scholarly journals BACH FLOWS OF PRODUCT MANIFOLDS

2012 ◽  
Vol 09 (05) ◽  
pp. 1250039 ◽  
Author(s):  
SANJIT DAS ◽  
SAYAN KAR
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Ricci Flow ◽  
Flow Characteristics ◽  
Geometric Flow ◽  
Flow Equations ◽  
First Order ◽  
Bach Tensor ◽  
Point Condition ◽  
Future Work

We investigate various aspects of a geometric flow defined using the Bach tensor. First, using a well-known split of the Bach tensor components for (2, 2) unwarped product manifolds, we solve the Bach flow equations for typical examples of product manifolds like S2 × S2, R2 × S2. In addition, we obtain the fixed-point condition for general (2, 2) manifolds and solve it for a restricted case. Next, we consider warped manifolds. For Bach flows on a special class of asymmetrically warped 4-manifolds, we reduce the flow equations to a first-order dynamical system, which is solved exactly to find the flow characteristics. We compare our results for Bach flow with those for Ricci flow and discuss the differences qualitatively. Finally, we conclude by mentioning possible directions for future work.

scholarly journals Laplacian Matrix-Based Power Flow Formulation for LVDC Grids with Radial and Meshed Configurations

Energies ◽  
2021 ◽  
Vol 14 (7) ◽  
pp. 1866
Author(s):  
Zahid Javid ◽  
Ulas Karaagac ◽  
Ilhan Kocar ◽  
Ka Wing Chan
Keyword(s):  
Fixed Point ◽  
Power Flow ◽  
Low Voltage ◽  
Mathematical Formulation ◽  
Distribution Networks ◽  
Load Flow ◽  
Banach Fixed Point Theorem ◽  
Loading Conditions ◽  
Flow Equations ◽  
Uniqueness Of The Solution

There is an increasing interest in low voltage direct current (LVDC) distribution grids due to advancements in power electronics enabling efficient and economical electrical networks in the DC paradigm. Power flow equations in LVDC grids are non-linear and non-convex due to the presence of constant power nodes. Depending on the implementation, power flow equations may lead to more than one solution and unrealistic solutions; therefore, the uniqueness of the solution should not be taken for granted. This paper proposes a new power flow solver based on a graph theory for LVDC grids having radial or meshed configurations. The solver provides a unique solution. Two test feeders composed of 33 nodes and 69 nodes are considered to validate the effectiveness of the proposed method. The proposed method is compared with a fixed-point methodology called direct load flow (DLF) having a mathematical formulation equivalent to a backward forward sweep (BFS) class of solvers in the case of radial distribution networks but that can handle meshed networks more easily thanks to the use of connectivity matrices. In addition, the convergence and uniqueness of the solution is demonstrated using a Banach fixed-point theorem. The performance of the proposed method is tested for different loading conditions. The results show that the proposed method is robust and has fast convergence characteristics even with high loading conditions. All simulations are carried out in MATLAB 2020b software.

Theoretical and Experimental Characterization of Emulsion Flow in a Cavity Transfer Mixer

2003 ◽  
Author(s):  
A. H. Raza ◽  
R. A. Lai-Fook ◽  
C. J. Lawrence
Keyword(s):  
Flow Characteristics ◽  
Axial Direction ◽  
Solution Procedure ◽  
Modes Of Operation ◽  
Flow Equations ◽  
Recirculating Flow ◽  
Rectangular Pattern ◽  
Mean Pressure ◽  
Emulsion Processing ◽  
Dependent Flow

A theoretical model of time-dependent flow based on Reynolds equation using emulsion processing in a Cavity Transfer Mixer (CTM) has been developed in Mathematica and is presented in this work. It is a continuum model, which allows the study of materials undergoing rapid deformation. The flow of a fluid in a CTM is examined using a finite difference analysis (FDA) to solve the flow equations for an unwound section with cavities arranged in a rectangular pattern. Periodic boundary conditions are included in the model to predict the pressure distribution, which allows subsequent determination of the flow field. The solution procedure gives a smooth function for the pressure field, with equal pressures at the boundaries in the y-direction and an overall mean pressure gradient in the x-direction. Once the pressure has been found, several flow properties follow directly. The flow in the downstream axial direction is seen to consist of purely pressure-driven flow. In contrast, the flow in the cross-cavity direction is a recirculating flow driven by the drag velocity of the moving rotor surface. These two flows taken together combine into a helical flow travelling through the cavity. Because of this, there is likely to bre a high degree of laminar and distributive flow in this type of machine. The experimental part of this work addresses the processing of an emulsion in the CTM when it is run under batch and continuous modes of operation. The flow characteristics have been studied for varying rotor speeds of 0 rpm, 16 rpm, 32 rpm, 48 rpm and 64 rpm. Also studied were the changes that the emulsion exhibits along the mixer length and with time in the mixer. The experiments indicate that increase in the rotational speed causes the viscosity to reduce systematically in both batch and continuous modes of operation.

TERBIUM MOLYBDATE: GROUP THEORY AND FLUCTUATIONS

1987 ◽  
Vol 01 (05n06) ◽  
pp. 239-244
Author(s):  
SERGE GALAM
Keyword(s):  
Fixed Point ◽  
Group Theory ◽  
Parameter Space ◽  
Landau Theory ◽  
Two Dimensional ◽  
Stable Fixed Point ◽  
Theory Analysis ◽  
Continuous Transition ◽  
Dimensional Parameter ◽  
First Order

A new mechanism to explain the first order ferroelastic—ferroelectric transition in Terbium Molybdate (TMO) is presented. From group theory analysis it is shown that in the two-dimensional parameter space ordering along either an axis or a diagonal is forbidden. These symmetry-imposed singularities are found to make the unique stable fixed point not accessible for TMO. A continuous transition even if allowed within Landau theory is thus impossible once fluctuations are included. The TMO transition is therefore always first order. This explanation is supported by experimental results.

μ-definable sets of integers

1993 ◽  
Vol 58 (1) ◽  
pp. 291-313 ◽  
Author(s):  
Robert S. Lubarsky
Keyword(s):  
Fixed Point ◽  
Free Variable ◽  
Order Logic ◽  
First Order Logic ◽  
Inductive Definition ◽  
Natural Class ◽  
First Order ◽  
Inductive Definitions ◽  
Definable Sets ◽  
The Universe

Inductive definability has been studied for some time already. Nonetheless, there are some simple questions that seem to have been overlooked. In particular, there is the problem of the expressibility of the μ-calculus.The μ-calculus originated with Scott and DeBakker [SD] and was developed by Hitchcock and Park [HP], Park [Pa], Kozen [K], and others. It is a language for including inductive definitions with first-order logic. One can think of a formula in first-order logic (with one free variable) as defining a subset of the universe, the set of elements that make it true. Then “and” corresponds to intersection, “or” to union, and “not” to complementation. Viewing the standard connectives as operations on sets, there is no reason not to include one more: least fixed point.There are certain features of the μ-calculus coming from its being a language that make it interesting. A natural class of inductive definitions are those that are monotone: if X ⊃ Y then Γ (X) ⊃ Γ (Y) (where Γ (X) is the result of one application of the operator Γ to the set X). When studying monotonic operations in the context of a language, one would need a syntactic guarantor of monotonicity. This is provided by the notion of positivity. An occurrence of a set variable S is positive if that occurrence is in the scopes of exactly an even number of negations (the antecedent of a conditional counting as a negation). S is positive in a formula ϕ if each occurrence of S is positive. Intuitively, the formula can ask whether x ∊ S, but not whether x ∉ S. Such a ϕ can be considered an inductive definition: Γ (X) = {x ∣ ϕ(x), where the variable S is interpreted as X}. Moreover, this induction is monotone: as X gets bigger, ϕ can become only more true, by the positivity of S in ϕ. So in the μ-calculus, a formula is well formed by definition only if all of its inductive definitions are positive, in order to guarantee that all inductive definitions are monotone.

scholarly journals Two periodic solutions of neutral difference equations modelling physiological processes

2006 ◽  
Vol 2006 ◽  
pp. 1-12 ◽  
Author(s):  
Jun Wu ◽  
Yicheng Liu
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Periodic Solutions ◽  
Point Theorem ◽  
Difference Equations ◽  
Neutral Difference Equations ◽  
First Order ◽  
Analysis Techniques ◽  
Physiological Processes

We establish existence, multiplicity, and nonexistence of periodic solutions for a class of first-order neutral difference equations modelling physiological processes and conditions. Our approach is based on a fixed point theorem in cones as well as some analysis techniques.

scholarly journals Chaotic Itinerancy Observed in Mutually Coupled Gaussian Maps

2018 ◽  
Vol 28 (04) ◽  
pp. 1830011
Author(s):  
Mio Kobayashi ◽  
Tetsuya Yoshinaga
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Chaotic Behavior ◽  
Periodic Points ◽  
Two Dimensional ◽  
Stable Fixed Point ◽  
Bifurcation Structure ◽  
Unstable Fixed Point ◽  
Gaussian Map ◽  
Gaussian Maps

A one-dimensional Gaussian map defined by a Gaussian function describes a discrete-time dynamical system. Chaotic behavior can be observed in both Gaussian and logistic maps. This study analyzes the bifurcation structure corresponding to the fixed and periodic points of a coupled system comprising two Gaussian maps. The bifurcation structure of a mutually coupled Gaussian map is more complex than that of a mutually coupled logistic map. In a coupled Gaussian map, it was confirmed that after a stable fixed point or stable periodic points became unstable through the bifurcation, the points were able to recover their stability while the system parameters were changing. Moreover, we investigated a parameter region in which symmetric and asymmetric stable fixed points coexisted. Asymmetric unstable fixed point was generated by the [Formula: see text]-type branching of a symmetric stable fixed point. The stability of the unstable fixed point could be recovered through period-doubling and tangent bifurcations. Furthermore, a homoclinic structure related to the occurrence of chaotic behavior and invariant closed curves caused by two-periodic points was observed. The mutually coupled Gaussian map was merely a two-dimensional dynamical system; however, chaotic itinerancy, known to be a characteristic property associated with high-dimensional dynamical systems, was observed. The bifurcation structure of the mutually coupled Gaussian map clearly elucidates the mechanism of chaotic itinerancy generation in the two-dimensional coupled map. We discussed this mechanism by comparing the bifurcation structures of the Gaussian and logistic maps.

STUDIES ON HOMOGENEOUS FIRST ORDER GAS REACTIONS: II. THE DECOMPOSITION OF THE ISOMERIC ESTERS BUTYLIDENE DIACETATE AND ETHYLIDENE DIPROPIONATE

1932 ◽  
Vol 6 (4) ◽  
pp. 417-427 ◽  
Author(s):  
C. C. Coffin
Keyword(s):  
Activation Energy ◽  
Degrees Of Freedom ◽  
Maximum Energy ◽  
Velocity Measurements ◽  
First Order ◽  
Gas Reactions ◽  
Points Of View ◽  
The Stability ◽  
Future Work ◽  
Internal Degrees Of Freedom

The gaseous decompositions of the esters butylidene diacetate and ethylidene dipropionate have been studied from points of view previously outlined in papers on the decomposition of ethylidene diacetate (2, 3). The decomposition velocities have been measured at initial pressures of from 5 to 56 cm. of mercury and at temperatures between 211 and 265 °C. The reactions are homogeneous and of the first order. They agree with the Arrhenius equation and give 100% yields (within experimental error) of an aldehyde and an anhydride. The preparation of the compounds and improvements in the technique of the velocity measurements are described.While the specific velocities of the three reactions at any temperature are somewhat different, their activation energies are the same. It is suggested that in the case of such simple reactions, which are strictly localized within the molecular structure, the activation energy can be identified as the maximum energy that the reactive bonds may possess and still exist; i.e., it may be taken as a measure of the stability of the bonds which are broken in the reaction. The suggestion is also made that for a series of reactions which have the same activation energy, the specific velocities can be taken as a relative measure of the number of internal degrees of freedom that contribute to the energy of activation. On the basis of these assumptions it becomes possible to use reaction-velocity measurements for the investigation of intramolecular energy exchange. The theoretical significance of the data is further discussed and the scope of future work in this connection is indicated.The monomolecular velocity constants (sec−1) of the decomposition of ethylidene diacetate, ethylidene dipropionate and butylidene diacetate are given respectively by the equations [Formula: see text], [Formula: see text], and [Formula: see text].

scholarly journals RENORMALIZATION GROUP FLOW EQUATIONS FOR THE SCALAR O(N) THEORY

2001 ◽  
Vol 16 (11) ◽  
pp. 2119-2124 ◽  
Author(s):  
B.-J. SCHAEFER ◽  
O. BOHR ◽  
J. WAMBACH
Keyword(s):  
Fixed Point ◽  
Renormalization Group ◽  
Three Dimensions ◽  
Leading Order ◽  
Renormalization Group Flow ◽  
Derivative Expansion ◽  
Scalar Theory ◽  
Flow Equations ◽  
Self Consistent ◽  
Group Flow

Self-consistent new renormalization group flow equations for an O(N)-symmetric scalar theory are approximated in next-to-leading order of the derivative expansion. The Wilson-Fisher fixed point in three dimensions is analyzed in detail and various critical exponents are calculated.

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