# Department of Mathematics

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Item Embargo Conics on Kummer quartics(Tohoku Daigaku Suugaku Kyoshitsu,Tohoku University, Mathematical Institute, 2023-09-25) Degtyarev, AlexShow more We classify the configurations of lines and conics in smooth Kummer quartics, assuming that all 16 Kummer divisors map to conics. We show that the number of conics on such a quartic is at most 800.Show more Item 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 MehmetShow more 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.Show more Item Open Access Vector invariants of permutation groups in characteristic zero(World Scientific Publishing Co. Pte. Ltd., 2023-12-21) Reimers, F.; Sezer, MüfitShow more We consider a finite permutation group acting naturally on a vector space V over a field k. A well-known theorem of G¨obel asserts that the corresponding ring of invariants k[V ] G is generated by the invariants of degree at most `dim V 2 ´ . In this paper, we show that if the characteristic of k is zero, then the top degree of vector coinvariants k[V m]G is also bounded above by `dim V 2 ´ , which implies the degree bound `dim V 2 ´ + 1 for the ring of vector invariants k[V m] G. So, G¨obel’s bound almost holds for vector invariants in characteristic zero as well.Show more Item Open Access Geometry of twisted products and applications on static perfect fluid spacetimes(Electronic Journal of Geometry, 2023-08-03) Güler, S.; De, U.C.; Ünal, BülentShow more In this paper, first we study the harmonicity of the functions and forms on the twisted products, and then we determine its sectional curvature. We explore some characteristics of static perfect fluid and static vacuum spacetimes on twisted product manifolds by proving the existence and obstructions on Ricci curvature. Finally, we study the problem of the existence static perfect fluid spacetime associated with the twisted generalized Robertson-Walker and standard static spacetime metrics.Show more Item Open Access Uncertainty principles in holomorphic function spaces on the unit ball(Cambridge University Press, 2023-07-10) Kaptanoğlu, Hakkı TurgayShow more On all Bergman–Besov Hilbert spaces on the unit disk, we find self-adjoint weighted shift operators that are differential operators of half-order whose commutators are the identity, thereby obtaining uncertainty relations in these spaces. We also obtain joint average uncertainty relations for pairs of commuting tuples of operators on the same spaces defined on the unit ball. We further identify functions that yield equality in some uncertainty inequalities.Show more Item Open Access Pseudo-projective tensor on sequential warped products(Birkhauser, 2023-01-29) Güler, S.; Ünal, BülentShow more The main objective of this paper is to study pseudo-projective tensor on sequential warped products and then to obtain necessary and sufficient conditions for a sequential warped product to be pseudo-projectively flat. Moreover, we also provide characterization of pseudo-projectively flat sequential generalized Robertson–Walker and pseudo-projectively flat sequential standard static spacetimes.Show more Item Open Access Planes in cubic fourfolds(European Mathematical Society Publishing House, 2023) Degtyarev, Alex; Itenberg, I.; Ottem, J. C.Show more We show that the maximal number of planes in a complex smooth cubic fourfold in P5 is 405, realized by the Fermat cubic only; the maximal number of real planes in a real smooth cubic fourfold is 357, realized by the so-called Clebsch–Segre cubic. Altogether, there are but three (up to projective equivalence) cubics with more than 350 planes © 2023,Algebraic Geometry. All Rights Reserved.Show more Item Open Access Synchronization in networks of anticipatory agents(IEEE - Institute of Electrical and Electronics Engineers, 2023-07-17) Dönmez, Bengi; Atay, Fatihcan M.Show more We consider a coupled Kuramoto system composed of agents that anticipate the future states of their neighbors based on past data and try to align their states accordingly. We show that this anticipatory behavior results in multiple synchronized solutions at different collective frequencies and different stability characteristics. We derive an exact condition for the stability of the synchronized states. We show that the system can exhibit multistability, converging to different synchronized solutions depending on the initial conditions.Show more Item Open Access Inversion sequences avoiding 021 and another pattern of length four(D M T C S, 2023-11-17) Toufik, M.; Yıldırım, GökhanShow more We study the enumeration of inversion sequences that avoid pattern 021 and another pattern of length four. We determine the generating trees for all possible pattern pairs and compute the corresponding generating functions. We introduce the concept of d-regular generating trees and conjecture that for any 021-avoiding pattern τ , the generating tree T ({021, τ }) is d-regular for some integer d.Show more Item Open Access Q-Learning for MDPs with general spaces: convergence and near optimality via quantization under weak continuity(Journal of Machine Learning Research, 2023-07-12) Kara, A. D.; Saldı, Naci; Yüksel, S.Show more Reinforcement learning algorithms often require finiteness of state and action spaces in Markov decision processes (MDPs) (also called controlled Markov chains) and various efforts have been made in the literature towards the applicability of such algorithms for continuous state and action spaces. In this paper, we show that under very mild regularity conditions (in particular, involving only weak continuity of the transition kernel of an MDP), Q-learning for standard Borel MDPs via quantization of states and actions (called Quantized Q-Learning) converges to a limit, and further-more this limit satisfies an optimality equation which leads to near optimality with either explicit performance bounds or which are guaranteed to be asymptotically optimal. Our approach builds on (i) viewing quantization as a measurement kernel and thus a quantized MDP as a partially observed Markov decision process (POMDP), (ii) utilizing near optimality and convergence results of Q-learning for POMDPs, and (iii) finally, near-optimality of finite state model approximations for MDPs with weakly continuous kernels which we show to correspond to the fixed point of the constructed POMDP. Thus, our paper presents a very general convergence and approximation result for the applicability of Q-learning for continuous MDPs.Show more Item Open Access Learning mean field games with discounted and average costs(Journal of Machine Learning Research, 2023-12-16) Anahtarcı, B; Karıksız, C. D.; Saldı, Naci; Alekh AgarwalShow more We consider learning approximate Nash equilibria for discrete-time mean-field games with stochastic nonlinear state dynamics subject to both average and discounted costs. To this end, we introduce a mean-field equilibrium (MFE) operator, whose fixed point is a mean-field equilibrium, i.e., equilibrium in the infinite population limit. We first prove that this operator is a contraction, and propose a learning algorithm to compute an approximate mean-field equilibrium by approximating the MFE operator with a random one. Moreover, using the contraction property of the MFE operator, we establish the error analysis of the proposed learning algorithm. We then show that the learned mean-field equilibrium constitutes an approximate Nash equilibrium for finite-agent games.Show more Item Open Access Decay rate estimates for the wave equation with subcritical semilinearities and locally distributed nonlinear dissipation(Springer, 2023-11-07) Cavalcanti, M. M.; Domingos Cavalcanti, V. N.; Gonzalez Martinez, V. H.; Özsarı, TürkerShow more We study the stabilization and the wellposedness of solutions of the wave equation with subcritical semilinearities and locally distributed nonlinear dissipation. The novelty of this paper is that we deal with the difficulty that the main equation does not have good nonlinear structure amenable to a direct proof of a priori bounds and a desirable observability inequality. It is well known that observability inequalities play a critical role in characterizing the long time behaviour of solutions of evolution equations, which is the main goal of this study. In order to address this, we truncate the nonlinearities, and thereby construct approximate solutions for which it is possible to obtain a priori bounds and prove the essential observability inequality. The treatment of these approximate solutions is still a challenging task and requires the use of Strichartz estimates and some microlocal analysis tools such as microlocal defect measures. We include an appendix on the latter topic here to make the article self contained and supplement details to proofs of some of the theorems which can be already be found in the lecture notes of Burq and Gérard (http://www.math.u-psud.fr/~burq/articles/coursX.pdf, 2001). Once we establish essential observability properties for the approximate solutions, it is not difficult to prove that the solution of the original problem also possesses a similar feature via a delicate passage to limit. In the last part of the paper, we establish various decay rate estimates for different growth conditions on the nonlinear dissipative effect. We in particular generalize the known results on the subject to a considerably larger class of dissipative effects.Show more Item Open Access Minimal models of some differential graded modules(Springer Science and Business Media B.V., 2023-01-03) Şentürk, B.; Ünlü, ÖzgünShow more Minimal models of chain complexes associated with free torus actions on spaces have been extensively studied in the literature. In this paper, we discuss these constructions using the language of operads. The main goal of this paper is to define a new Koszul operad that has projections onto several of the operads used in these minimal model constructions.Show more Item Open Access A simplicial category for higher correspondences(Springer Science and Business Media B.V., 2022-12-27) Haderi, RediShow more 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.Show more Item Open Access An algorithmic approach based on generating trees for enumerating pattern-avoiding inversion sequences(Academic Press Ltd- Elsevier Science Ltd, 2023-05-18) Kotsireas, I.; Mansour, T.; Yıldırım, GökhanShow more We introduce an algorithmic approach based on a generating tree method for enumerating the inversion sequences with various pattern-avoidance restrictions. For a given set of patterns, we propose an algorithm that outputs either an accurate description of the succession rules of the corresponding generating tree or an ansatz. By using this approach, we determine the generating trees for the pattern classes In(000, 021), In(100, 021), In(110, 021), In(102, 021), In(100, 012), In(011, 201), In(011, 210) and In(120, 210). Then we use the kernel method, obtain generating functions of each class, and find enumerating formulas. Lin and Yan studied the classification of the Wilf-equivalences for inversion sequences avoiding pairs of length-three patterns and showed that there are 48 Wilf classes among 78 pairs. In this paper, we solve six open cases for such pattern classes. Moreover, we extend the algorithm to restricted growth sequences and apply it to several classes. In particular, we present explicit formulas for the generating functions of the restricted growth sequences that avoid either {12313, 12323}, {12313, 12323, 12333}, or {123 ··· 1}.Show more Item Open Access Israel-Wilson-Perjes metrics in a theory with a dilaton field(American Physical Society, 2023-07-25) Gürses, Metin; Şişman, T. Ç.; Tekin, B.Show more We are interested in the charged dust solutions of the Einstein field equations in stationary and axially symmetric spacetimes and inquire if the naked singularities of the Israel-Wilson-Perjes (IWP) metrics can be removed. The answer is negative in four dimensions. We examine whether this negative result can be avoided by adding scalar or dilaton fields. We show that IWP metrics also arise as solutions of the Einstein-Maxwell system with a stealth dilaton field. We determine the IWP metrics completely in terms of one complex function satisfying the Laplace equation. With the inclusion of the stealth dilaton field, it is now possible to add a perfect fluid source. In this case, the field equations reduce to a complex cubic equation. Hence, this procedure provides interior solutions to each IWP metric, and it is possible to cover all naked singularities inside a compact surface where there is matter distribution.Show more Item Open Access Hairy Kiselev black hole solutions(American Physical Society, 2023-08-28) Heydarzade, Yaghoub; Misyura, M.; Vertogradov, V.Show more In the realm of astrophysics, black holes exist within nonvacuum cosmological backgrounds, making it crucial to investigate how these backgrounds influence the properties of black holes. In this work, we first introduce a novel static spherically-symmetric exact solution of Einstein field equations representing a surrounded hairy black hole. This solution represents a generalization of the hairy Schwarzschild solution recently derived using the extended gravitational decoupling method. Then, we discuss how the new induced modification terms attributed to the primary hairs and various background fields affect the geodesic motion in comparison to the conventional Schwarzschild case. Although these modifications may appear insignificant in most cases, we identify specific conditions where they can be comparable to the Schwarzschild case for some particular background fields.Show more Item Open Access Analytic relationship of relative synchronizability to network structure and motifs(National Academy of Sciences, 2023-09-05) Lizier, J. T.; Bauer, F.; Atay, Fatihcan Mehmet; Jost, J.Show more Synchronization phenomena on networks have attracted much attention in studies of neural, social, economic, and biological systems, yet we still lack a systematic understanding of how relative synchronizability relates to underlying network structure. Indeed, this question is of central importance to the key theme of how dynamics on networks relate to their structure more generally. We present an analytic technique to directly measure the relative synchronizability of noise-driven time-series processes on networks, in terms of the directed network structure. We consider both discrete-time autoregressive processes and continuous-time Ornstein–Uhlenbeck dynamics on networks, which can represent linearizations of nonlinear systems. Our technique builds on computation of the network covariance matrix in the space orthogonal to the synchronized state, enabling it to be more general than previous work in not requiring either symmetric (undirected) or diagonalizable connectivity matrices and allowing arbitrary self-link weights. More importantly, our approach quantifies the relative synchronization specifically in terms of the contribution of process motif (walk) structures. We demonstrate that in general the relative abundance of process motifs with convergent directed walks (including feedback and feedforward loops) hinders synchronizability. We also reveal subtle differences between the motifs involved for discrete or continuous-time dynamics. Our insights analytically explain several known general results regarding synchronizability of networks, including that small-world and regular networks are less synchronizable than random networks.Show more Item Open Access Dynamical wormhole solutions in Rastall theory(Springer , 2023-08-10) Heydarzade, Yaghoub; Ranjbar, M.Show more Wormhole configurations in Einstein’s general theory of relativity (GR) require exotic matter sources violating the weak energy condition (WEC). Rastall’s theory is a generalization of GR in its matter source considering a nonconserved energy momentum (EM) tensor. Hence, on the one hand, the nature of this generalization of the matter source of field equations and, on the other hand, the possibility of respecting energy conditions for dynamical wormholes in contrast with static ones motivates us to study the possibility of existence of wormhole configurations respecting energy conditions or minimizing the violations of them in Rastall’s modified theory. We derive general analytical solutions considering a constant redshift functions and a particular equation of state for energy density and pressure profiles. We show that because of the modifications in EM source of the field equations, there exist solutions respecting the WEC in the vicinity of the wormhole’s throat for a specified values of the parameters. Some particular solutions are discussed in detail.Show more Item Open Access On construction of darboux integrable discrete models(Elsevier Ltd, 2023-12) Zheltukhin, Kostyantyn; Zheltukhina, NatalyaShow more The problem of discretization of Darboux integrable equations is considered. Given a Darboux integrable continuous equation, one can obtain a Darboux integrable differential-discrete equation, using the integrals of the continuous equation. In the present paper, the discretization of the differential-discrete equations is done using the corresponding characteristic algebras. New examples of integrable discrete equations are obtained.Show more