Universal crack closure integral for SIF estimation

Numerical analysis of cracked structures often involves numerical estimation of stress intensity factors (SIFs) at a crack tip/front. A newly developed formulation called universal crack closure integral (UCCI) for the evaluation of potential energy release rates (PERRs) and the corresponding SIFs is presented in this paper. Unlike the existing element dedicated forms of crack closure integrals (MCCI, VCCI) with application limited to finite element analysis, this new numerical SIF/PERR estimation technique is independent of the basic stress analysis procedure, making it universally applicable. The second merit of this procedure is that it avoids the generally error-producing zones close to the crack tip/front singularity. The UCCI procedure, based on Irwin's original CCI, is formulated and explored using a simple 2D problem of a straight crack in an infinite sheet. It is then applied to some three-dimensional crack geometries with the stresses and displacements obtained from a boundary element program.

Efficient Output-Sensitive Construction of Reeb Graphs

The Reeb graph tracks topology changes in level sets of a scalar function and finds applications in scientific visualization and geometric modeling. This paper describes a near-optimal two-step algorithm that constructs the Reeb graph of a Morse function defined over manifolds in any dimension. The algorithm first identifies the critical points of the input manifold, and then connects these critical points in the second step to obtain the Reeb graph. A simplification mechanism based on topological persistence aids in the removal of noise and unimportant features. A radial layout scheme results in a feature-directed drawing of the Reeb graph. Experimental results demonstrate the efficiency of the Reeb graph construction in practice and its applications.

On the cubicity of bipartite graphs

A unit cube in k-dimension (or a k-cube) is defined as the Cartesian product R-1 x R-2 x ... x R-k, where each R-i is a closed interval on the real line of the form [a(j), a(i), + 1]. The cubicity of G, denoted as cub(G), is the minimum k such that G is the intersection graph of a collection of k-cubes. Many NP-complete graph problems can be solved efficiently or have good approximation ratios in graphs of low cubicity. In most of these cases the first step is to get a low dimensional cube representation of the given graph. It is known that for graph G, cub(G) <= left perpendicular2n/3right perpendicular. Recently it has been shown that for a graph G, cub(G) >= 4(Delta + 1) In n, where n and Delta are the number of vertices and maximum degree of G, respectively. In this paper, we show that for a bipartite graph G = (A boolean OR B, E) with |A| = n(1), |B| = n2, n(1) <= n(2), and Delta' = min {Delta(A),Delta(B)}, where Delta(A) = max(a is an element of A)d(a) and Delta(B) = max(b is an element of B) d(b), d(a) and d(b) being the degree of a and b in G, respectively , cub(G) <= 2(Delta' + 2) bar left rightln n(2)bar left arrow. We also give an efficient randomized algorithm to construct the cube representation of G in 3 (Delta' + 2) bar right arrowIn n(2)bar left arrow dimension. The reader may note that in general Delta' can be much smaller than Delta.

Cubicity of threshold graphs

We show that the cubicity of a connected threshold graph is equal to inverted right perpendicularlog(2) alpha inverted left perpendicular, where alpha is its independence number.

Characterisation of k-Variegated Graphs, k greater-than-or-equal-to 3

A graph is said to be k-variegated if its vertex set can be partitioned into k equal parts such that each vertex is adjacent to exactly one vertex from every other part not containing it. Bednarek and Sanders [1] posed the problem of characterizing k-variegated graphs. V.N. Bhat-Nayak, S.A. Choudum and R.N. Naik [2] gave the characterization of 2-variegated graphs. In this paper we characterize k-variegated graphs for k greater-or-equal, slanted 3.

Analytical studies of gain optimization in CO2-N2 gasdynamic lasers employing two-dimensional wedge nozzles

An analytical method has been proposed to optimise the small-signaloptical gain of CO2-N2 gasdynamic lasers (gdl) employing two-dimensional (2D) wedge nozzles. Following our earlier work the equations governing the steady, inviscid, quasi-one-dimensional flow in the wedge nozzle of thegdl are reduced to a universal form so that their solutions depend on a single unifying parameter. These equations are solved numerically to obtain similar solutions for the various flow quantities, which variables are subsequently used to optimize the small-signal-gain. The corresponding optimum values like reservoir pressure and temperature and 2D nozzle area ratio also have been predicted and graphed for a wide range of laser gas compositions, with either H2O or He as the catalyst. A large number of graphs are presented which may be used to obtain the optimum values of small signal gain for a wide range of laser compositions without further computations.

Representation of functional dependencies in relational databases using linear graphs

Functional dependencies in relational databases are investigated. Eight binary relations, viz., (1) dependency relation, (2) equipotence relation, (3) dissidence relation, (4) completion relation, and dual relations of each of them are described. Any one of these eight relations can be used to represent the functional dependencies in a database. Results from linear graph theory are found helpful in obtaining these representations. The dependency relation directly gives the functional dependencies. The equipotence relation specifies the dependencies in terms of attribute sets which functionally determine each other. The dissidence relation specifies the dependencies in terms of saturated sets in a very indirect way. Completion relation represents the functional dependencies as a function, the range of which turns out to be a lattice. Depletion relation which is the dual of the completion relation can also represent functional dependencies and similarly can the duals of dependency, equipotence, and dissidence relations. The class of depleted sets, which is the dual of saturated sets, is defined and used in the study of depletion relations.

Efficient algorithms for computing Reeb graphs

The Reeb graph tracks topology changes in level sets of a scalar function and finds applications in scientific visualization and geometric modeling. We describe an algorithm that constructs the Reeb graph of a Morse function defined on a 3-manifold. Our algorithm maintains connected components of the two dimensional levels sets as a dynamic graph and constructs the Reeb graph in O(nlogn+nlogg(loglogg)3) time, where n is the number of triangles in the tetrahedral mesh representing the 3-manifold and g is the maximum genus over all level sets of the function. We extend this algorithm to construct Reeb graphs of d-manifolds in O(nlogn(loglogn)3) time, where n is the number of triangles in the simplicial complex that represents the d-manifold. Our result is a significant improvement over the previously known O(n2) algorithm. Finally, we present experimental results of our implementation and demonstrate that our algorithm for 3-manifolds performs efficiently in practice.

Logic as the Universal Science : Bertrand Russell's Early Conception of Logic and Its Philosophical Context

Bertrand Russell (1872 1970) introduced the English-speaking philosophical world to modern, mathematical logic and foundational study of mathematics. The present study concerns the conception of logic that underlies his early logicist philosophy of mathematics, formulated in The Principles of Mathematics (1903). In 1967, Jean van Heijenoort published a paper, Logic as Language and Logic as Calculus, in which he argued that the early development of modern logic (roughly the period 1879 1930) can be understood, when considered in the light of a distinction between two essentially different perspectives on logic. According to the view of logic as language, logic constitutes the general framework for all rational discourse, or meaningful use of language, whereas the conception of logic as calculus regards logic more as a symbolism which is subject to reinterpretation. The calculus-view paves the way for systematic metatheory, where logic itself becomes a subject of mathematical study (model-theory). Several scholars have interpreted Russell s views on logic with the help of the interpretative tool introduced by van Heijenoort,. They have commonly argued that Russell s is a clear-cut case of the view of logic as language. In the present study a detailed reconstruction of the view and its implications is provided, and it is argued that the interpretation is seriously misleading as to what he really thought about logic. I argue that Russell s conception is best understood by setting it in its proper philosophical context. This is constituted by Immanuel Kant s theory of mathematics. Kant had argued that purely conceptual thought basically, the logical forms recognised in Aristotelian logic cannot capture the content of mathematical judgments and reasonings. Mathematical cognition is not grounded in logic but in space and time as the pure forms of intuition. As against this view, Russell argued that once logic is developed into a proper tool which can be applied to mathematical theories, Kant s views turn out to be completely wrong. In the present work the view is defended that Russell s logicist philosophy of mathematics, or the view that mathematics is really only logic, is based on what I term the Bolzanian account of logic . According to this conception, (i) the distinction between form and content is not explanatory in logic; (ii) the propositions of logic have genuine content; (iii) this content is conferred upon them by special entities, logical constants . The Bolzanian account, it is argued, is both historically important and throws genuine light on Russell s conception of logic.

On the Cubicity of Interval Graphs

A k-cube (or ``a unit cube in k dimensions'') is defined as the Cartesian product R-1 x . . . x R-k where R-i (for 1 <= i <= k) is an interval of the form [a(i), a(i) + 1] on the real line. The k-cube representation of a graph G is a mapping of the vertices of G to k-cubes such that the k-cubes corresponding to two vertices in G have a non-empty intersection if and only if the vertices are adjacent. The cubicity of a graph G, denoted as cub(G), is defined as the minimum dimension k such that G has a k-cube representation. An interval graph is a graph that can be represented as the intersection of intervals on the real line - i. e., the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. We show that for any interval graph G with maximum degree Delta, cub(G) <= inverted right perpendicular log(2) Delta inverted left perpendicular + 4. This upper bound is shown to be tight up to an additive constant of 4 by demonstrating interval graphs for which cubicity is equal to inverted right perpendicular log(2) Delta inverted left perpendicular.

A universal test to determine the integrity of RNA, and its suitability for reverse transcription, in animal tissue laboratory specimens

Degradation of RNA in diagnostic specimens can cause false-negative test results and potential misdiagnosis when tests rely on the detection of specific RNA sequence. Current molecular methods of checking RNA integrity tend to be host species or group specific, necessitating libraries of primers and reaction conditions. The objective here was to develop a universal (multi-species) quality assurance tool for determining the integrity of RNA in animal tissues submitted to a laboratory for analyses. Ribosomal RNA (16S rRNA) transcribed from the mitochondrial 16S rDNA was used as template material for reverse transcription to cDNA and was amplified using polymerase chain reaction (PCR). As mitochondrial DNA has a high level of conservation, the primers used were shown to reverse transcribe and amplify RNA from every animal species tested. Deliberate degradation of rRNA template through temperature abuse of samples resulted in no reverse transcription and amplification. Samples spiked with viruses showed that single-stranded viral RNA and rRNA in the same sample degraded at similar rates, hence reverse transcription and PCR amplification of 16S rRNA could be used as a test of sample integrity and suitability for analysis that required the sample's RNA, including viral RNA. This test will be an invaluable quality assurance tool for determination of RNA integrity from tissue samples, thus avoiding erroneous test results that might occur if degraded target RNA is used unknowingly as template material for reverse transcription and subsequent PCR amplification.

Boxicity of Halin graphs

A k-dimensional box is the Cartesian product R-1 x R-2 x ... x R-k where each R-i is a closed interval on the real line. The boxicity of a graph G, denoted as box(G) is the minimum integer k such that G is the intersection graph of a collection of k-dimensional boxes. Halin graphs are the graphs formed by taking a tree with no degree 2 vertex and then connecting its leaves to form a cycle in such a way that the graph has a planar embedding. We prove that if G is a Halin graph that is not isomorphic to K-4, then box(G) = 2. In fact, we prove the stronger result that if G is a planar graph formed by connecting the leaves of any tree in a simple cycle, then box(G) = 2 unless G is isomorphic to K4 (in which case its boxicity is 1).

A note on acyclic edge coloring of complete bipartite graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic (2-colored) 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). Let Delta = Delta(G) denote the maximum degree of a vertex in a graph G. A complete bipartite graph with n vertices on each side is denoted by K-n,K-n. Alon, McDiarmid and Reed observed that a'(K-p-1,K-p-1) = p for every prime p. In this paper we prove that a'(K-p,K-p) <= p + 2 = Delta + 2 when p is prime. Basavaraju, Chandran and Kummini proved that a'(K-n,K-n) >= n + 2 = Delta + 2 when n is odd, which combined with our result implies that a'(K-p,K-p) = p + 2 = Delta + 2 when p is an odd prime. Moreover we show that if we remove any edge from K-p,K-p, the resulting graph is acyclically Delta + 1 = p + 1-edge-colorable. (C) 2009 Elsevier B.V. All rights reserved.

Acyclic Edge Coloring of Graphs with Maximum Degree 4

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 that for any simple and finite graph G, a'(G) <= Delta+2, where Delta=Delta(G) denotes the maximum degree of G. We prove the conjecture for connected graphs with Delta(G)<= 4, with the additional restriction that m <= 2n-1, where n is the number of vertices and m is the number of edges in G. Note that for any graph G, m <= 2n, when Delta(G)<= 4. It follows that for any graph G if Delta(G)<= 4, then a'(G) <= 7.