Scholarly Publications - Mathematics

Permanent URI for this collectionhttps://hdl.handle.net/11693/115503

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Open Access
Steenrod closed parameter ideals in the mod-2 cohomology of A₄ and SO(3)
(Elsevier B.V., 2025-02-05) Rüping, Henrik; Stephan, Marc; Yalçın, Ergün
In this paper, we classify the parameter ideals in H∗(BA₄; F₂) and in the Dickson algebra H∗(BSO(3); F₂) that are closed under Steenrod operations. Consequently, we obtain restrictions on the dimensions n,m for which A₄ (and SO(3)) can act freely on Sₙ × Sₘ.
Open Access
Dynamical wormhole solutions in f(R, T) gravity
(Springer, 2025-07-19) Heydarzade, Yaghoub; Ranjbar, Maryam
A class of $f(R, T)$ theories extends the Einstein-Hilbert action by incorporating a general function of $R$ and $T$, the Ricci scalar and the trace of the ordinary energy-momentum tensor Tμν, respectively, thereby introducing a specific modification to the Einstein’s field equations based on matter fields. Given that this modification is intrinsically tied to an energy-momentum tensor $T_{μν}$ that a priori respects energy conditions, we explore the potential of $f(R, T)$ theories admitting wormhole configurations satisfying energy conditions, unlike General Relativity, which typically necessitates exotic matter sources. Consequently, we investigate the existence of dynamical wormhole geometries that either uphold energy conditions or minimize their violations within the framework of trace of energy-momentum tensor squared gravity. To ensure the generality of our study, we consider two distinct equations of state for the matter content and systematically classify possible solutions based on constraints related to the wormhole’s throat size, the coupling parameter of the theory, and the equation of state parameters.
Open Access
Characterizations of relativistic magneto-fluid spacetimes admitting Einstein soliton
(University of Nis, 2025) Roy, Soumendu; Dey, Santu; Ünal, Bülent
In this paper, we study some geometric properties of relativistic magneto-fluid spacetime admitting Einstein soliton. Here, we expose the nature of soliton when a relativistic magneto-fluid spacetime and $W_2$-flat relativistic magneto-fluid spacetime and Q-flat relativistic magneto-fluid spacetime satisfying Einstein field equation. Also we explore magneto-fluid warped product spacetimes that admits magneto-fluid generalized Robertson-Walker spacetimes and magneto-fluid standard static spacetimes in terms of Einstein soliton. Finally, we discuss some results on different types of magneto-fluid spacetimes admitting Einstein soliton.
Open Access
Robustness and approximation of discrete-time mean-field games under discounted cost criterion
(Institute for Operations Research and the Management Sciences, 2026-02) Aydın, Uğur; Saldi, Naci
In this paper, we investigate the robustness of stationary mean-field equilibria in the presence of model uncertainties, specifically focusing on infinite-horizon discounted cost functions. To achieve this, we initially establish convergence conditions for value iterationbased algorithms in mean-field games. Subsequently, utilizing these results, we demonstrate that the mean-field equilibrium obtained through this value iteration algorithm remains robust even in the face of system dynamics misspecifications. We then apply these robustness findings to the finite model approximation problem in mean-field games, showing that if the state space quantization is fine enough, the mean-field equilibrium for the finite model closely approximates the nominal one.
Open Access
Linear mean-field games with discounted cost
(Institute for Operations Research and the Management Sciences (INFORMS), 2026-02-17) Saldı, Naci
In this paper, we introduce discrete-time linear mean-field games subject to an infinite-horizon discounted-cost optimality criterion. At every time, each agent is randomly coupled with another agent via their dynamics and one-stage cost function, where this randomization is generated via the empirical distribution of their states (i.e., the mean-field term). Therefore, the transition probability and the one-stage cost function of each agent depend linearly on the mean-field term, which is the key distinction between classical mean-field games and linear mean-field games. Under mild assumptions, we show that the policy obtained from infinite population equilibrium is 𝜀⁡(𝑁)-Nash when the number of agents N is sufficiently large, where 𝜀⁡(𝑁) is an explicit function of N. Then, using the linear programming formulation of Markov decision processes (MDPs) and the linearity of the transition probability in the mean-field term, we formulate the game in the infinite population limit as a generalized Nash equilibrium problem (GNEP) and establish an algorithm for computing equilibrium with a convergence guarantee.
Open Access
Graph entropy, degree assortativity, and hierarchical structures in networks
(American Physical Society, 2025-12-24) Atay, Fatihcan Mehmet; Bıyıkoğlu, Türker
We connect several notions relating the structural and dynamical properties of a graph. Among them are the topological entropy coming from the vertex shift, which is related to the spectral radius of the graph's adjacency matrix, the Randić index, and the degree assortativity. We show that, among all connected graphs with the same degree sequence, the graph having maximum entropy is characterized by a hierarchical structure; namely, it satisfies a breadth-first search ordering with decreasing degrees (BFD ordering for short). Consequently, the maximum-entropy graph necessarily has high degree assortativity; furthermore, for such a graph the degree centrality and eigenvector centrality coincide. Moreover, the notion of assortativity is related to the general Randić index. We prove that the graph that maximizes the Randić index satisfies a BFD ordering. For trees, the converse holds as well. We also define a normalized Randić function and show that its maximum value equals the difference of Shannon entropies of two probability distributions defined on the edges and vertices of the graph based on degree correlations.
Open Access
Extremal simplicial distributions on cycle scenarios with arbitrary outcomes
(Institute of Physics, 2025-09-15) Kharoof, Aziz; İpek, Selman; Okay, Cihan
Cycle scenarios are a significant class of contextuality scenarios, with the Clauser–Horne–Shimony–Holt scenario being a notable example. While binary outcome measurements in these scenarios are well understood, the generalization to arbitrary outcomes remains less explored, except in specific cases. In this work, we employ homotopical methods in the framework of simplicial distributions to characterize all contextual vertices of the non-signaling polytope corresponding to cycle scenarios with arbitrary outcomes. Additionally, our techniques utilize the bundle perspective on contextuality and the decomposition of measurement spaces. This enables us to extend beyond scenarios formed by gluing cycle scenarios and describe contextual extremal simplicial distributions in these generalized contexts.
Embargo
Moving null curves and integrability
(Wiley, 2025-12-09) Gürses, Metin; Pekcan, Aslı
We study the null curves and their motion in 3-dimensional flat space-time M₃ . We show that when the motion of nullcurves forms two surfaces in M₃ , the integrability conditions lead to the well-known Ablowitz–Kaup–Newell–Segur(AKNS) hierarchy. In this case, we obtain all the geometrical quantities of the surfaces arising from the whole hier-archy but we particularly focus on the surfaces of the modified Korteweg-de Vries (MKdV) and KdV equations. Weobtain one- and two-soliton surfaces associated with the MKdV equation and show that the Gauss and mean curva-tures of these surfaces develop singularities in finite time. We also show that the triad vectors on the curves satisfythe spin vector equation in the ferromagnetism model of Heisenberg.
Open Access
Quantum Markov decision processes: general theory, approximations, and classes of policies
(Society for Industrial and Applied Mathematics, 2025-04-11) Saldı, Naci; Sanjari, Sina; Yüksel, Serdar
In this paper, the aim is to develop a quantum counterpart to classical Markov decision processes (MDPs). First, we provide a very general formulation of quantum MDPs with state and action spaces in the quantum domain, quantum transitions, and cost functions. Once we formulate the quantum MDP (q-MDP), our focus shifts to establishing the verification theorem that proves the sufficiency of Markovian quantum control policies and provides a dynamic programming principle. Subsequently, a comparison is drawn between our q-MDP model and previously established quantum MDP models (referred to as QOMDPs) found in the literature. Furthermore, approximations of q-MDPs are obtained via finite-action models, which can be formulated as QOMDPs. Finally, classes of open-loop and classical-state-preserving closed-loop policies for q-MDPs are introduced, along with structural results for these policies. In summary, we present a novel quantum MDP model aiming to introduce a new framework, algorithms, and future research avenues. We hope that our approach will pave the way for a new research direction in discrete-time quantum control.
Open Access
Non vanishing of cubic Dirichlet L-functions over the Eisenstein field
(American Mathematical Society, 2025-03-03) Güloǧlu, Ahmet Muhtar
We establish an asymptotic formula for the first moment and derive an upper bound for the second moment of L-functions associated with the complete family of primitive cubic Dirichlet characters defined over the Eisenstein field. Our results are unconditional, and indicate that there are infinitely many characters within this family for which the L-function L(s, χ) does not vanish at the central point s = 1/2.
Open Access
Interpolating projections in Fréchet algebras
(Springer Cham, 2026-02-01) Goncharov, Alexander; Duman, Oktay; Erkus-Duman, Esra
Suppose we are given a Fréchet algebra X of functions deﬁned on a set D. A sequence Z of distinct points (zk)k∞=1. in D deﬁnes an ideal J of functions equal to zero on Z. We are interested in the geometric characterization for the splitting of the short exact sequence 0 −→ J −→i X −π→ X/J −→ 0.. Here the quotient space is a sequence space and the right inverse of the epimorphism (if exists) is naturally represented as an interpolating operator. We review some known cases when X is a Banach algebra of analytic functions, namely the Hardy space of bounded holomorphic functions on the unit disk and the disc algebra. Then new results are presented for the algebra of inﬁnitely differentiable functions and Whitney algebra. Interestingly, the geometric conditions for the continuity of the corresponding interpolation projections in these cases are opposite in the following sense. For the spaces of analytic functions, arbitrary rapid convergence of (zk)k∞=1. to a boundary point is allowed, whereas there is an upper limit on the rate of convergence for such sequences in the second case. Also, some linear topological properties of t he corresponding restriction spaces and Whitney spaces are analyzed.
Open Access
Embedding of lattices and K3-covers of an Enriques surface
(Walter de Gruyter GmbH, 2025-09-11) Sonel, Serkan
In this study, we establish necessary conditions for the embeddings of lattices and apply these conditions to the problem of characterizing algebraic 𝐾⁢3 surfaces that cover an Enriques surface. By refining existing criteria and providing a more elementary approach, we offer a new perspective on the structure of such surfaces. Our results apply to any lattices, extending beyond specific cases and offering a comprehensive framework for understanding the embedding conditions in terms of Gram matrices.
Open Access
On the stability of chaos synchronization in networks of anticipatory agents
(Elsevier, 2025-07-21) Atay, Fatihcan Mehmet; Murat, Nazira
We consider networks of coupled nonlinear systems in discrete time where the units anticipate the states of their neighbors and try to align their states accordingly. Anticipation is done using past state information, and hence introduces a memory effect in the form of a time delay. We study the stability of synchronized states in the presence of such delays and show that anticipation can induce synchronization in networks of chaotic maps.
Open Access
Counting lines with Vinberg's algorithm
(EMS Press, 2025-10-21) Degtyarev, Alex; Rams, Sławomir
We combine classical Vinberg's algorithms with the lattice-theoretic/arithmetic approach from Degtyarev (2019) to give a method of classifying large line configurations on complex quasi-polarized K3-surfaces. We apply our method to classify all complex K3-octic surfaces with at worst Du Val singularities and at least 32 lines. The upper bound on the number of lines is 36, as in the smooth case, with at most 32 lines if the singular locus is non-empty.
Open Access
Representing locally Hilbert spaces and functional models for locally normal operators
(Birkhaeuser Science, 2025-12-18) Gheondea, Aurelian
The aim of this article is to explore in all remaining aspects the spectral theory of locally normal operators. In a previous article we proved the spectral theorem in terms of locally spectral measures. Here we prove the spectral theorem in terms of projective limits of certain multiplication operators with locally L$^∞$ functions. In order to do this, we first investigate strictly inductive systems of measure spaces and point out the concept of representing locally Hilbert space for which we obtain a functional model as a strictly inductive limit of L$^2$ type spaces. Then, we first obtain a functional model for locally normal operators on representing locally Hilbert spaces combined with a spectral multiplicity model on a pseudo-concrete functional model for the underlying locally Hilbert space, under the technical condition on the directed set to be sequentially finite. Finally, under the same technical condition on the directed set, we derive the spectral theorem for locally normal operators in terms of projective limits of certain multiplication operators with locally L$^∞$ functions. As a consequence of the main result we sketch the direct integral representation of locally normal operators under the same technical assumptions and the separability of the locally Hilbert space. Examples of strictly inductive systems of measure spaces involving the Hata tree-like self-similar set, which justify the technical condition on a relevant case and which may open a connection with analysis on fractal sets, are included
Open Access
On p-permutation equivalences between direct products of blocks
(Hacettepe University, 2025-06-24) Yılmaz, Deniz
We extend the notion of a p-permutation equivalence to an equivalence between direct products of block algebras. We prove that a p-permutation equivalence between direct products of blocks gives a bijection between the factors and induces a p-permutation equivalence between corresponding blocks.
Open Access
Well-posedness of the higher-order nonlinear Schrödinger equation on a finite interval
(Birkhauser, 2026-01-16) Mayo, Chris; Mantzavinos, Dionyssios; Özsarı, Türker
We establish the local Hadamard well-posedness of a certain third-order nonlinear Schrödinger equation with a multi-term linear part and a general power nonlinearity known as the higher-order nonlinear Schrödinger equation, formulated on a finite interval with a combination of nonzero Dirichlet and Neumann boundary conditions. Specifically, for initial and boundary data in suitable Sobolev spaces that are related to one another through the time regularity induced by the equation, we prove the existence of a unique solution as well as the continuous dependence of that solution on the data. The precise choice of solution space depends on the value of the Sobolev exponent and is dictated both by the linear estimates associated with the forced linear counterpart of the nonlinear initial-boundary value problem and, in the low-regularity setting below the Sobolev algebra property threshold, by certain nonlinear estimates that control the Sobolev norm of the power nonlinearity. In particular, as usual in Schrödinger-type equations, in the case of low regularity it is necessary to derive Strichartz estimates in suitable Lebesgue/Bessel potential spaces. The proof of well-posedness is based on a contraction mapping argument combined with the aforementioned linear estimates, which are established by employing the explicit solution formula for the forced linear problem derived via the unified transform of Fokas. Due to the nature of the finite interval problem, this formula involves contour integrals in the complex Fourier plane with corresponding integrands that contain differences of exponentials in their denominators, thus requiring delicate handling through appropriate contour deformations. It is worth noting that, in addition to various linear and nonlinear results obtained for the finite interval problem, novel time regularity results are established here also for the relevant half-line problem.
Open Access
Low-regularity solutions of the nonlinear Schrödinger equation on the spatial quarter-plane
(Society for Industrial and Applied Mathematics, 2025-12-03) Mantzavinos, Dionyssios; Özsarı, Türker
The Hadamard well-posedness of the nonlinear Schrödinger equation with power nonlinearity formulated on the spatial quarter-plane is established in a low-regularity setting with Sobolev initial data and Dirichlet boundary data in appropriate Bourgain-type spaces. As both of the spatial variables are restricted to the half-line, a different approach is needed than the one previously used for the well-posedness of other initial-boundary value problems. In particular, now the solution of the forced linear initial-boundary problem is estimated directly, both in Sobolev spaces and in Strichartz-type spaces, i.e., without a linear decomposition that would require estimates for the associated homogeneous and nonhomogeneous initial value problems. In the process of deriving the linear estimates, the function spaces for the boundary data are identified as the intersections of certain modified Bourgain-type spaces that involve spatial half-line Fourier transforms instead of the usual whole-line Fourier transform found in the definition of the standard Bourgain space associated with the one-dimensional initial value problem. The fact that the quarter-plane has a corner at the origin poses an additional challenge, as it requires one to expand the validity of certain Sobolev extension results to the case of a domain with a nonsmooth (Lipschitz) and noncompact boundary.
Open Access
Quantum Markov decision processes: dynamic and semi-definite programs for optimal solutions
(Springer New York LLC, 2026-02-17) Saldi, Naci; Sanjari, Sina; Yüksel, Serdar
In this paper, building on the formulation of quantum Markov decision processes (q-MDPs) presented in our previous work [N. Saldi, S. Sanjari, and S. Yüksel, Quantum Markov Decision Processes: General Theory, Approximations, and Classes of Policies, SIAM Journal on Control and Optimization, 2024], our focus shifts to the development of semi-definite programming approaches for optimal policies and value functions of both open-loop and classical-state-preserving closed-loop policies. First, by using the duality between the dynamic programming and the semi-definite programming formulations of any q-MDP with open-loop policies, we establish that the optimal value function is linear and there exists a stationary optimal policy among open-loop policies. Then, using these results, we establish a method for computing an approximately optimal value function and formulate computation of optimal stationary open-loop policy as a bi-linear program. Next, we turn our attention to classical-state-preserving closed-loop policies. Dynamic programming and semi-definite programming formulations for classical-state-preserving closed-loop policies are established, where duality of these two formulations similarly enables us to prove that the optimal policy is linear and there exists an optimal stationary classical-state-preserving closed-loop policy. Then, similar to the open-loop case, we establish a method for computing the optimal value function and pose computation of optimal stationary classical-state-preserving closed-loop policies as a bi-linear program.
Open Access
The operadic theory of convexity
(Springer Dordrecht, 2025-05-24) Haderi, Redi; Okay, Cihan; Stern, Walker H.
In this paper, we present an operadic characterization of convexity utilizing a PROP which governs convex structures and derive several convex Grothendieck constructions. Our main focus is a Grothendieck construction which simultaneously captures convex structures and monoidal structures on categories. Our proof of this Grothendieck construction makes heavy use of both our operadic characterization of convexity and operads governing monoidal structures. We apply these new tools to two key concepts: entropy in information theory and quantum contextuality in quantum foundations. In the former, we explain that Baez, Fritz, and Leinster’s categorical characterization of entropy has a more natural formulation in terms of continuous, convex-monoidal functors out of convex Grothendieck constructions; and in the latter we show that certain convex monoids used to characterize contextual distributions naturally arise as convex Grothendieck constructions.