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Article Citation - WoS: 16Citation - Scopus: 20Common Fixed Point Theorems in Cone Banach Spaces(Hacettepe Univ, Fac Sci, 2011) Abdeljawad, Thabet; Karapinar, Erdal; Tas, Kenan; MathematicsRecently, E. Karapinar (Fixed Point Theorems in Cone Banach Spaces, Fixed Point Theory Applications, Article ID 609281, 9 pages, 2009) presented some fixed point theorems for self-mappings satisfying certain contraction principles on a cone Banach space. Here we will give some generalizations of this theorem.Article Citation - WoS: 6Citation - Scopus: 6Dynamic Reliability and Performance Evaluation of Multi-State Systems With Two Components(Hacettepe Univ, Fac Sci, 2011) Eryilmaz, Serkan; Industrial EngineeringIn this paper we study multi-state systems consisting of two components when the number of system states and the number of states of each component are the same, i.e. the systems under consideration are homogeneous multi-state systems. In particular we evaluate multi-state series and cold standby systems assuming that the degradation in their components follow a Markov process. The behaviour of systems with respect to degradation rates is also investigated in terms of stochastic ordering.Article Citation - WoS: 4Citation - Scopus: 5Construction of a Complex Jacobi Matrix From Two-Spectra(Hacettepe Univ, Fac Sci, 2011) Guseinov, Gusein Sh; MathematicsIn this paper we study the inverse spectral problem for two-spectra of finite order complex Jacobi matrices (tri-diagonal matrices). The problem is to reconstruct the matrix using two sets of eigenvalues, one for the original Jacobi matrix and one for the matrix obtained by deleting the first column and the first row of the Jacobi matrix. An explicit procedure of reconstruction of the matrix from the two-spectra is given.Article Citation - WoS: 1Citation - Scopus: 1On Some Permutation Trinomials in Characteristic Three(Hacettepe Univ, Fac Sci, 2025) Temür, Burcu Gülmez; Özkaya, BuketIn this paper, we determine the permutation properties of the polynomial x3 +xq+2 −x4q−1 over the finite field Fq2 in characteristic three. Moreover, we consider the trinomials of the form x4q−1 + x2q+1 ± x3. In particular, we first show that x3 + xq+2 − x4q−1 permutes Fq2 with q = 3m if and only if m is odd. This enables us to show that the sufficient condition in [34, Theorem 4] is also necessary. Next, we prove that x4q−1 + x2q+1 − x3 permutes Fq2 with q = 3m if and only if m ̸≡ 0 (mod 4). Consequently, we prove that the sufficient condition in [20, Theorem 3.2] is also necessary. Finally, we investigate the trinomial x4q−1 + x2q+1 + x3 and show that it is never a permutation polynomial of Fq2 in any characteristic. All the polynomials considered in this work are not quasi-multiplicative equivalent to any known class of permutation trinomials.Article Citation - WoS: 1Citation - Scopus: 3LS-14 Test Suite for Long Sequences(Hacettepe Univ, Fac Sci, 2024) Akcengiz, Ziya; Aslan, Melis; Doğanaksoy, Ali; Sulak, Fatih; Uguz, MuhiddinRandom number sequences are used in many branches of science. Because of many techni- cal reasons and their practicality, pseudo random sequences are usually employed in place of true number sequences. Whether a sequence generated through a deterministic process is a pseudo random, in other words, random-looking sequence or it contains certain pat- terns, can be determined with the help of statistics and mathematics. Although, in the literature there are many statistical randomness tests for this purpose, there is no much work on test suites specialized for long sequences, that is sequences of length 1,000,000 bits or more. Most of the randomness tests for long sequences use some mathematical ap- proximations to compute expected values of the random variables and hence their results contain some errors. Another approach to evaluate randomness criteria of long sequences is to partition the long sequence into a collection short sequences and evaluate the collec- tion for the ran- domness using statistical goodness of fit tests. The main advantage of this approach is, as the individual sequences are short, there is no need to use mathematical approximations. On the other hand when the second approach is preferred, partition the long sequence into a collection of fixed length subsequences and this approach causes a loss of information in some cases. Hence the idea of dynamic partition should be included to perform a more reliable test suite. In this paper, we propose three new tests, namely the entire R2 run, dynamic saturation point, and dynamic run tests. Moreover, we in- troduce a new test suite, called LS-14, consisting of 14 tests to evaluate randomness of long sequences. As LS-14 employs all three approaches: testing the entire long sequence, testing the collection of fixed length partitions of it, and finally, testing the collection obtained by the dynamic partitions of it, the proposed LS-14 test suit differs from all existing suites. Mutual comparisons of all 14 tests in the LS-14 suite, with each other are computed. Moreover, results obtained from the proposed test suite and NIST SP800-22 suite are compared. Examples of sequences with certain patterns which are not observed by NIST SP800-22 suite but detected by the proposed test suite are given.Article Unit and Idempotent Additive Maps Over Countable Linear Transformations(Hacettepe Univ, Fac Sci, 2024) Gümüsel, Günselı; Kosan, Tamer; Zemlicka, JanLet V be a countably generated right vector space over a field F and σ ∈ End(VF ) be a shift operator. We show that there exist a unit u and an idempotent e in End(VF ) such that 1−u,σ−u are units in End(VF) and 1−e,σ−e are idempotents in End(VF). We also obtain that if D is a division ring D Z2, Z3 and VD is a D-module, then for every α ∈ End(VD) there exists a unit u ∈ End(VD) such that 1−u,α−u are units in End(VD).
