied with the variables. We note that Gotzmann sets in the quotient R := F[x1, . . . , xn]/(x a 1 ) arise from certain Gotzmann sets in S. Then we partition the monomials in a Gotzmann set in S with respect to the multiplicity of xi and obtain bounds on the size of a component in the partition depending on neighboring components. We show that if the growth of the size of a component is larger than the size of a neighboring component, then this component is a multiple of a Gotzmann set in F[x1, . . . , xi−1, xi+1, . . . xn]. We also adopt some properties of the minimal growth of the Hilbert function in S to R.
Open Access
Reproducing kernel Hilbert spaces
(2005) Okutmuştur, Baver
In this thesis we make a survey of the theory of reproducing kernel Hilbert spaces associated with positive definite kernels and we illustrate their applications for interpolation problems of Nevanlinna-Pick type. Firstly we focus on the properties of reproducing kernel Hilbert spaces, generation of new spaces and relationships between their kernels and some theorems on extensions of functions and kernels. One of the most useful reproducing kernel Hilbert spaces, the Bergman space, is studied in details in chapter 3. After giving a brief definition of Hardy spaces, we dedicate the last part for applications of interpolation problems of NevanlinnaPick type with three main theorems: interpolation with a finite number of points, interpolation with an infinite number of points and interpolation with points on the boundary. Finally we include an Appendix that contains a brief recall of the main results from functional analysis and operator theory.
Open Access
Some criteria of selfadjointness for unbounded operators in Hilbert spaces
(2013) Özkaraca, Mustafa İsmail
This is a detailed presentation of some criteria of selfadjointness for unbounded operators in a Hilbert space, through operator Cauchy problems. We also include detailed preliminary results on unbounded linear operators in Hilbert spaces, the spectral theory of selfadjoint operators in Hilbert spaces, as well as the theory of extensions of Hermitian operators. The material of this thesis is classical, it was presented in the Operator Theory Seminar during the last two years, and contains material that can be found scattered through the textbooks cited in the bibliography list.