Given a stream of edge additions and deletions, how can we estimate the count of triangles in it? If we can store only a subset of the edges, how can we obtain unbiased estimates with small ...variances?
Counting triangles (i.e., cliques of size three) in a graph is a classical problem with applications in a wide range of research areas, including social network analysis, data mining, and databases. Recently, streaming algorithms for triangle counting have been extensively studied since they can naturally be used for large dynamic graphs. However, existing algorithms cannot handle edge deletions or suffer from low accuracy.
Can we handle edge deletions while achieving high accuracy? We propose T
hink
D, which accurately estimates the counts of global triangles (i.e., all triangles) and local triangles associated with each node in a fully dynamic graph stream with additions and deletions of edges. Compared to its best competitors, T
hink
D is
(a) Accurate:
up to
4.3
×
more accurate
within the same memory budget,
(b) Fast:
up to
2.2
×
faster
for the same accuracy requirements, and
(c) Theoretically sound:
always maintaining estimates with zero bias (i.e., the difference between the true triangle count and the expected value of its estimate) and small variance. As an application, we use T
hink
D to detect suddenly emerging dense subgraphs, and we show its advantages over state-of-the-art methods.
This study addresses high school students' conceptions of mathematical definitions of congruent and similar triangles. The findings indicate that many of the participants differentiated between ...definitions and theorems and did not always accept the congruent and similar triangles theorems as formal definitions of congruency and similarity. Based on the participants' explanations of their responses and from the interviews performed, it appears that two issues prevented some participants from accepting or preferring these theorems as definitions. The first was a concern for uniformity: there is only one known and accepted definition of each concept. The second was a focus on the essence of the concepts: the essence of the concepts of similarity and congruency lies primarily in the lengths of the sides of a triangle. The students who accepted these theorems as formal definitions explained their reaction as arising from the equivalence and from the theorems including necessary and sufficient attributes. Students' difficulties in understanding the characteristics and roles of mathematical definitions of geometric concepts affected their understandings of mathematical and geometric definitions. This behaviour indicates a tendency to interpret the content of theorems incorrectly and an inability to unpack the logical structure of the theorem.
The Heilbronn triangle problem is a classical geometrical problem that asks for a placement of n$$ n $$ points in the unit square 0,12$$ {\left0,1\right}^2 $$ that maximizes the smallest area of a ...triangle formed by three of those points. This problem has natural generalizations. For an integer k≥3$$ k\ge 3 $$ and a set 𝒫 of n$$ n $$ points in 0,12$$ {\left0,1\right}^2 $$, let Ak(𝒫) be the minimum area of the convex hull of k$$ k $$ points in 𝒫. Here, instead of considering the supremum of Ak(𝒫) over all such choices of 𝒫, we consider its average value, Δ˜k(n)$$ {\tilde{\Delta}}_k(n) $$, when the n$$ n $$ points in 𝒫 are chosen independently and uniformly at random in 0,12$$ {\left0,1\right}^2 $$. We prove that Δ˜k(n)=Θn−kk−2$$ {\tilde{\Delta}}_k(n)=\Theta \left({n}^{\frac{-k}{k-2}}\right) $$, for every fixed k≥3$$ k\ge 3 $$.
Based on students' geometry knowledge, there are contrast and less relevant skills goals being prepared for the students. Students are more dominant in learning how to apply procedural knowledge so ...there is a need for students to use the Pythagorean theorem when facing a triangle properties problem. Therefore, this case study aims to analyze how students' conceptual knowledge depends on the Pythagorean theorem. The analysis uses a cognitive diagnostic assessment framework through the three parallel design of abstraction problem. This study was conducted for students at the senior high school. The findings are the Pythagorean theorem as a result of thinking abstraction at least two of the three design problem formations, including for the effect of claims and metacognitive knowledge them. There is a disconnected conceptual system between the products of thought and the claims elicited so that abstraction is not optimal. Development for in-depth understanding of conceptual experience is needed in the instructional intervention so that more adequate reasoning.
Since the 1990s, regional organizations of the United Nations and international financial institutions have adopted a new dynamic of transnational integration, within the framework of the ...regionalization process of globalization. In place of the growth triangles of the 1970s, a strategy based on transnational economic corridors has changed the scale of regionalization.
Thanks to the initiative of the Asian Development Bank, Southeast Asia provides two of the most advanced examples of such a process in East Asia with, on the one hand, the Greater Mekong Subregion, structured by continental corridors, and on the other, the Malacca Straits, combining maritime and land corridors. This book compares, after two decades, the effects of these developing networks on transnational integration in both subregions.
After presenting the general issue of economic corridors, the work deals with the characteristics and structures peculiar to these two regions, followed by a study of national strategies mobilizing actors at different levels of state organization. There follows a study of the emergence of new urban nodes on corridors at land and sea borders, and the impact of these corridors on the local societies. This approach makes it possible to compare the effects of transnational integration processes on the spatial and urban organization of the two subregions and on the increasing diversity of the stakeholders involved.
Hyperbolic Lunes Dray, Tevian
The College mathematics journal,
09/2023, Letnik:
54, Številka:
4
Journal Article
Recenzirano
SummaryThe formula for the area of a hyperbolic triangle in terms of its angle defect is derived using hyperbolic lunes, in analogy with the argument using (elliptic) lunes to express the area of an ...elliptic triangle in terms of its angle excess. Several pedagogical features of this construction are then discussed.
Non‐canonical interactions in DNA remain under‐explored in DNA nanotechnology. Recently, many structures with non‐canonical motifs have been discovered, notably a hexagonal arrangement of typically ...rhombohedral DNA tensegrity triangles that forms through non‐canonical sticky end interactions. Here, we find a series of mechanisms to program a hexagonal arrangement using: the sticky end sequence; triangle edge torsional stress; and crystallization condition. We showcase cross‐talking between Watson–Crick and non‐canonical sticky ends in which the ratio between the two dictates segregation by crystal forms or combination into composite crystals. Finally, we develop a method for reconfiguring the long‐range geometry of formed crystals from rhombohedral to hexagonal and vice versa. These data demonstrate fine control over non‐canonical motifs and their topological self‐assembly. This will vastly increase the programmability, functionality, and versatility of rationally designed DNA constructs.
In DNA nanotechnology, the programmability of non‐canonical interactions is a key area of exploration. Here, we present three methods for programming a non‐canonical hexagonal arrangement of normally rhombohedral DNA tensegrity triangles. We also determine cross‐talking and reconfiguration between hexagonal and rhombohedral DNA crystals. These methods allow for additional versatility and programmability of non‐canonical 3D DNA constructs.
The dimensionality of an electronic quantum system is decisive for its properties. In one dimension electrons form a Luttinger liquid and in two dimensions they exhibit the quantum Hall effect. ...However, very little is known about the behavior of electrons in non-integer, or fractional dimensions1. Here, we show how arrays of artificial atoms can be defined by controlled positioning of CO molecules on a Cu (111) surface2-4, and how these sites couple to form electronic Sierpiński fractals. We characterize the electron wave functions at different energies with scanning tunneling microscopy and spectroscopy and show that they inherit the fractional dimension. Wave functions delocalized over the Sierpiński structure decompose into self-similar parts at higher energy, and this scale invariance can also be retrieved in reciprocal space. Our results show that electronic quantum fractals can be artificially created by atomic manipulation in a scanning tunneling microscope. The same methodology will allow future study to address fundamental questions about the effects of spin-orbit interaction and a magnetic field on electrons in non-integer dimensions. Moreover, the rational concept of artificial atoms can readily be transferred to planar semiconductor electronics, allowing for the exploration of electrons in a well-defined fractal geometry, including interactions and external fields.
The kagome lattice is a two-dimensional network of corner-sharing triangles that is known to host exotic quantum magnetic states. Theoretical work has predicted that kagome lattices may also host ...Dirac electronic states that could lead to topological and Chern insulating phases, but these states have so far not been detected in experiments. Here we study the d-electron kagome metal Fe
Sn
, which is designed to support bulk massive Dirac fermions in the presence of ferromagnetic order. We observe a temperature-independent intrinsic anomalous Hall conductivity that persists above room temperature, which is suggestive of prominent Berry curvature from the time-reversal-symmetry-breaking electronic bands of the kagome plane. Using angle-resolved photoemission spectroscopy, we observe a pair of quasi-two-dimensional Dirac cones near the Fermi level with a mass gap of 30 millielectronvolts, which correspond to massive Dirac fermions that generate Berry-curvature-induced Hall conductivity. We show that this behaviour is a consequence of the underlying symmetry properties of the bilayer kagome lattice in the ferromagnetic state and the atomic spin-orbit coupling. This work provides evidence for a ferromagnetic kagome metal and an example of emergent topological electronic properties in a correlated electron system. Our results provide insight into the recent discoveries of exotic electronic behaviour in kagome-lattice antiferromagnets and may enable lattice-model realizations of fractional topological quantum states.
An auxetic metamaterial based on a semirotation system is designed herein by combining rigid units of two different shapes, such that each unit cell consists of one nonrotating rhombus and four ...rotating right triangles. The on‐axes Poisson's ratios are obtained for both infinitesimal and finite deformations by geometrical consideration. The on‐axes Young's moduli of this metamaterial are established by matching the spring potential energy stored at the hinged joints with the strain energy of deformation for the homogenized continuum of the metamaterial. Plotted results describe the interlacing effect of the shape descriptor, the original separation angles, and the change in the separation angle on the on‐axes Poisson's ratio and Young's modulus. The latter is directly proportional to the equivalent spring stiffness per unit cell. In addition to attaining auxetic behavior, this metamaterial can also be designed to achieve specific Young's modulus behavior including extreme stiffness in one of the directions either at infinitesimal or finite deformation.
This Aztec‐inspired metamaterial exhibits auxetic behavior for the shown range via rotating triangles and nonrotating rhombi as a result of on‐axes loading. In terms of area, 50% of the metamaterial is nonrotating.
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