Symmetric Neural Networks and Propositional Logic Satisfiability

1991 ◽  
Vol 3 (2) ◽  
pp. 282-291 ◽  
Author(s):  
Gadi Pinkas
Keyword(s):  
Energy Function ◽  
Quadratic Function ◽  
Nonmonotonic Reasoning ◽  
Propositional Calculus ◽  
High Order ◽  
Satisfiability Problem ◽  
Problem Solver ◽  
Energy Functions ◽  
Connectionist Network ◽  
Wide Range

Connectionist networks with symmetric weights (like Hopfield networks and Boltzmann Machines) use gradient descent to find a minimum for quadratic energy functions. We show equivalence between the problem of satisfiability in propositional calculus and the problem of minimizing those energy functions. The equivalence is in the sense that for any satisfiable well-formed formula (WFF) we can find a quadratic function that describes it, such that the set of solutions that minimizes the function is equal to the set of truth assignments that satisfy the WFF. We also show that in the same sense every quadratic energy function describes some satisfiable WFF. Algorithms are given to transform any propositional WFF into an energy function that describes it and vice versa. High-order models that use sigma-pi units are shown to be equivalent to the standard quadratic models with additional hidden units. An algorithm to convert high-order networks to low-order ones is used to implement a satisfiability problem-solver on a connectionist network. The results give better understanding of the role of hidden units and of the limitations and capabilities of symmetric connectionist models. The techniques developed for the satisfiability problem may be applied to a wide range of other problems, such as associative memories, finding maximal consistent subsets, automatic deduction, and even nonmonotonic reasoning.

scholarly journals Modelling the Inflation and Elastic Instabilities of Rubber-Like Spherical and Cylindrical Shells Using a New Generalised Neo-Hookean Strain Energy Function

2021 ◽  
Author(s):  
Afshin Anssari-Benam ◽  
Andrea Bucchi ◽  
Giuseppe Saccomandi
Keyword(s):  
Experimental Data ◽  
Energy Function ◽  
Limit Point ◽  
Strain Energy ◽  
Cylindrical Shells ◽  
Strain Energy Function ◽  
Model Parameters ◽  
Wide Range ◽  
Cylindrical Tubes ◽  
Gent Model

AbstractThe application of a newly proposed generalised neo-Hookean strain energy function to the inflation of incompressible rubber-like spherical and cylindrical shells is demonstrated in this paper. The pressure ($P$ P ) – inflation ($\lambda $ λ or $v$ v ) relationships are derived and presented for four shells: thin- and thick-walled spherical balloons, and thin- and thick-walled cylindrical tubes. Characteristics of the inflation curves predicted by the model for the four considered shells are analysed and the critical values of the model parameters for exhibiting the limit-point instability are established. The application of the model to extant experimental datasets procured from studies across 19th to 21st century will be demonstrated, showing favourable agreement between the model and the experimental data. The capability of the model to capture the two characteristic instability phenomena in the inflation of rubber-like materials, namely the limit-point and inflation-jump instabilities, will be made evident from both the theoretical analysis and curve-fitting approaches presented in this study. A comparison with the predictions of the Gent model for the considered data is also demonstrated and is shown that our presented model provides improved fits. Given the simplicity of the model, its ability to fit a wide range of experimental data and capture both limit-point and inflation-jump instabilities, we propose the application of our model to the inflation of rubber-like materials.

scholarly journals FAO-56 Penman-Monteith DailyETofrom Linear Regression Calibrated Hargreaves Equation with Wind Terms in Tropics with Limited Data

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Eric Kra
Keyword(s):  
Linear Regression ◽  
Quadratic Function ◽  
Constant Term ◽  
Calibration Method ◽  
Temperature Data ◽  
Limited Data ◽  
Wide Range ◽  
Hargreaves Equation ◽  
Historical Temperature ◽  
Monteith Equation

Hargreaves equation (HG), which lacks a wind speed (u2) term, was modified, through a linear regression calibration method, into LHGu which hasu2terms. LHGu is effectively a simplified method for approximating FAO-56 Penman-Monteith equation (FPM) daily reference evapotranspiration (ETo) in tropics with only temperature data. In LHGu, the “0.0023” constant term in HG was calibrated as a shifted power function ofu2, and the calibration constant was parametrized as a quadratic function ofu2. LHGu was developed using simulated constantu2data and historical temperature data for four sites in West Africa: Abidjan, Accra, Daloa, and Lome. LHGu matched FPMETobetter than HG over a wide range ofu2: for Accra, foru2range 0.5–6.0 m/s, the modified coefficient of efficiency,E1, varied narrowly (0.83–0.98) for LHGu but widely (0.14–0.95) for HG optimized foru2=2.0 m/s; the corresponding MBE ranges were −0.05–0.01 mm/d for LHGu and 0.02–0.63 mm/d for HG which cannot respond to varying dailyu2. LHGu is useful for quickly computing practically accurate estimates of FPMETofor varying dailyu2where only temperature data are available.

MODELS OF HOPFIELD-TYPE QUATERNION NEURAL NETWORKS AND THEIR ENERGY FUNCTIONS

2005 ◽  
Vol 15 (01n02) ◽  
pp. 129-135 ◽  
Author(s):  
MITSUO YOSHIDA ◽  
YASUAKI KUROE ◽  
TAKEHIRO MORI
Keyword(s):  
Neural Networks ◽  
Information Processing ◽  
Energy Function ◽  
Point Of View ◽  
Complex Numbers ◽  
Energy Functions ◽  
Fully Connected ◽  
Complex Valued

Recently models of neural networks that can directly deal with complex numbers, complex-valued neural networks, have been proposed and several studies on their abilities of information processing have been done. Furthermore models of neural networks that can deal with quaternion numbers, which is the extension of complex numbers, have also been proposed. However they are all multilayer quaternion neural networks. This paper proposes models of fully connected recurrent quaternion neural networks, Hopfield-type quaternion neural networks. Since quaternion numbers are non-commutative on multiplication, some different models can be considered. We investigate dynamics of these proposed models from the point of view of the existence of an energy function and derive their conditions for existence.

scholarly journals An experimental decomposition of nonlinear forces on a surface-piercing column: Stokes-type expansions of the force harmonics

2018 ◽  
Vol 848 ◽  
pp. 42-77 ◽  
Author(s):  
L. F. Chen ◽  
J. Zang ◽  
P. H. Taylor ◽  
L. Sun ◽  
G. C. J. Morgan ◽  
...  
Keyword(s):  
High Order ◽  
Quality Data ◽  
Wave Group ◽  
Wave Loading ◽  
Harmonic Force ◽  
Wave Basin ◽  
Wide Range ◽  
Time Histories ◽  
Focused Wave ◽  
Sloping Bed

Wave loading on marine structures is the major external force to be considered in the design of such structures. The accurate prediction of the nonlinear high-order components of the wave loading has been an unresolved challenging problem. In this paper, the nonlinear harmonic components of hydrodynamic forces on a bottom-mounted vertical cylinder are investigated experimentally. A large number of experiments were conducted in the Danish Hydraulic Institute shallow water wave basin on the cylinder, both on a flat bed and a sloping bed, as part of a European collaborative research project. High-quality data sets for focused wave groups have been collected for a wide range of wave conditions. The high-order harmonic force components are separated by applying the ‘phase-inversion’ method to the measured force time histories for a crest focused wave group and the same wave group inverted. This separation method is found to work well even for locally violent nearly-breaking waves formed from bidirectional wave pairs. It is also found that the $n$th-harmonic force scales with the $n$th power of the envelope of both the linear undisturbed free-surface elevation and the linear force component in both time variation and amplitude. This allows estimation of the higher-order harmonic shapes and time histories from knowledge of the linear component alone. The experiments also show that the harmonic structure of the wave loading on the cylinder is virtually unaltered by the introduction of a sloping bed, depending only on the local wave properties at the cylinder. Furthermore, our new experimental results reveal that for certain wave cases the linear loading is actually less than 40 % of the total wave loading and the high-order harmonics contribute more than 60 % of the loading. The significance of this striking new result is that it reveals the importance of high-order nonlinear wave loading on offshore structures and means that such loading should be considered in their design.

scholarly journals Generating Bessel beams with broad depth-of-field by using phase-only acoustic holograms

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Sergio Jiménez-Gambín ◽  
Noé Jiménez ◽  
José M. Benlloch ◽  
Francisco Camarena
Keyword(s):  
Field Distribution ◽  
Focal Length ◽  
Bessel Beam ◽  
Transverse Field ◽  
High Order ◽  
Depth Of Field ◽  
Uniform Field ◽  
Axial Distribution ◽  
Bessel Beams ◽  
Wide Range

AbstractWe report zero-th and high-order acoustic Bessel beams with broad depth-of-field generated using acoustic holograms. While the transverse field distribution of Bessel beams generated using traditional passive methods is correctly described by a Bessel function, these methods present a common drawback: the axial distribution of the field is not constant, as required for ideal Bessel beams. In this work, we experimentally, numerically and theoretically report acoustic truncated Bessel beams of flat-intensity along their axis in the ultrasound regime using phase-only holograms. In particular, the beams present a uniform field distribution showing an elongated focal length of about 40 wavelengths, while the transverse width of the beam remains smaller than 0.7 wavelengths. The proposed acoustic holograms were compared with 3D-printed fraxicons, a blazed version of axicons. The performance of both phase-only holograms and fraxicons is studied and we found that both lenses produce Bessel beams in a wide range of frequencies. In addition, high-order Bessel beam were generated. We report first order Bessel beams that show a clear phase dislocation along their axis and a vortex with single topological charge. The proposed method may have potential applications in ultrasonic imaging, biomedical ultrasound and particle manipulation applications using passive lenses.

A Model for the Coupled Lift and Drag on a Circular Cylinder

2003 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Farouk Owis ◽  
Muhammad R. Hajj
Keyword(s):  
Circular Cylinder ◽  
Stokes Equations ◽  
Quadratic Function ◽  
Phase Relations ◽  
Suitable Model ◽  
Van Der Pol ◽  
Spectral Moments ◽  
Drag Coefficients ◽  
Wide Range ◽  
Lift And Drag

The time-varying coupled lift and drag coefficients acting on a circular cylinder are modeled. Data used for the model are obtained by numerically solving the unsteady Reynolds-Averaged Navier Stokes equations over a wide range of Reynolds numbers. Using spectral moments, we determine the frequency components in the lift and drag coefficients and their phase relations. Using a perturbation technique, we obtain approximate solutions of both the van der Pol and Rayleigh equations. By fitting the amplitude and phase relations, we find that the van der Pol equation is the suitable model for the lift. The Rayleigh equation fails to give the correct phase relation. Because the major frequency in the drag component is twice that of the lift, the drag component is modeled as a quadratic function of the lift. Through analysis with higher-order spectral moments, the correct quadratic relation of the lift that yields the drag is determined. The model and results presented here are a first step in the development of a reduced-order model for vortex-induced vibrations, which includes the motions of the cylinder.

A New Method to Detect P-Wave Based on Quadratic Function

2011 ◽  
Vol 267 ◽  
pp. 462-467
Author(s):  
Nan Quan Zhou
Keyword(s):  
Quadratic Function ◽  
Least Square Method ◽  
Detection Algorithm ◽  
Least Square ◽  
P Wave ◽  
Good Effect ◽  
Rate Variation ◽  
Optimal Interval ◽  
Wave Detection ◽  
Wide Range

The paper presents a P-wave detection algorithm based on fitting function in the optimal interval. In the algorithm we used quadratic function to fit the P wave by this means of least square method in every interval, which was shifted in local range. Then we found the optimal fitting interval of P wave by comparing the error of fitting. Finally, we obtained the characteristic points of P wave by using the fitting function to fit P wave in the optimal interval. The performance of the algorithm tested using the records of the MIT-BIH database was effective and accurate. The algorithm on the wide range of heart rate variation and small P wave of ECG P-wave detection has good effect. Also it has some capabilities of anti-interference, particularly the false dismissal probability is quite low.

Finding the Constitutive Relation for a Specific Elastomer

1993 ◽  
Vol 115 (3) ◽  
pp. 329-336 ◽  
Author(s):  
Yun Ling ◽  
Peter A. Engel ◽  
Wm. L. Brodskey ◽  
Yifan Guo
Keyword(s):  
Experimental Data ◽  
Energy Function ◽  
Strain Energy ◽  
Pure Shear ◽  
Strain Energy Function ◽  
Invariant Polynomial ◽  
Energy Functions ◽  
Finite Element Program ◽  
Biaxial Extension ◽  
Strain Energy Functions

The main purpose of this study was to determine a suitable strain energy function for a specific elastomer. A survey of various strain energy functions proposed in the past was made. For natural rubber, there were some specific strain energy functions which could accurately fit the experimental data for various types of deformations. The process of determining a strain energy function for the specific elastomer was then described. The second-order invariant polynomial strain energy function (James et al., 1975) was found to give a good fit to the experimental data of uniaxial tension, uniaxial compression, equi-biaxial extension, and pure shear. A new form of strain energy function was proposed; it yielded improved results. The equi-biaxial extension experiment was done in a novel way in which the moire techniques (Pendleton, 1989) were used. The obtained strain energy functions were then utilized in a finite element program to calculate the load-deflection relation of an electrometric spring used in an electrical connector.

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