Department of Mathematics

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Open Access
2 + 1 KdV(N) equations
(American Institute of Physics, 2011) Gürses, M.; Pekcan, A.
We present some nonlinear partial differential equations in 2 + 1-dimensions derived from the KdV equation and its symmetries. We show that all these equations have the same 3-soliton solution structures. The only difference in these solutions are the dispersion relations. We also show that they possess the Painlevé property. © 2011 American Institute of Physics.
Open Access
(2 + 1)-dimensional AKNS(−N) systems II
(Elsevier BV, 2021-06) Gürses, Metin; Pekcan, Aslı
In our previous work (Gürses and Pekcan, 2019, [40]) we started to investigate negative AKNS(−N) hierarchy in (2 + 1)-dimensions. We were able to obtain only the first three, N = 0, 1, 2, members of this hierarchy. The main difficulty was the nonexistence of the Hirota formulation of the AKNS(N) hierarchy for N ≥ 3. Here in this work we overcome this difficulty for N = 3, 4 and obtain Hirota bilinear forms of (2 + 1)-dimensional AKNS(−N) equations for these members. We study the local and nonlocal reductions of these systems of equations and obtain several new integrable local and nonlocal equations in (2 + 1)- dimensions. We also give one-, two-, and three-soliton solutions of the reduced equations
Open Access
(2+1)-dimensional local and nonlocal reductions of the negative AKNS system: soliton solutions
(Elsevier, 2018) Gürses, Metin; Pekcan, A.
Wefirstconstructa(2+1)dimensionalnegativeAKNShierarchyandthenwegiveallpossiblelocaland(discrete)nonlocalreductionsoftheseequations.WefindHirotabilinearformsofthenegativeAKNShierarchyandgiveone-andtwo-solitonsolutions.ByusingthesolitonsolutionsofthenegativeAKNShierarchywefindone-solitonsolutionsofthelocalandnonlocalreducedequations.
Open Access
2-Killing vector fields on warped product manifolds
(World Scientific Publishing, 2015) Shenawy, S.; Ünal, B.
This paper provides a study of 2-Killing vector fields on warped product manifolds as well as characterization of this structure on standard static and generalized Robertson-Walker space-times. Some conditions for a 2-Killing vector field on a warped product manifold to be parallel are obtained. Moreover, some results on the curvature of a warped product manifolds in terms of 2-Killing vector fields are derived. Finally, we apply our results to describe 2-Killing vector fields of some well-known warped product space-time models. © 2015 World Scientific Publishing Company.
Open Access
800 conics on a smooth quartic surface
(Elsevier BV * North-Holland, 2022-03-10) Degtyarev, Alex
We construct an example of a smooth spatial quartic surface that contains 800 irreducible conics. © 2022 Elsevier B.V.
Open Access
A mathematical interpretation for outbreaks of bacterial meningitis under the effect of time-dependent transmission parameters
(Springer Science and Business Media B.V., 2023-06-02) Türkün, C.; Gölgeli, M.; Atay, Fatihcan Mehmet
We consider a SIR-type compartmental model divided into two age classes to explain the seasonal exacerbations of bacterial meningitis, especially among children outside of the meningitis belt. We describe the seasonal forcing through time-dependent transmission parameters that may represent the outbreak of the meningitis cases after the annual pilgrimage period (Hajj) or uncontrolled inflows of irregular immigrants. We present and analyse a mathematical model with time-dependent transmission. We consider not only periodic functions in the analysis but also general non-periodic transmission processes. We show that the long-time average values of transmission functions can be used as a stability marker of the equilibrium. Furthermore, we interpret the basic reproduction number in case of time-dependent transmission functions. Numerical simulations support and help visualize the theoretical results. © 2023, The Author(s), under exclusive licence to Springer Nature B.V.
Open Access
A simplicial category for higher correspondences
(Springer Science and Business Media B.V., 2022-12-27) Haderi, Redi
In this work we propose a realization of Lurie’s prediction that inner fibrations p : X → A are classified by A-indexed diagrams in a “higher category” whose objects are ∞-categories, morphisms are correspondences between them and higher morphisms are higher correspondences.We will obtain this as a corollary of a more general result which classifies all simplicial maps between ordinary simplicial sets in a similar fashion. Correspondences between simplicial sets (and ∞-categories) are a generalization of the concept of profunctor (or bimodule) pertaining to categories. While categories, functors and profunctors are organized in a double category, we will exhibit simplicial sets, simplicial maps, and correspondences as part of a simplicial category. This allows us to make precise statements and provide proofs. Our main tool is the language of double categories, which we use in the context of simplicial categories as well.
Open Access
About curvature, conformal metrics and warped products
(Institute of Physics Publishing Ltd., 2007) Dobarro, F.; Ünal, B.
We consider the curvature of a family of warped products of two pseduo-Riemannian manifolds (B, gB) and (F, gF) furnished with metrics of the form c2gB ⊕ w2g F and, in particular, of the type w2μgB ⊕ w2gF, where c, w:B → (0, ∞) are smooth functions and μ is a real parameter. We obtain suitable expressions for the Ricci tensor and scalar curvature of such products that allow us to establish results about the existence of Einstein or constant scalar curvature structures in these categories. If (B, gB) is Riemannian, the latter question involves nonlinear elliptic partial differential equations with concave-convex nonlinearities and singular partial differential equations of the Lichnerowicz-York-type among others. © 2007 IOP Publishing Ltd.
Open Access
Absence of phase transitions in one-dimensional antiferromagnetic models with long-range interactions
(Kluwer Academic Publishers-Plenum Publishers, 1993) Kerimov, A.
The absence of phase transitions in a one-dimensional model with long-range antiferromagnetic potential is established at low temperatures when the ground states have a rational density. A description of the set of all ground states and typical configurations is given. © 1993 Plenum Publishing Corporation.
Open Access
Absolute continuity for operator valued completely positive maps on C *-algebras
(2009) Gheondea, A.; Kavruk, A. Ş.
Motivated by applicability to quantum operations, quantum information, and quantum probability, we investigate the notion of absolute continuity for operator valued completely positive maps on C* -algebras, previously introduced by Parthasarathy [in Athens Conference on Applied Probability and Time Series Analysis I (Springer-Verlag, Berlin, 1996), pp. 34-54]. We obtain an intrinsic definition of absolute continuity, we show that the Lebesgue decomposition defined by Parthasarathy is the maximal one among all other Lebesgue-type decompositions and that this maximal Lebesgue decomposition does not depend on the jointly dominating completely positive map, we obtain more flexible formulas for calculating the maximal Lebesgue decomposition, and we point out the nonuniqueness of the Lebesgue decomposition as well as a sufficient condition for uniqueness. In addition, we consider Radon-Nikodym derivatives for absolutely continuous completely positive maps that, in general, are unbounded positive self-adjoint operators affiliated to a certain von Neumann algebra, and we obtain a spectral approximation by bounded Radon-Nikodym derivatives. An application to the existence of the infimum of two completely positive maps is indicated, and formulas in terms of Choi's matrices for the Lebesgue decomposition of completely positive maps in matrix algebras are obtained. © 2009 American Institute of Physics.
Open Access
Accelerated Born-Infeld metrics in Kerr-Schild geometry
(2003) Gürses, M.; Sarioǧlu Ö.
We consider Einstein Born-Infeld theory with a null fluid in Kerr-Schild geometry. We find accelerated charge solutions of this theory. Our solutions reduce to the Plebański solution when the acceleration vanishes and to the Bonnor-Vaidya solution as the Born-Infeld parameter b goes to infinity. We also give the explicit form of the energy flux formula due to the acceleration of the charged sources.
Open Access
Accelerated charge Kerr-Schild metrics in D dimensions
(Institute of Physics Publishing Ltd., 2002) Gürses M.; Sarioǧlu, Ö.
We consider the D-dimensional Einstein-Maxwell theory with a null fluid in Kerr-Schild geometry. We obtain a complete set of differential conditions that are necessary for finding the solutions. We examine the case of vanishing pressure and cosmological constant in detail. For this specific case, we give the metric, the electromagnetic vector potential and the fluid energy density. This is, in fact, the generalization of the well-known Bonnor-Vaidya solution to arbitrary D dimensions. We show that due to the acceleration of charged sources, there is an energy flux in D ≥ 4 dimensions and we give the explicit form of this energy flux formula.
Open Access
Accelerated Levi-Civita-Bertotti-Robinson metric in D dimensions
(Springer New York LLC, 2005) Gürses, M.; Sarıoǧlu, Ö.
A conformally flat accelerated charge metric is found in an arbitrary dimension D. It is a solution of the Einstein-Maxwell-null fluid equations with a cosmological constant in D ≥ 4 dimensions. When the acceleration is zero, our solution reduces to the Levi-Civita-Bertotti-Robinson metric. We show that the charge loses its energy, for all dimensions, due to the acceleration.
Open Access
Accerelated Born-Infield Metrics in Kerr-Schild Geometry
(Institute of Physics Publishing Ltd., 2003) Gürses M.; Sarıoğlu, Ö.
We consider Einstein Born–Infeld theory with a null fluid in Kerr–Schild geometry. We find accelerated charge solutions of this theory. Our solutions reduce to the Plebanski solution when the acceleration vanishes and to the ´ Bonnor–Vaidya solution as the Born–Infeld parameter b goes to infinity. We also give the explicit form of the energy flux formula due to the acceleration of the charged sources.
Open Access
AdS waves as exact solutions to quadratic gravity
(American Physical Society, 2011-04-08) Güllü, I.; Gürses, M.; Sişman, T.C.; Tekin, B.
We give an exact solution of the quadratic gravity in D dimensions. The solution is a plane-fronted wave metric with a cosmological constant. This metric solves not only the full quadratic gravity field equations but also the linearized ones which include the linearized equations of the recently found critical gravity. A subset of the solutions change the asymptotic structure of the anti-de Sitter space due to their logarithmic behavior. © 2011 American Physical Society.
Open Access
AdS-Plane wave and pp-wave solutions of generic gravity theories
(American Physical Society, 2014-12-02) Gürses M.; Şişman, T. Ç.; Tekin, B.
We construct the anti–de Sitter-plane wave solutions of generic gravity theory built on the arbitrary powers of the Riemann tensor and its derivatives in analogy with the pp-wave solutions. In constructing the wave solutions of the generic theory, we show that the most general two-tensor built from the Riemann tensor and its derivatives can be written in terms of the traceless Ricci tensor. Quadratic gravity theory plays a major role; therefore, we revisit the wave solutions in this theory. As examples of our general formalism, we work out the six-dimensional conformal gravity and its nonconformal deformation as well as the tricritical gravity, the Lanczos-Lovelock theory, and string-generated cubic curvature theory.
Open Access
Alcahestic subalgebras of the alchemic algebra and a correspondence of simple modules
(Elsevier Inc., 2008) Coşkun, Olcay
The unified treatment of the five module-theoretic notions, transfer, inflation, transport of structure by an isomorphism, deflation and restriction, is given by the theory of biset functors, introduced by Bouc. In this paper, we construct the algebra realizing biset functors as representations. The algebra has a presentation similar to the well-known Mackey algebra. We adopt some natural constructions from the theory of Mackey functors and give two new constructions of simple biset functors. We also obtain a criterion for semisimplicity in terms of the biset functor version of the mark homomorphism. The criterion has an elementary generalization to arbitrary finite-dimensional algebras over a field.
Open Access
The Alexander module of a trigonal curve
(Eurpean Mathematical Society, 2014) Degtyarev, A.
We describe the Alexander modules and Alexander polynomials (both over ℚ and over finite fields Fp) of generalized trigonal curves. The rational case is completely resolved; in the case of characteristic p > 0, a few points remain open. The results obtained apply as well to plane curves with deep singularities. © European Mathematical Society.
Open Access
Almost p-structures on vector-bundles
(Cambridge University Press, 2003) Dibag, I.
For p ≥ 2 we introduce the notion of an almost p-structure on vector-bundles which generalizes the notion of an almost-complex structure and investigate the existence of almost p-structures on spheres and complex projective spaces.
Open Access
Almost unit-clean rings
(Editura Academiei Romane, 2019) Chen, H.; Köse, H.; Kurtulmaz, Yosum
A ring R is almost unit-clean provided that every element in R is equivalent to the sum of an idempotent and a regular element. We investigate conditions under which a ring is almost unit-clean. We prove that every ring in which every zero-divisor is strongly _-regular is almost unit-clean and every matrix ring of elementary divisor domains is almost unit-clean. Furthermore, it is shown that the trivial extension R(M) of a commutative ring R and an R-module M is almost unit-clean if and only if each x 2 R can be written in the form ux = r+e where u 2 U(R); r 2 R 􀀀 (Z(R) [ Z(M)) and e 2 Id(R). We thereby construct many examples of such rings.