Liouville integrability of the reduction of the associativity equations on the set of stationary points of an integral in the case of three primary fields

2019 ◽  
Vol 74 (2) ◽  
pp. 369-371
Author(s):  
O. I. Mokhov ◽  
N. A. Strizhova
Keyword(s):  
Stationary Points ◽  
Liouville Integrability ◽  
Associativity Equations

scholarly journals LIOUVILLE INTEGRABLE REDUCTIONS OF THE ASSOCIATIVITY EQUATIONS ON THE SET OF STATIONARY POINTS OF AN INTEGRAL IN THE CASE OF THREE PRIMARY FIELDS

2019 ◽  
Vol 47 (1) ◽  
pp. 88-90
Author(s):  
O.I. Mokhov ◽  
N.A. Strizhova
Keyword(s):  
Stationary Points ◽  
Evolutionary System ◽  
Liouville Integrability ◽  
Russian Science ◽  
Partial Derivatives ◽  
First Order ◽  
System Of Hydrodynamic Type ◽  
Associativity Equations ◽  
Liouville Integrable ◽  
Hamiltonian Property

In this work, in the case of three primary fields, a reduction of the associativity equations (the Witten–Dijkgraaf–Verlinde–Verlinde system, see (Witten, 1990, Dijkgraaf et al., 1991, Dubrovin, 1994) with antidiagonal matrix ηij on the set of stationary points of a nondegenerate integral quadratic with respect to the first-order partial derivatives is constructed in an explicit form and its Liouville integrability is proved. In Mokhov’s paper (Mokhov, 1995, Mokhov, 1998), these associativity equations were presented in the form of an integrable nondiagonalizable system of hydrodynamic type. In the papers (Ferapontov, Mokhov, 1996, Ferapontov et al., 1997, Mokhov, 1998), a bi-Hamiltonian representation for these equations and a nondegenerate integral quadratic with respect to the first-order partial derivatives were found. Using Mokhov’s construction on canonical Hamiltonian property of an arbitrary evolutionary system on the set of stationary points of its nondegenerate integral of the papers (Mokhov, 1984, Mokhov, 1987), we construct explicitly the reduction for the integral quadratic with respect to the first-order partial derivatives, found explicitly the Hamiltonian of the corresponding canonical Hamiltonian system. We also found three functionally-independent integrals in involution with respect to the canonical Poisson bracket on the phase space for the constructed reduction of the associativity equations and thus proved the Liouville integrability of this reduction. This work is supported by the Russian Science Foundation under grant No. 18-11-00316.

scholarly journals ON THE HAMILTONIAN REDUCTION OF THE ASSOCIATIVITY EQUATIONS IN THE CASE OF FOUR PRIMARY FIELDS

2019 ◽  
Vol 47 (1) ◽  
pp. 118-122
Author(s):  
N.A. Strizhova
Keyword(s):  
Quantum Cohomology ◽  
Mathematical Physics ◽  
Fundamental Principle ◽  
Stationary Points ◽  
Hamiltonian Reduction ◽  
Enumerative Geometry ◽  
Associativity Equations ◽  
Topological Field ◽  
Hamiltonian Property ◽  
Arbitrary Flow

The associativity equations arose in the papers of Witten (Witten, 1990) and Dijkgraaf, Verlinde, (Dijkgraaf et al., 1991) on two-dimensional topological field theories and subsequently they became to play a key role in many other important domains of mathematics and mathematical physics: in quantum cohomology, Gromov–Witten invariants, enumerative geometry, theory of submanifolds and so on. In Mokhov’s papers (Mokhov, 1984), (Mokhov, 1987) a general fundamental principle stating a canonical Hamiltonian property for the restriction of an arbitrary flow on the set of stationary points of its nondegenerate integral was proposed and proved. In this paper the Hamiltonians of the reductions of the associativity equations with antidiagonal matrix ηij in the case of four primary fields according to Mokhov`s construction is found in an explicit form.

scholarly journals Enhancement of the transverse thermoelectric conductivity originating from stationary points in nodal lines

2020 ◽  
Vol 102 (20) ◽  
Author(s):  
Susumu Minami ◽  
Fumiyuki Ishii ◽  
Motoaki Hirayama ◽  
Takuya Nomoto ◽  
Takashi Koretsune ◽  
...  
Keyword(s):  
Stationary Points ◽  
Nodal Lines

scholarly journals Direct Kinetic Measurements and Master Equation Modelling of the Unimolecular Decomposition of Resonantly-Stabilized CH2CHCHC(O)OCH3 Radical and an Upper Limit Determination for CH2CHCHC(O)OCH3 + O2 Reaction

2020 ◽  
Vol 234 (7-9) ◽  
pp. 1251-1268 ◽  
Author(s):  
Satya Prakash Joshi ◽  
Prasenjit Seal ◽  
Timo Theodor Pekkanen ◽  
Raimo Sakari Timonen ◽  
Arrke J. Eskola
Keyword(s):  
Master Equation ◽  
Density Functional ◽  
Rate Coefficient ◽  
Decomposition Kinetics ◽  
Basis Sets ◽  
Stationary Points ◽  
Unimolecular Decomposition ◽  
Kinetic Measurements ◽  
Point Energy ◽  
Upper Limit

AbstractMethyl-Crotonate (MC, (E)-methylbut-2-enoate, CH3CHCHC(O)OCH3) is a potential component of surrogate fuels that aim to emulate the combustion of fatty acid methyl ester (FAME) biodiesels with significant unsaturated FAME content. MC has three allylic hydrogens that can be readily abstracted under autoignition and combustion conditions to form a resonantly-stabilized CH2CHCHC(O)OCH3 radical. In this study we have utilized photoionization mass spectrometry to investigate the O2 addition kinetics and thermal unimolecular decomposition of CH2CHCHC(O)OCH3 radical. First we determined an upper limit for the bimolecular rate coefficient of CH2CHCHC(O)OCH3 + O2 reaction at 600 K (k ≤ 7.5 × 10−17 cm3 molecule−1 s−1). Such a small rate coefficient suggest this reaction is unlikely to be important under combustion conditions and subsequent efforts were directed towards measuring thermal unimolecular decomposition kinetics of CH2CHCHC(O)OCH3 radical. These measurements were performed between 750 and 869 K temperatures at low pressures (<9 Torr) using both helium and nitrogen bath gases. The potential energy surface of the unimolecular decomposition reaction was probed at density functional (MN15/cc-pVTZ) level of theory and the electronic energies of the stationary points obtained were then refined using the DLPNO-CCSD(T) method with the cc-pVTZ and cc-pVQZ basis sets. Master equation simulations were subsequently carried out using MESMER code along the kinetically important reaction pathway. The master equation model was first optimized by fitting the zero-point energy corrected reaction barriers and the collisional energy transfer parameters $\Delta{E_{{\text{down}},\;{\text{ref}}}}$ and n to the measured rate coefficients data and then utilize the constrained model to extrapolate the decomposition kinetics to higher pressures and temperatures. Both the experimental results and the MESMER simulations show that the current experiments for the thermal unimolecular decomposition of CH2CHCHC(O)OCH3 radical are in the fall-off region. The experiments did not provide definite evidence about the primary decomposition products.

scholarly journals Topological Approach to Mathematical Programs with Switching Constraints

2021 ◽  
Author(s):  
Vladimir Shikhman
Keyword(s):  
Stationary Point ◽  
Mountain Pass Theorem ◽  
Strong Stability ◽  
Complete Characterization ◽  
Stationary Points ◽  
Mathematical Programs ◽  
Linear Independence Constraint Qualification ◽  
Point Set ◽  
Stationary Level

AbstractWe study mathematical programs with switching constraints (for short, MPSC) from the topological perspective. Two basic theorems from Morse theory are proved. Outside the W-stationary point set, continuous deformation of lower level sets can be performed. However, when passing a W-stationary level, the topology of the lower level set changes via the attachment of a w-dimensional cell. The dimension w equals the W-index of the nondegenerate W-stationary point. The W-index depends on both the number of negative eigenvalues of the restricted Lagrangian’s Hessian and the number of bi-active switching constraints. As a consequence, we show the mountain pass theorem for MPSC. Additionally, we address the question if the assumption on the nondegeneracy of W-stationary points is too restrictive in the context of MPSC. It turns out that all W-stationary points are generically nondegenerate. Besides, we examine the gap between nondegeneracy and strong stability of W-stationary points. A complete characterization of strong stability for W-stationary points by means of first and second order information of the MPSC defining functions under linear independence constraint qualification is provided. In particular, no bi-active Lagrange multipliers of a strongly stable W-stationary point can vanish.

scholarly journals Stability of stationary points for one-dimensional Willmore energy with spatially heterogeneous term

2020 ◽  
pp. 132812
Author(s):  
Masaaki Uesaka ◽  
Ken-Ichi Nakamura ◽  
Keiichi Ueda ◽  
Masaharu Nagayama
Keyword(s):  
Stationary Points ◽  
One Dimensional ◽  
Willmore Energy

scholarly journals A study of Babylonian planetary theory III. The planet Mercury

2021 ◽  
Author(s):  
Teije de Jong
Keyword(s):  
Direct Method ◽  
Step Function ◽  
Computational Effort ◽  
Stationary Points ◽  
Planetary Motion ◽  
Time Intervals ◽  
Outer Planets ◽  
Step Functions ◽  
System A ◽  
Computational Systems

AbstractIn this series of papers I attempt to provide an answer to the question how the Babylonian scholars arrived at their mathematical theory of planetary motion. Papers I and II were devoted to system A theory of the outer planets and of the planet Venus. In this third and last paper I will study system A theory of the planet Mercury. Our knowledge of the Babylonian theory of Mercury is at present based on twelve Ephemerides and seven Procedure Texts. Three computational systems of Mercury are known, all of system A. System A1 is represented by nine Ephemerides covering the years 190 BC to 100 BC and system A2 by two Ephemerides covering the years 310 to 290 BC. System A3 is known from a Procedure Text and from Text M, an Ephemeris of the last evening visibility of Mercury for the years 424 to 403 BC. From an analysis of the Babylonian observations of Mercury preserved in the Astronomical Diaries and Planetary Texts we find: (1) that dates on which Mercury reaches its stationary points are not recorded, (2) that Normal Star observations on or near dates of first and last appearance of Mercury are rare (about once every twenty observations), and (3) that about one out of every seven pairs of first and last appearances is recorded as “omitted” when Mercury remains invisible due to a combination of the low inclination of its orbit to the horizon and the attenuation by atmospheric extinction. To be able to study the way in which the Babylonian scholars constructed their system A models of Mercury from the available observational material I have created a database of synthetic observations by computing the dates and zodiacal longitudes of all first and last appearances and of all stationary points of Mercury in Babylon between 450 and 50 BC. Of the data required for the construction of an ephemeris synodic time intervals Δt can be directly derived from observed dates but zodiacal longitudes and synodic arcs Δλ must be determined in some other way. Because for Mercury positions with respect to Normal Stars can only rarely be determined at its first or last appearance I propose that the Babylonian scholars used the relation Δλ = Δt −3;39,40, which follows from the period relations, to compute synodic arcs of Mercury from the observed synodic time intervals. An additional difficulty in the construction of System A step functions is that most amplitudes are larger than the associated zone lengths so that in the computation of the longitudes of the synodic phases of Mercury quite often two zone boundaries are crossed. This complication makes it difficult to understand how the Babylonian scholars managed to construct System A models for Mercury that fitted the observations so well because it requires an excessive amount of computational effort to find the best possible step function in a complicated trial and error fitting process with four or five free parameters. To circumvent this difficulty I propose that the Babylonian scholars used an alternative more direct method to fit System A-type models to the observational data of Mercury. This alternative method is based on the fact that after three synodic intervals Mercury returns to a position in the sky which is on average only 17.4° less in longitude. Using reduced amplitudes of about 14°–25° but keeping the same zone boundaries, the computation of what I will call 3-synarc system A models of Mercury is significantly simplified. A full ephemeris of a synodic phase of Mercury can then be composed by combining three columns of longitudes computed with 3-synarc step functions, each column starting with a longitude of Mercury one synodic event apart. Confirmation that this method was indeed used by the Babylonian astronomers comes from Text M (BM 36551+), a very early ephemeris of the last appearances in the evening of Mercury from 424 to 403 BC, computed in three columns according to System A3. Based on an analysis of Text M I suggest that around 400 BC the initial approach in system A modelling of Mercury may have been directed towards choosing “nice” sexagesimal numbers for the amplitudes of the system A step functions while in the later final models, dating from around 300 BC onwards, more emphasis was put on selecting numerical values for the amplitudes such that they were related by simple ratios. The fact that different ephemeris periods were used for each of the four synodic phases of Mercury in the later models may be related to the selection of a best fitting set of System A step function amplitudes for each synodic phase.

scholarly journals Identification of mechanical properties of arteries with certification of global optimality

2021 ◽  
Author(s):  
Jan-Lucas Gade ◽  
Carl-Johan Thore ◽  
Jonas Stålhand
Keyword(s):  
Global Solution ◽  
Branch And Bound ◽  
Clinical Data ◽  
Minimization Problem ◽  
Branch And Bound Algorithm ◽  
Stationary Points ◽  
Model Parameters ◽  
Global Optimality ◽  
Clinical Value ◽  
Non Linear

AbstractIn this study, we consider identification of parameters in a non-linear continuum-mechanical model of arteries by fitting the models response to clinical data. The fitting of the model is formulated as a constrained non-linear, non-convex least-squares minimization problem. The model parameters are directly related to the underlying physiology of arteries, and correctly identified they can be of great clinical value. The non-convexity of the minimization problem implies that incorrect parameter values, corresponding to local minima or stationary points may be found, however. Therefore, we investigate the feasibility of using a branch-and-bound algorithm to identify the parameters to global optimality. The algorithm is tested on three clinical data sets, in each case using four increasingly larger regions around a candidate global solution in the parameter space. In all cases, the candidate global solution is found already in the initialization phase when solving the original non-convex minimization problem from multiple starting points, and the remaining time is spent on increasing the lower bound on the optimal value. Although the branch-and-bound algorithm is parallelized, the overall procedure is in general very time-consuming.

On Interior Logarithmic Smoothing and Strongly Stable Stationary Points

2010 ◽  
Vol 20 (5) ◽  
pp. 2137-2156
Author(s):  
Hubertus Th. Jongen ◽  
Jan-J. Rückmann
Keyword(s):  
Stationary Points
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