Department of Mathematics

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  • ItemOpen Access
    Erratum: On the extremal points of the Λ-Polytopes and classical simulatıon of quantum computation with magic states
    (2022-05) Cihan, Okay; Zurel, Michael; Raussendorf, Robert; Cihan, Okay
    We will fix an error in the proof of Theorem 2 of the work On the extremal points of the $\Lambda $-polytopes and classical simulation of quantum computation with magic states by the current authors, published in Quantum Information and Computation Vol.21 No.13\&14, 1533-7146 (2021). The theorem as it is stated is still correct, however there is a gap in the proof that needs to be filled.
  • ItemOpen Access
    Tritangents to smooth sextic curves
    (Association des Annales de l'Institut Fourier, 2022-10-21) Degtyarev, Alex; Degtyarev, Alex
    We prove that a smooth plane sextic curve can have at most 72 tritangents, whereas a smooth real sextic may have at most 66 real tritangents. © 2022 Association des Annales de l'Institut Fourier. All rights reserved.
  • ItemOpen Access
    Optimizing foreign exchange reserves: Protection against external shocks in Ghana
    (Frontiers Media S.A., 2022-11-02) Abdul-Rahaman, Abdul-Rashid; Hongxing, Yao; Alhassan Alolo Akeji, Abdul-Rasheed; Ayamba, Emmanuel Caesar; Bernard Pea-Assounga, Jean Baptiste; Alhassan, Mohammed Kamil; Alhassan, Mohammed Kamil
    Using Least Square Residual Minimization techniques, this paper develops an optimal reserve model, known as the OPREM model, which is essential in optimizing the costs of reserve holding. The paper also sets-out to test and compare the relative predictions of economic trends of the OPREM model as well as the predictions of alternative models in literature. Establishing the predictive accuracy of economic trends of these models are crucial for the gradual and cost-effective accumulation of reserves. The research concludes that, the decision to optimize the cost of reserves under a stable currency environment is reliant on the gold impact factor and not on inflation or interest rates. We also found on further analysis of the OPREM that the OPREM model is better positioned to eliminate the procyclicality and perverse rush in reserve build-ups experienced in developing and emerging countries by effectively setting the reserve stock against economic trends. The research fixes the optimal reserves around a benchmark of 0.7–1.2 of previous year's optimal value. However, in the absence of past optimal values, a benchmark between 2 and 6 times of average inflows for short-term analysis or analysis with small data observations. However, for long-term analysis or analysis with large data frequency (i.e., exceeding 13 data observations), the reserve stock should be fixed on a benchmark of 2–9 times of the average inflows. Copyright © 2022 Abdul-Rahaman, Hongxing, Alhassan Alolo Akeji, Ayamba, Bernard Pea-Assounga and Alhassan.
  • ItemOpen Access
    Dispersion estimates for the boundary integral operator associated with the fourth order Schrödinger equation posed on the half line
    (Element d.o.o., 2021-09) Özsarı, Türker; Alkan, Kıvılcım; Kalimeris, Konstantinos; Özsarı, Türker
    In this paper, we prove dispersion estimates for the boundary integral operator associated with the fourth order Schr¨odinger equation posed on the half line. Proofs of such estimates for domains with boundaries are rare and generally require highly technical approaches, as opposed to our simple treatment which is based on constructing a boundary integral operator of oscillatory nature via the Fokas method. Our method is uniform and can be extended to other higher order partial differential equations where the main equation possibly involves more than one spatial derivatives.
  • ItemOpen Access
    Geometry of information structures, strategic measures and associated stochastic control topologies
    (Institute of Mathematical Statistics, 2022) Saldı, Naci; Yüksel, Serdar; Saldı, Naci
    In many areas of applied mathematics, decentralization of information is a ubiquitous attribute affecting how to approach a stochastic optimization, decision and estimation, or control problem. In this review article, we present a general formulation of information structures under a probability theoretic and geometric formulation. We define information structures, place various topologies on them, and study closedness, compactness and convexity properties on the strategic measures induced by information structures and decentralized control/decision policies under varying degrees of relaxations with regard to access to private or common randomness. Ultimately, we present existence and tight approximation results for optimal decision/control policies. We discuss various lower bounding techniques, through relaxations and convex programs ranging from classically realizable and classically non-realizable (such as quantum theoretic and non-signaling) relaxations. For each of these, we establish closedness and convexity properties and also a hierarchy of correlation structures. As a further theme, we review and introduce various topologies on decision/ control strategies defined independent of information structures, but for which information structures determine whether the topologies entail utility in arriving at existence, compactness, convexification or approximation results. These approaches, which we term as the strategic measures approach and the control topology approach, lead to complementary results on existence, approximations and upper and lower bounds in optimal decentralized stochastic decision, estimation, and control problems © 2022. Probability Surveys.All Rights Reserved.
  • ItemOpen Access
    The interior-boundary Strichartz estimate for the Schrödinger equation on the half-line revisited
    (TÜBİTAK, 2022-01-01) Köksal, Bilge; Özsarı, Türker; Köksal, Bilge; Özsarı, Türker
    In recent papers, it was shown for the biharmonic Schrödinger equation and 2D Schrödinger equation that Fokas method-based formulas are capable of defining weak solutions of associated nonlinear initial boundary value problems (ibvps) below the Banach algebra threshold. In view of these results, we revisit the theory of interiorboundary Strichartz estimates for the Schrödinger equation posed on the right half line, considering both Dirichlet and Neumann cases. Finally, we apply these estimates to obtain low regularity solutions for the nonlinear Schrödinger equation (NLS) with Neumann boundary condition and a coupled system of NLS equations defined on the half line with Dirichlet/Neumann boundary conditions. © This work is licensed under a Creative Commons Attribution 4.0 International License.
  • ItemOpen Access
    800 conics on a smooth quartic surface
    (Elsevier BV * North-Holland, 2022-03-10) Degtyarev, Alex; Degtyarev, Alex
    We construct an example of a smooth spatial quartic surface that contains 800 irreducible conics. © 2022 Elsevier B.V.
  • ItemOpen Access
    Optimality of independently randomized symmetric policies for exchangeable stochastic teams with infinitely many decision makers
    (Institute for Operations Research and the Management Sciences (INFORMS), 2022-08-24) Sanjari, S.; Saldi, Naci; Yüksel, S.; Saldi, Naci
  • ItemOpen Access
    Conjectural invariance with respect to the fusion system of an almost-source algebra
    (Walter de Gruyter GmbH, 2022-03-23) Barker, Laurence; Gelvin, Matthew; Barker, Laurence; Gelvin, Matthew
    We show that, given an almost-source algebra 𝐴 of a 𝑝-block of a finite group 𝐺, then the unit group of 𝐴 contains a basis stabilized by the left and right multiplicative action of the defect group if and only if, in a sense to be made precise, certain relative multiplicities of local pointed groups are invariant with respect to the fusion system. We also show that, when 𝐺 is 𝑝-solvable, those two equivalent conditions hold for some almost-source algebra of the given 𝑝-block. One motive lies in the fact that, by a theorem of Linckelmann, if the two equivalent conditions hold for 𝐴, then any stable basis for 𝐴 is semicharacteristic for the fusion system.
  • ItemOpen Access
    Integrability of three dimensional gravity field equations
    (IOP Publishing Ltd, 2022-02-10) Gurses, Metin; Gurses, Metin
    We show that the tree dimensional Einstein vacuum feld equations with cosmological constant are integrable. Using the sl(2, R) valued soliton connections we obtain the metric of the spacetime in terms of the dynamical variables of the integrable nonlinear partial diferential equations. © 2021 Published under licence by IOP Publishing Ltd.
  • ItemOpen Access
    Some characterizations of Pseudo Z symmetric spacetimes
    (Univerzitet u Nisu,University of Nis, 2022) Dea, Uday Chand; Ünal, Bülent; Srivastava, Sudhir Kumar; Ünal, Bülent
    The motive of this work is to investigate pseudo Z symmetric spacetimes. At first we present some basic properties of pseudo Z symmetric spacetimes showing that the 1-forms A and B and the scalars a and b associated with the spacetime agree with a specific relation. Next we explore conditions under which a pseudo Z symmetric spacetime to be a GRW spacetime and a quasi-Einstein spacetime respectively. Also we provide some results on pseudo Ricci symmetric spacetimes.
  • ItemOpen Access
    Hedging portfolio for a market model of degenerate diffusions
    (Taylor & Francis, 2022-11-30) Çağlar, M.; Demirel, İ.; Üstünel, Süleyman; Üstünel, Süleyman
    We consider a semimartingale market model when the underlying diffusion has a singular volatility matrix and compute the hedging portfolio for a given payoff function. Recently, the representation problem for such degenerate diffusions as a stochastic integral with respect to a martingale has been completely settled. This representation and Malliavin calculus established further for the functionals of a degenerate diffusion process constitute the basis of the present work. Using the Clark–Hausmann–Bismut–Ocone type representation formula derived for these functionals, we prove a version of this formula under an equivalent martingale measure. This allows us to derive the hedging portfolio as a solution of a system of linear equations. The uniqueness of the solution is achieved by a projection idea that lies at the core of the martingale representation at the first place. We demonstrate the hedging strategy as explicitly as possible with some examples of the payoff function such as those used in exotic options, whose value at maturity depends on the prices over the entire time horizon.
  • ItemOpen Access
    Analysis of calculus textbook problems via Bloom's taxonomy
    (Taylor & Francis Inc., 2022-04-15) Pehlivan, L.; Karaali, Gizem; Karaali, Gizem
    In calculus courses, instructors often use the end-of-section problems in a textbook in homework assignments or other course assessments. As a result, these problems influence the teaching and learning of calculus. In this study, we examine the levels of cognitive demand of these problems in a mainstream calculus textbook and classify them within the framework of Bloom's Taxonomy. We provide examples of the types of problems assigned to each of the six categories in this taxonomy and share some of the deliberations that led us to these assignments. Finally, we discuss the implications of our results for teaching calculus courses. We believe that our analysis will help calculus instructors be more cognizant of the cognitive demand of problems when assigning them for homework and, as a result, help them to appropriately support, assess, and enhance their students' understanding of the topics.
  • ItemOpen Access
    Robustness and delay margin analysis of a gene regulatory network model
    (Elsevier, 2022-10-10) Öztürk, Dilan; Özbay, Hitay; Atay, Fatihcan M.; Öztürk, Dilan; Özbay, Hitay; Atay, Fatihcan M.
    In the past, a special type of nonlinear delay differential system structure was proposed for gene regulatory networks. For this cyclic dynamical system, stability analysis was done using various tools from systems theory. This paper investigates robust stability of an extended gene regulatory network model with time delayed negative feedback. Specifically, the delay margin analysis is done for this system under multiplicative uncertainty. The effect of uncertainty on the delay margin is determined. It is also shown that for particular higher order extensions of the model it is possible to improve the delay margin.
  • ItemOpen Access
    A decomposition of column-convex polyominoes and two vertex statistics
    (Springer, 2022-04-27) Cakić, N.; Mansour, T.; Yıldırım, Gökhan; Yıldırım, Gökhan
    We introduce a decomposition method for column-convex polyominoes and enumerate them in terms of two statistics: the number of internal vertices and the number of corners in the boundary. We first find the generating function for the column-convex polyominoes according to the horizontal and vertical half-perimeter, and the number of interior vertices. In particular, we show that the average number of interior vertices over all column-convex polyominoes of perimeter 2n is asymptotic to αon3 / 2 where αo≈ 0.57895563 …. We also find the generating function for the column-convex polyominoes according to the horizontal and vertical half-perimeter, and the number of corners in the boundary. In particular, we show that the average number of corners over all column-convex polyominoes of perimeter 2n is asymptotic to α1n where α1≈ 1.17157287 …. © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
  • ItemOpen Access
    Variations on a theme of Mirsky
    (World Scientific Publishing, 2022-07-05) Akbal, Yıldırım; Güloğlu, Ahmet; Güloğlu, Ahmet
    Let k and r be non-zero integers with r≥2. An integer is called r-free if it is not divisible by the rth power of a prime. A result of Mirsky states that there are infinitely many primes p such that p+k is r-free. In this paper, we study an additive Goldbach-type problem and prove two uniform distribution results using these primes. We also study certain properties of primes p such that p+a1,…,p+aℓ are simultaneously r-free, where a1,…,aℓ are non-zero integers and ℓ≥1 .
  • ItemOpen Access
    Möbius-invariant harmonic function spaces on the unit disc
    (Springer, 2022-03) Kaptanoğlu, Hakkı Turgay; Kaptanoğlu, Hakkı Turgay
    We investigate and identify Möbius-invariant harmonic function spaces on the unit disc. We derive their fundamental properties and establish connections among various topologies on them. We show that the harmonic Bloch space b∞ is the “maximal” and the Besov space b−21 is the minimal invariant complete seminormed space. There is only one invariant semi-Hilbert space and it is the harmonic Dirichlet space b−22. © 2021, Akadémiai Kiadó, Budapest.
  • ItemOpen Access
    On the rank and exponent of the fixed points of coprime actions
    (Springer, 2022-02) Kızmaz, Muhammet Yasir; Kızmaz, Muhammet Yasir
    Let A be a group acting on a p-group P coprimely. We show that if A centralizes some specified abelian subgroups of P, then A acts trivially on P. As a consequence of this, we obtain that the special rank of CP(A) is strictly less than that of P unless the action of A on P is trivial. Secondly, we prove that if A acts on a group G coprimely and [G,A]=G, then the exponent of CG(A)/(CG(A))′ divides |G:CG(A)|.
  • ItemOpen Access
    Higher limits over the fusion orbit category
    (Elsevier, 2022-06-09) Yalçın, Ergün; Yalçın, Ergün
    The fusion orbit category F‾C(G) of a discrete group G over a collection C is the category whose objects are the subgroups H in C, and whose morphisms H→K are given by the G-maps G/H→G/K modulo the action of the centralizer group CG(H). We show that the higher limits over F‾C(G) can be computed using the hypercohomology spectral sequences coming from the Dwyer G-spaces for centralizer and normalizer decompositions for G. If G is the discrete group realizing a saturated fusion system F, then these hypercohomology spectral sequences give two spectral sequences that converge to the cohomology of the centric orbit category Oc(F). This allows us to apply our results to the sharpness problem for the subgroup decomposition of a p-local finite group. We prove that the subgroup decomposition for every p-local finite group is sharp (over F-centric subgroups) if it is sharp for every p-local finite group with nontrivial center. We also show that for every p-local finite group (S,F,L), the subgroup decomposition is sharp if and only if the normalizer decomposition is sharp.
  • ItemOpen Access
    Partially observed discrete-time risk-sensitive mean field games
    (Birkhaeuser Science, 2022-06-07) Saldi, Naci; Başar, T.; Raginsky, M.; Saldi, Naci
    In this paper, we consider discrete-time partially observed mean-field games with the risk-sensitive optimality criterion. We introduce risk-sensitivity behavior for each agent via an exponential utility function. In the game model, each agent is weakly coupled with the rest of the population through its individual cost and state dynamics via the empirical distribution of states. We establish the mean-field equilibrium in the infinite-population limit using the technique of converting the underlying original partially observed stochastic control problem to a fully observed one on the belief space and the dynamic programming principle. Then, we show that the mean-field equilibrium policy, when adopted by each agent, forms an approximate Nash equilibrium for games with sufficiently many agents. We first consider finite-horizon cost function and then discuss extension of the result to infinite-horizon cost in the next-to-last section of the paper.