Twisted Cube Research Articles

Analysis of carbide transformation in MC-M23C6 and its effect on mechanical properties of Ni-based superalloy

An investigation based on electron microscopy and crystallographic analysis was employed to gain insights into the carbide transformation in MC-M 23 C 6 complex carbide of Haynes282; nanoindentation test and first-principles calculations were performed to study its effect on mechanical properties of the alloy. The results show that MC(M=Ti, Mo)-M 23 C 6 (M=Cr, Mo) complex carbide, which exhibits as bull’s eye structure, shows an orientation relationship of ( 13 1 ̅ ) MC// ( 4 ̅ 20 ) M 23 C 6 , [ 1 1 ̅ 2 ̅ ] MC// [ 123 ] M 23 C 6 . Besides, intermediate phase M 2 C forms at partial MC/M 23 C 6 interface. During creep, Cr diffuses from M 23 C 6 to MC and substitutes Ti under the driving force of concentration gradient. The substitution induces thermal expansion anisotropy in polyhedral clusters of MC, which will trigger the flipping of diagonals and transform MC into M 23 C 6 through the transformation of coordination polyhedron of C atoms from octahedron to twisted cube. The hybridizations, especially the low-energy hybridizations, between C-p, Ni-d, Cr-d and Ti-d orbitals of the atoms located at M 23 C 6 /Ni and M 23 C 6 /MC interfaces make great contribution to the plentiful chemical bonds formed at the interfaces. These chemical bonds result in the high bonding strength of the two interfaces and thus the beneficial effects of the MC-M 23 C 6 carbides on mechanical properties of the alloy. Moreover, nanoindentation test indicates that MC-M 23 C 6 carbides do a better job of strengthening than individual MC or M 23 C 6 carbide. Therefore, the transformation from MC to M 23 C 6 is detrimental to the mechanical properties of the alloy. The demonstrated concept shed light on the rational design of higher property superalloys for various potential applications. • MC would in-situ transform into M 23 C 6 as the substitution of Ti in MC by Cr atoms. • Mechanism of the carbide transformation was studied by crystallographic analysis. • Transformation from MC to M 23 C 6 is detrimental to the mechanical properties of the alloy. • It proposed a promising method to improve the mechanical properties of Ni-based superalloy.

Cyclic connectivity, edge-elimination, and the twisted Isaacs graphs

Edge-elimination is an operation of removing an edge of a cubic graph together with its endvertices and suppressing the resulting 2-valent vertices. We study the effect of this operation on the cyclic connectivity of a cubic graph. Disregarding a small number of cubic graphs with no more than six vertices, this operation cannot decrease cyclic connectivity by more than two. We show that apart from three exceptional graphs (the cube, the twisted cube, and the Petersen graph) every 2-connected cubic graph on at least eight vertices contains an edge whose elimination decreases cyclic connectivity by at most one. The proof reveals an unexpected behaviour of connectivity 6, which requires a detailed structural analysis featuring the Isaacs flower snarks and their natural generalisation, the twisted Isaacs graphs, as forced structures. A complete characterisation of this family, which includes the Heawood graph as a sporadic case, serves as the main tool for excluding the existence of exceptional graphs in connectivity 6. As an application we show that every cyclically 5-edge-connected cubic graph has a decycling set of vertices whose removal leaves a tree and the set itself has at most one edge between its vertices. This strengthens a classical result of Payan and Sakarovitch (1975) about the structure of minimum decycling sets in cyclically 4-edge-connected graphs.

Grossberg–Karshon Twisted Cubes and Hesitant Jumping Walk Avoidance

Let $G$ be a complex simply-laced semisimple algebraic group of rank $r$ and $B$ a Borel subgroup. Let $\mathbf i \in [r]^n$ be a word and let $\boldsymbol{\ell} = (\ell_1,\dots,\ell_n)$ be a sequence of non-negative integers. Grossberg and Karshon introduced a virtual lattice polytope associated to $\mathbf i$ and $\boldsymbol{\ell}$ called a twisted cube, whose lattice points encode the character of a $B$-representation. More precisely, lattice points in the twisted cube, counted with sign according to a certain density function, yields the character of the generalized Demazure module determined by $\mathbf i$ and $\boldsymbol{\ell}$. In recent work, the author and Harada described precisely when the Grossberg–Karshon twisted cube is untwisted, i.e., the twisted cube is a closed convex polytope, in the situation when the integer sequence $\boldsymbol{\ell}$ comes from a weight $\lambda$ of $G$. However, not every integer sequence $\boldsymbol{\ell}$ comes from a weight of $G$. In the present paper, we interpret the untwistedness of Grossberg–Karshon twisted cubes associated with any word $\mathbf i$ and any integer sequence $\boldsymbol{\ell}$ using the combinatorics of $\mathbf i$ and $\boldsymbol{\ell}$. Indeed, we prove that the Grossberg–Karshon twisted cube is untwisted precisely when $\mathbf i$ is hesitant-jumping-$\boldsymbol{\ell}$-walk-avoiding.

Reliability Evaluation of Generalized ExchangedX-Cubes Based on the Condition ofg-Good-Neighbor

In the cloud computing environment with massive information services and decision-making resources, the accuracy and reliability of information are more important than previous single closed systems. Therefore, ensuring the reliability of information and the stable operation of the system are the core problems in the research fields such as the Internet Plus and the Internet of Things. The connectivity and diagnosability are two important measures for the fault tolerance of multiprocessor systems. Theg-good-neighbor conditional connectivity (Rg-connectivity) is the minimum number of nodes that make the graph disconnected, and each node has at leastgneighbors in every remaining component. Theg-good-neighbor conditional diagnosability (g-GNCD) is the maximum number of faulty processors that has been correctly identified in a system, and any fault-free processor has no less thangfault-free neighbors. ExchangedX-cubes are a class of irregular networks, obtained by deleting links from hypercubes and some variant networks of hypercubes (X-cubes). They not only combine the advantages ofX-cubes but also reduce the interconnection complexity. ExchangedX-cubes classify its nodes into two different classes clusters with a unique connecting rule. In this paper, we propose the generalized exchangedX-cubes framework so that architecture can be constructed by different connecting rules. Furthermore, we study theRg-connectivity andg-GNCD of generalized exchangedX-cubes under the PMC and MM∗models. As applications, theRg-connectivity andg-GNCD of generalized exchanged hypercubes, dual-cube-like networks, generalized exchanged crossed cubes, and locally generalized exchanged twisted cubes are determined, respectively.

Conditional diagnosability of component-composition graphs under the PMC model

Diagnosability is an important metric for measuring the reliability of multiprocessor systems. The conditional diagnosability, which is more general than the classical diagnosability, is to measure the diagnosability of a multiprocessor system under the assumption that all of the neighbors of any node in the system cannot fail at the same time. This paper adopts the PMC model and outlines the common properties of a wide class of interconnection networks, called component-composition graphs (CCGs), to determine their conditional diagnosability by using the derived properties. As applications, the diagnosability of hypercube-like networks (including hypercubes, crossed cubes, Möbius cubes, twisted cubes, locally twisted cubes, generalized twisted cubes, and recursive circulants), and bubble-sort graphs, all of which belong to CCGs can be readily obtained.

Structural Refinement and Density Functional Theory Study of Synthetic Ge-Akaganéite (β-FeOOH)

In this study, we propose a revised structural model for highly ordered synthetic Ge-akaganéite, a stable analogue of tunnel-type Fe-oxyhydroxide, based on the Rietveld refinement of synchrotron X-ray diffraction data and density functional theory with dispersion correction (DFT-D) calculations. In the proposed crystal structure of Ge-akaganéite, Ge is found not only in the tunnel sites as GeO(OH)3− tetrahedra, but also 4/5 of total Ge atoms are in the octahedral sites substituting 1/10 of Fe. In addition, the tunnel structures are stabilized by the presence of hydrogen bonds between the framework OH and Cl− species, forming a twisted cube structure and the GeO(OH)3− tetrahedra corner oxygen, forming a conjugation bond. The chemical formula of the synthetic Ge-akaganéite was determined to be (Fe7.2Ge0.8)O8.8(OH)7.2Cl0.8(Ge(OH)4)0.2.

Theg-Good-Neighbor Conditional Diagnosability of Locally Exchanged Twisted Cubes

Connectivity and diagnosability are important parameters in measuring the fault tolerance and reliability of interconnection networks. The $R^g$-vertex-connectivity of a connected graph $G$ is the minimum cardinality of a faulty set $X\subseteq V(G)$ such that $G-X$ is disconnected and every fault-free vertex has at least $g$ fault-free neighbors. The $g$-good-neighbor conditional diagnosability is defined as the maximum cardinality of a $g$-good-neighbor conditional faulty set that the system can guarantee to identify. The interconnection network considered here is the locally exchanged twisted cube $LeTQ(s,t)$. For $1\leq s\leq t$ and $0\leq g\leq s$, we first determine the $R^g$-vertex-connectivity of $LeTQ(s,t)$, then establish the $g$-good neighbor conditional diagnosability of $LeTQ(s,t)$ under the PMC model and MM$^*$ model, respectively.

A t/k diagnosis algorithm on hypercube‐like networks

SummaryProcessor fault diagnosis takes a key role in fault‐tolerant computing on multiprocessor systems. The t/k diagnosis strategy which is a generalization of the precise and pessimistic diagnosis strategies can significantly improve the self‐diagnosing capability of the system. Using this tool, it is possible to deal with large faults in the system. This paper presents a t/k diagnosis algorithm on n‐dimensional hypercube‐like networks (include Hypercubes, Crossed cubes, Möbius cubes, Locally Twisted cubes, and Twisted cubes) for any k∈[0,n−2]. The algorithm can correctly identify all nodes except at most k nodes undiagnosed. It runs in O(N) time, where N=2n is the total number of nodes of n‐dimensional hypercube‐like networks. To the best of our knowledge, in the case k≥4, there is no known t/k diagnosis algorithm for general diagnosable system or any specific system.

Conditional diagnosability of a class of matching composition networks under the comparison model

Fault diagnosis of interconnection networks is an important consideration in the design and maintenance of multiprocessor systems. Herein, we study fault diagnosis, which is the identification of faulty processors in high speed parallel processing systems. Conditional diagnosability, proposed by Lai et al. [22], assumes that no fault set can contain all the neighbors of any processor in a system; this is a well-accepted and general measure of the diagnosis ability of an interconnection network of multiprocessor systems. The diagnosability and conditional diagnosability of many interconnection networks have been studied using various diagnosis models. In this paper we study the conditional diagnosability of matching composition networks under the comparison model (MM* model). In [31] Yang determined a set of sufficient conditions for a network G to be conditionally (3n−3−C(G))-diagnosable. Our main contribution in this paper is to extend Yang's result by determining a larger class of networks that are conditionally (3n−3−C(G))-diagnosable. Yang's result [31] and earlier results for the hypercube, the crossed cube, the twisted cube and the Möbius cube [18,32,33] all become corollaries of our main result. Thus this paper extends the state of the art in the area of conditional diagnosability of multiprocessor systems.

A parallel algorithm for constructing independent spanning trees in twisted cubes

A long-standing conjecture mentions that a k-connected graph G admits k independent spanning trees (ISTs for short) rooted at an arbitrary node of G. An n-dimensional twisted cube, denoted by TQn, is a variation of hypercube with connectivity n and has many features superior to those of hypercube. Yang (2010) first proposed an algorithm to construct n edge-disjoint spanning trees in TQn for any odd integer n⩾3 and showed that half of them are ISTs. At a later stage, Wang et al. (2012) inferred that the above conjecture in affirmative for TQn by providing an O(NlogN) time algorithm to construct n ISTs, where N=2n is the number of nodes in TQn. However, this algorithm is executed in a recursive fashion and thus is hard to be parallelized. In this paper, we revisit the problem of constructing ISTs in twisted cubes and present a non-recursive algorithm. Our approach can be fully parallelized to make the use of all nodes of TQn as processors for computation in such a way that each node can determine its parent in all spanning trees directly by referring its address and tree indices in O(logN) time.

Conditional edge-fault hamiltonian-connectivity of restricted hypercube-like networks

A graph G is considered conditional k-edge-fault hamiltonian-connected if, after k faulty edges are removed from G, under the assumption that each node is incident to at least three fault-free edges, a hamiltonian path exists between any two distinct nodes in the resulting graph. This paper focuses on the conditional edge-fault hamiltonian-connectivity of a wide class of interconnection networks called restricted hypercube-like networks (RHLs). An n-dimensional RHL (RHLn) is proved to be conditional (2n−7)-edge-fault hamiltonian-connected for n≥5. The technical theorem proposed in this paper is then applied to show that several multiprocessor systems, including n-dimensional crossed cubes, n-dimensional twisted cubes for odd n, n-dimensional locally twisted cubes, n-dimensional generalized twisted cubes, n-dimensional Möbius cubes, and recursive circulants G(2n,4) for odd n, are all conditional (2n−7)-edge-fault hamiltonian-connected.

Newton-Okounkov bodies of Bott-Samelson varieties and Grossberg-Karshon twisted cubes

We describe, under certain conditions, the Newton-Okounkov body of a Bott-Samelson variety as a lattice polytope defined by an explicit list of inequalities. The valuation that we use to define the Newton-Okounkov body is different from that used previously in the literature. The polytope that arises is a special case of the Grossberg-Karshon twisted cubes studied by Grossberg and Karshon in connection to character formulae for irreducible $G$-representations and also studied previously by the authors in relation to certain toric varieties associated to Bott-Samelson varieties.

Grossberg–Karshon twisted cubes and hesitant walk avoidance

Let $G$ be a complex semisimple simply connected linear algebraic group. Let $\lambda$ be a dominant weight for $G$ and $\mathcal{I} = (i_1, i_2, \ldots, i_n)$ a word decomposition for an element $w = s_{i_1} s_{i_2} \cdots s_{i_n}$ of the Weyl group of $G$, where the $s_i$ are the simple reflections. In the 1990s, Grossberg and Karshon introduced a virtual lattice polytope associated to $\lambda$ and $\mathcal{I}$, which they called a twisted cube, whose lattice points encode (counted with sign according to a density function) characters of representations of $G$. In recent work, the first author and Jihyeon Yang prove that the Grossberg-Karshon twisted cube is untwisted (so the support of the density function is a closed convex polytope) precisely when a certain torus-invariant divisor on a toric variety, constructed from the data of $\lambda$ and $\mathcal{I}$, is basepoint-free. This corresponds to the situation in which the Grossberg-Karshon character formula is a true combinatorial formula in the sense that there are no terms appearing with a minus sign. In this note, we translate this toric-geometric condition to the combinatorics of $\mathcal{I}$ and $\lambda$. More precisely, we introduce the notion of hesitant $\lambda$-walks and then prove that the associated Grossberg-Karshon twisted cube is untwisted precisely when $\mathcal{I}$ is hesitant-$\lambda$-walk-avoiding.

GROSSBERG-KARSHON TWISTED CUBES AND BASEPOINT-FREE DIVISORS

Let $G$ be a complex semisimple simply connected linear algebraic group. The main result of this note is to give several equivalent criteria for the untwistedness of the twisted cubes introduced by Grossberg and Karshon. In certain cases arising from representation theory, Grossberg and Karshon obtained a Demazure-type character formula for irreducible $G$-representations as a sum over lattice points (counted with sign according to a density function) of these twisted cubes. A twisted cube is untwisted when it is a "true" (i.e. closed, convex) polytope; in this case, Grossberg and Karshon's character formula becomes a purely positive formula with no multiplicities, i.e. each lattice point appears precisely once in the formula, with coefficient $+1$. One of our equivalent conditions for untwistedness is that a certain divisor on the special fiber of a toric degeneration of a Bott-Samelson variety, as constructed by Pasquier, is basepoint-free. We also show that the strict positivity of some of the defining constants for the twisted cube, together with convexity (of its support), is enough to guarantee untwistedness. Finally, in the special case when the twisted cube arises from the representation-theoretic data of $\lambda$ an integral weight and $\underline{w}$ a choice of word decomposition of a Weyl group element, we give two simple necessary conditions for untwistedness which is stated in terms of $\lambda$ and $\underline{w}$.

DCube: A family of network structures for containerized data centers using dual-port servers

A fundamental goal of datacenter networking is to efficiently interconnect a large number of servers in a cost-effective way. Inspired by the commodity servers in today’s data centers that come with dual-port, we consider how to design low-cost, robust, and symmetrical network structures for containerized data centers with dual-port servers and low-end switches. In this paper, we propose a family of such network structure called a DCube, including H-DCube and M-DCube. The DCube consists of one or multiple interconnected sub-networks, each of which is a compound graph made by interconnecting a certain number of basic building blocks by means of a hypercube-like graph. More precisely, the H-DCube and M-DCube utilize the hypercube and 1-möbius cube, respectively, while the M-DCube achieves a considerably higher aggregate bottleneck throughput compared to H-DCube. Mathematical analysis and simulation results show that the DCube exhibits graceful performance degradation as the failure rate of server or switch increases. Moreover, the DCube significantly reduces the required wires and switches compared to the BCube and fat-tree. In addition, the DCube achieves a higher speedup than the BCube does for the one-to-several traffic patterns. The proposed methodologies in this paper can be applied to the compound graph of the basic building block and other hypercube-like graphs, such as Twisted cube, Flip MCube, and fastcube.

Diagnosability of Component-Composition Graphs in the MM* Model

Diagnosability is an important metric for measuring the reliability of multiprocessor systems. This article adopts the MM* model and outlines the common properties of a wide class of interconnection networks, called component-composition graphs (CCGs), to determine their diagnosability by using their obtained properties. By applying the results to multiprocessor systems, the diagnosability of hypercube-like networks (including hypercubes, crossed cubes, Möbius cubes, twisted cubes, locally twisted cubes, generalized twisted cubes, and recursive circulants), star graphs, pancake graphs, bubble-sort graphs, and burnt pancake graphs, all of which belong to the class of CCGs, can also be computed.

Reliability analysis of bijective connection networks in terms of the extra edge-connectivity

Reliability evaluation of interconnection network is of significant importance to the design and maintenance of multiprocessor systems. The extra edge-connectivity (also called restricted edge-connectivity) is an important parameter for the reliability evaluation of interconnection network. In this paper, we find a method to study the h-extra edge-connectivity of an n-dimensional bijective connection network (in brief, BC network, also called hypercube-like network). As an application, we determine h-extra edge-connectivity of an n-dimensional BC network is a constant 2n-1 for 2n-1+2f3⩽h≤2n-1,n⩾4, where f=0 when n is even, and f=1 when n is odd. We also show that the lower bound of h is sharp. This paper also obtains the exact value of h-extra edge-connectivity of an n-dimensional BC network for 1⩽h⩽2⌊n2⌋+1,n⩾4. Besides, since the BC network includes several well-known network models, such as, hypercubes, twisted cubes, crossed cubes, Möbius cubes, locally twisted cubes, generalized twisted cubes and Mcubes, so the previous results on the h-extra edge-connectivity of these networks are our corollaries.

On the [formula omitted]-diagnosability of BC networks

To increase the degree of the diagnosability of a system under system-level diagnosis strategies, Somani and Peleg introduced a measure, so called t/k-diagnosis strategy, in which the identified fault-set is allowed to contain at most k fault-free processors. Using this diagnosis strategy, the degree of diagnosability of the BC graphs (include hypercubes, crossed cubes, Möbius cubes, and twisted cubes, etc. as the subfamily) increases greatly as the number of the fault-free processors in the fault-set increases for 1⩽k⩽n. In this paper, we extend the result in [6] on the t/k-diagnosability of BC graphs for 1⩽k⩽n by showing that every n-dimensional BC graph is t(n,k)/k-diagnosable when n⩾5 and n+1⩽k⩽2n-1, where t(n,k)=-12(k+1)2+(2n-32)(k+1)-(n2-2) for n+1⩽k⩽2n-1. The result implies the crossed cube, the Möbius cube, the twisted cube and the hypercube have the same t/k-diagnosability.

Two Systematic Methods for Generating Paths in Twisted Cube Networks

The twisted cubeTQnis an alternative to the popular hypercube network and some interesting properties of aTQnwere investigated recently. The problem of how to embed paths into a host graph has attracted a great attention in recent years. However, there are few systematic methods proposed to generate the desired paths in aTQn. In this paper, we provide two kinds of systematic methods of embedding paths into aTQn.

Formula omitted]-Extraconnectivities of hypercube-like networks

A subset of vertices X is said to be a cutset if G−X is not connected. A cutset X is called an Rg-cutset if every component of G−X has at least g+1 vertices. If G has at least one Rg-cutset, the g-extraconnectivity of G is then defined as the minimum cardinality over all Rg-cutsets of G. In this paper, we first show that the 2-extraconnectivity of an n-dimensional hypercube-like network is 3n−5 for n⩾5. This improves on the previously best known result, which showed that the 2-extraconnectivity of an n-dimensional hypercube-like network is 3n−5 for n⩾8. We further demonstrate that the 3-extraconnectivity of an n-dimensional hypercube-like network is 4n−9 for n⩾6. Based on the above results, the 2-extraconnectivity and 3-extraconnectivity of several interconnection networks, including hypercubes, twisted cubes, crossed cubes, Möbius cubes, locally twisted cubes, generalized twisted cubes, recursive circulants, and Mcubes, can be determined efficiently.