Search

The Dirac equation provides a fully relativistic covariant equation which can be used to calculate relativistic transition rates but only for one-electron systems. For the two-electron case, one can either use approximate relativistic wave functions or obtain equivalent nonrelativistic operators that can be used with Schr{\"o}dinger wave functions; an approach that is preferred for low atomic number (Z) atoms. By using equivalent nonrelativistic operators obtained from the Foldy-Wouthuysen transformation and relativistically corrected Schr{\"o}dinger wave functions, we show that we obtain the same transition amplitude as in Dirac Theory up to order $\alpha^2$, where $\alpha$ is the fine structure constant. We show this for the one-electron case and provide a theoretical framework for the two-electron case. For the one-electron case we obtain analytic first order corrected wave functions for the $2p$ states which have not been published before. For the two-electron case we obtain first order corrected wave functions using a variational method and compare two different Sturmian basis sets, which we label triangular and linear basis sets. We show that the triangular basis set provides a significant advantage over the linear basis set, increasing the precision by two orders of magnitude. We also compare the wave functions obtained using pseudostates with those obtained analytically and give some suggestions to improve the agreement near zero.

We give a detailed account of an $\it{ab}$ $\it{initio}$ spectral approach for the calculation of energy spectra of two active electron atoms in a system of hyperspherical coordinates. In this system of coordinates, the Hamiltonian has the same structure as the one of atomic hydrogen with the Coulomb potential expressed in terms of a hyperradius and the nuclear charge replaced by an angle dependent effective charge. The simplest spectral approach consists in expanding the hyperangular wave function in a basis of hyperspherical harmonics. This expansion however, is known to be very slowly converging. Instead, we introduce new hyperangular sturmian functions. These functions do not have an analytical expression but they treat the first term of the multipole expansion of the electron-electron interaction potential, namely the radial electron correlation, exactly. The properties of these new functions are discussed in detail. For the basis functions of the hyperradius, several choices are possible. In the present case, we use Coulomb sturmian functions of half integer angular momentum. We show that, in the case of H$^-$, the accuracy of the energy and the width of the resonance states obtained through a single diagonalization of the Hamiltonian, is comparable to the values given by state-of-the-art methods while using a much smaller basis set. In addition, we show that precise values of the electric-dipole oscillator strengths for $S\rightarrow P$ transitions in helium are obtained thereby confirming the accuracy of the bound state wave functions generated with the present method., Comment: 28 pages, 4 figures

The previous paper in this series introduced a class of infinite binary strings, called two-pattern strings, that constitute a significant generalization of, and include, the much-studied Sturmian strings. The class of two-pattern strings is a union of a sequence of increasing (with respect to inclusion) subclasses Tλ of two-pattern strings of scope λ, λ=1,2,…. Prefixes of two-pattern strings are interesting from the algorithmic point of view (their recognition, generation, and computation of repetitions and near-repetitions) and since they include prefixes of the Fibonacci and the Sturmian strings, they merit investigation of how many finite two-pattern strings of a given size there are among all binary strings of the same length. In this paper we first consider the frequency fλ(n) of occurrence of two-pattern strings of length n and scope λ among all strings of length n on {a,b}: we show that limn→∞fλ(n)=0, but that for strings of lengths n⩽2λ, two-pattern strings of scope λ constitute more than one-quarter of all strings. Since the class of Sturmian strings is a subset of two-pattern strings of scope 1, it was natural to focus the study of the substring complexity of two-pattern strings to those of scope 1. Though preserving the aperiodicity of the Sturmian strings, the generalization to two-pattern strings greatly relaxes the constrained substring complexity (the number of distinct substrings of the same length) of the Sturmian strings. We derive upper and lower bounds on C1(k) (the number of distinct substring of length k) of two-pattern strings of scope 1, and we show that it can be considerably greater than that of a Sturmian string. In fact, we describe circumstances in which limk→∞(C1(k)−k)=∞.

Three new characterisations of balanced words are presented. Each of these characterisations is based on the ordering of a shift orbit, either lexicographically or with respect to the norm |·|1 (which counts the number of occurrences of the symbol 1).

We consider the following open question in the spirit of Ramsey theory: Given an aperiodic infinite word w, does there exist a finite coloring of its factors such that no factorization of w is monochromatic? We show that such a coloring always exists whenever w is a Sturmian word or a standard episturmian word. © 2013 Springer-Verlag.

06.03.2018 tarihli ve 30352 sayılı Resmi Gazetede yayımlanan “Yükseköğretim Kanunu İle Bazı Kanun Ve Kanun Hükmünde Kararnamelerde Değişiklik Yapılması Hakkında Kanun” ile 18.06.2018 tarihli “Lisansüstü Tezlerin Elektronik Ortamda Toplanması, Düzenlenmesi ve Erişime Açılmasına İlişkin Yönerge” gereğince tam metin erişime açılmıştır. x, elemanları özel kurulmuş ve alfabe olarak adlandırılan harflerin bir dizisidir. Bu noktada harflerin kelimeyi, kelimelerin de diziyi oluşturduğu görülmektedir.şeklinde iki uzunluğunda, ikili alfabe üzerinde durularak Sturmian diziler tanımlanır. alfabesi üzerinde tanımlı olan bir dizisinin Sturmian dizi olması için gerek ve yeter şart 'in dengeli olması ve sonunda periyodik olmamasıdır.Sonsuz Sturmian dizisinin boştan farklı sonlu bir alt dizisine sonlu Sturmian dizi denir. Fibonacci dizileri, Sturmian dizilerinin en iyi bilinen özel bir durumudur. Sturmian diziler, düzlemdeki doğrularla da ilişkilidirler.Merkezi kelimelerin kümesi PER şeklinde gösterilir ve her Sturmian kelime PER deki kelimenin alt dizisidir. Bu yüzden PER Sturmian kelimelerin çekirdeğidir. Bu noktada PER'den yararlanarak Sturmian dizilerin kombinasyonel özellikleri hakkında bilgi sahibi olunabilir.Sturmian diziyi açıklamak ve tanımlamak için farklı yollar bu tezde çalışılmıştır.Anahtar Kelimeler: Kelime, Dizi, Sturmian, Fibonacci Dizisi A string is a sequence of letters which are the elements of a specified set called alphabet. At this point we can see letters generate the word, words generate the string obviously.Sturmian Strings are defined on binary alphabet which is called and size 2. string, which is difined on alphabet, can be Sturmian String if and only if is balanced and not ultimately periodic.A finite Sturmian string is any finite non-empty substring of an infinite Sturmian string. A Fibonacci string is a special case of a sturmian string that is well-known. Sturmian strings are related to straight lines.Set of all central words denote by PER and every Sturmian word is a substring of a word in PER. Hence the set PER is a kernel of the Sturmian words. At this point it can be make knowledge about combinational properties of the Sturmian strings by using the PER.In this thesis, it is worked on different ways for defined and explaining of Sturmian string.Key Words: Word, String, Sturmian, Fibonacci Strings.

International audience; We say $x \in \{ 0,1,2 \}^{\NN}$ is a word with Sturmian erasures if for any $a\in \{ 0,1,2 \}$ the word obtained erasing all $a$ in $x$ is a Sturmian word. A large family of such words is given coding trajectories of balls in the game of billiards in the cube. We prove that the monoid of morphisms mapping all words with Sturmian erasures to words with Sturmian erasures is not finitely generated.

In this paper we apply a simple representation of Sturmian strings, which we call a “reduction sequence”, to three algorithms. The first algorithm accepts as input a given finite string x and determines in time O(|x|) whether or not x is Sturmian. The second algorithm is a modification of the first that, in the case that x is Sturmian, outputs a reduction sequence for a superstring u of x that is a prefix of an infinite Sturmian string. The third algorithm uses the reduction sequence of u to compute all the repetitions in u in time Θ(|u|), thus extending a recent result for Fibonacci strings. The third algorithm is also based on a characterization of the repetitions in a Sturmian string that describes them compactly in terms of “runs”. Finally, for every integer r⩾4, we show how to construct an infinite Sturmian string that contains maximal repetitions of exponents 2,3,…,r−1, but none of exponent r.

This paper introduces a new class of strings on {a,b}, called two-pattern strings, that constitute a substantial generalization of Sturmian strings while at the same time sharing many of their nice properties. In particular, we show in this paper that two-pattern strings can be recognized in time proportional to their length. This result is a prelude to showing that the repetitions in these substrings can also be computed in linear time, further that two-pattern strings occur in some sense frequently in the class of all strings on {a,b}.