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Publication Metadata only On regular embedding of H-designs into G-designs(Utilitas Mathematica, 2013) Quattrocchi, Gaetano; Department of Mathematics; Department of Mathematics; Department of Mathematics; Küçükçifçi, Selda; Smith, Benjamin R.; Yazıcı, Emine Şule; Faculty Member; Researcher; Faculty Member; Department of Mathematics; College of Sciences; College of Sciences; College of Sciences; 105252; N/A; 27432The graph H is embedded in the graph G, if H is a subgraph of G. An H-design is a decomposition of a complete graph into edge disjoint copies of the graph H, called blocks. An H-i-design with k blocks, say H-1, H-2, ...H-k is embedded in a G-design if for every H-i, there exists a distinct block, say G(i), in the G-design that embeds H-i. If G(i) are all isomorphic for 1 <= i <= k then the embedding is called regular. This paper solves the problem of the regular embedding of H-designs into G-designs when G has at most four vertices and four edges.Publication Metadata only The full metamorphosis of lambda-fold block designs with block size four into lambda-fold kite systems(Utilitas Mathematica Publishing, 2013) N/A; Department of Mathematics; Department of Mathematics; Department of Mathematics; Küçükçifçi, Selda; Smith, Benjamin R.; Yazıcı, Emine Şule; Faculty Member; Researcher; Faculty Member; Department of Mathematics; College of Sciences; College of Sciences; College of Sciences; 105252; N/A; 27432Let(X,B)be a λ-fold block design with block size 4. If a path of length two is removed from each block of B the resulting collection of kites K is a partial λ-fold kite system(X,K). If the deleted edges can be arranged into a collection of kites D,then(X,K ∪ D)is a λ-fold kite system [5]. Now for each block 6 ∈ B let {P1(6),P 2(b),P3(b)} be a partition of b into paths of length two and define for each i = 1,2,3, sets Ki and Di as follows: for each b ∈ B,put the kite b\Pi(b)in Ki and the two edges belonging to the path Pi(b)in Di. If the edges in Di can be arranged into a collection of kites Di * then Mi =(X,Ki∪Di *)is a λ-fold kite system,called the ith metamorphosis of(X,B). The full metamorphosis is the set of three metamorphoses {M 1,M2,M3}. We give a complete solution of the following problem: for which n and A does there exist a λ-fold block design with block size 4 having a full metamorphosis into a λ-fold kite system?Publication Metadata only The metamorphosis of lambda-fold block designs with block size four into maximum packings of lambda K-n with kites(Util Math Publ Inc, 2005) N/A; Department of Mathematics; Küçükçifçi, Selda; Faculty Member; Department of Mathematics; College of Sciences; 105252Let (X, B) be a lambda-fold block design with block size four and define sets B(K) and E(K-4 \ K) as follows: for each block b is an element of B, remove a path of length two, obtain a kite (a triangle with a tail), and place the kites in B(K) and the paths of length 2 in E(K-4 \ K). If we can reassemble the edges belonging to E(K-4 \ K) into a collection of kites E(K) with leave L, then (X, B(K) boolean OR E(K), L) is a packing of lambda K-n with kites. If vertical bar L vertical bar is as small as possible, then (X, B(K) boolean OR E(K), L) is called a metamorphosis of the lambda-fold block design (X, B) into a maximum packing of lambda K-n with kites. In this paper we give a complete solution of the metamorphosis problem for lambda-fold block designs with block size four into a maximum packing of lambda K-n with kites for all lambda. That is, for each lambda we determine the set of all n such that there exists a lambda-fold block design of order n having a metamorphosis into a maximum packing of lambda K-n with kites.Publication Metadata only The parallel genetic algorithm for designing dna randomizations in a combinatorial protein experiment(Springer-Verlag Berlin, 2006) Blazewicz, Jacek; Dziurdza, Beniamin; Markiewicz, Wojciech T; Department of Industrial Engineering; Oğuz, Ceyda; Faculty Member; Department of Industrial Engineering; College of Engineering; 6033Evolutionary methods of protein engineering such as phage display have revolutionized drug design and the means of studying molecular binding. In order to obtain the highest experimental efficiency, the distributions of constructed combinatorial libraries should be carefully adjusted. The presented approach takes into account diversity- completeness trade-off and tries to maximize the number of new amino acid sequences generated in each cycle of the experiment. In the paper, the mathematical model is introduced and the parallel genetic algorithm for the defined optimization problem is described. Its implementation on the SunFire 6800 computer proves a high efficiency of the proposed approach.Publication Metadata only Balanced and strongly balanced 4-kite designs(Utilitas Mathematica Publishing, 2013) Gionfriddo, Mario; Milazzo, Lorenzo; Department of Mathematics; Küçükçifçi, Selda; Faculty Member; Department of Mathematics; College of Sciences; 105252A G-design is called balanced if the degree of each vertex x is a constant. A G-design is called strongly balanced if for every i = 1, 2, ⋯, h, there exists a constant Ci such that dAi(x)= Ci for every vertex x, where AiS are the orbits of the automorphism group of G on its vertex-set and dAi(x) of a vertex is the number of blocks of containing x as an element of Ai. We say that a G-design is simply balanced if it is balanced, but not strongly balanced. In this paper we determine the spectrum of simply balanced and strongly balanced 4-kite designs.Publication Metadata only The full metamorphosis of lambda-fold K-4-designs into lambda-fold 3-star systems(Util Math Publ Inc, 2019) N/A; Department of Mathematics; Küçükçifçi, Selda; Faculty Member; Department of Mathematics; College of Sciences; 105252Let (X, B) be a lambda-fold K-4-design. If a triangle is removed from each block of B the resulting collection of 3-stars; S, is a partial lambda-fold 3-star system; (X, S). If the edges belonging to the deleted triangles can be arranged into a collection of 3-stars T*, then (X, S boolean OR T*) is a lambda-fold 3-star system, called a metamorphosis of the lambda-fold K-4-design (X, B) into a lambda-fold 3-star system. Label the elements of each block b with b(1), b(2), b(3) and b(4) (in any manner). For each i = 1,2, 3, 4 define a set of triangles T-i and a set of stars S-i as follows: for each block b = [b(1), b(2), b(3), b(4)] belonging to B, partition b into a star centered at b(i) and the triangle b\b(i), then place the star in S-i and the triangle in T-i. (X, S-i) forms a partial lambda-fold 3-star system. Now if the edges belonging to the triangles in T-i can be arranged into a collection of stars T-i* then (X, S-i boolean OR T-i*) is a lambda-fold 3-star system and we say that M-i = (X, S-i boolean OR T-i*) is the ith metamorphosis of (X, B). The full metamorphosis of (X, B) is the set of four metamorphoses {M-1, M-2, M-3, M-4}. The purpose of this work is to give a complete solution of the following problem: For which n and lambda does there exist a lambda-fold K-4-design having a full metamorphosis into lambda-fold 3-star systems?Publication Metadata only Computation of the canonical lifting via division polynomials(Union Matematica Argentina, 2015) Department of Mathematics; Erdoğan, Altan; Teaching Faculty; Department of Mathematics; College of Sciences; N/AWe study the canonical lifting of ordinary elliptic curves over the ring of Witt vectors. We prove that the canonical lifting is compatible with the base field of the given ordinary elliptic curve which was first proved in Finotti, J. Number Theory 130 (2010), 620-638. We also give some results about division polynomials of elliptic curves de fined over the ring of Witt vectors.