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Permanent URI for this collectionhttps://hdl.handle.net/20.500.14288/3
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Publication Metadata only On maximal partial Latin hypercubes(Springer, 2023) Donovan, Diane M.; Grannell, Mike J.; Department of Mathematics; Yazıcı, Emine Şule; Department of Mathematics; College of SciencesA lower bound is presented for the minimal number of filled cells in a maximal partial Latin hypercube of dimension d and order n. The result generalises and extends previous results for d= 2 (Latin squares) and d= 3 (Latin cubes). Explicit constructions show that this bound is near-optimal for large n> d . For d> n , a connection with Hamming codes shows that this lower bound gives a related upper bound for the same quantity. The results can be interpreted in terms of independent dominating sets in certain graphs, and in terms of codes that have covering radius 1 and minimum distance at least 2.Publication Metadata only Can Nth order Born approximation be exact?(IOP Publishing Ltd, 2024) Loran, Farhang; Department of Mathematics; Mostafazadeh, Ali; Department of Mathematics; College of SciencesFor the scattering of scalar waves in two and three dimensions and electromagnetic waves in three dimensions, we identify a condition on the scattering interaction under which the Nth order Born approximation gives the exact solution of the scattering problem for some N >= 1.Publication Metadata only QC-LDPC codes from difference matrices and difference covering arrays(IEEE-Inst Electrical Electronics Engineers Inc, 2023) Donovan, Diane M.; Rao, Asha; Üsküplü, Elif; Department of Mathematics; Yazıcı, Emine Şule; Department of Mathematics; ; College of Sciences;We give a framework that generalizes LDPC code constructions using transversal designs or related structures such as mutually orthogonal Latin squares. Our constructions offer a broader range of code lengths and codes rates. Similar earlier constructions rely on the existence of finite fields of order a power of a prime, which significantly restricts the functionality of the resulting codes. In contrast, the LDPC codes constructed here are based on difference matrices and difference covering arrays, structures that are available for any order a, resulting in LDPC codes across a broader class of parameters, notably length a(a - 1), for all even a. Such values are not possible with earlier constructions, thus establishing the novelty of these new constructions. Specifically the codes constructed here satisfy the RC constraint and for a odd, have length a(2) and rate 1 - (4a - 3)/a(2), and for a even, length a(2) - a and rate at least 1 - (4a - 6)/(a(2 )- a). When 3 does not divide a, these LDPC codes have stopping distance at least 8. When a is odd and both 3 and 5 do not divide a, our construction delivers an infinite family of QC-LDPC codes with minimum distance at least 10. We also determine lower bounds for the stopping distance of the code. Further we include simulation results illustrating the performance of our codes. The BER and FER performance of our codes over AWGN (via simulation) is at least equivalent to codes constructed previously.Publication Metadata only Uniform syndeticity in multiple recurrence(CAMBRIDGE UNIV PRESS, 2024) Pan, Minghao; Department of Mathematics; Jamneshan, Asgar; Department of Mathematics; College of SciencesThe main theorem of this paper establishes a uniform syndeticity result concerning the multiple recurrence of measure-preserving actions on probability spaces. More precisely, for any integers d, l >= 1 and any epsilon > 0, we prove the existence of delta > 0 and K >= 1 (dependent only on d, l, and epsilon) such that the following holds: Consider a solvable group Gamma of derived length l, a probability space (X, mu), and d pairwise commuting measure-preserving Gamma-actions T-1, & mldr;, T-d on (X, mu). Let E be a measurable set in X with mu(E) >= epsilon. Then, K many (left) translates of {gamma is an element of Gamma: mu (T-1(gamma-1 )(E)boolean AND T-2(gamma-1)degrees T-1(gamma-1 )(E) boolean AND center dot center dot center dot boolean AND T-d(gamma-1 )degrees T-d-1(gamma-1 )degrees center dot center dot center dot degrees T-1(gamma-1 )(E)) >= delta} cover Gamma. This result extends and refines uniformity results by Furstenberg and Katznelson. As a combinatorial application, we obtain the following uniformity result. For any integers d, l >= 1 and any epsilon>0, there are delta>0 and K >= 1 (dependent only on d, l, and epsilon) such that for all finite solvable groups G of derived length l and any subset E subset of G(d) with m(circle times d)(E) >= epsilon (where m is the uniform measure on G), we have that K-many (left) translates of {g is an element of G:m(circle times d)({(a(1), & mldr;, a(n)) is an element of G(d): (a(1), & mldr;, a(n)), (ga(1), a(2), & mldr;, a(n)), & mldr;, (ga(1), ga(2), & mldr;, ga(n)) is an element of E}) >= delta} cover G. The proof of our main result is a consequence of an ultralimit version of Austin's amenable ergodic Szemeredi theorem.Publication Metadata only Consistent treatment of quantum systems with a time-dependent Hilbert space(MDPI, 2024) Department of Mathematics; Mostafazadeh, Ali; Department of Mathematics; College of SciencesWe consider some basic problems associated with quantum mechanics of systems having a time-dependent Hilbert space. We provide a consistent treatment of these systems and address the possibility of describing them in terms of a time-independent Hilbert space. We show that in general the Hamiltonian operator does not represent an observable of the system even if it is a self-adjoint operator. This is related to a hidden geometric aspect of quantum mechanics arising from the presence of an operator-valued gauge potential. We also offer a careful treatment of quantum systems whose Hilbert space is obtained by endowing a time-independent vector space with a time-dependent inner product. © 2024 by the author.Publication Metadata only Infinitesimal dilogarithm on curves over truncated polynomial rings(Mathematical Sciences Publishers, 2024) Department of Mathematics; Ünver, Sinan; Department of Mathematics; College of SciencesWe construct infinitesimal invariants of thickened one dimensional cycles in three dimensional space, which are the simplest cycles that are not in the Milnor range. This generalizes Park’s work on the regulators of additive cycles. The construction also allows us to prove the infinitesimal version of the strong reciprocity conjecture for thickenings of all orders. Classical analogs of our invariants are based on the dilogarithm function and our invariant could be seen as their infinitesimal version. Despite this analogy, the infinitesimal version cannot be obtained from their classical counterparts through a limiting process.Publication Metadata only Deformations of Bloch groups and Aomoto dilogarithms in characteristic p(Academic Press Inc Elsevier Science, 2011) Department of Mathematics; Ünver, Sinan; Faculty Member; Department of Mathematics; College of Sciences; 177871In this paper, we study the Bloch group B-2(F[epsilon](2)) over the ring of dual numbers of the algebraic closure of the field with p elements, for a prime p >= 5. We show that a slight modification of Kontsevich's 11/2-logarithm defines a function on B-2(F[epsilon](2)). Using this function and the characteristic p version of the additive dilogarithm function that we previously defined, we determine the structure of the infinitesimal part of B-2(Ff[epsilon](2)) completely. This enables us to define invariants on the group of deformations of Aomoto dilogarithms and determine its structure. This final result might be viewed as the analog of Hilbert's third problem in characteristic p.Publication Metadata only Asymptotic behavior of the irrational factor(Springer, 2008) Ledoan, A. H.; Zaharescu, Alexandru; Department of Mathematics; Alkan, Emre; Faculty Member; Department of Mathematics; College of Sciences; 32803We study the irrational factor function I(n) introduced by Atanassov and defined by I(n) = Pi(k)(k=1)p(v)(1/alpha v), where n = Pi(k)(v=1) p(v)(alpha v) is the prime factorization of n. We show that the sequence {G(n)/n}(n >= 1), where G(n) = Pi(n)(v=1) I(v)(1/n), is covergent; this answers a question of Panaitopol. We also establish asymptotic formulas for averages of the function I(n).Publication Metadata only Pseudoduality and complex geometry in sigma models(World Scientific Publ Co Pte Ltd, 2013) Department of Mathematics; Sarısaman, Mustafa; Researcher; Department of Mathematics; College of Sciences; 217652We study the pseudoduality transformations in two-dimensional N = (2, 2) sigma models on Kahler manifolds. We show that structures on the target space can be transformed into the pseudodual manifolds by means of (anti) holomorphic preserving mapping. This map requires that torsions related to individual spaces and riemann connection on pseudodual manifold must vanish. We also consider holomorphic isometries which puts additional constraints on the pseudoduality.Publication Metadata only Statistical origin of pseudo-Hermitian supersymmetry and pseudo-hermitian fermions(Iop Publishing Ltd, 2004) Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231We show that the metric operator for a pseudo-supersymmetric Hamiltonian that has at least one negative real eigenvalue is necessarily indefinite. We introduce pseudo-Hermitian fermion (phermion) and abnormal phermion algebras and provide a pair of basic realizations of the algebra of N = 2 pseudo-supersymmetric quantum mechanics in which pseudo-supersymmetry is identified with either a boson-phermion or a boson-abnormal-phermion exchange symmetry. We further establish the physical equivalence (nonequivalence) of phermions (abnormal phermions) with ordinary fermions, describe the underlying Lie algebras and study multi-particle systems of abnormal phermions. The latter provides a certain bosonization of multifermion systems.