An interpolation inequality involving $L\log L$ spaces and application to the characterization of blow‐up behavior in a two‐dimensional Keller–Segel–Navier–Stokes system - Wang, Yulan; Winkler, Michael (Journal of the London Mathematical Society)
Abstract In a smoothly bounded two‐dimensional domain and for a given nondecreasing positive unbounded , for each and the inequality is shown to hold for any positive fulfilling This is thereafter applied to nonglobal solutions of the Keller–Segel system coupled to the incompressible Navier–Stokes equations through transport and buoyancy, and it is seen that in any such blow‐up event the corresponding population density cannot remain uniformly integrable over near its explosion time.
Keywords: General Mathematics

Pairwise Disjoint Perfect Matchings in r-Edge-Connected r-Regular Graphs - Ma, Yulai; Mattiolo, Davide; Steffen, Eckhard; Wolf, Isaak Hieronymus (SIAM Journal on Discrete Mathematics)
Keywords: General Mathematics

Dequantizing the Quantum Singular Value Transformation: Hardness and Applications to Quantum Chemistry and the Quantum PCP Conjecture - Gharibian, Sevag; Le Gall, François (SIAM Journal on Computing)
Keywords: General Mathematics; General Computer Science

Error analysis for the numerical approximation of the harmonic map heat flow with nodal constraints - Bartels, Sören; Kovács, Balázs; Wang, Zhangxian (IMA Journal of Numerical Analysis)
Abstract An error estimate for a canonical discretization of the harmonic map heat flow into spheres is derived. The numerical scheme uses standard finite elements with a nodal treatment of linearized unit-length constraints. The analysis is based on elementary approximation results and only uses the discrete weak formulation.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Relaxation in a Keller-Segel-consumption system involving signal-dependent motilities - Li, Genglin; Winkler, Michael (Communications in Mathematical Sciences)
Keywords: Applied Mathematics; General Mathematics

Global generalized solvability in a strongly degenerate taxis-type parabolic system modeling migration–consumption interaction - Winkler, Michael (Zeitschrift für angewandte Mathematik und Physik)
Abstract The parabolic problem $$\begin{aligned} \left\{ \begin{array}{l} u_t=\Delta \big (u\phi (v)\big ), \\ v_t=\Delta v-uv, \end{array} \right. \end{aligned}$$ u t = Δ ( u ϕ ( v ) ) , v t = Δ v - u v , is considered in smoothly bounded subdomains of $${\mathbb {R}}^n$$ R n with arbitrary $$n\ge 1$$ n ≥ 1 . Under the assumptions that $$\phi \in C^0([0,\infty )) \cap C^3((0,\infty ))$$ ϕ ∈ C 0 ( [ 0 , ∞ ) ) ∩ C 3 ( ( 0 , ∞ ) ) is positive on $$(0,\infty )$$ ( 0 , ∞ ) and satisfies $$\begin{aligned} \liminf _{\xi \searrow 0} \frac{\phi (\xi )}{\xi ^\alpha }>0 \quad {\text{ and }} \quad \limsup _{\xi \searrow 0} \big \{ \xi ^\beta |\phi '(\xi )| \big \}<\infty \end{aligned}$$ lim inf ξ ↘ 0 ϕ ( ξ ) ξ α > 0 and lim sup ξ ↘ 0 { ξ β | ϕ ′ ( ξ ) | } < ∞ with some $$\alpha >0$$ α > 0 and $$\beta >0$$ β > 0 , for all reasonably regular initial data an associated no-flux type initial-boundary value problem is shown to admit a global solution in an appropriately generalized sense. This extends previously developed solution theories on problems of this form, which either concentrated on non-degenerate or weakly degenerate cases corresponding to the choices $$\alpha =0$$ α = 0 and $$\alpha \in (0,2)$$ α ∈ ( 0 , 2 ) , or were restricted to low-dimensional settings by requiring that $$n\le 2$$ n ≤ 2 .
Keywords: Applied Mathematics; General Physics and Astronomy; General Mathematics

Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type - Winkler, Michael (Open Mathematics)
Abstract The Cauchy problem in R n {{\mathbb{R}}}^{n} , n ≥ 2 n\ge 2 , for u t = Δ u − ∇ ⋅ ( u S ⋅ ∇ v ) , 0 = Δ v + u , ( ⋆ ) \begin{array}{r}\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}{u}_{t}=\Delta u-\nabla \cdot \left(uS\cdot \nabla v),\\ 0=\Delta v+u,\end{array}\right.\hspace{2.0em}\hspace{2.0em}\hspace{2.0em}\left(\star )\end{array} is considered for general matrices S ∈ R n × n S\in {{\mathbb{R}}}^{n\times n} . A theory of local-in-time classical existence and extensibility is developed in a framework that differs from those considered in large parts of the literature by involving bounded classical solutions. Specifically, it is shown that for all non-negative initial data belonging to BUC ( R n ) ∩ L p ( R n ) {\rm{BUC}}\left({{\mathbb{R}}}^{n})\cap {L}^{p}\left({{\mathbb{R}}}^{n}) with some p ∈ [ 1 , n ) p\in \left[1,n) , there exist T max ∈ ( 0 , ∞ ] {T}_{\max }\in \left(0,\infty ] and a uniquely determined u ∈ C 0 ( [ 0 , T max ) ; BUC ( R n ) ) ∩ C 0 ( [ 0 , T max ) ; L p ( R n ) ) ∩ C ∞ ( R n × ( 0 , T max ) ) u\in {C}^{0}\left(\left[0,{T}_{\max });\hspace{0.33em}{\rm{BUC}}\left({{\mathbb{R}}}^{n}))\cap {C}^{0}\left(\left[0,{T}_{\max });\hspace{0.33em}{L}^{p}\left({{\mathbb{R}}}^{n}))\cap {C}^{\infty }\left({{\mathbb{R}}}^{n}\times \left(0,{T}_{\max })) such that with v ≔ Γ ⋆ u v:= \Gamma \star u , and with Γ \Gamma denoting the Newtonian kernel on R n {{\mathbb{R}}}^{n} , the pair ( u , v ) \left(u,v) forms a classical solution of ( ⋆ \star ) in R n × ( 0 , T max ) {{\mathbb{R}}}^{n}\times \left(0,{T}_{\max }) , which has the property that if T max < ∞ , then both limsup t ↗ T max ‖ u ( ⋅ , t ) ‖ L ∞ ( R n ) = ∞ and limsup t ↗ T max ‖ ∇ v ( ⋅ , t ) ‖ L ∞ ( R n ) = ∞ . \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}{T}_{\max }\lt \infty ,\hspace{1.0em}\hspace{0.1em}\text{then both}\hspace{0.1em}\hspace{0.33em}\mathop{\mathrm{limsup}}\limits_{t\nearrow {T}_{\max }}\Vert u\left(\cdot ,t){\Vert }_{{L}^{\infty }\left({{\mathbb{R}}}^{n})}=\infty \hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\mathop{\mathrm{limsup}}\limits_{t\nearrow {T}_{\max }}\Vert \nabla v\left(\cdot ,t){\Vert }_{{L}^{\infty }\left({{\mathbb{R}}}^{n})}=\infty . An exemplary application of this provides a result on global classical solvability in cases when ∣ S + 1 ∣ | S+{\bf{1}}| is sufficiently small, where 1 = diag ( 1 , … , 1 ) {\bf{1}}={\rm{diag}}\hspace{0.33em}\left(1,\ldots ,1) .
Keywords: General Mathematics

Keller–Segel–Stokes interaction involving signal‐dependent motilities - Tian, Yu; Winkler, Michael (Mathematical Methods in the Applied Sciences)
The chemotaxis‐Stokes system is considered along with homogeneous boundary conditions of no‐flux type for and , and of Dirichlet type for , in a smoothly bounded domain . Under the assumption that , that is bounded on each of the intervals with arbitrary , and that with some and , we have It is shown that for any suitably regular initial data, an associated initial‐boundary value problem admits a global very weak solution.
Keywords: General Engineering; General Mathematics

New examples of harmonic maps to the hyperbolic plane via Bäcklund transformation - Polychrou, G.; Papageorgiou, Efthymia; Fotiadis, A.; Daskaloyannis, C. (Revista Matemática Complutense)
Abstract We study harmonic maps from a subset of the complex plane to a subset of the hyperbolic plane. In Fotiadis and Daskaloyannis (Nonlinear Anal 214, 112546, 2022), harmonic maps are related to the sinh-Gordon equation and a Bäcklund transformation is introduced, which connects solutions of the sinh-Gordon and sine-Gordon equation. We develop this machinery in order to construct new harmonic maps to the hyperbolic plane.
Keywords: General Mathematics

Implementing profession orientation as a design principle for overcoming Klein’s second discontinuity – preservice teacher’s perspectives on interface activities in the context of a geometry course - Hoffmann, Max; Biehler, Rolf (ZDM – Mathematics Education)
Intending to counteract Klein’s second discontinuity in teacher education, we explored and applied the innovation of “interface ePortfolio” in the context of a geometry course for preservice teachers (PSTs). The tool offers the possibility of implementing the design principle of profession orientation. In the article, we theoretically clarify what we understand by this principle and locate our innovative concept against this theoretical background. We empirically investigate the extent to which counteraction against the second discontinuity is successful by analyzing reflection texts created in the interface ePortfolio, focusing on PSTs’ perspectives. Our qualitative content analysis shows that most of them perceive the innovation as helpful in the intended sense and indicates that the course concept, in general, and the interface ePortfolio, in particular, have helped establish relevant links between the course content and their later work as teachers.
Keywords: General Mathematics; Education

Characterizing external visualization in mathematics education research: a scoping review - Schoenherr, Johanna; Schukajlow, Stanislaw (ZDM – Mathematics Education)
Abstract External visualization (i.e., physically embodied visualization) is central to the teaching and learning of mathematics. As external visualization is an important part of mathematics at all levels of education, it is diverse, and research on external visualization has become a wide and complex field. The aim of this scoping review is to characterize external visualizations in recent mathematics education research in order to develop a common ground and guide future research. A qualitative content analysis of the full texts of 130 studies published between 2018 and 2022 applied a deductive-inductive coding procedure to assess four dimensions: visualization product or process, type of visualization, media, and purpose. The analysis revealed different types of external visualizations including visualizations with physical resemblance ranging from pictorial to abstract visualizations as well as three types of visualizations with structural resemblance: length, area, and relational visualizations. Future research should include measures of visualization products or processes to help explain the demands and affordances that different types of visualizations present to learners and teachers.
Keywords: General Mathematics; Education

Intersection theorems for finite general linear groups - Ernst, Alena; Schmidt, Kai-Uwe (Mathematical Proceedings of the Cambridge Philosophical Society)
Keywords: General Mathematics

The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds - Cekić, Mihajlo; Delarue, Benjamin; Dyatlov, Semyon; Paternain, Gabriel P. (Inventiones mathematicae)
Abstract We show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold $$Σ $$ Σ with Betti number $$b_1$$ b 1 , the order of vanishing of the Ruelle zeta function at zero equals $$4-b_1$$ 4 - b 1 , while in the hyperbolic case it is equal to $$4-2b_1$$ 4 - 2 b 1 . This is in contrast to the 2-dimensional case where the order of vanishing is a topological invariant. The proof uses the microlocal approach to dynamical zeta functions, giving a geometric description of generalized Pollicott–Ruelle resonant differential forms at 0 in the hyperbolic case and using first variation for the perturbation. To show that the first variation is generically nonzero we introduce a new identity relating pushforwards of products of resonant and coresonant 2-forms on the sphere bundle $$SΣ $$ S Σ with harmonic 1-forms on $$Σ $$ Σ .
Keywords: General Mathematics

Numerical fatigue life prediction of corroded and non-corroded clinched joints - Harzheim, Sven; Hofmann, Martin; Wallmersperger, Thomas (Mechanics of Advanced Materials and Structures)
Mechanical clinching is used to create lightweight hybrid structures. In order to estimate the service life of clinched components, its fatigue properties need to be known under different mechanical loading conditions. In addition to fatigue, corrosion is another factor that affects the fatigue life of clinched joints. In the literature, many corrosion and high-cycle fatigue damage models exist. However, little is known about how both phenomena interact in clinched joints. In this article, the influence of galvanic corrosion on clinched EN AW-6014/HCT590X + Z sheets on the fatigue life is investigated by means of numerical simulations and experimental results. An accurate prediction of the Wöhler lines of non-corroded and pre-corroded clinched specimens is shown.
Keywords: Mechanical Engineering; Mechanics of Materials; General Materials Science; General Mathematics; Civil and Structural Engineering

Ruelle–Pollicott resonances for manifolds with hyperbolic cusps - Guedes Bonthonneau, Yannick; Weich, Tobias (Journal of the European Mathematical Society)
Keywords: Applied Mathematics; General Mathematics

First-year university students' self-regulated learning during the COVID-19 pandemic: a qualitative longitudinal study - Liebendörfer, Michael; Kempen, Leander; Schukajlow, Stanislaw (ZDM – Mathematics Education)
Abstract When the COVID-19 pandemic began, many universities switched to fully online teaching. This unexpected switching to online teaching was challenging for both teachers and students, and restrictions that were put in place because of pandemic made this challenge even greater. However, new ways of teaching might also open new opportunities for students’ learning. The research question driving our study was as follows: how do students regulate their learning and specifically their choice of resources and peer learning in university mathematics classes that are fully taught online as offered during the COVID-19 pandemic? We report on a longitudinal, qualitative study in which students recorded a brief audio diary twice a week over one whole semester (14 weeks). We focused on three students who completed 70 interviews in total and finished the semester with varying degrees of success. The results show how the students structured their studying (e.g., the roles that deadlines or synchronous teaching events played). They illustrate the strengths and limitations of digital materials provided by the lecturer and the use of complementary media. Further, the pandemic uncovered the double-edged role of simple, often anonymous exchanges (e.g., via Discord servers), with few binding forces for either side, and the significance of stable learning partnerships for students’ success. Our research highlights aspects that should be focal points when comparing traditional instruction and online instruction during the pandemic from a self-regulatory perspective. Practical implications refer to how these aspects can be combined sensibly in fully online courses, but also in blended learning contexts.
Keywords: General Mathematics; Education

Two-Colour Spectrally Multimode Integrated SU(1,1) Interferometer - Ferreri, Alessandro; Sharapova, Polina R. (Symmetry)
Multimode integrated interferometers have great potential for both spectral engineering and metrological applications. However, the material dispersion of integrated platforms constitutes an obstacle that limits the performance and precision of such interferometers. At the same time, two-colour nonlinear interferometers present an important tool for metrological applications, when measurements in a certain frequency range are difficult. In this manuscript, we theoretically developed and investigated an integrated multimode two-colour SU(1,1) interferometer operating in a supersensitive mode. By ensuring the proper design of the integrated platform, we suppressed the dispersion, thereby significantly increasing the visibility of the interference pattern. The use of a continuous wave pump laser provided the symmetry between the spectral shapes of the signal and idler photons concerning half the pump frequency, despite different photon colours. We demonstrate that such an interferometer overcomes the classical phase sensitivity limit for wide parametric gain ranges, when up to 3×104 photons are generated.
Keywords: Physics and Astronomy (miscellaneous); General Mathematics; Chemistry (miscellaneous); Computer Science (miscellaneous)

Quantum generalizations of the polynomial hierarchy with applications to QMA(2) - Gharibian, Sevag; Santha, Miklos; Sikora, Jamie; Sundaram, Aarthi; Yirka, Justin (Computational Complexity)
Keywords: Computational Mathematics; Computational Theory and Mathematics; General Mathematics; Theoretical Computer Science

ℓ-torsion bounds for the class group of number fields with an ℓ-group as Galois group - Klüners, Jürgen; Wang, Jiuya (Proceedings of the American Mathematical Society)
We describe the relations among the ℓ-torsion conjecture, a conjecture of Malle giving an upper bound for the number of extensions, and the discriminant multiplicity conjecture. We prove that the latter two conjectures are equivalent in some sense. Altogether, the three conjectures are equivalent for the class of solvable groups. We then prove the ℓ-torsion conjecture for ℓ-groups and the other two conjectures for nilpotent groups.
Keywords: Applied Mathematics; General Mathematics

Binary Hermitian Lattices over Number Fields - Kirschmer, Markus; Nebe, Gabriele (Experimental Mathematics)
In a previous paper the authors developed an algorithm to classify certain quaternary quadratic lattices over totally real fields. The present article applies this algorithm to the classification of binary Hermitian lattices over totally imaginary fields. We use it in particular to classify the 48-dimensional extremal even unimodular lattices over the integers that admit a semilarge automorphism.
Keywords: General Mathematics

Alte Fotografien, neue Erkenntnisse - Liszt-Rohlf, Verena; Wochnik, Markus; Schwabl, Franziska (Zeitschrift für Berufs- und Wirtschaftspädagogik)
Keywords: Applied Mathematics; General Mathematics

About the structure of the discrete and continuous Eringen’s nonlocal elastica - Cresson, Jacky; Hariz-Belgacem, Khaled (Mathematics and Mechanics of Solids)
The Eringen’s nonlocal elastica equation does not possess a Lagrangian formulation. In this article, we find a variational integrating factor which enables us to provide a Lagrangian and Hamiltonian structure associated to this equation. Explicit expressions of the solutions in terms of elliptic integrals of the first kind are then deduced. We then derive discrete version of the Eringen’s nonlocal elastica preserving the Lagrangian and Hamiltonian structure and compare it with Challamel’s and co-worker definition of a discrete Eringen’s nonlocal elastica.
Keywords: Mechanics of Materials; General Materials Science; General Mathematics

About the structure of the discrete and continuous Eringen’s nonlocal elastica - Cresson, Jacky; Hariz Belgacem, Khaled (Mathematics and Mechanics of Solids)
The Eringen’s nonlocal elastica equation does not possess a Lagrangian formulation. In this article, we find a variational integrating factor which enables us to provide a Lagrangian and Hamiltonian structure associated to this equation. Explicit expressions of the solutions in terms of elliptic integrals of the first kind are then deduced. We then derive discrete version of the Eringen’s nonlocal elastica preserving the Lagrangian and Hamiltonian structure and compare it with Challamel’s and co-worker definition of a discrete Eringen’s nonlocal elastica.
Keywords: Mechanics of Materials; General Materials Science; General Mathematics

Prospective Teachers’ Diagnostic Thinking on Students’ Understanding of Multi-Digit Multiplication: A Content-Related Analysis on Unpacking of Knowledge Elements - Dröse, Jennifer; Prediger, Susanne (Journal für Mathematik-Didaktik)
Abstract Teachers’ in-depth diagnostic thinking has been shown to be crucial for student-centered teaching as they need to perceive and interpret students’ understanding for well-informed decision-making on adaptive teaching practices. The paper presents a content-related approach to analyzing diagnostic thinking processes with respect to the mathematical knowledge elements that prospective teachers identify as students’ resources and obstacles. Prospective teachers’ challenge is that some relevant knowledge elements first have to be unpacked, because compact concepts (such as the place value concept) or procedures (such as for multi-digit multiplication) comprise several smaller knowledge elements (such as the positional property) that have to be made explicit for students to foster their learning processes adequately. Our study examines what knowledge elements prospective teachers perceive and interpret in a transcript vignettes on multi-digit multiplication (of decimal and natural numbers) and its underlying basic arithmetic concepts (place value understanding and meaning of multiplication) in written diagnostic judgments on students’ resources and obstacles ( N = 196). A comparative design within the vignette is used to investigate how far the process of perceiving can be supported by thematic cues. The analysis reveals that those knowledge elements cued in the vignette by being already unpacked and explicitly addressed are perceived and interpreted more often (but with lower correctness) than those that are uncued and therefore have to be unpacked by the prospective teachers themselves. This confirms the need to prepare prospective teachers for unpacking mathematical concepts themselves.
Keywords: Education; General Mathematics

Maximal regularity of backward difference time discretization for evolving surface PDEs and its application to nonlinear problems - Kovács, Balázs; Li, Buyang (IMA Journal of Numerical Analysis)
Abstract Maximal parabolic $L^p$-regularity of linear parabolic equations on an evolving surface is shown by pulling back the problem to the initial surface and studying the maximal $L^p$-regularity on a fixed surface. By freezing the coefficients in the parabolic equations at a fixed time and utilizing a perturbation argument around the freezed time, it is shown that backward difference time discretizations of linear parabolic equations on an evolving surface along characteristic trajectories can preserve maximal $L^p$-regularity in the discrete setting. The result is applied to prove the stability and convergence of time discretizations of nonlinear parabolic equations on an evolving surface, with linearly implicit backward differentiation formulae characteristic trajectories of the surface, for general locally Lipschitz nonlinearities. The discrete maximal $L^p$-regularity is used to prove the boundedness and stability of numerical solutions in the $L^∞ (0,T;W^{1,∞ })$ norm, which is used to bound the nonlinear terms in the stability analysis. Optimal-order error estimates of time discretizations in the $L^∞ (0,T;W^{1,∞ })$ norm is obtained by combining the stability analysis with the consistency estimates.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Bulk–surface Lie splitting for parabolic problems with dynamic boundary conditions - Altmann, Robert; Kovács, Balázs; Zimmer, Christoph (IMA Journal of Numerical Analysis)
Abstract This paper studies bulk–surface splitting methods of first order for (semilinear) parabolic partial differential equations with dynamic boundary conditions. The proposed Lie splitting scheme is based on a reformulation of the problem as a coupled partial differential–algebraic equation system, i.e., the boundary conditions are considered as a second dynamic equation that is coupled to the bulk problem. The splitting approach is combined with bulk–surface finite elements and an implicit Euler discretization of the two subsystems. We prove first-order convergence of the resulting fully discrete scheme in the presence of a weak CFL condition of the form $τ \leqslant c h$ for some constant $c>0$. The convergence is also illustrated numerically using dynamic boundary conditions of Allen–Cahn type.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Error estimates for a splitting integrator for abstract semilinear boundary coupled systems - Csomós, Petra; Farkas, Bálint; Kovács, Balázs (IMA Journal of Numerical Analysis)
Abstract We derive a numerical method, based on operator splitting, to abstract parabolic semilinear boundary coupled systems. The method decouples the linear components that describe the coupling and the dynamics in the abstract bulk- and surface-spaces, and treats the nonlinear terms similarly to an exponential integrator. The convergence proof is based on estimates for a recursive formulation of the error, using the parabolic smoothing property of analytic semigroups, and a careful comparison of the exact and approximate flows. This analysis also requires a deep understanding of the effects of the Dirichlet operator (the abstract version of the harmonic extension operator), which is essential for the stable coupling in our method. Numerical experiments, including problems with dynamic boundary conditions, reporting on convergence rates are presented.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion System - Winkler, Michael (International Mathematics Research Notices)
Abstract The Neumann problem for (0.1)$$ \begin{align}& V_t = \Delta V-aV+f(x,t) \end{align}$$is considered in bounded domains $\Omega \subset {\mathbb {R}}^n$ with smooth boundary, where $n\ge 1$ and $a\in {\mathbb {R}}$. By means of a variational approach, a statement on boundedness of the quantities $$ \begin{eqnarray*} \sup_{t\in (0,T)} \int_\Omega \big|\nabla V(\cdot,t)\big|^p L^{\frac{n+p}{n+2}} \Big( \big|\nabla V(\cdot,t)\big| \Big) \end{eqnarray*}$$in dependence on the expressions (0.2)$$ \begin{align}& \sup_{t\in (0,T-\tau)} \int_t^{t+\tau} \int_\Omega |f|^{\frac{(n+2)p}{n+p}} L\big( |f|\big) \end{align}$$is derived for $p\ge 2$, $\tau>0$, and $T\ge 2\tau $, provided that $L\in C^0([0,\infty ))$ is positive, strictly increasing, unbounded, and slowly growing in the sense that $\limsup _{s\to \infty } \frac {L(s^{\lambda _0})}{L(s)} <\infty $ for some $\lambda _0>1$. In the particular case when $p=n\ge 2$, an additional condition on growth of $L$, particularly satisfied by $L(\xi ):=\ln ^\alpha (\xi +b)$ whenever $b>0$ and $\alpha>\frac {(n+2)(n-1)}{2n}$, is identified as sufficient to ensure that as a consequence of the above, bounds for theintegrals in (0.2) even imply estimates for the spatio-temporal modulus of continuity of solutions to (0.1). A subsequent application to the Keller–Segel system $$ \begin{eqnarray*} \left\{ \begin{array}{l} u_t = \nabla \cdot \big( D(v)\nabla u\big) - \nabla \cdot \big( uS(v)\nabla v\big) + ru - \mu u^2, \\[1mm] v_t = \Delta v-v+u, \end{array} \right. \end{eqnarray*}$$shows that when $n=2$, $r\in {\mathbb {R}}$, $0<D\in C^2([0,\infty ))$, and $S\in C^2([0,\infty )) \cap W^{1,\infty }((0,\infty ))$ and thus especially in the presence of arbitrarily strong diffusion degeneracies implied by rapid decay of $D$, any choice of $\mu>0$ excludes blowup in the sense that for all suitably regular nonnegative initial data, an associated initial-boundary value problem admits a global bounded classical solution.
Keywords: General Mathematics

Arbitrarily fast grow-up rates in quasilinear Keller–Segel systems - Winkler, Michael (Communications in Contemporary Mathematics)
The chemotaxis system [Formula: see text] is considered in a ball [Formula: see text], [Formula: see text], where the positive function [Formula: see text] reflects suitably weak diffusion by satisfying [Formula: see text] for some [Formula: see text]. It is shown that whenever [Formula: see text] is positive and satisfies [Formula: see text] as [Formula: see text], one can find a suitably regular nonlinearity [Formula: see text] with the property that at each sufficiently large mass level [Formula: see text] there exists a globally defined radially symmetric classical solution to a Neumann-type boundary value problem for (⋆) which satisfies [Formula: see text]
Keywords: Applied Mathematics; General Mathematics

Finite-time blow-up in a repulsive chemotaxis-consumption system - Wang, Yulan; Winkler, Michael (Proceedings of the Royal Society of Edinburgh: Section A Mathematics)
In a ball $\Omega \subset \mathbb {R}^{n}$ with $n\ge 2$ , the chemotaxis system \[ \left\{ \begin{array}{@{}l} u_t = \nabla \cdot \big( D(u)\nabla u\big) + \nabla\cdot \big(\dfrac{u}{v} \nabla v\big), \\ 0=\Delta v - uv \end{array} \right. \] is considered along with no-flux boundary conditions for $u$ and with prescribed constant positive Dirichlet boundary data for $v$ . It is shown that if $D\in C^{3}([0,\infty ))$ is such that $0< D(\xi ) \le {K_D} (\xi +1)^{-\alpha }$ for all $\xi >0$ with some ${K_D}>0$ and $\alpha >0$ , then for all initial data from a considerably large set of radial functions on $\Omega$ , the corresponding initial-boundary value problem admits a solution blowing up in finite time.
Keywords: General Mathematics

Exponential grow-up rates in a quasilinear Keller–Segel system - Winkler, Michael (Asymptotic Analysis)
The chemotaxis system ( ⋆ ) u t = ∇ · ( D ( u ) ∇ u ) − ∇ · ( u S ( u ) ∇ v ) , 0 = Δ v − μ + u , μ = 1 | Ω | ∫ Ω u , is considered in a ball Ω = B R ( 0 ) ⊂ R n . It is shown that if S ∈ C 2 ( [ 0 , ∞ ) ) suitably generalizes the prototype given by S ( ξ ) = χ ξ + 1 , ξ ⩾ 0 , with some χ > 0, and if diffusion is suitably weak in the sense that 0 < D ∈ C 2 ( ( 0 , ∞ ) ) is such that there exist K D > 0 and m ∈ ( − ∞ , 1 − 2 n ) fulfilling D ( ξ ) ⩽ K D ξ m − 1 for all ξ > 0 , then for appropriate choices of sufficiently concentrated initial data, an associated no-flux initial-boundary value problem admits a global classical solution ( u , v ) which blows up in infinite time and satisfies 1 C e χ t ⩽ ‖ u ( · , t ) ‖ L ∞ ( Ω ) ⩽ C e χ t for all t > 0 . A major part of the proof is based on a comparison argument involving explicitly constructed subsolutions to a scalar parabolic problem satisfied by mass accumulation functions corresponding to solutions of (⋆).
Keywords: General Mathematics

Does Leray’s structure theorem withstand buoyancy-driven chemotaxis-fluid interaction? - Winkler, Michael (Journal of the European Mathematical Society)
Keywords: Applied Mathematics; General Mathematics

Application of the Moser–Trudinger inequality in the construction of global solutions to a strongly degenerate migration model - Winkler, Michael (Bulletin of Mathematical Sciences)
A no-flux initial-boundary value problem for the cross-diffusion system [Formula: see text] is considered in smoothly bounded domains [Formula: see text] with [Formula: see text]. It is shown that whenever [Formula: see text] is positive on [Formula: see text] and such that [Formula: see text] for some [Formula: see text], for all suitably regular positive initial data a global very weak solution, particularly preserving mass in its first component, can be constructed. This extends previous results which either concentrate on non-degenerate analogs, or are restricted to the special case [Formula: see text]. To appropriately cope with the considerably stronger cross-degeneracies thus allowed through [Formula: see text] when [Formula: see text] is large, in its core part the analysis relies on the use of the Moser–Trudinger inequality in controlling the respective diffusion rates [Formula: see text] from below.
Keywords: General Mathematics

Scaffolding Close Reading of Mathematical Text in Pre-service Primary Teacher Education at the Tertiary Level: Design and Evaluation - Rezat, Sebastian; Malik, Sara Naseem; Leifeld, Markus (International Journal of Science and Mathematics Education)
Abstract Even in the digital age, learning mathematics at an academic level still requires much reading of mathematical text. Research has shown that reading mathematical text requires readers to engage with all the structures of the book and with its pedagogical voice, making connections, and plausible reasoning. Specific practices and strategies that support the close reading of mathematical text have been suggested; however, descriptions and empirical evaluations of materials designed to support these activities are rare. We present the design and first evaluation cycle of materials developed in a design research project that aims to scaffold close reading of mathematical text. The materials were designed and evaluated in a German university course on elementary geometry for first-year teacher education students who study mathematics to become primary teachers. The reading strategies were explained and modeled for students in reading-strategy videos. Additionally, close reading of mathematical text was scaffolded by close-reading tasks and homework tasks and problems that build on the reading strategies and were specifically designed to foster understanding of the mathematical text. Survey data were collected from 296 students to evaluate their use of and attitude toward the different materials. The quantitative results indicate that students used the materials and were generally able to learn the course content by themselves. From all provided materials, they found the close-reading tasks most helpful. A qualitative analysis of answers to open questions revealed issues with different materials, particularly with the script, and requests for additional materials. The issues with the script were categorized inductively. The categories are presented as a qualitative result of the study and discussed.
Keywords: General Mathematics; Education

Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces - Küster, Benjamin; Weich, Tobias (International Mathematics Research Notices)
Abstract For a compact Riemannian locally symmetric space $\mathcal M$ of rank 1 and an associated vector bundle $\mathbf V_{τ }$ over the unit cosphere bundle $S^{* }\mathcal M$, we give a precise description of those classical (Pollicott–Ruelle) resonant states on $\mathbf V_{τ }$ that vanish under covariant derivatives in the Anosov-unstable directions of the chaotic geodesic flow on $S^{* }\mathcal M$. In particular, we show that they are isomorphically mapped by natural pushforwards into generalized common eigenspaces of the algebra of invariant differential operators $D(G,σ )$ on compatible associated vector bundles $\mathbf W_{σ }$ over $\mathcal M$. As a consequence of this description, we obtain an exact band structure of the Pollicott–Ruelle spectrum. Further, under some mild assumptions on the representations $τ$ and $σ$ defining the bundles $\mathbf V_{τ }$ and $\mathbf W_{σ }$, we obtain a very explicit description of the generalized common eigenspaces. This allows us to relate classical Pollicott–Ruelle resonances to quantum eigenvalues of a Laplacian in a suitable Hilbert space of sections of $\mathbf W_{σ }$. Our methods of proof are based on representation theory and Lie theory.
Keywords: General Mathematics

Increasing the effect of a remedial mathematics course by switching to an online format during the COVID-19 crisis: evidence from a German university - Büchele, Stefan; Liebendörfer, Michael; Lankeit, Elisa (Teaching Mathematics and its Applications: An International Journal of the IMA)
Abstract Due to the COVID-19 crisis, many courses have been offered digitally. Using data from n = 1,173 business students participating in a preparatory mathematics course at a German university that covered the same content as in 2018, 2019 and 2020, we examine how students’ participation and the effect of the preparatory course changed. The data show that the participation rate has fallen slightly, but students’ participation is rather similar to preceding years. Interestingly, students have participated more intensively. There are clear signs of dishonesty in the self-test (use of a calculator) and significant changes in predictors of performance. In particular, the effect of students’ engagement in the course on their performance substantially increased. Further, we found a gender gap in performance affecting women. Finally, the data show that digital courses can be as effective as on-campus courses.
Keywords: Education; General Mathematics

University students’ fully digital study of mathematics: an identification of student-groups via their resources usage and a characterization by personal and affective characteristics - Kempen, Leander; Liebendörfer, Michael (Teaching Mathematics and its Applications: An International Journal of the IMA)
Abstract We investigated university students’ study of mathematics in the digital setting context of the COVID-19 pandemic. We gathered data from a survey of 89 students enrolled in a ‘Linear Algebra 1’ course including affective variables, learning strategies, social relatedness and resources considered useful. The results indicate students’ high effort and self-regulation and a high variation in affective characteristics and social relatedness. All the traditional aspects of mathematics teaching (lecture, tutorials and lecture notes) were rated as particularly useful. In addition, the videos from external resources were rated as equally useful as the teaching team’s videos. In contrast, traditional literature such as textbooks was rarely considered useful. The most useful resource rated was communication with peers, underlining the important role of social learning despite fully digital learning environments. Finally, a cluster analysis based on students’ rated usefulness of the resources led to three different user-types. Whereas the ‘digitals’ find the external digital resources very useful (videos, webpages, etc.), the ‘traditionalists’ rate the digitalized traditional resources best (lecture, tutorials, etc.). All resources receive uniformly good ratings from a third group (‘all resource users’). We reflect on our findings in light of the pandemic and describe directions for future research.
Keywords: Education; General Mathematics

Sonine Formulas and Intertwining Operators in Dunkl Theory - Rösler, Margit; Voit, Michael (International Mathematics Research Notices)
Abstract Let $V_k$ denote Dunkl’s intertwining operator associated with some root system $R$ and multiplicity $k$. For two multiplicities $k, k^{′ }$ on $R$, we study the intertwiner $V_{k^{′ },k} = V_{k^{′ }}° V_k^{-1}$ between Dunkl operators with multiplicities $k$ and $k^{′ }.$ It has been a long-standing conjecture that $V_{k^{′ },k}$ is positive if $k^{′ } ≥ k ≥ 0.$ We disprove this conjecture by constructing counterexamples for root system $B_n$. This matter is closely related to the existence of Sonine-type integral representations between Dunkl kernels and Bessel functions with different multiplicities. In our examples, such Sonine formulas do not exist. As a consequence, we obtain necessary conditions on Sonine formulas for Heckman–Opdam hypergeometric functions of type $BC_n$ and conditions for positive branching coefficients between multivariable Jacobi polynomials.
Keywords: General Mathematics

Positive intertwiners for Bessel functions of type B - Rösler, Margit; Voit, Michael (Proceedings of the American Mathematical Society)
Keywords: Applied Mathematics; General Mathematics

Relaxed parameter conditions for chemotactic collapse in logistic-type parabolic–elliptic Keller–Segel systems - Black, Tobias; Fuest, Mario; Lankeit, Johannes (Zeitschrift für angewandte Mathematik und Physik)
Keywords: Applied Mathematics; General Physics and Astronomy; General Mathematics

Prescribed signal concentration on the boundary: Weak solvability in a chemotaxis-Stokes system with proliferation - Black, Tobias; Wu, Chunyan (Zeitschrift für angewandte Mathematik und Physik)
Keywords: Applied Mathematics; General Physics and Astronomy; General Mathematics

Selbstreguliertes Lernen im berufsschulischen Übergangssystem - Daniel-Söltenfuß, Desiree; Schwabl, Franziska (Zeitschrift für Berufs- und Wirtschaftspädagogik)
Keywords: Applied Mathematics; General Mathematics

Geometrisches Begriffsverständnis in der Grundschule am Beispiel der Begriffe Viereck, Rechteck und Quadrat - Bruns, Julia; Unterhauser, Elisabeth; Gasteiger, Hedwig (Journal für Mathematik-Didaktik)
Zusammenfassung Zu den ersten geometrischen Begriffen, die Kinder bereits im Elementar- und Primarbereich lernen, zählen u. a. Viereck, Rechteck und Quadrat. Studien zeigen, dass Lernende bereits früh individuelle Vorstellungen, sog. individuelle Begriffskonzepte, zu diesen Begriffen aufbauen. Zwar wird die Entwicklung von Begriffsverständnis in verschiedenen mathematikdidaktischen Stufenmodellen dargestellt, diese sind jedoch generisch und beschreiben nicht explizit die Entwicklung der ersten individuellen Begriffskonzepte von Lernenden zu Viereck, Rechteck und Quadrat. Aus empirischer Sicht liegen verschiedene Studien vor, die einzelne Aspekte der individuellen Begriffskonzepte von Lernenden unterschiedlicher Altersgruppen zu diesen Begriffen ausleuchten. Um Begriffsbildungsprozesse aus empirischer Sicht detaillierter entlang der jeweils vorherrschenden individuellen Begriffskonzepte zu beschreiben, fehlen insbesondere Studien in der Grundschule, die alle vier Klassenstufen betrachten und dabei differenzierte Erkenntnisse zu verschiedenen theoretischen Indikatoren des Begriffsverständnisses liefern. Daher geht die vorliegende Studie der Frage nach, welches Verständnis der Begriffe Viereck, Rechteck und Quadrat Schülerinnen und Schüler der Jahrgangsstufen 1, 2, 3 und 4 zeigen. Dazu wurde eine Quasi-Längsschnittstudie mit N = 456 Grundschulkindern (ca. 100 pro Jahrgangsstufe) durchgeführt. Die Ergebnisse geben detaillierte Einblicke in die individuellen Begriffskonzepte der Lernenden und zeigen, dass Lernende zunehmend Eigenschaften der Figuren berücksichtigen, jedoch individuelle Begriffskonzepte über lange Zeit auch prototypisch geprägt sind. Implikationen dieser Ergebnisse für Forschung und Praxis werden diskutiert.
Keywords: Education; General Mathematics

A convergent finite element algorithm for generalized mean curvature flows of closed surfaces - Binz, Tim; Kovács, Balázs (IMA Journal of Numerical Analysis)
Abstract An algorithm is proposed for generalized mean curvature flow of closed two-dimensional surfaces, which include inverse mean curvature flow and powers of mean and inverse mean curvature flow. Error estimates are proved for semidiscretizations and full discretizations for the generalized flow. The algorithm proposed and studied here combines evolving surface finite elements, whose nodes determine the discrete surface, and linearly implicit backward difference formulae for time integration. The numerical method is based on a system coupling the surface evolution to nonlinear second-order parabolic evolution equations for the normal velocity and normal vector. A convergence proof is presented in the case of finite elements of polynomial degree at least 2 and backward difference formulae of orders 2 to 5. The error analysis combines stability estimates and consistency estimates to yield optimal-order $H^1$-norm error bounds for the computed surface position, velocity, normal vector, normal velocity and therefore for the mean curvature. The stability analysis is performed in the matrix–vector formulation and is independent of geometric arguments, which only enter the consistency analysis. Numerical experiments are presented to illustrate the convergence results and also to report on monotone quantities, e.g. Hawking mass for inverse mean curvature flow, and complemented by experiments for nonconvex surfaces.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Error estimates for the Cahn–Hilliard equation with dynamic boundary conditions - Harder, Paula; Kovács, Balázs (IMA Journal of Numerical Analysis)
Abstract A proof of convergence is given for a bulk–surface finite element semidiscretisation of the Cahn–Hilliard equation with Cahn–Hilliard-type dynamic boundary conditions in a smooth domain. The semidiscretisation is studied in an abstract weak formulation as a second-order system. Optimal-order uniform-in-time error estimates are shown in the $L^2$- and $H^1$-norms. The error estimates are based on a consistency and stability analysis. The proof of stability is performed in an abstract framework, based on energy estimates exploiting the anti-symmetric structure of the second-order system. Numerical experiments illustrate the theoretical results.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Mathematics textbooks and curriculum resources as instruments for change - Rezat, Sebastian; Fan, Lianghuo; Pepin, Birgit (ZDM – Mathematics Education)
Abstract In this survey paper we aim to provide an overview of research on mathematics textbooks and, more broadly, curriculum resources as instruments for change related to mathematical content, instructional goals and practices, and student learning of mathematics. In particular, we elaborate on the following themes: (1) The role of curriculum resources as instruments for change from a theoretical perspective; (2) The design of curriculum resources to mediate the implementation of reform ideas and innovative practice; (3) Teachers’ influence on the implementation of change through curriculum resources; (4) Students’ influence on the implementation of change through curriculum resources; and (5) Evidence of curriculum resources yielding changes in student-related factors or variables. We claim that, whilst textbooks and curriculum resources are influential, they alone cannot change teachers’ teaching nor students’ learning practices in times of curricular change. Moreover, more knowledge is needed about features of curriculum resources that support the implementation of change. We contend that curriculum innovations are likely to be successful, if teachers and students are supported to co- and re-design the relevant curriculum trajectories and materials in line with the reform efforts and their own individual needs.
Keywords: General Mathematics; Education

How automated feedback from a digital mathematics textbook affects primary students’ conceptual development: two case studies - Rezat, Sebastian (ZDM Mathematics Education)
Abstract One of the most prevalent features of digital mathematics textbooks, compared to traditional ones, is the provision of automated feedback on students’ solutions. Since feedback is regarded as an important factor that influences learning, this is often seen as an affordance of digital mathematics textbooks. While there is a large body of mainly quantitative research on the effectiveness of feedback in general, very little is known about how feedback actually affects students’ individual content specific learning processes and conceptual development. A theoretical framework based on Rabardel’s theory of the instrument and Vergnaud’s theory of conceptual fields is developed to study qualitatively how feedback actually functions in the learning process. This framework was applied in a case study of two elementary school students’ learning processes when working on a probability task from a German 3rd grade digital textbook. The analysis allowed detailed reconstruction of how students made sense of the information provided by the feedback and adjusted their behavior accordingly. This in-depth analysis unveiled that feedback does not necessarily foster conceptual development in the desired way, and a correct solution does not always coincide with conceptual understanding. The results point to some obstacles that students face when working individually on tasks from digital mathematics textbooks with automated feedback, and indicate that feedback needs to be developed in design-based research cycles in order to yield the desired effects.
Keywords: General Mathematics; Education

The process of actively building a model for a randomization test – insights into learners’ modeling activities based on a case study - Podworny, Susanne; Biehler, Rolf (Mathematical Thinking and Learning)
Keywords: Developmental and Educational Psychology; Education; General Mathematics

Sprachsensible Formulierung von Erhebungsinstrumenten - Schwabl, Franziska; Janssen, Elmar A.; Sloane, Peter F. E. (Zeitschrift für Berufs- und Wirtschaftspädagogik)
Keywords: Applied Mathematics; General Mathematics

Sprachsensible Formulierung von Erhebungsinstrumenten - Schwabl, Franziska; Janssen, Elmar; Sloane, Peter F. E. (Zeitschrift für Berufs- und Wirtschaftspädagogik)
Keywords: Applied Mathematics; General Mathematics