Self-Assembled Pd(II) Metallocycles Using an Ambidentate Donor and the Study of Square-Triangle Equilibria

The self-assembly reaction of a cis-blocked 90° square planar metal acceptor with a symmetrical linear flexible linker is expected to yield a [4 + 4] self-assembled square, a [3 + 3] assembled triangle, or a mixture of these.However, if the ligand is a nonsymmetrical ambidentate, it is expected to form a complex mixture comprising several linkage isomeric squares and triangles as a result of different connectivities of the ambidentate linker. We report instead that the reaction of a 90° acceptor cis-(dppf)Pd(OTf)2 [where dppf ) 1,1′-bis(diphenylphosphino)- ferrocene] with an equimolar amount of the ambidentate unsymmetrical ligand Na-isonicotinate unexpectedly yields a mixture of symmetrical triangles and squares in the solution. An analogous reaction using cis-(tmen)Pd(NO3)2 instead of cis-(dppf)Pd(OTf)2 also produced a mixture of symmetrical triangles and squares in the solution. In both cases the square was isolated as the sole product in the solid state, which was characterized by a single crystal structure analysis. The equilibrium between the triangle and the square in the solution is governed by the enthalpic and entropic contributions. The former parameter favors the formation of the square due to less strain in the structure whereas the latter one favors the formation of triangles due to the formation of more triangles from the same number of starting linkers. The effects of temperature and concentration on the equilibria have been studied by NMR techniques. This represents the first report on the study of square-triangle equilibria obtained using a nonsymmetric ambidentate linker. Detail NMR spectroscopy along with the ESI-mass spectrometry unambiguously identified the components in the mixture while the X-ray structure analysis determined the solid-state structure.

Acyclic Edge Coloring of Triangle-Free Planar Graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a'(G). It was conjectured by Alon, Sudakov and Zaks (and much earlier by Fiamcik) that a'(G) ? ? + 2, where ? = ?(G) denotes the maximum degree of the graph. If every induced subgraph H of G satisfies the condition |E(H)| ? 2|V(H)|-1, we say that the graph G satisfies Property A. In this article, we prove that if G satisfies Property A, then a'(G) ? ? + 3. Triangle-free planar graphs satisfy Property A. We infer that a'(G) ? ? + 3, if G is a triangle-free planar graph. Another class of graph which satisfies Property A is 2-fold graphs (union of two forests). (C) 2011 Wiley Periodicals, Inc. J Graph Theory

Space vector-based analysis of overmodulation in triangle-comparison based PWM for voltage source inverter

The equivalence of triangle-comparison-based pulse width modulation (TCPWM) and space vector based PWM (SVPWM) during linear modulation is well-known. This paper analyses triangle-comparison based PWM techniques (TCPWM) such as sine-triangle PWM (SPWM) and common-mode voltage injection PWM during overmodulation from a space vector point of view. The average voltage vector produced by TCPWM during overmodulation is studied in the stationary (a-b) reference frame. This is compared and contrasted with the average voltage vector corresponding to the well-known standard two-zone algorithm for space vector modulated inverters. It is shown that the two-zone overmodulation algorithm itself can be derived from the variation of average voltage vector with TCPWM. The average voltage vector is further studied in a synchronously revolving (d-q) reference frame. The RMS value of low-order voltage ripple can be estimated, and can be used to compare harmonic distortion due to different PWM methods during overmodulation. The measured values of the total harmonic distortion (THD) in the line currents are presented at various fundamental frequencies. The relative values of measured current THD pertaining to different PWM methods tally with those of analytically evaluated RMS voltage ripple.

Nodal domains of the equilateral triangle billiard

We characterize the eigenfunctions of an equilateral triangle billiard in terms of its nodal domains. The number of nodal domains has a quadratic form in terms of the quantum numbers, with a non-trivial number-theoretic factor. The patterns of the eigenfunctions follow a group-theoretic connection in a way that makes them predictable as one goes from one state to another. Extensive numerical investigations bring out the distribution functions of the mode number and signed areas. The statistics of the boundary intersections is also treated analytically. Finally, the distribution functions of the nodal loop count and the nodal counting function are shown to contain information about the classical periodic orbits using the semiclassical trace formula. We believe that the results belong generically to non-separable systems, thus extending the previous works which are concentrated on separable and chaotic systems.

Fixed-orientation equilateral triangle matching of point sets

Given a point set P and a class C of geometric objects, G(C)(P) is a geometric graph with vertex set P such that any two vertices p and q are adjacent if and only if there is some C is an element of C containing both p and q but no other points from P. We study G(del)(P) graphs where del is the class of downward equilateral triangles (i.e., equilateral triangles with one of their sides parallel to the x-axis and the corner opposite to this side below that side). For point sets in general position, these graphs have been shown to be equivalent to half-Theta(6) graphs and TD-Delaunay graphs. The main result in our paper is that for point sets P in general position, G(del)(P) always contains a matching of size at least vertical bar P vertical bar-1/3] and this bound is tight. We also give some structural properties of G(star)(P) graphs, where is the class which contains both upward and downward equilateral triangles. We show that for point sets in general position, the block cut point graph of G(star)(P) is simply a path. Through the equivalence of G(star)(P) graphs with Theta(6) graphs, we also derive that any Theta(6) graph can have at most 5n-11 edges, for point sets in general position. (C) 2013 Elsevier B.V. All rights reserved.

Lower bounds for testing triangle-freeness in Boolean functions

Given a Boolean function , we say a triple (x, y, x + y) is a triangle in f if . A triangle-free function contains no triangle. If f differs from every triangle-free function on at least points, then f is said to be -far from triangle-free. In this work, we analyze the query complexity of testers that, with constant probability, distinguish triangle-free functions from those -far from triangle-free. Let the canonical tester for triangle-freeness denotes the algorithm that repeatedly picks x and y uniformly and independently at random from , queries f(x), f(y) and f(x + y), and checks whether f(x) = f(y) = f(x + y) = 1. Green showed that the canonical tester rejects functions -far from triangle-free with constant probability if its query complexity is a tower of 2's whose height is polynomial in . Fox later improved the height of the tower in Green's upper bound to . A trivial lower bound of on the query complexity is immediate. In this paper, we give the first non-trivial lower bound for the number of queries needed. We show that, for every small enough , there exists an integer such that for all there exists a function depending on all n variables which is -far from being triangle-free and requires queries for the canonical tester. We also show that the query complexity of any general (possibly adaptive) one-sided tester for triangle-freeness is at least square root of the query complexity of the corresponding canonical tester. Consequently, this means that any one-sided tester for triangle-freeness must make at least queries.

Heterochromatic paths in edge colored graphs without small cycles and heterochromatic-triangle-free graphs

Conditions for the existence of heterochromatic Hamiltonian paths and cycles in edge colored graphs are well investigated in literature. A related problem in this domain is to obtain good lower bounds for the length of a maximum heterochromatic path in an edge colored graph G. This problem is also well explored by now and the lower bounds are often specified as functions of the minimum color degree of G - the minimum number of distinct colors occurring at edges incident to any vertex of G - denoted by v(G). Initially, it was conjectured that the lower bound for the length of a maximum heterochromatic path for an edge colored graph G would be 2v(G)/3]. Chen and Li (2005) showed that the length of a maximum heterochromatic path in an edge colored graph G is at least v(G) - 1, if 1 <= v(G) <= 7, and at least 3v(G)/5] + 1 if v(G) >= 8. They conjectured that the tight lower bound would be v(G) - 1 and demonstrated some examples which achieve this bound. An unpublished manuscript from the same authors (Chen, Li) reported to show that if v(G) >= 8, then G contains a heterochromatic path of length at least 120 + 1. In this paper, we give lower bounds for the length of a maximum heterochromatic path in edge colored graphs without small cycles. We show that if G has no four cycles, then it contains a heterochromatic path of length at least v(G) - o(v(G)) and if the girth of G is at least 4 log(2)(v(G)) + 2, then it contains a heterochromatic path of length at least v(G) - 2, which is only one less than the bound conjectured by Chen and Li (2005). Other special cases considered include lower bounds for the length of a maximum heterochromatic path in edge colored bipartite graphs and triangle-free graphs: for triangle-free graphs we obtain a lower bound of 5v(G)/6] and for bipartite graphs we obtain a lower bound of 6v(G)-3/7]. In this paper, it is also shown that if the coloring is such that G has no heterochromatic triangles, then G contains a heterochromatic path of length at least 13v(G)/17)]. This improves the previously known 3v(G)/4] bound obtained by Chen and Li (2011). We also give a relatively shorter and simpler proof showing that any edge colored graph G contains a heterochromatic path of length at least (C) 2015 Elsevier Ltd. All rights reserved.

Analysis of sine-triangle and zero-sequence injection modulation schemes for split-phase induction motor drive

A split-phase induction motor is fed from two three-phase voltage source inverters for speed control. This study analyses carrier-comparison based pulse width modulation (PWM) schemes for a split-phase motor drive, from a space-vector perspective. Sine-triangle PWM, one zero-sequence injection PWM where the same zero-sequence signal is used for both the inverters, and another zero-sequence injection PWM where different zero-sequence signals are employed for the two inverters are considered. The set of voltage vectors applied, the sequence in which the voltage vectors are applied, and the resulting current ripple vector are analysed for all the PWM methods. Besides all the PWM methods are compared in terms of dc bus utilisation. For the same three-phase sine reference, the PWM method with different zero-sequence signals for the two inverters is found to employ a set of vectors different from the other methods. Both analysis and experimental results show that this method results in lower total harmonic distortion and higher dc bus utilisation than the other two PWM methods.

Flow through a plateau border of cellular foam

Experimentally measured average velocities through plateau borders of stationary cellular foam, when compared with those calculated with the assumption of rigid Plateau Border walls, show that the assumption of rigid walls severely underestimates the velocities. An analysis of the situation wherein plateau border walls have velocities, as decided by the surface viscosity of the system, is presented here. The plateau border is idealized as a pipe of equilateral triangular cross-section with vertices of the triangle having zero velocity. The pertinent form of Navier-Stoke's equations with inhomogeneous boundary conditions and its solution through a procedure of successive approximations is presented in dimensionless form. The solution reduces to the known solution of slow steady flow through a triangular pipe, when surface viscosity is infinite. Results indicate that the assumption of rigid plateau border walls is valid only when value of the inverse of dimensionless surface viscosity is less than 0.044. Beyond that the assumption severely underestimates the flow and the effect of nonrigidity of the wall must be considered.

Topology-Guided Tessellation Of Quadratic Elements

Topology-based methods have been successfully used for the analysis and visualization of piecewise-linear functions defined on triangle meshes. This paper describes a mechanism for extending these methods to piecewise-quadratic functions defined on triangulations of surfaces. Each triangular patch is tessellated into monotone regions, so that existing algorithms for computing topological representations of piecewise-linear functions may be applied directly to the piecewise-quadratic function. In particular, the tessellation is used for computing the Reeb graph, a topological data structure that provides a succinct representation of level sets of the function.

Optimum choice of wavelengths in the anomalous scattering technique with synchrotron radiation

A formula has been derived for the mean-square error in the phases of crystal reflections determined through the multiwavelength anomalous scattering method.The error is written in terms of a simple function of the positions in the complex plane of the 'centres' corresponding to the different wavelengths. For the case of three centres, the mean-square error is inversely proportional to the area of the triangle formed by them.

Phase relations in the system Ta-Rh-O and thermodynamic properties of TaRhO4

Phase relations in the system Ta-Rh-O were determined by analysis of quenched samples corresponding to thirteen compositions inside the ternary triangle after equilibration at 1273 K. All the Ta-Rh alloys were found to be in equilibrium with Ta2O5. Only one ternary oxide TaRhO4 was detected. Based on phase relations in the ternary system, a solid-state electrochemical cell, incorporating calcia-stabilized zirconia as the electrolyte, was designed to measure the standard Gibbs energy of formation (Delta G degrees, J mol(-1)) of TaRhO4 in the temperature range from 900 to 1300 K. For the reaction, 1/2 beta-Ta2O5 + 1/2 Rh2O3(ortho) -> TaRhO4 Delta G degrees = -42993 + 5.676T (+/- 85) The calculated decomposition temperatures of TaRhO4 are 1644 +/- 5K in pure O-2 and 1543 +/- 5K in air at a total pressure p(o) = 0.1 MPa. Thermodynamic properties of TaRhO4 at 298.15K have been evaluated from the results. The limited experimental thermodynamic data for Rh-rich alloys available in the literature are in fair accord with Miedema's model. The Gibbs energies of formation of the different phases in the binary system Ta-Rh were estimated based on these inputs, consistent with the binary phase diagram. Based on the thermodynamic information on the stability of various phases, an oxygen potential diagram for the system Ta-Rh-O at 1273K was constructed. Also presented are temperature-composition diagrams for the ternary system at constant oxygen partial pressures (po(2)/p(o) = 0.212 and 10(-6)) calculated form the basic data.

Approximation algorithm for maximum independent set in planar traingle-free graphs

The maximum independent set problem is NP-complete even when restricted to planar graphs, cubic planar graphs or triangle free graphs. The problem of finding an absolute approximation still remains NP-complete. Various polynomial time approximation algorithms, that guarantee a fixed worst case ratio between the independent set size obtained to the maximum independent set size, in planar graphs have been proposed. We present in this paper a simple and efficient, O(|V|) algorithm that guarantees a ratio 1/2, for planar triangle free graphs. The algorithm differs completely from other approaches, in that, it collects groups of independent vertices at a time. Certain bounds we obtain in this paper relate to some interesting questions in the theory of extremal graphs.