Set Of Planes Research Articles

Synthesis, crystal structures and magnetic properties of cyanide-bridged heterobimetallic trinuclear Cr 2 III MnII complexes based on the cis-dicyanidemetalate [Cr(2,2′-bipy)2(CN)2]ClO4 building block

A structurally characterized cis-dicyanidemetalate [Cr(2,2′-bipy)2(CN)2]ClO4 (1) building block and two mononuclear seven-coordinate macrocyclic manganese(II) complexes have been employed to assemble two cyanide-bridged heterometallic CrIII–MnIII complexes: {[Mn(L1)][Cr(2,2′-bipy)2(CN)2]2}{ClO4}4·H2O (2) and {[Mn(L2)][Cr(2,2′-bipy)2(CN)2]2}{ClO4}4·2H2O (3) (L1 = 2,13-dimethyl-3,6,9,12,18-pentaazabicyclo[12.3.1]octadeca-1(18),2,12,14,16-pentaene, L2 = 2,13-dimethyl-6,9-dioxa-3,12,18-triazabicyclo[12.3.1]octadeca-1(18),2,12,14,16-pentaene). Single-crystal X-ray diffraction analysis shows that 2 and 3 have similar cyanide-bridged cationic trinuclear Cr2Mn structures with free ClO4 − as balancing anions, in which the cyanide precursor acts as a monodentate ligand to connect the macrocyclic manganese(II) units. The coordination geometry of the manganese(II) centers in the two complexes is slightly distorted pentagonal-bipyramidal, with two cyanide nitrogen atoms at the trans-positions and N5 or N3O2 donor sets in the equatorial plane provided by the macrocyclic ligand. Investigation of the magnetic properties reveals antiferromagnetic magnetic coupling between the cyanide-bridged Cr(III) and Mn(II) centers. A best-fit analysis of the magnetic susceptibility leads to the magnetic coupling constants J = −1.15 and −1.08 cm−1 for complexes 2 and 3, respectively.

Connecting a set of circles with minimum sum of radii

We consider the problem of assigning radii to a given set of points in the plane, such that the resulting set of disks is connected, and the sum of radii is minimized. We prove that the problem is NP-hard in planar weighted graphs if there are upper bounds on the radii and sketch a similar proof for planar point sets. For the case when there are no upper bounds on the radii, the complexity is open; we give a polynomial-time approximation scheme. We also give constant-factor approximation guarantees for solutions with a bounded number of disks; these results are supported by lower bounds, which are shown to be tight in some of the cases. Finally, we show that the problem is polynomially solvable if a connectivity tree is given, and we conclude with some experimental results.

$${\varvec{k}}$$ k -Bisectors of Finite Planar Sets

Let V be a finite set of points in the plane, not all on one line, and let l be a line that contains at least 2 points of V. We say that l is a k -bisector of V if there are at least k points of V on each one of the two open half-planes bounded by l. For $$k \ge 6$$ we construct planar sets of $$2k+4$$ points having no k-bisector (this might be best possible). Furthermore, we show that if $$|V| > 3 k$$ , then in every triangulation of convV with vertex set V there is an edge whose loading line is a k-bisector of V. This is best possible for all k.

On distance sets, box-counting and Ahlfors-regular sets

We obtain box-counting estimates for the pinned distance sets of (dense subsets of) planar discrete Ahlfors-regular sets of exponent $s>1$. As a corollary, we improve upon a recent result of Orponen, by showing that if $A$ is Ahlfors-regular of dimension $s>1$, then almost all pinned distance sets of $A$ have lower box-counting dimension $1$. We also show that if $A,B\subset\mathbb{R}^2$ have Hausdorff dimension $>1$ and $A$ is Ahlfors-regular, then the set of distances between $A$ and $B$ has modified lower box-counting dimension $1$, which taking $B=A$ improves Orponen's result in a different direction, by lowering packing dimension to modified lower box-counting dimension. The proofs involve ergodic-theoretic ideas, relying on the theory of CP-processes and projections.

Numerical analysis of evaluation methods and influencing factors for dynamic stability of bedding rock slope

As the inclination of a bedding surface is consistent with the inclination of a slope, the stability of a bedding rock slope is relatively poor, especially under dynamic loads such as earthquake and blasting. In the dynamic stability analysis of slope, the evaluation methods and influence factors of slope stability are two important concerns. Therefore, two typical bedding rock slopes are respectively established by FLAC3D to study the above concerns. The pseudo-static method, dynamic time-history method and dynamic strength reduction method is used to evaluate the dynamic stability of the model slope, and the applicability of the three methods is compared. The influence of five parameters including dynamic load frequency, slope angle, slope height, strata inclination and strata thickness on the dynamic stability is considered in the model slope with a set of bedding planes. The results show that the dynamic strength reduction method has good suitability for the stability evaluation of a bedding rock slope due to its good solution in the instability judgment and evaluation index. The dynamic stability of a slope becomes worse when the load frequency is close to the natural frequency of the slope. Due to the “elevation effect” and “bedding surface effect” in the dynamic slope response, the slope stability decreases with the increase of slope height and the reduction of strata thickness. The slope stability decreases with the increase of strata inclination and slope angle, and the strata inclination is the most sensitive parameter influencing the slope stability. When the slope angle and height increase to a certain value, the downward trend of slope stability gradually become gentle. For the model slope in this paper, when the slope angle reaches 55° and the slope height reaches 200 m, the reduction of slope stability will be no longer obvious with the increase of a slope angle and slope height.

Comparing Rock Discontinuity Measurements Using Geological Compass, Smartphone Application, and Laser Scanning Methods

To determine stability of rock slopes, discontinuity orientations must be measured accurately. Several methods exist for taking field measurements of discontinuity orientations. The objective of this study was to compare the traditional method of hand measuring discontinuity orientations using a Brunton compass to the more modern methods of measuring discontinuities using smartphone applications and ground-based Lidar laser scanning. The field site is a rock outcrop along Stroubles Road on Price Mountain, in Montgomery County, Virginia. The Price Mountain structure is a doubly plunging anticline exposed in a window in the Pulaski Thrust Sheet. The rock outcrop consists of a Mississippian sandstone mapped as the Upper Price Formation. The site features a set of bedding planes that dip steeply into the cut slope face, yielding toppling failures. In addition, two distinct sets of joints exist creating both planar and wedge failures in the cut slope. Using window mapping, discontinuity orientations were measured along 200 feet (61 meters) of outcrop using a Brunton compass and a smartphone application, along with a laser scan. These measurements were compared using stereonet analyses to determine the relative accuracy of the different methods. The results show a strong agreement between measurements taken with the Brunton compass and the smartphone application. However, the laser scan shows that scanner data needs calibration with field measurements and observations to yield equally good results. Remote sensing methods like laser scanning, such as terrestrial photogrammetry, cannot be used completely independent of traditional field characterization and input from experienced professionals.

Counting and enumerating crossing-free geometric graphs

We describe a framework for constructing data structures which allow fast counting and enumeration of various types of crossing-free geometric graphs on a planar point set. The framework generalizes ideas of Alvarez and Seidel, who used them to count triangulations in time $O(2^nn^2)$ where $n$ is the number of points. The main idea is to represent geometric graphs as source-sink paths in a directed acyclic graph. The following results will emerge. The number of all crossing-free geometric graphs can be computed in time $O(c^nn^4)$ for some $c < 2.83929$. The number of crossing-free convex partitions can be computed in time $O(2^nn^4)$. The number of crossing-free perfect matchings can be computed in time $O(2^nn^4)$. The number of convex subdivisions can be computed in time $O(2^nn^4)$. The number of crossing-free spanning trees can be computed in time $O(c^nn^4)$ for some $c < 7.04313$. The number of crossing-free spanning cycles can be computed in time $O(c^nn^4)$ for some $c < 5.61804$. Moreover, after a preprocessing phase with the same time bounds as above, we can enumerate the respective classes efficiently. For example, after $O(2^nn^4)$ time of preprocessing we can enumerate the set of all crossing-free perfect matchings using polynomial time per enumerated object. For crossing-free perfect matchings and convex partitions we further obtain enumeration algorithms where the time delay for each (in particular, the first) output is bounded by a polynomial in $n$. All described algorithms are comparatively simple, both in terms of their analysis and implementation.

A New Perspective for a Global QoS-based Internet

Several Quality of Service (QoS) architectures and mechanisms have been proposed by standardisation bodies.However, those proposals were not always aligned with the spirit of IP networks. Furthermore, most research activities have focused to date on supporting QoS only within a singleadministrative domain. In this paper, we demonstrate that it is possible to provision QoS-enabled services that maintain the ability to easily connect any pair of users worldwide. For this purpose, we introduce the notion of Meta-QoS-Class and demonstrate its relevance for building QoS-enabled services across multiple domains. We then show how a QoS-enhanced Border Gateway Protocol (q-BGP) can be used together with Meta-QoS-Classes, for building a set of parallel Internet planes with different QoS capabilities. This concept opens up a new perspective for a global QoS-based Internet.

ESTIMATION OF PARAMETERS IN A PLANAR SEGMENT PROCESS WITH A BIOLOGICAL APPLICATION

The paper deals with modeling of segment systems in a bounded planar set (a cell) by means of random segment processes. Two models with a density with respect to the Poisson process are presented. In model I interactions are given by the number of intersections, model II includes the length distribution and takes into account distances from the centre of the cell. The estimation of parameters of the models is suggested based on Takacz-Fiksel method. The method is tested first using simulated data. Further the real data from fluorescence imaging of stress fibres in mesenchymal human stem cells are evaluated. We apply model II which is inhomogeneous. The degree-of-fit testing of the model using various characteristics yields quite satisfactory results.

Weak ferromagnetic order breaking the threefold rotational symmetry of the underlying kagome lattice in CdCu3(OH)6(NO3)2·H2O

Novel magnetic phases are expected to occur in highly frustrated spin systems. Here, we study the structurally perfect kagome antiferromagnet $\mathrm{CdC}{\mathrm{u}}_{3}{(\mathrm{OH})}_{6}{({\mathrm{NO}}_{3})}_{2}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{H}}_{2}\mathrm{O}$ by magnetization, magnetic torque, and heat capacity measurements using single crystals. An antiferromagnetic order accompanied by a small spontaneous magnetization that surprisingly is confined in the kagome plane sets in at ${T}_{\mathrm{N}}\ensuremath{\sim}4\phantom{\rule{0.16em}{0ex}}\mathrm{K}$, well below the nearest-neighbor exchange interaction $J/{k}_{\mathrm{B}}=45\phantom{\rule{0.16em}{0ex}}\mathrm{K}$. This suggests that a unique ``$\mathbf{q}=0$'' type ${120}^{\ensuremath{\circ}}$ spin structure with ``negative'' (downward) vector chirality, which breaks the underlying threefold rotational symmetry of the kagome lattice and thus allows a spin canting within the plane, is exceptionally realized in this compound rather than a common one with ``positive'' (upward) vector chirality. The origin is discussed in terms of the Dzyaloshinskii-Moriya interaction.

More Distinct Distances Under Local Conditions

We establish the following result related to Erdős’s problem on distinct distances. Let V be an n-element planar point set such that any p members of V determine at least \(\left( {\begin{array}{*{20}{c}} p 2 \end{array}} \right) - p + 6\) distinct distances. Then V determines at least \(n^{\tfrac{8} {7} - o(1)}\) distinct distances, as n tends to infinity.

On the noise-perturbed spatial Julia set generated by Lorenz system

This paper extends the study of noise-perturbed planar Julia sets to the spatial case by studying the Julia set of a complex Lorenz system. We give the definition of the Julia set of a complex Lorenz system and visualize its spatial fractal structure. Then, the symmetry property of the 3-D slice of the CLS Julia set is proved. Finally, the influence of additive and multiplicative noises on the CLS Julia set are analyzed with respect to structural damages and changes in symmetry separately.

Submatrix Maximum Queries in Monge Matrices and Partial Monge Matrices, and Their Applications

We describe a data structure for submatrix maximum queries in Monge matrices or partial Monge matrices, where a query seeks the maximum element in a contiguous submatrix of the given matrix. The structure, for an n × n Monge matrix, takes O ( n log n ) space and O ( n log n ) preprocessing time, and answers queries in O (log 2 n ) time. For partial Monge matrices, the space grows by α( n ), the preprocessing grows by α( n )log n (α( n ) is the inverse Ackermann function), and the query remains O (log 2 n ). Our design exploits an interpretation of the column maxima in a Monge (partial Monge, respectively) matrix as an upper envelope of pseudo-lines (pseudo-segments, respectively). We give two applications: (1) For a planar set of n points in an axis-parallel rectangle B , we build a data structure, in O ( n α( n )log 4 n ) time and O ( n α( n )log 3 n ) space, that returns, for a query point p , the largest-area empty axis-parallel rectangle contained in B and containing p , in O (log 4 n ) time. This improves substantially the nearly quadratic storage and preprocessing obtained by Augustine et al. [2010]. (2) Given an n -node arbitrarily weighted planar digraph, with possibly negative edge weights, we build, in O ( n log 2 n /log log n ) time, a linear-size data structure that supports edge-weight updates and graph-distance queries between arbitrary pairs of nodes in O ( n 2/3 log 5/3 n ) time per operation. This improves a previous algorithm of Fakcharoenphol and Rao [2006]. Our data structure has already been applied in a recent maximum flow algorithm for planar graphs in Borradaile et al. [2011].

On Lebesgue constants for interpolation points on a quasiconformal arc

Using the theory of quasiconformal mappings, we simplify the proof of the recent result by Taylor and Totik (2010) on the behavior of the Lebesgue constants for interpolation points on a compact set in the complex plane.

A GENERALISATION OF THE CLUNIE–SHEIL-SMALL THEOREM II

We study properties of the simply connected sets in the complex plane, which are finite unions of domains convex in the horizontal direction. These considerations allow us to state new univalence criteria for complex-valued local homeomorphisms. In particular, we apply our results to planar harmonic mappings obtaining generalisations of the shear construction theorem due to Clunie and Sheil-Small [‘Harmonic univalent functions’, Ann. Acad. Sci. Fenn. Ser. A. I. Math.9 (1984), 3–25].

On multiple Borsuk numbers in normed spaces

Hujter and Lángi defined the k-fold Borsuk number of a set S in Euclidean n-space of diameter d &gt; 0 as the smallest cardinality of a family F of subsets of S, of diameters strictly less than d, such that every point of S belongs to at least k members of F. We investigate whether a k-fold Borsuk covering of a set S in a finite dimensional real normed space can be extended to a completion of S. Furthermore, we determine the k-fold Borsuk number of sets in not angled normed planes, and give a partial characterization for sets in angled planes.

On three measures of non-convexity

The invisibility graph $I(X)$ of a set $X \subseteq \mathbb{R}^d$ is a (possibly infinite) graph whose vertices are the points of $X$ and two vertices are connected by an edge if and only if the straight-line segment connecting the two corresponding points is not fully contained in $X$. We consider the following three parameters of a set $X$: the clique number $\omega(I(X))$, the chromatic number $\chi(I(X))$ and the convexity number $\gamma(X)$, which is the minimum number of convex subsets of $X$ that cover $X$. We settle a conjecture of Matou\v{s}ek and Valtr claiming that for every planar set $X$, $\gamma(X)$ can be bounded in terms of $\chi(I(X))$. As a part of the proof we show that a disc with $n$ one-point holes near its boundary has $\chi(I(X)) \ge \log\log(n)$ but $\omega(I(X))=3$. We also find sets $X$ in $\mathbb{R}^5$ with $\chi(X)=2$, but $\gamma(X)$ arbitrarily large.

STEM moiré analysis for 2D strain measurements.

A moiré pattern is created in a scanning transmission electron microscope (STEM) when the scan step is close to a crystalline periodicity. Usually, fringes are visible in only one direction, corresponding to a single set of lattice planes, but fringes can be formed in two directions or more. Using an accurate independent calibration, the strains in silicon devices have been determined from the spacing and orientation of one-directional STEM moiré fringes. In this report, we first discuss the origin of the STEM moiré, and then we show how an accurate calibration of the scan step can be obtained from the STEM moiré pattern itself, providing that we know initially only an approximate scan step and the planar spacing. The new calibration scheme also makes the STEM moiré experiments easier, since it can be applied for the moiré where the scan direction is not precisely aligned with the crystalline lattice. Finally, we show how the two-dimensional strain information will be readily extracted from two one-directional moiré patterns using the concept of geometric phase.

On Some Sequences of the Sets of Uniform Convergence for Two-Dimensional Continued Fractions

For two-dimensional continued fractions whose elements belong to some rectangular sets of a complex plane, we establish the truncation error bound of their figured approximants. It is shown that the two-dimensional continued fractions are uniformly convergent with respect to a sequence of these rectangular sets.

Terahertz dual-band dipole antenna with novel small flat quartz-copper reflector

In this paper, we have proposed a Terahertz dual-band dipole antenna with a novel small flat quartz-copper reflector. The dimensions of reflector, which is located at a 80μm distance from antenna, are 300×160×45μm. The antenna supports maximum gain of 12.4dB at 5.2THz frequency with two main resonant frequencies at 2.58 and 5.33THz. Also, the proposed Terahertz antenna is used as an array element in linear and planar sets of elements, which are evaluated by using array method. An array Terahertz antenna, which has maximum gain of 18.7dB at 5.2THz, is proposed.