The purpose of this study is to provide an in-depth analysis of the geometric construction strategies and characteristics of geometric perception demonstrated by elementary gifted students during example-generating activities exploring tangential quadrilaterals. To this end, a qualitative case study was conducted with 79 fourth-grade students enrolled in a gifted education center in City S. A three-stage example-generating task integrating paper-and-pencil activities and a Dynamic Geometry Environment (DGE) was implemented. The results revealed that while students initially relied on visual intuition and prototypical shapes to construct examples, their strategies evolved into structural approaches focusing on the relationships among geometric elements through an iterative process of examining counter-examples and modifying conditions. The strengthening of task constraints and the utilization of DGE served as key mechanisms in facilitating the flexible and multifaceted evolution of students' strategies. Specifically, the dynamic manipulation and measurement features of the DGE transformed visual conjectures into logical verifications, thereby promoting the structural generalization of conditions. Overall, these results suggest that example-generating activities can serve as a meaningful approach to enable mathematically gifted elementary students to actively construct the boundaries and structures of geometric concepts.

A proper Euler’s magic matrix is an integer n×n matrix M∈Zn×n such that M⋅Mt=γ⋅I for some nonzero constant γ, the sum of the squares of the entries along each of the two main diagonals equals γ, and the squares of all entries in M are pairwise distinct. Euler constructed such matrices for n=4. In this work, we use multiplication matrices of the octonions to construct examples for n=8, and prove that no such matrix exists for n=3.

In this article, we investigate the regularization and qualitative properties of parabolic Ginzburg–Landau equations in variable exponent Herz spaces. These spaces capture both local and global behavior, providing a natural framework for our analysis. We establish boundedness results for fractional Bessel–Riesz operators, construct examples highlighting their advantage over classical Riesz potentials, and recover several known theorems as special cases. As an application, we analyze a parabolic Ginzburg–Landau operator with VMO coefficients, showing that our estimates ensure the boundedness and continuity of solutions.

Abstract We study infinite superelliptic curves as translation surfaces and explore their Veech groups. These objects are branched covering of the complex plane, branching over infinitely many points. We provide a criterion for isomorphism between a special family of infinite superelliptic curves. We describe the geometry of saddle connections and holonomy vectors on these infinite superelliptic curves. In addition, we prove that the Veech group of an infinite superelliptic curve consists of matrices arising from the differentials of the affine mappings from C {\mathbb{C}} to itself, which permutes the branched points. We obtain necessary and sufficient conditions to guarantee that the Veech group of an infinite superelliptic curve is uncountable. Furthermore, we establish a trichotomy on the holonomy vector set and precisely characterize certain countable groups that can appear as Veech groups of an infinite superelliptic curve. Finally, we also construct and study several examples of interesting infinite superelliptic curves illustrating our results.

This paper aims to explore causes of difficulties while learning linear algebra course at master level and find possible ways to mitigate these difficulties. I applied case study methods and selected two cases responded purposively. I used semi-structured interview guidelines and performance test to explore the causes of difficulties and used general inductive approach to analyze the data. The result shows that the students at master level are experiencing linear algebra course difficult for conceptual and procedural understanding because of having poor English language, unavailability of appropriate learning resources, having poor perquisites, abstract nature of concepts, unable to construct examples and counter examples, and the nature of the curriculum. These difficulties can mitigate by developing strong prerequisites before introducing new concepts, using examples of abstract concepts, choosing varieties of techniques of proofs for theorems, connecting abstraction with real-life examples, and other software and videos of the concepts in linear algebra while teaching and learning at master level.

Abstract Let be an exact sequence where is the fundamental group of a closed surface of genus greater than one, is hyperbolic, and is finitely generated free. The aim of this paper is to provide sufficient conditions to prove that is cubulable and construct examples satisfying these conditions. The main result may be thought of as a combination theorem for virtually special hyperbolic groups when the amalgamating subgroup is not quasiconvex. Ingredients include the theory of tracks, the quasiconvex hierarchy theorem of Wise, the distance estimates in the mapping class group from subsurface projections due to Masur–Minsky, and the model geometry for doubly degenerate Kleinian surface groups used in the proof of the ending lamination theorem. An appendix to this paper by Manning, Mj, and Sageev proves a reduction theorem by showing that cubulability of follows from the existence of an essential incompressible quasiconvex track in a surface bundle over a graph with fundamental g\roup .

Despite the fact that copulas are commonly considered as analytically smooth/regular objects, derivatives of copulas have to be handled with care. Triggered by a recently published result characterizing multivariate copulas via (d−1)-increasingness of their partial derivative we study the bivariate setting in detail and show that the set of non-differentiability points of a copula may be quite large. We first construct examples of copulas C whose first partial derivative ∂1C(x,y) is pathological in the sense that for almost every x∈(0,1) it does not exist on a dense subset of y∈(0,1), and then show that the family of these copulas is dense. Since in commonly considered subfamilies more regularity might be typical, we then focus on bivariate Extreme Value copulas (EVCs) and show that a topologically typical EVC is not absolutely continuous but has degenerated discrete component, implying that in this class typically ∂1C(x,y) exists in full (0,1)2.Considering that regularity of copulas is closely related to their mass distributions we then study mass distributions of topologically typical copulas and prove the surprising fact that topologically typical bivariate copulas are mutually completely dependent with full support. Furthermore, we use the characterization of EVCs in terms of their associated Pickands dependence measures ϑ on [0,1], show that regularity of ϑ carries over to the corresponding EVC and prove that the subfamily of all EVCs whose absolutely continuous, discrete and singular component has full support is dense in the class of all EVCs.

We characterize the 2-Killing vector fields on a multiply twisted product manifold, with a special view towards generalized spacetimes. More precisely, we determine the nonlinear differential equations that completely describe them and the twisted functions, give particular solutions, and construct examples.

We show various criteria to verify if a given nested fractal has a good labeling property, inter alia we present a characterization of GLP for fractals with an odd number of essential fixed points. We show a convenient reduction of the area to be investigated in the verification of GLP and give examples that further reduction is impossible. We prove that if the number of essential fixed points is a power of two, then a fractal must have GLP and that there are no values other than primes or powers of two guaranteeing GLP. For all other numbers of essential fixed points, we are able to construct examples having and others not having GLP.

N-functions and their growth and regularity properties are crucial in order to introduce and study Orlicz classes and Orlicz spaces. We consider N-functions which are given in terms of so-called associated weight functions. These functions are frequently appearing in the theory of ultradifferentiable function classes and in this setting additional information is available since associated weight functions are defined in terms of a given weight sequence. We express and characterize several known properties for N-functions purely in terms of weight sequences which allows to construct (counter-) examples. Moreover, we study how for abstractly given N-functions this framework becomes meaningful and finally we establish a connection between the complementary N-function and the recently introduced notion of the so-called dual sequence.

On moments and symmetrical sequences

Abstract In this paper, we present a constructive and proof-relevant development of graph theory, including the notion of maps, their faces and maps of graphs embedded in the sphere, in homotopy type theory (HoTT). This allows us to provide an elementary characterisation of planarity for locally directed finite and connected multigraphs that takes inspiration from topological graph theory, particularly from combinatorial embeddings of graphs into surfaces. A graph is planar if it has a map and an outer face with which any walk in the embedded graph is walk-homotopic to another. A result is that this type of planar maps forms a homotopy set for a graph. As a way to construct examples of planar graphs inductively, extensions of planar maps are introduced. We formalise the essential parts of this work in the proof assistant Agda with support for HoTT.

We study injective homomorphisms between big mapping class groups of infinite-type surfaces. First, we construct (uncountably many) examples of surfaces without boundary whose (pure) mapping class groups are not co-Hopfian; these are first such examples of injective endomorphisms of mapping class groups that fail to be surjective. We then prove that, subject to some topological conditions on the domain surface, any continuous injective homomorphism between (arbitrary) big mapping class groups that sends Dehn twists to Dehn twists is induced by a subsurface embedding. Finally, we explore the extent to which, in stark contrast to the finite-type case, superinjective maps between curve graphs impose no topological restrictions on the underlying surfaces.

In this paper, we relate the theory of quasi-conformal maps to the regularity of the solutions to nonlinear thin-obstacle problems; we prove that the contact set is locally a finite union of intervals and apply this result to the solutions of one-phase Bernoulli free boundary problems with geometric constraint. We also introduce a new conformal hodograph transform, which allows to obtain the precise expansion at branch points of both the solutions to the one-phase problem with geometric constraint and a class of symmetric solutions to the two-phase problem, as well as to construct examples of free boundaries with cusp-like singularities.

Dilation and Birkhoff-James orthogonality

Mohiudddine and Alotaibi [25] introduced the notion of intuitionistic generalized fuzzy metric space to extend the generalized fuzzy metric space. Choi et al.[Structure for 1-Metric Spaces and Related Fixed Point Theorems, arXiv preprint arXiv:1804.03651, (2018)] has recently proposed the notion of 1-metric as a generalized notion of the distance function. Employing the idea in this paper, we first put forth the notion of Intuitionistic fuzzy G-metric space with order n as a generalization of intuitionistic fuzzy metric space. We describe some properties of this novel space and construct examples based on it. Then, we propose the concepts of statistical convergence, statistical limit points and statistical cluster points of sequences in this space and establish theorems in their regard by providing appropriate examples in support of them.

A precise and timely classification of particulate matter 2.5 concentration levels is important for the design of air quality regulatory measures in a contemporaneous context characterized by the transition to a low-carbon economy. This study uses a well-known air quality dataset retrieved from the University of California at Irvine repository, which consists of a multivariate time series covering particulate matter 2.5 concentration levels in the city of Beijing for a period of 5 years. We train, test, and validate several deep learning architectures for a multinomial classification of the target variable in the period of 24 h ahead from the contemporaneous moment of action relying on historical information about the last 168 h and considering a sliding window of 24 h to construct examples. Results indicate that the internationally patented Variable Split Convolutional Attention model exhibits the best accuracy. The main novelty of this model consists of introducing bidimensional convolutional operations inside the attention block to capture the relative attention weight given to patterns of contiguous segments within different time-steps for each input variable. Therefore, a valuable deep learning architecture is presented to properly classify particulate matter 2.5 concentration levels in the atmosphere.

Hamza and Klebaner (2007) [10] posed the problem of constructing martingales with one-dimensional Brownian marginals that differ from Brownian motion, so-called fake Brownian motions. Besides its theoretical appeal, this problem represents the quintessential version of the ubiquitous fitting problem in mathematical finance where the task is to construct martingales that satisfy marginal constraints imposed by market data.Non-continuous solutions to this challenge were given by Madan and Yor (2002) [22], Hamza and Klebaner (2007) [10], Hobson (2016) [11] and Fan et al. (2015) [8], whereas continuous (but non-Markovian) fake Brownian motions were constructed by Oleszkiewicz (2008) [23], Albin (2008) [1], Baker et al. (2006) [4], Hobson (2013) [14], Jourdain and Zhou (2020) [16]. In contrast, it is known from Gyöngy (1986) [9], Dupire (1994) [7] and ultimately Lowther (2008) [17] and Lowther (2009) [20] that Brownian motion is the unique continuous strong Markov martingale with one-dimensional Brownian marginals.We took this as a challenge to construct examples of a “barely fake” Brownian motion, that is, continuous Markov martingales with one-dimensional Brownian marginals that miss out only on the strong Markov property.

We construct and study several examples of continuous families of AdS4 mathcal{N} = 1 solutions of four-dimensional maximal gauged supergravity. These backgrounds provide the holographic descriptions of conformal manifolds of the dual 3d mathcal{N} = 1 SCFTs. The solutions we study can be uplifted to type IIB supergravity where they arise from D3-branes wrapping an S1 with an S-duality twist. We find the spectrum of low lying operators in the 3d mathcal{N} = 1 SCFTs as a function of the exactly marginal coupling and discuss the structure of the corresponding superconformal multiplets. Using string theory techniques we also study additional examples of continuous families of mathcal{N} = 1 AdS4 vacua in type IIB and massive type IIA supergravity.

Abstract For a probability measure μ on [0, 1] without discrete component, the best possible order of approximation by a finite point set in terms of the star-discrepancy is &inline as has been proven relatively recently. However, if μ contains a discrete component no non-trivial lower bound holds in general because it is straightforward to construct examples without any approximation error in this case. This might explain, why the approximation of discrete measures on [0, 1] by finite point sets has so far not been completely covered in the existing literature. In this note, we close the gap by giving a complete description for discrete measures. Most importantly, we prove that for any discrete measures (not supported on one point only) the best possible order of approximation is for infinitely many N bounded from below by &inline for some constant 6 ≥ c> 2 which depends on the measure. This implies, that for a finitely supported discrete measure on [0, 1]d the known possible order of approximation &inline is indeed the optimal one.