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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 A characterization of heaviness in terms of relative symplectic cohomology(Wiley, 2024) Mak, Cheuk Yu; Sun, Yuhan; Department of Mathematics; Varolgüneş, Umut; Department of Mathematics; College of SciencesFor a compact subset K$K$ of a closed symplectic manifold (M,omega)$(M, \omega)$, we prove that K$K$ is heavy if and only if its relative symplectic cohomology over the Novikov field is nonzero. As an application, we show that if two compact sets are not heavy and Poisson commuting, then their union is also not heavy. A discussion on superheaviness together with some partial results is also included.Publication Metadata only Orthogonal cycle systems with cycle length less than 10(John Wiley and Sons Inc, 2024) Department of Mathematics; Küçükçifçi, Selda; Yazıcı, Emine Şule; Department of Mathematics; College of SciencesAn H-decomposition of a graph G is a partition of the edge set of G into subsets, where each subset induces a copy of the graph H. A k-orthogonal H-decomposition of G is a set of kH-decompositions of G such that any two copies of H in distinct H-decompositions intersect in at most one edge. When G = K-v, we call the H-decomposition an H-system of order v. In this paper, we consider the case H is an l-cycle and construct a pair of orthogonal l-cycle systems for all admissible orders when l is an element of {5, 6, 7, 8, 9}, except when l = v.Publication Metadata only Complex vs. convex Morse functions and geodesic open books(World Scientific, 2024) Dehornoy, Pierre; Department of Mathematics; Özbağcı, Burak; Department of Mathematics; College of SciencesSuppose that Sigma is a closed and oriented surface equipped with a Riemannian metric. In the literature, there are three seemingly distinct constructions of open books on the unit (co)tangent bundle of Sigma, having complex, contact and dynamical flavors, respectively. Each one of these constructions is based on either an admissible divide or an ordered Morse function on Sigma. We show that the resulting open books are pairwise isotopic provided that the ordered Morse function is adapted to the admissible divide on Sigma. Moreover, we observe that if Sigma has positive genus, then none of these open books are planar and furthermore, we determine the only cases when they have genus one pages.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 Stopping levels for a spectrally negative Markov additive process(Springer Science and Business Media Deutschland GmbH, 2024) Vardar-Acar, C.; Department of Mathematics; Çağlar, Mine; Department of Mathematics; College of SciencesThe optimal stopping problem for pricing Russian options in finance requires taking the supremum of the discounted reward function over all finite stopping times. We assume the logarithm of the asset price is a spectrally negative Markov additive process with finitely many regimes. The reward function is given by the exponential of the running supremum of the price process. Previous work on Russian optimal stopping problem suggests that the optimal stopping time would be an upcrossing time of the drawdown at a certain level for each regime. We derive explicit formulas for identifying the stopping levels and computing the corresponding value functions through a recursive algorithm. A numerical is provided for finding these stopping levels and their value functions.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 The triangle intersection problem for K4 - E designs(Utilitas Mathematica Publishing Inc., 2007) Billington, Elizabeth J.; Lindner, C. C.; Department of Mathematics; Yazıcı, Emine Şule; Faculty Member; Department of Mathematics; College of Sciences; 27432An edge-disjoint decomposition of the complete graph Kn into copies of K4 - e, the simple graph with four vertices and five edges, is known to exist if and only if n ≡ 0 or 1 (mod 5) and n ≥ 6 (Bermond and Schönheim, Discrete Math. 19 (1997)). The intersection problem for K4 - e designs has also been solved (Billington, M. Gionfriddo and Lindner, J. Statist. Planning Inference 58 (1997)); this problem finds the number of common K4 - e blocks which two K4 - e designs on the same set may have. Here we answer the question: how many common triangles may two K4 - e designs on the same set have? Since it is possible for two K4 - e designs on the same set to have no common K4 - e blocks and yet some positive number of common triangles, this problem is largely independent of the earlier K4 - e intersection result.Publication Metadata only Counterexamples to regularity of Mane projections in the theory of attractors(Institute of Physics (IOP) Publishing, 2013) Eden, Alp; Zelik, Sergey V.; Department of Mathematics; Kalantarov, Varga; Faculty Member; Department of Mathematics; College of Sciences; 117655This paper is a study of global attractors of abstract semilinear parabolic equations and their embeddings in finite-dimensional manifolds. As is well known, a sufficient condition for the existence of smooth (at least C-1-smooth) finite-dimensional inertial manifolds containing a global attractor is the so-called spectral gap condition for the corresponding linear operator. There are also a number of examples showing that if there is no gap in the spectrum, then a C-1-smooth inertial manifold may not exist. on the other hand, since an attractor usually has finite fractal dimension, by Mane's theorem it projects bijectively and Holder-homeomorphically into a finite-dimensional generic plane if its dimension is large enough. It is shown here that if there are no gaps in the spectrum, then there exist attractors that cannot be embedded in any Lipschitz or even log-Lipschitz finite-dimensional manifold. Thus, if there are no gaps in the spectrum, then in the general case the inverse Mane projection of the attractor cannot be expected to be Lipschitz or log-Lipschitz. Furthermore, examples of attractors with finite Hausdorff and infinite fractal dimension are constructed in the class of non-linearities of finite smoothness.Publication Metadata only Addendum to “On the mean square average of special values of L-functions” [J. Number Theory 131 (8) (2011) 1470–1485](Elsevier, 2011) Department of Mathematics; Alkan, Emre; Faculty Member; Department of Mathematics; College of Sciences; 32803