Scholarly Publications - Mathematics

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

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Open Access
Pawłucki–Pleśniak extension operator for non-Markov sets
(Polska Akademia Nauk * Instytut Matematyczny, Polish Academy of Sciences, Institute of Mathematics, 2024-01-29) Goncharov, Alexander; Paksoy, Yaman
We show that Pawłucki–Pleśniak’s operator is bounded for some non-Markov sets.
Open Access
Addendum - vector invariants of permutation groups in characteristic zero
(World Scientific Publishing Co. Pte. Ltd., 2025-04) Reimers, Fabian; Sezer, Müfit
Open Access
Dynamics of neural fields with exponential temporal kernel
(Springer, 2024-03-09) Shamsara, Elham; Yamakou, Marius E.; Atay, Fatihcan M.; Jost, Jürgen
We consider the standard neural field equation with an exponential temporal kernel. We analyze the time-independent (static) and time-dependent (dynamic) bifurcations of the equilibrium solution and the emerging spatiotemporal wave patterns. We show that an exponential temporal kernel does not allow static bifurcations such as saddle-node, pitchfork, and in particular, static Turing bifurcations. However, the exponential temporal kernel possesses the important property that it takes into account the finite memory of past activities of neurons, which Green’s function does not. Through a dynamic bifurcation analysis, we give explicit bifurcation conditions. Hopf bifurcations lead to temporally non-constant, but spatially constant solutions, but Turing–Hopf bifurcations generate spatially and temporally non-constant solutions, in particular, traveling waves. Bifurcation parameters are the coefficient of the exponential temporal kernel, the transmission speed of neural signals, the time delay rate of synapses, and the ratio of excitatory to inhibitory synaptic weights.
Embargo
Homotopical characterization of strongly contextual simplicial distributions on cone spaces
(Elsevier BV, 2024-07-01) Kharoof, Aziz; Okay, Cihan
This paper offers a novel homotopical characterization of strongly contextual simplicial distributions with binary outcomes, specifically those defined on the cone of a 1-dimensional space. In the sheaf-theoretic framework, such distributions correspond to non-signaling distributions on measurement scenarios where each context contains 2 measurements with binary outcomes. To establish our results, we employ a homotopical approach that includes collapsing measurement spaces and introduce categories associated with simplicial distributions that can detect strong contextuality.
Embargo
Simplicial techniques for operator solutions of linear constraint systems
(Elsevier BV, 2024-05-01) Chung, Ho Yiu; Okay, Cihan; Sikora, Igor
A linear constraint system is specified by linear equations over the group Zd of integers modulo d. Their operator solutions play an important role in the study of quantum contextuality and non-local games. In this paper, we use the theory of simplicial sets to develop a framework for studying operator solutions of linear systems. Our approach refines the well-known group-theoretical approach based on solution groups by identifying these groups as algebraic invariants closely related to the fundamental group of a space. In this respect, our approach also makes a connection to the earlier homotopical approach based on cell complexes. Within our framework, we introduce a new class of linear systems that come from simplicial sets and show that any linear system can be reduced to one of that form. Then we specialize in linear systems that are associated with groups. We provide significant evidence for a conjecture stating that for odd devery linear system admitting a solution in a group admits a solution in Zd.
Open Access
Essential program features identified by students working toward a doctorate in mathematics education
(The International Group for the Psychology of Mathematics Education, 2024-07-21) Courtney, Scott A.; Alexander, Anita N.
What are the essential components of a doctorate program in mathematics education or didactics of mathematics concerning research, coursework, seminars, and collaboration? The purpose of this study was to learn from doctoral students across the world about how their programs in mathematics education are preparing them for research and teaching in mathematics education; how their programs provide academic research and writing support; and what they view as missing from their experiences. Online surveys, along with follow-up interviews from a subset of survey respondents, indicated that doctoral students from 17 different countries stressed the importance of international collaboration, examining fundamental theories of learning mathematics, and identified a need for more support with academic writing. © 2024, Psychology of Mathematics Education (PME). All rights reserved.
Open Access
Common information approach for static team problems with polish spaces and existence of optimal policies
(Natural Sciences Publishing Corporation, 2024) Saldı, Naci
In this paper, we demonstrate the existence of team-optimal strategies for static teams under observation-sharing information structures. Assuming that agents can access shared observations, we begin by converting the team problem into an equivalent centralized stochastic control problem through the introduction of a topology on policies. We subsequently apply conventional methods from stochastic control to prove the existence of team-optimal strategies. This study expands upon the widely recognized common information approach for team problems, originally designed for discrete scenarios, and adapts it to a more abstract continuous framework. The primary difficulty in this context is to establish the appropriate topology on policies..
Open Access
Stabilisation of linear waves with inhomogeneous Neumann boundary conditions
(Taylor & Francis, 2024-10-21) Özsarı, Türker; Susuzlu, İdem
We study linear damped and viscoelastic wave equations evolving on a bounded domain. For both models, we assume that waves are subject to an inhomogeneous Neumann boundary condition on a portion of the domain's boundary. The analysis of these models presents additional interesting features and challenges compared to their homogeneous counterparts. In the present context, energy depends on the boundary trace of velocity. It is not clear in advance how this quantity should be controlled based on the given data, due to regularity issues. However, we establish global existence and also prove uniform stabilisation of solutions with decay rates characterised by the Neumann input. We supplement these results with numerical simulations in which the data do not necessarily satisfy the given assumptions for decay. These simulations provide, at a numerical level, insights into how energy could possibly change in the presence of, for example, improper data.
Open Access
The group of splendid morita equivalences of principal 2-blocks with dihedral and generalised quaternion defect groups
(International Electronic Journal of Algebra, 2024-01-09) Karagüzel, Çisil; Yılmaz, Deniz
Let $k$ be an algebraically closed field of characteristic $2$, let $G$ be a finite group and let $B$ be the principal $2$-block of $kG$ with a dihedral or a generalised quaternion defect group $P$. Let also $\calT(B)$ denote the group of splendid Morita auto-equivalences of $B$. We show that $$\begin{align*} \calT(B)\cong \Out_P(A)\rtimes \Out(P,\calF), \end{align*}$$ where $\Out(P,\calF)$ is the group of outer automorphisms of $P$ which stabilize the fusion system $\calF$ of $G$ on $P$ and $\Out_P(A)$ is the group of algebra automorphisms of a source algebra $A$ of $B$ fixing $P$ modulo inner automorphisms induced by ($A^P)^\times$.
Open Access
Positive semidefinite maps on ∗ and linearisations
(Birkhaeuser Science, 2024-09-11) Gheondea, Aurelian
Motivated by current investigations in dilation theory, in both operator theory and operator algebras, and the theory of groupoids, we obtain a generalisation of the Sz-Nagy’s Dilation Theorem for opera- tor valued positive semidefinite maps on ∗-semigroupoids with unit, with varying degrees of aggregation, firstly by ∗-representations with unbounded operators and then we characterise the existence of the cor- responding ∗-representations by bounded operators. By linearisation of these constructions, we obtain similar results for operator valued posi- tive semidefinite maps on ∗-algebroids with unit and then, for the special case of B∗-algebroids with unit, we obtain a generalisation of the Stine- spring’s Dilation Theorem. As an application of the generalisation of the Stinespring’s Dilation Theorem, we show that some natural questions on C∗-algebroids are equivalent.
Open Access
Kerr-Vaidya type radiating black holes in semiclassical gravity with conformal anomaly
(American Physical Society, 2024-07-02) Gürses, Metin; Tekin, Bayram
Static black holes in the conformal anomaly-sourced semiclassical general relativity in four dimensions were recently extended to rotating, stationary solutions. These quantum-corrected black holes show different features compared to the Kerr black hole and need for further extensions. Here we remove the condition of stationarity and find radiating (Kerr-Vaidya-type) solutions in the same theory augmented with a cosmological constant. As long as the coupling constant α of the A-type trace anomaly is nonzero, we show that: (i) the cosmological constant is bounded from above, i.e., Λ ≤ 3/4 α; (ii) static black holes exist but they may not be unique; (iii) static black holes do not satisfy the second law of black hole thermodynamics; (iv) static black holes may have unstable inner horizons; (v) in the nonstationary and axially symmetric case, stability of the event horizon and the second law of thermodynamics for black holes are problematic.
Open Access
Geometric perfect fluids and the dark side of the universe
(American Physical Society, 2024-07-26) Gürses, Metin; Heydarzade, Yaghoub; Şentürk, Çetin
Recently, we showed that in Friedman-Lemaître-Robertson-Walker (FLRW) cosmology, the contribution from higher curvature terms in any generic metric gravity theory to the energy-momentum tensor is of the perfect fluid form. Such a geometric perfect fluid can be interpreted as a fluid remaining from the beginning of the Universe, where string theory is thought to be effective. Just a short time after the beginning of the Universe, it is known that the Einstein-Hilbert action is assumed to be modified by adding all possible curvature invariants. We propose that the observed late-time accelerating expansion of the Universe can be solely driven by this geometric fluid. To support our claim, we specifically study the quadratic gravity field equations in D dimensions. We show that the field equations of this theory for the FLRW metric possess a geometric perfect fluid source containing two critical parameters σ₁ and σ₂. To analyze this theory concerning its parameter space (σ₁, σ₂), we obtain the general second-order nonlinear differential equation governing the late-time dynamics of the deceleration parameter q. Hence, using some present-day cosmological data as our initial conditions, our findings for the σ₂ = 0 case are as follows: (i) To have a positive energy density for the geometric fluid ρᵍ, the parameter σ₁ must be negative for all dimensions up to D = 11. (ii) For a suitable choice of σ₁, the deceleration parameter experiences signature changes in the past and future, and in the meantime, it lies within a negative range, which means that the current observed accelerated expansion phase of the Universe can be driven solely by the curvature of spacetime. (iii) q experiences a signature change, and as the dimension D of spacetime increases, this signature change happens at earlier and later times, in the past and future, respectively.
Open Access
Wave metrics in the Cotton and conformal Killing gravity theories
(American Physical Society, 2024-05-30) Gürses, Metin; Heydarzade, Yaghoub; Şentürk, Çetin
We study wave metrics in the context of Cotton gravity and conformal Killing gravity. First, we consider pp-wave metrics with flat and nonflat wave surfaces and show that they are exact solutions to the field equations of these theories. More explicitly, the field equations reduce to inhomogeneous Laplace and Helmholtz differential equations, depending on the curvature of the two-dimensional geometry of the wave surfaces. An interesting point here is that the ones with nonflat wave surfaces are not present in classical GR, which manifests a crucial distinction between these theories and GR. Moreover, we investigate Kerr-Schild-Kundt metrics in the context of these theories and show that, from among these metrics, only the AdS wave metrics solve the field equations of these theories. However, AdS spherical and dS hyperbolic wave metrics do not solve the field equations of these theories, which is in contrast to the classical GR. In the case of AdS wave metrics, the field equations of these theories reduce to an inhomogeneous Klein-Gordon equation. We give all the necessary and sufficient conditions for the metric function V to solve these field equations.
Open Access
Finite dimensional backstepping controller design
(IEEE, 2025-06) Kalantarov, Varga K.; Özsarı, Türker; Yılmaz, K. Cem
We introduce a finite-dimensional version of backstepping controller design for stabilizing solutions of partial differential equations (PDEs) from boundary. Our controller uses only a finite number of Fourier modes of the state of solution, as opposed to the classical backstepping controller which uses all (infinitely many) modes. We apply our method to the reaction-diffusion equation, which serves only as a canonical example but the method is applicable also to other PDEs whose solutions can be decomposed into a slow finite-dimensional part and a fast tail, where the former dominates the evolution in large time. One of the main goals is to estimate the sufficient number of modes needed to stabilize the plant at a prescribed rate. In addition, we find the minimal number of modes that guarantee the stabilization at a certain (unprescribed) decay rate. Theoretical findings are supported with numerical solutions.
Open Access
Minimal Einstein-Aether theory
(Springer, 2024-09-18) Gürses, Metin; Şentürk, Çetin; Tekin, Bayram
We show that there is a phenomenologically and theoretically consistent limit of the generic Einstein-Aether theory in which the Einstein-Aether field equations reduce to Einstein field equations with a perfect fluid distribution sourced by the aether field. This limit is obtained by taking three of the coupling constants of the theory to be zero but keeping the expansion coupling constant to be nonzero. We then consider the further reduction of this limited version of Einstein-Aether theory by taking the expansion of the aether field to be constant (possibly zero), and thereby we introduce the Minimal Einstein-Aether theory that supports the Einstein metrics as solutions with a reduced cosmological constant. The square of the expansion of the unit-timelike aether field shifts the bare cosmological constant and thus provides, via local Lorentz symmetry breaking inherent in the Einstein-Aether theories, a novel mechanism for reconciling the observed, small cosmological constant (or dark energy) with the large theoretical prediction coming from quantum field theories. The crucial point here is that minimal Einstein-Aether theory does not modify the well-tested aspects of General Relativity such as solar system tests and black hole physics including gravitational waves.
Open Access
A note on the distribution of d3(n) in arithmetic progressions
(Springer New York LLC, 2024-10-14) Parry, Tomos
Nguyen has shown that on averaging over a=1,...,q the 3-fold divisor function has exponent of distribution 2/3, following Banks et al. (Int Math Res Not 1:1-25, 2005). We follow (Blomer, in: Q J Math 59:275-286, 2008) which leads to stronger bounds.
Open Access
No quantum solutions to linear constraint systems in odd dimension from pauli group and diagonal cliffords
(Verein zur Foerderung des Open Access Publizierens in den Quantenwissenschaften, 2025-01-08) Frembs, Markus; Okay, Cihan; Chung, Ho Yiu
Linear constraint systems (LCS) have proven to be a surprisingly prolific tool in the study of non-classical correlations and various related issues in quantum foundations. Many results are known for the Boolean case, yet the generalisation to systems of odd dimension is largely open. In particular, it is not known whether there exist LCS in odd dimension, which admit finite-dimensional quantum, but no classical solutions. In recent work, [J. Phys. A, 53, 385304 (2020)] have shown that unlike in the Boolean case, where the n-qubit Pauli group gives rise to quantum solutions of LCS such as the Mermin-Peres square, the n-qudit Pauli group never gives rise to quantum solutions of a LCS in odd dimension. Here, we generalise this result towards the Clifford hierarchy. More precisely, we consider tensor products of groups generated by (single-qudit) Pauli and diagonal Clifford operators. © 2025 Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften.
Open Access
Hidden variable model for quantum computation with magic states on qudits of any dimension
(Verein zur Foerderung des Open Access Publizierens in den Quantenwissenschaften, 2024-04-30) Zurel, Michael; Okay, Cihan; Raussendorf, Robert; Heimendahl, Arne
It was recently shown that a hidden variable model can be constructed for universal quantum computation with magic states on qubits. Here we show that this result can be extended, and a hidden variable model can be defined for quantum computation with magic states on qudits with any Hilbert space dimension. This model leads to a classical simulation algorithm for universal quantum computation.
Open Access
Numerical computation of Neumann controls for the heat equation on a finite interval
(IEEE, 2024-01) Kalimeris, Konstantinos; Özsarı, Türker; Dikaios, Nikolaos
This article presents a new numerical method, which approximates Neumann type null controls for the heat equation and is based on the Fokas method. This is a direct method for solving problems originating from the control theory, which allows the realization of an efficient numerical algorithm that requires small computational effort for determining the null control with exponentially small error. Furthermore, the unified character of the Fokas method makes the extension of the numerical algorithm to a wide range of other linear partial differential equations and different type of boundary conditions straightforward.
Open Access
Correction to: Asymptotically optimal Bayesian sequential change detection and identification rules
(2024) Dayanık, Savaş; Powell, W. B.; Yamazaki, K.
The article Asymptotically optimal Bayesian sequential change detection and identification rules, written by Savas Dayanik, Warren B. Powell, Kazutoshi Yamazaki, was originally published electronically on the publisher’s internet portal on 12 April, 2012 without open access. With the author(s)’ decision to opt for Open Choice the copyright of the article changed on 02 Sept, 2021 to © The Author(s). Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.