Congruent Transformations Research Articles

Relaxed local nonquadratic stabilization for nonhomogeneous hidden Markov jump fuzzy systems over non-closed system operating range

This paper is dedicated to investigating the problem of relaxed local nonquadratic stabilization of nonhomogeneous hidden Markov jump fuzzy systems (HMJFSs) over a non-closed system operating range in the continuous-time domain. Within the framework of the mode-dependent nonquadratic Lyapunov function, the proposed method formulates the local stabilization conditions in a less conservative manner. Different from other recent studies, our formulation process is carried out without assumptions on the additional state restrictions required to establish a closed operating range, which opens the possibility to further increase the size of the domain of attraction. Furthermore, this method provides a systematic way to solve the nonconvex problem arising from the asynchronism between hidden and observed modes through augmented state-based congruent transformations. Not only that, significant progress is made in further reducing conventional conservatism and computational complexity by removing unnecessary auxiliary conditions. Finally, two numerical examples are given to show the validity of the proposed method.

General Synthesis Method for Dispersively Coupled Resonator Filters With Cascaded Topologies

This article presents a new synthesis technique for circuits that may include dispersively coupled resonators and admitting an overall cascaded topology. A decomposition technique of the Darlington type is first introduced to split the original response S of the filter, taken as its scattering matrix, into m subresponses S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> ,⋯, S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> corresponding to each subblock of the cascaded circuital structure. Each individual subresponse Sk is then synthesized separately. In this article, the state space equations governing the model of dispersively coupled resonators are detailed. An extension to the case of dispersive coupling of the shortest path rule, which determines the maximum number of finite TZs realizable by a given topology, is then introduced. Congruent transformations that extend the concept of rotations or similarity transformations while preserving the filter response are exploited to reduce the individual synthesis problems to the determination of a basis of vectors verifying certain orthogonality relations. A direct synthesis technique for dispersive building blocks, such as duplets, triplets, and quadruplets, is then given in the form of an orthogonalization procedure used for the computation of the desired basis. This approach is then combined with the aforementioned decomposition technique to produce a versatile algorithm able to synthesize hybrid circuits made of cascaded subblocks of different orders and types that implement each a subset of the overall TZs by means of coupling topologies containing a mixture of dispersive and nondispersive couplings. The first synthesis example is detailed where two dispersive duplets are combined with a classical quadruplet to realize a symmetric 6-4 response. A hardware implementation of the synthesized circuit is presented in combline technology. The second example proposes a slightly more involved coupling topology able to realize 10-8 asymmetric responses by means of four cascaded basic dispersive blocks.

Students' Ability to Solve Geometry Problems with Emphasis on Interdisciplinary Relations

In this paper we focus on the interesting relationship of geometry and music from the perspective of problem-based learning. This teaching has to be designed with the intention of supporting students' self-education. We are looking for a connection between teaching geometry, especially congruent transformations in plane and selected musical works. We also provide specific examples of these works and analysing the occurrence of congruent transformations in them. Subsequently, we created tasks for students from this field. Our interpretations of the findings from student solutions are also presented.

On graphs with exactly two positive eigenvalues

The inertia of a graph G is defined to be the triplet In( G ) = ( p ( G ), n ( G ), η ( G )) , where p ( G ) , n ( G ) and η ( G ) are the numbers of positive, negative and zero eigenvalues (including multiplicities) of the adjacency matrix A ( G ) , respectively. Traditionally p ( G ) (resp. n ( G ) ) is called the positive (resp. negative) inertia index of G . In this paper, we introduce three types of congruent transformations for graphs that keep the positive inertia index and negative inertia index. By using these congruent transformations, we determine all graphs with exactly two positive eigenvalues and one zero eigenvalue.

Simultaneous band reduction of two symmetric matrices

In this paper, we consider simultaneous band reduction of two dense symmetric matrices by congruent transformations. The ideas of simultaneous tridiagonalization are generalized to propose an efficient algorithm for the simultaneous band reduction. In contrast to the algorithms of simultaneous tridiagonalization which are mainly based on matrix–vector operations, the proposed algorithm of simultaneous band reduction has the advantage that matrix–matrix operations can be fully used to achieve better performance on modern computer architecture. Numerical results are presented to illustrate the effectiveness of our proposed algorithm.

Seeing a straight line on a curved surface: decoupling of patterns from surfaces by single IT neurons.

We have no difficulty seeing a straight line on a curved piece of paper, but in fact, doing so requires decoupling the shape of the surface from the pattern itself. Here we report a novel form of invariance in the visual cortex: single neurons in monkey inferior temporal cortex respond similarly to congruent transformations of patterns and surfaces, in effect decoupling patterns from the surface on which they are overlaid.

An efficient blocking M2L translation for low-frequency fast multipole method in three dimensions

We propose an efficient scheme to perform the multipole-to-local (M2L) translation in the three-dimensional low-frequency fast multipole method (LFFMM). Our strategy is to combine a group of matrix–vector products associated with M2L translation into a matrix–matrix product in order to diminish the memory traffic. For this purpose, we first developed a grouping method (termed as internal blocking) based on the congruent transformations (rotational and reflectional symmetries) of M2L-translators for each target box in the FMM hierarchy (adaptive octree). Next, we considered another method of grouping (termed as external blocking) that was able to handle M2L translations for multiple target boxes collectively by using the translational invariance of the M2L translation. By combining these internal and external blockings, the M2L translation can be performed efficiently whilst preservingthe numerical accuracy exactly. We assessed the proposed blocking scheme numerically and applied it to the boundary integral equation method to solve electromagnetic scattering problems for perfectly electrical conductor. From the numerical results, it was found that the proposed M2L scheme achieved a few times speedup compared to the non-blocking scheme.

Novel Repair-by-Transfer Codes and Systematic Exact-MBR Codes with Lower Complexities and Smaller Field Sizes

The <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$(n,k,d)$</tex> <mathgraphic fileref="chung-ieq1-2297109.gif" graphicformat="GIF"/></formula> regenerating code is a class of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$(n,k)$</tex><mathgraphic fileref="chung-ieq2-2297109.gif" graphicformat="GIF"/> </formula> erasure codes with the capability to recover a lost code fragment from other <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$d$</tex><mathgraphic fileref="chung-ieq3-2297109.gif" graphicformat="GIF"/></formula> existing code fragments. In this paper, we focus on the design of exact regenerating codes at minimum bandwidth regenerating (MBR) points. For <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$d=n-1$</tex><mathgraphic fileref="chung-ieq4-2297109.gif" graphicformat="GIF"/> </formula> , a class of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$(n,k,d=n-1)$</tex> <mathgraphic fileref="chung-ieq5-2297109.gif" graphicformat="GIF"/></formula> exact-MBR codes, termed as repair-by-transfer codes, have been developed in prior work to avoid arithmetic operations in node repairing process. The first result of this paper presents a new class of repair-by-transfer codes via congruent transformations. As compared with prior works, the advantages of proposed codes include: i) the minimum of field size is significantly reduced from <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n \atopwithdelims ()2$</tex> <mathgraphic fileref="chung-ieq6-2297109.gif" graphicformat="GIF"/></formula> to <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n$ </tex><mathgraphic fileref="chung-ieq7-2297109.gif" graphicformat="GIF"/></formula> ; ii) the encoding complexity is decreased from <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n^4$</tex> <mathgraphic fileref="chung-ieq8-2297109.gif" graphicformat="GIF"/></formula> to <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX"> $n^3$</tex><mathgraphic fileref="chung-ieq9-2297109.gif" graphicformat="GIF"/></formula> . Our simulation results show that the proposed code achieves faster operations than the prior approach does under large <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n$</tex><mathgraphic fileref="chung-ieq10-2297109.gif" graphicformat="GIF"/> </formula> . The second result of this paper presents a new form of coding matrix for product-matrix exact-MBR codes. The proposed coding matrix includes the following advantages: i) the minimum of finite field size is reduced from <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n-k+d$</tex><mathgraphic fileref="chung-ieq11-2297109.gif" graphicformat="GIF"/> </formula> to <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n$</tex> <mathgraphic fileref="chung-ieq12-2297109.gif" graphicformat="GIF"/></formula> ; ii) the fast Reed-Solomon erasure coding algorithms can be applied on the proposed exact-MBR codes to reduce the time complexities.

Braiding statistics and congruent invariance of twist defects in bosonic bilayer fractional quantum Hall states

We describe the braiding statistics of topological twist defects in fractional quantum Hall (FQH) states. In particular, we study Abelian bosonic $(mmn)$ FQH states with bilayer symmetries. Twist defects are staircases that lift orbiting quasiparticles from one layer to another. These point defects exhibit semiclassical non-Abelian characteristics. We develop a procedure to evaluate consistent basis transformations ($F$ symbols) of the defect fusion theory. Unlike quantum anyonic excitations of a topological phase, the exchange and braiding statistics of twist defects is not characterized by modular transformations, but instead a subcollection known as congruent transformations. We also characterize the projective nature of unitary braiding operations and relate it to the global ${\mathbb{Z}}_{2}$ bilayer symmetry.

A Unified View to Greedy Geometric Routing Algorithms in Ad Hoc Networks

The main aim of this paper is to give a unified view to greedy geometric routing algorithms in ad hoc networks. For this, we firstly present a general form of greedy routing algorithm using a class of objective functions which are invariant under congruent transformations of a point set. We show that some known greedy routing algorithms such as Greedy Routing, Compass Routing, and Midpoint Routing can be regarded as special cases of the generalized greedy routing algorithm. In addition, inspired by the unified view of greedy routing, we propose three new greedy routing algorithms. We then derive a sufficient condition for our generalized greedy routing algorithm to guarantee packet delivery on every Delaunay graph. This condition makes it easier to check whether a given routing algorithm guarantees packet delivery, and the class of objective functions with this condition is closed under convex combination. We show that Greedy Routing, Midpoint Routing, and the three new greedy routing algorithms proposed in this paper satisfy the sufficient condition, i.e., they guarantee packet delivery on Delaunay graphs, and then compare the merits and demerits of these methods.

Extended and reduced POG dynamic model of an automatic corking machine for threaded plastic caps

The aim of this work is to show how the modeling of an electromechanical system can be addressed using the energy-based graphical modeling technique named “Power-Oriented Graphs” (POG). Differences and analogies of POG against Bond Graphs modeling technique are discussed. The paper presents the POG dynamic model of an automatic corking machine for threaded plastic caps: the system is composed by two electric motors moving a ball screw/spline that realizes the linear/rotary motion necessary to screw a plastic cap on a bottle. First an extended POG model is presented, together with the equivalent Bond Graph model, then some proper congruent state space transformations and a POG-based graphical method are introduced to transform and reduce the system dynamic model. Simulation and experimental results are finally presented and compared.

The Ni-Re-V (Nickel-Rhenium-Vanadium) System

The Ni-Re system [Massalski 2] (Fig. 1) is a simple peritectic system, the peritectic reaction L + eM c occurs at 1620 C where e and c are the terminal solid solutions (Re) and (Ni), respectively. The Ni-V system [Massalski 2] (Fig. 2) has five intermediate phases: Ni8V(f), Ni3V(p), Ni2V(q), r, and NiV3(d). The f, p, and q phases form from the solid solution c through congruent transformations at 405, 1045, and 922 C, respectively. The r phase appears to have two polymorphic forms r¢ and r. The r¢ phase forms through a peritectic reaction L + aMr¢ at 1280 C, where a is the terminal solid solution (V). The r phase occurs below 790 C, the probable reaction at the high Ni side is a peritectoid reaction r¢ + qM r at 790 C and a eutectoid reaction r¢Mr + d probably occurs at the high V end. The d phase forms through a peritectoid reaction L + aM d at 900 C. A eutectoid reaction LM c + r¢ occurs at 1202 C and two eutectoid reactions cM p + q and cM q + r¢ occur at 908 and 890 C. The Re-V system [Massalski 2] has two intermediate phases r and d. The r phase forms through a pertiectic reaction L + eMr at 2460 C. The d phase forms through a peritectoid reaction r + aM d at 2280 C. The a phase extends from V to 56 at.% Re and has a congruent melting temperature of 2400 C at 47 at.% V. A eutectic reaction LMr + a occurs at 2380 C. The r and d phases exist only at high temperatures and undergo eutectoid transformations rM e + d at 1970 C and dM e + a at 1470 C.

Passive Reduced-Order Macromodeling Algorithm for Microelectromechanical Systems

In this paper, we present a passive reduced-order macromodeling algorithm for second-order dynamic systems of linear microelectromechanical systems (MEMS) devices. The proposed reduction algorithm is based on congruent transformations. The system equations of MEMS devices, given by finite-element methods (FEMs), are converted to state-space forms that are compatible with passive Krylov subspace methods. To achieve this, a modified matrix equation is proposed for second-order MEMS dynamics. In addition, the generalized procedure is provided for first-order heat transfer problems and second-order structure dynamic problems to ensure that the discretized FEM system satisfies all the necessary conditions to guarantee the passivity of the reduced-order system. Finally, numerical examples are provided to demonstrate the validity of the proposed passive reduction technique.

Realization of mechanical systems from second-order models

Congruent coordinate transformations are used to convert second-order models to a form in which the mass, damping, and stiffness matrices can be interpreted as a passive mechanical system. For those systems which can be constructed from interconnected mass, stiffness, and damping elements, it is shown that the input–output preserving transformations can be parametrized by an orthogonal matrix whose dimension corresponds to the number of internal masses—those masses at which an input is not applied nor an output measured. Only a subset of these transformations results in mechanically realizable models. For models with a small number of internal masses, complete discrete mapping of the transformation space is possible, permitting enumeration of all mechanically realizable models sharing the original model’s input–output behavior. When the number of internal masses is large, a nonlinear search of transformation space can be employed to identify mechanically realizable models. Applications include scale model vibration testing of complicated structures and the design of electromechanical filters.

Experimental determination of NH4NO3-KNO3 binary phase diagram

The solid-state phase transitions in ammonium nitrate (AN)-potassium nitrate (KN) system, and the equilibrium AN-KN phase diagram have been determined by using differential scanning calorimetry and high-temperature in situ x-ray diffractometry. Sample preparation was performed in a special “dry room” with very low humidity. A single phase region (AN III) with no phase transitions to 373 K was observed in the composition range 5 to 20% KN; this is critical for use in air bag gas generators. The high-temperature KN phase (KN I) has a wide range of stability from 20 to 100 wt.% KN. There are one eutectic, two eutectoid, three peritectoid, and one congruent transformations in this phase diagram. Two new nonstoichiometric phases were found at lower temperatures in the mid-composition range between the AN and KN terminal solid solutions. Details of the phase equilibria are presented.

Differential thermal analysis under quasi-isothermal, quasi-isobaric conditions (Q-DTA): Part III. Mechanism of congruent and incongruent phase transformations of salt hydrates

By using the “ simultaneous Q-DTA, Q-TG measuring technique” elaborated recently, conditions near to the requirements of thermodynamics can be created, thus the “normalized” course of curves taken by this method, their characteristic temperatures or the values measured for the enthalpy change can be considered to be near to the theoretical values. For testing this new method, CuSO 4·5H 2O, MnSO 4·5H 2O, Mg(NO 3) 2·6H 2O and Na 2SO 4·10H 2O have been investigated. These model substances can be prepared in stoichiometric composition free of impurities, their decomposition mechanism is well-known, thus the effect of the altered experimental conditions by the simultaneous Q-DTA, Q-TG method on the course of their dehydration can be calculated.

P–T–x–y phase diagram of the Cd–Zn–Te system

The p–T and T–x–y projections of the p–T–x–y phase diagram and its T–x–y isobars and x–y isobar–isothermal sections for the Cd–Zn–Te system were constructed. The character of the sublimation of the Cd1−xZnxTe (CZT) solid solution was discussed. It was shown that quasi–binary approximation is insufficient for CZT congruent transformations.

Singular Equilibria in Paragonite Blueschists, Amphibolites and Eclogites

Chemographic constraints from high-pressure mafic rocks of the INTRODUCTION Nevado–Filabride Complex (Betic–Rif mountain belts, SE Spain) The relevance of singular equilibria has been largely and from worldwide high-pressure terrains have been used to unravel recognized in chemical systems of interest for the Earth phase relationships between amphibole, garnet, clinopyroxene, and Sciences. After the early experiments of Konovalov on albite in the system Na2O–CaO–FeO–MgO–Al2O3–SiO2–H2O azeotropic or congruent transformations, Ricci (1951) (NCFMASH) at quartz-, paragonite-, clinozoisiteand waterexhaustively discussed the fundamentals of singularities. saturated conditions. Extensive analysis of element partitioning Two types of singularities were recognized: the first, between amphibole and clinopyroxene suggests that the Fe–Mg called type m and most relevant, is associated with azeopartition coefficient is low at low-temperature and high-pressure tropic points, and the second, called type k, is related to conditions (‘high-temperature’ blueschists–low-temperature eclogites) the intersection of solvi with compositional loops, as and a progressive increase is achieved by an increase in temperature, occurs in the alkali feldspar system at low pressures or by a decrease in pressure, or by a decrease in the sodium content (Morse, 1970). Examples and discussion of both type m of the clinopyroxene. Mass-balance analysis of paragonite eclogites and type k singular points in igneous systems have been and amphibolites suggests that at least two singular reactions are given by Morse (1980, e.g. see chapters 16 and 18). The possible in NCFMASH: clinopyroxene+ paragonite+ quartz= widespread occurrence of singularities in igneous systems amphibole + albite + clinozoisite + water (garnet-absent), and is obviously related to the extremely large compositional amphibole+ clinozoisite= garnet+ clinopyroxene+ paragonite range of silicate liquids. The well-known concept of + quartz + water (albite-absent). Albite-absent singular equithe ‘thermal barrier’ or ‘thermal divide’ implies the librium is relevant for the blueschist–eclogite transition, whereas occurrence of singular equilibria, and their importance garnet-absent equilibrium may control the eclogite to amphibolite in the petrogenesis of granites [e.g. modelled in the binary transition. A complex arrangement of divariant reactions, singular system kalsilite–silica (Lindsley, 1966)] or basalts [e.g. in reactions and conjugate non-degenerate reactions arises in topologies the binary system forsterite–silica or in the ternary system at the blueschist–eclogite–amphibolite transitions. Schreinemakers’ forsterite–silica–diopside or forsterite–silica–anorthite analysis in P–T diagrams, and mixed-variable T–XM′–XC′ and (Presnall et al., 1979)] has been widely discussed. P–XM′–XC′ diagrams suggest that complex pressure–temperature In contrast, the significance and implications of singular paths are not mandatory if a recurrent appearance and disappearance point nets at subsolidus conditions have been largely of similar phase assemblages is observed in paragonite-bearing mafic undervalued. Although Korzhinskii (1963) analysed azeorocks. The P–T location of the singular curve net is discussed in tropic transformations in subsolidus open systems, only the context of Nevado–Filabride Complex mafic rocks. recently have applications to metamorphic assemblages

BEC: A 2-node fast converging shear-deformable isotropic and composite beam element based on 6 rigid-body and 6 straining modes

We present a 2-node isotropic and composite shear-deformable beam element in space—the element BEC. The formulation is based on the natural mode finite element method whereby the 12 Cartesian degrees of freedom (6 per node) are transformed to 6 natural rigid-body and 6 natural straining modes which describe the element's kinematics. Only a sparse 6 × 6 natural stiffness matrix needs to be formed while congruent transformations, performed on the computer, transform the natural stiffness to the local and global Cartesian coordinates with significant impact on computational efficiency and economy. Initial load due to temperature is also given. In addition, simplified and partly simplified geometrical stiffnesses are established which permit the study of nonlinear phenomena such as buckling and large displacements. Throughout the formulation no locking or any other parasitic phenomena were encountered. Although a general formulation is adopted, the model is applied here to a solid beam section. The theory is implemented in our structural analysis code SANI (Structural Analysis and Information). Following validation with reference solutions, applications are made to isotropic and composite beams, frames and three-dimensional composite beam structures. The theory presented indicates a new direction in finite element analysis and structural mechanics in general.

A Formal Basis For Removing Goto Statements

Goto statements detract from the quality of imperative programs. They tend to make control-structures difficult to understand and, at the same time, introduce the risk of non-termination and other correctness problems. A new, formal, generally applicable procedure for removing all goto statements from program structures is presented. This method is based on formal semantics and congruent equivalence transformations. Not only does the method logically simplify program structures; it also detects a range of defects including a class of non-termination problems, unreachable code and redundancy problems. The method can also be used to eliminate recursion.