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Статті в журналах з теми "BV solutions"

Автор: Grafiati

Опубліковано: 17 серпня 2022

Оновлено: 18 вересня 2022

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1

Krejčí, Pavel, and Vincenzo Recupero. "$\rm BV$ solutions of rate independent differential inclusions." Mathematica Bohemica 139, no. 4 (2014): 607–19. http://dx.doi.org/10.21136/mb.2014.144138.

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2

YOUNG, ROBIN. "NONUNIQUENESS OF BV SOLUTIONS OF QUASILINEAR HYPERBOLIC SYSTEMS." Journal of Hyperbolic Differential Equations 09, no. 04 (December 2012): 555–70. http://dx.doi.org/10.1142/s021989161250018x.

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Анотація:

We construct examples of one-dimensional quasilinear hyperbolic systems for which the constant state and Riemann problem are unstable under O(1) perturbations in the space BV. We generate a system and solution consisting of three compressions which focus at the same point, resulting in a nonlinear interaction from which no waves emerge. Since this solution is time-reversible, this leads to nontrivial solutions of the system with constant initial data. These solutions are classical in the sense that they do not contain shocks and are discontinuous only at the origin. We then find some elementary necessary conditions for a system in conservative form to exhibit this type of behavior.

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3

Ancona, Fabio, Laura Caravenna, and Andrea Marson. "On the structure of BV entropy solutions for hyperbolic systems of balance laws with general flux function." Journal of Hyperbolic Differential Equations 16, no. 02 (June 2019): 333–78. http://dx.doi.org/10.1142/s0219891619500139.

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The paper describes the qualitative structure of BV entropy solutions of a general strictly hyperbolic system of balance laws with characteristic field either piecewise genuinely nonlinear or linearly degenerate. In particular, we provide an accurate description of the local and global wave-front structure of a BV solution generated by a fractional step scheme combined with a wave-front tracking algorithm. This extends the corresponding results in [S. Bianchini and L. Yu, Global structure of admissible BV solutions to piecewise genuinely nonlinear, strictly hyperbolic conservation laws in one space dimension, Comm. Partial Differential Equations 39(2) (2014) 244–273] for strictly hyperbolic system of conservation laws.

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4

NOLAN, B. C. "WEAK SOLUTIONS FOR WEAK SINGULARITIES." International Journal of Modern Physics A 17, no. 20 (August 10, 2002): 2769. http://dx.doi.org/10.1142/s0217751x02011965.

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We revisit the problem of the development of singularities in the gravitational collapse of an inhomogeneous dust sphere. As shown by Yodzis et al1, naked singularities may occur at finite radius where shells of dust cross one another. These singularities are gravitationally weak 2, and it has been claimed that at these singularities, the metric may be written in continuous form 2, with locally L∞ connection coefficients 3. We correct these claims, and show how the field equations may be reformulated as a first order, quasi-linear, non-conservative, non-strictly hyperbolic system. We discuss existence and uniqueness of generalized solutions of this system using bounded functions of bounded variation (BV) 4, where the product of a BV function and the derivative of another BV function may be interpreted as a locally finite measure. The solutions obtained provide a dynamical extension to the future of the singularity.

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5

Bianchini, Stefano. "BV Solutions of the Semidiscrete Upwind Scheme." Archive for Rational Mechanics and Analysis 167, no. 1 (March 2003): 1–81. http://dx.doi.org/10.1007/s00205-002-0237-2.

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6

Knees, Dorothee. "Convergence analysis of time-discretisation schemes for rate-independent systems." ESAIM: Control, Optimisation and Calculus of Variations 25 (2019): 65. http://dx.doi.org/10.1051/cocv/2018048.

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Анотація:

It is well known that rate-independent systems involving nonconvex energy functionals in general do not allow for time-continuous solutions even if the given data are smooth. In the last years, several solution concepts were proposed that include discontinuities in the notion of solution, among them the class of global energetic solutions and the class of BV-solutions. In general, these solution concepts are not equivalent and numerical schemes are needed that reliably approximate that type of solutions one is interested in. In this paper, we analyse the convergence of solutions of three time-discretisation schemes, namely an approach based on local minimisation, a relaxed version of it and an alternate minimisation scheme. For all three cases, we show that under suitable conditions on the discretisation parameters discrete solutions converge to limit functions that belong to the class of BV-solutions. The proofs rely on a reparametrisation argument. We illustrate the different schemes with a toy example.

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7

Anzellotti, G. "BV solutions of quasilinear PDEs in divergence form." Communications in Partial Differential Equations 12, no. 1 (January 1987): 77–122. http://dx.doi.org/10.1080/03605308708820485.

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8

Bressan, Alberto, and Wen Shen. "Small BV Solutions of Hyperbolic Noncooperative Differential Games." SIAM Journal on Control and Optimization 43, no. 1 (January 2004): 194–215. http://dx.doi.org/10.1137/s0363012903425581.

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9

Lerner, Nicolas, and Ferruccio Colombini. "Uniqueness of continuous solutions for BV vector fields." Duke Mathematical Journal 111, no. 2 (February 2002): 357–84. http://dx.doi.org/10.1215/s0012-7094-01-11126-5.

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10

Minh, Mach Nguyet. "BV solutions constructed using the epsilon-neighborhood method." ESAIM: Control, Optimisation and Calculus of Variations 22, no. 1 (January 2016): 188–207. http://dx.doi.org/10.1051/cocv/2015001.

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11

Bugajewski, Dariusz. "On BV-Solutions of Some Nonlinear Integral Equations." Integral Equations and Operator Theory 46, no. 4 (August 2003): 387–98. http://dx.doi.org/10.1007/s00020-001-1146-8.

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12

Bugajewski, Dariusz, and Donal O'Regan. "Existence Results for BV-Solutions of Nonlinear Integral Equations." Journal of Integral Equations and Applications 15, no. 4 (December 2003): 343–57. http://dx.doi.org/10.1216/jiea/1181074981.

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13

Dafermos, Constantine M., and Ronghua Pan. "Global $BV$ Solutions for theP-System with Frictional Damping." SIAM Journal on Mathematical Analysis 41, no. 3 (January 2009): 1190–205. http://dx.doi.org/10.1137/080735126.

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14

Luo, Tao, Roberto Natalini, and Tong Yang. "Global BV Solutions to a p-System with Relaxation." Journal of Differential Equations 162, no. 1 (March 2000): 174–98. http://dx.doi.org/10.1006/jdeq.1999.3697.

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15

Bianchini, Stefano, and Alberto Bressan. "BV solutions for a class of viscous hyperbolic systems." Indiana University Mathematics Journal 49, no. 4 (2000): 0. http://dx.doi.org/10.1512/iumj.2000.49.1776.

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16

Mielke, Alexander, Riccarda Rossi, and Giuseppe Savaré. "BV solutions and viscosity approximations of rate-independent systems." ESAIM: Control, Optimisation and Calculus of Variations 18, no. 1 (December 23, 2010): 36–80. http://dx.doi.org/10.1051/cocv/2010054.

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17

Cao, Wentao. "Global BV entropy solutions to the Gauss–Codazzi system." Journal of Mathematical Analysis and Applications 444, no. 2 (December 2016): 1015–26. http://dx.doi.org/10.1016/j.jmaa.2016.07.006.

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18

Zhan, Huashui, and Shuping Chen. "The BV solution of the parabolic equation with degeneracy on the boundary." Open Mathematics 14, no. 1 (January 1, 2016): 272–82. http://dx.doi.org/10.1515/math-2016-0025.

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Анотація:

AbstractConsider a parabolic equation which is degenerate on the boundary. By the degeneracy, to assure the well-posedness of the solutions, only a partial boundary condition is generally necessary. When 1 ≤ α < p – 1, the existence of the local BV solution is proved. By choosing some kinds of test functions, the stability of the solutions based on a partial boundary condition is established.

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19

Yang, Xinyu, Mengxia Wei, Hao Tian, Tingting Liu, and Lei Yang. "Enrichment and Purification of Aucubin from Eucommia ulmoides Ionic Liquid Extract Using Macroporous Resins." Materials 11, no. 9 (September 18, 2018): 1758. http://dx.doi.org/10.3390/ma11091758.

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Анотація:

Aiming to address the shortcomings of high-concentration ethanol or methanol extraction solutions that need to be diluted and concentrated prior to use in conventional macroporous resin adsorption approaches, an efficient approach for enrichment and purification of aucubin from the ionic liquid extraction solution of samaras of Eucommia ulmoides was proposed. Among the nine kinds of macroporous resins investigated, the HPD850 resin was found to be the most suitable. Equilibrium adsorption tests were investigated and found to be better fitted by the Langmuir isotherm model. After the dynamic tests on a column packed with HPD850, the optimum operational conditions were as follows: for the absorption process, an initial aucubin concentration of 9.87 mg/L, a sample volume of 13 bed volumes (BV), and a flow rate of 2 BV/h; for the water washing process, 5 BV of deionized water and a flow rate of 3 BV/h; for the ethanol desorption process, a 10–80% ethanol volume fraction as the eluent, 2 BV for each ethanol volume fraction, and a flow rate of 3 BV/h. The 40–80% ethanol volume fraction eluent was collected and concentrated to produce the final products, resulting in an aucubin purity and recovery of 79.41% and 72.92%, respectively.

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20

Kano, F., K. Ijichi, N. Ashida, Y. Watanabe, S. Sakata та H. Machida. "5′-O-Alkyl and Acyl Prodrugs of 1-β-D-Arabinofuranosyl-E-5-(2-Bromovinyl)Uracil". Antiviral Chemistry and Chemotherapy 5, № 2 (квітень 1994): 74–82. http://dx.doi.org/10.1177/095632029400500203.

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Анотація:

5′- O-Alkyl derivatives of 1-β-D-arabinofuranosyl- E-5-(2-bromovinyl)uracil (BV-araU) were synthesized by selective alkylation and deprotection of 2′,3′-bis- O-tetrahydropyranyl BV-araU to enhance metabolic stability and evaluated for efficacy as oral prodrugs of BV-araU. For comparison, their acyl congeners, and 3′- O- and 2′- O-ethyl BV-araU, were also prepared by direct acylation of BV-araU and by selective protection, alkylation, and deprotection, respectively. The 5′- O-alkyl prodrugs were stable in acidic solutions, whereas the 5′- O-acyl analogues were unstable under the same conditions. When incubated with enterobacteria, the 5′- O-acyl derivatives resulted in the formation of BV-uracil through non-enzymatic hydrolysis of BV-araU, but the 5′- O-alkyl prodrugs did not. 5′- O-Short-chain aliphatic alkyl (not longer than butyl) and generally acyl prodrugs gave higher blood concentrations of BV-araU than the aromatic derivatives. Plasma concentrations of BV-araU were equal or slightly higher than those after equivalent oral doses of BV-araU. 5′- O-Ethyl BV-araU was effective against intracerebral, intraperitoneal, and cutaneous infections with herpes simplex virus type 1 in mice. 5′- O-Short-chain aliphatic alkyl derivatives may prove to be useful oral prodrugs of BV-araU because of increased metabolic stability.

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21

AOYAMA, S., and S. VANDOREN. "THE BATALIN-VILKOVISKY FORMALISM ON FERMIONIC KÄHLER MANIFOLDS." Modern Physics Letters A 08, no. 39 (December 21, 1993): 3773–84. http://dx.doi.org/10.1142/s0217732393003512.

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We show that the Kähler structure can be naturally incorporated in the Batalin-Vilkovisky formalism. The phase space of the BV formalism becomes a fermionic Kähler manifold. By introducing an isometry we explicitly construct the fermionic irreducible hermitian symmetric space. We then give some solutions of the master equation in the BV formalism.

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22

Yong, Ls, and Haibo Yan. "Several Dynamical Properties for a Nonlinear Shallow Water Equation." Mathematical Problems in Engineering 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/162472.

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Анотація:

A nonlinear third order dispersive shallow water equation including the Degasperis-Procesi model is investigated. The existence of weak solutions for the equation is proved in the spaceL1(R)∩BV (R)under certain assumptions. The Oleinik type estimate andL2N(R) (Nis a natural number) estimate for the solution are obtained.

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23

Mielke, Alexander, Riccarda Rossi, and Giuseppe Savaré. "Balanced Viscosity (BV) solutions to infinite-dimensional rate-independent systems." Journal of the European Mathematical Society 18, no. 9 (2016): 2107–65. http://dx.doi.org/10.4171/jems/639.

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24

Luo, Tao, and Roberto Natalini. "BV solutions and relaxation limit for a model in viscoelasticity." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 128, no. 4 (1998): 775–95. http://dx.doi.org/10.1017/s0308210500021776.

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Анотація:

Global existence of uniformly bounded BV entropy solutions to a 2 × 2 quasilinear system with relaxation, arising in viscoelasticity, is established by using special fractional step versions of Godunov and Glimm schemes for arbitrarily large initial data. Thanks to the uniform estimates obtained, we prove the convergence of solutions to the corresponding equilibrium limit as the relaxation parameter tends to zero.

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25

Kong, Linghai, Zhongdan Huan, and Boling Guo. "BV solutions to a degenerate parabolic equation for image denoising." Acta Mathematica Scientia 27, no. 1 (January 2007): 169–79. http://dx.doi.org/10.1016/s0252-9602(07)60015-0.

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26

Samsonyuk, Olga N., and Sergey A. Timoshin. "BV solutions of rate independent processes driven by impulsive controls." IFAC-PapersOnLine 51, no. 32 (2018): 361–66. http://dx.doi.org/10.1016/j.ifacol.2018.11.410.

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27

Christoforou, Cleopatra. "BV weak solutions to Gauss–Codazzi system for isometric immersions." Journal of Differential Equations 252, no. 3 (February 2012): 2845–63. http://dx.doi.org/10.1016/j.jde.2011.08.046.

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28

Dafermos, Constantine M. "BV solutions for hyperbolic systems of balance laws with relaxation." Journal of Differential Equations 255, no. 8 (October 2013): 2521–33. http://dx.doi.org/10.1016/j.jde.2013.07.002.

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29

Dafermos, Constantine M. "Periodic BV solutions of hyperbolic balance laws with dissipative source." Journal of Mathematical Analysis and Applications 428, no. 1 (August 2015): 405–13. http://dx.doi.org/10.1016/j.jmaa.2015.03.026.

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30

Edmond, Jean Fenel, and Lionel Thibault. "BV solutions of nonconvex sweeping process differential inclusion with perturbation." Journal of Differential Equations 226, no. 1 (July 2006): 135–79. http://dx.doi.org/10.1016/j.jde.2005.12.005.

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31

Dafermos, C. M., and X. Geng. "Generalised characteristics in hyperbolic systems of conservation laws with special coupling." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 116, no. 3-4 (1990): 245–78. http://dx.doi.org/10.1017/s0308210500031504.

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Анотація:

SynopsisUsing the theory of generalised characteristics, we study the structure of BV solutions of genuinely nonlinear systems of two conservation laws whose shock and rarefaction wave curves of the first family are straight lines. We also establish a priori estimates on the variation of the solution similar to those obtained earlier by Glimm and Lax.

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32

Yong, Wen-An. "A difference scheme for a stiff system of conservation laws." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 128, no. 6 (1998): 1403–14. http://dx.doi.org/10.1017/s0308210500027396.

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Анотація:

A stiff system of conservation laws is analysed using a difference method. The existence of entropy-satisfying BV-solutions to the initial value problems is established. Furthermore, we show that the solutions converge to the solutions of the corresponding equilibrium system as the relaxation time tends to zero.

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33

Dafermos, Constantine M. "Asymptotic behavior of BV solutions to the equations of nonlinear viscoelasticity." Communications in Information and Systems 13, no. 2 (2013): 201–9. http://dx.doi.org/10.4310/cis.2013.v13.n2.a4.

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34

Evje, Steinar, and Kenneth Hvistendahl Karlsen. "Monotone Difference Approximations Of BV Solutions To Degenerate Convection-Diffusion Equations." SIAM Journal on Numerical Analysis 37, no. 6 (January 2000): 1838–60. http://dx.doi.org/10.1137/s0036142998336138.

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35

Karimie, M. T. "Synthetic light curve solutions for the W UMa-system BV draconis." Astrophysics and Space Science 115, no. 2 (1985): 251–55. http://dx.doi.org/10.1007/bf00653803.

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36

Evje, Steinar, and Kenneth Hvistendahl Karlsen. "Discrete approximations of BV solutions to doubly nonlinear degenerate parabolic equations." Numerische Mathematik 86, no. 3 (September 1, 2000): 377–417. http://dx.doi.org/10.1007/s002110000156.

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37

Bourdarias, C., M. Gisclon, and S. Junca. "Fractional BV spaces and applications to scalar conservation laws." Journal of Hyperbolic Differential Equations 11, no. 04 (December 2014): 655–77. http://dx.doi.org/10.1142/s0219891614500209.

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Анотація:

We obtain new fine properties of entropy solutions to scalar nonlinear conservation laws. For this purpose, we study the "fractional BV spaces" denoted by BVs(ℝ) (for 0 < s ≤ 1), which were introduced by Love and Young in 1937 and closely related to the critical Sobolev space Ws,1/s(ℝ). We investigate these spaces in connection with one-dimensional scalar conservation laws. The BVs spaces allow one to work with less regular functions than BV functions and appear to be more natural in this context. We obtain a stability result for entropy solutions with BVs initial data. Furthermore, for the first time, we get the maximal Ws,p smoothing effect conjectured by Lions, Perthame and Tadmor for all nonlinear (possibly degenerate) convex fluxes.

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38

Vasin, Vladimir, and Vladimir Belyaev. "THE MODIFIED TIKHONOV REGULARIZATION METHOD WITH THE SMOOTHED TOTAL VARIATION FOR SOLVING THE LINEAR ILL-POSED PROBLEMS." Eurasian Journal of Mathematical and Computer Applications 9, no. 2 (June 2021): 88–100. http://dx.doi.org/10.32523/2306-6172-2021-9-2-88-100.

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Анотація:

We investigate a linear operator equation of the first kind that is ill-posed in the Hadamard sence. It is assumed that its solution is representable as a sum of smooth and discontinuous components. To construct a stable approximate solutions, we use the modified Tikhonov method with the stabilizing functional as a sum of the Lebesgue norm for the smooth component and a smoothed BV-norm for the discontinuous component. Theorems of exis- tence, uniqueness, and convergence both the regularized solutions and its finite-dimentional approximations are proved. Also, results of numerical experiments are presented.

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39

Li, Jinghua, Tao Ke, Chao He, Wei Cao, Mengqi Wei, Lei Zhang, Jin-Xia Zhang, et al. "The Anti-Arthritic Effects of Synthetic Melittin on the Complete Freund's Adjuvant-Induced Rheumatoid Arthritis Model in Rats." American Journal of Chinese Medicine 38, no. 06 (January 2010): 1039–49. http://dx.doi.org/10.1142/s0192415x10008457.

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Bee venom (BV) has been used for millennia in Chinese traditional medicine to treat rheumatoid arthritis (RA). However, its components and mechanism remain unclear, which has hampered its development and application for the treatment of RA. In this study, we examined the anti-arthritis effects of melittin, which composes nearly 50% of the dry weight of whole BV, on the complete Freund's adjuvant-induced (CFA-induced) RA model in rats. The RA animal models were treated with solutions of BV, melittin, and saline by injection into a specific acupoint (Zusanli). The BV and melittin treatments statistically diminished the thickness of the arthroses in the injected side of the paw, compared to the saline treatment. Melittin therapy also significantly reduced arthritis-induced nociceptive behaviors, as assessed by the thermal hyperalgesia test. In addition, CFA-induced Fos expression in the superficial layer of the lumbar spinal cord was significantly suppressed by the BV and melittin treatments, compared to the saline treatment. These results indicate that melittin is an effective anti-arthritis component of whole bee venom, making it a promising candidate as an anti-arthritis drug.

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40

Christoforou, Cleopatra. "A remark on the Glimm scheme for inhomogeneous hyperbolic systems of balance laws." Journal of Hyperbolic Differential Equations 12, no. 04 (December 2015): 787–97. http://dx.doi.org/10.1142/s021989161550023x.

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Анотація:

General hyperbolic systems of balance laws with inhomogeneous flux and source are studied. Global existence of entropy weak solutions to the Cauchy problem is established for small BV data under appropriate assumptions on the decay of the flux and the source with respect to space and time. There is neither a hypothesis about equilibrium solution nor about the dependence of the source on the state vector as previous results have assumed.

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41

ROBERTS, JOSEPH. "STEADY AND SELF-SIMILAR FULL EULER FLOW." Journal of Hyperbolic Differential Equations 10, no. 04 (December 2013): 799–840. http://dx.doi.org/10.1142/s021989161350029x.

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Анотація:

We consider solutions to the full (non-isentropic) two-dimensional Euler equations that are constant in time and along rays emanating from the origin. We prove that for a polytropic equation of state, entropy admissible solutions in L∞ with non-vanishing velocity, density, and internal energy must be BV. Moreover, we obtain some results concerning the structure of such solutions.

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42

YAN, HAO, and WEN-AN YONG. "WEAK ENTROPY SOLUTIONS OF NONLINEAR REACTION–HYPERBOLIC SYSTEMS for AXONAL TRANSPORT." Mathematical Models and Methods in Applied Sciences 21, no. 10 (October 2011): 2135–54. http://dx.doi.org/10.1142/s0218202511005684.

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Анотація:

This paper is concerned with a class of nonlinear reaction–hyperbolic systems as models for axonal transport in neuroscience. We show the global existence of entropy-satisfying BV-solutions to the initial-value problems by using hyperbolic-type methods. Moreover, we rigorously justify the limit as the biochemical processes are much faster than the transport ones.

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43

Knees, Dorothee, and Chiara Zanini. "Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads." Discrete & Continuous Dynamical Systems - S 14, no. 1 (2021): 121–49. http://dx.doi.org/10.3934/dcdss.2020332.

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44

Amadori, Debora, and Graziano Guerra. "Global BV solutions and relaxation limit for a system of conservation laws." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 131, no. 1 (February 2001): 1–26. http://dx.doi.org/10.1017/s0308210500000767.

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Анотація:

We consider the Cauchy problem for the (strictly hyperbolic, genuinely nonlinear) system of conservation laws with relaxation Assume there exists an equilibrium curve A(u), such that r(u,A(u)) = 0. Under some assumptions on σ and r, we prove the existence of global (in time) solutions of bounded variation, uε, υε, for ε > 0 fixed.As ε → 0, we prove the convergence of a subsequence of uε, υε to some u, υ that satisfy the equilibrium equations

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45

Amadori, Debora, and Wen Shen. "Global Existence of Large BV Solutions in a Model of Granular Flow." Communications in Partial Differential Equations 34, no. 9 (August 24, 2009): 1003–40. http://dx.doi.org/10.1080/03605300902892279.

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46

Hsiao, Ling, Tao Luo, and Tong Yang. "Global BV Solutions of Compressible Euler Equations with Spherical Symmetry and Damping." Journal of Differential Equations 146, no. 1 (June 1998): 203–25. http://dx.doi.org/10.1006/jdeq.1998.3427.

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47

Zhang, Linan, and Hayden Schaeffer. "Stability and error estimates of BV solutions to the Abel inverse problem." Inverse Problems 34, no. 10 (August 1, 2018): 105003. http://dx.doi.org/10.1088/1361-6420/aad1c7.

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48

Jingxue, Yin. "On The Uniqueness and Stability of BV Solutions for Nonlinear Diffusion Equations." Communications in Partial Differential Equations 15, no. 12 (January 1990): 54–67. http://dx.doi.org/10.1080/03605309908820743.

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49

Amadori, Debora, Laurent Gosse, and Graziano Guerra. "Global BV Entropy Solutions and Uniqueness for Hyperbolic Systems of Balance Laws." Archive for Rational Mechanics and Analysis 162, no. 4 (May 1, 2002): 327–66. http://dx.doi.org/10.1007/s002050200198.

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50

Balz, R., U. Brändle, E. Kämmerer, D. Köhnlein, O. Lutz, and A. Nolle. "11B and 10B NMR Investigations in Aqueous Solutions." Zeitschrift für Naturforschung A 41, no. 5 (May 1, 1986): 737–42. http://dx.doi.org/10.1515/zna-1986-0508.

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Анотація:

11B NMR chemical shifts and linewidths have been measured in very dilute aqueous solutions of boric acid and borates. The results can be explained bv taking pH dependent weighted averages over the species B(OH)3 and B(OH)4−. The11B−10B primary isotope effect on the magnetic shielding is smaller then 3 · 10-8. The H2O - D2O solvent isotope effect on T1 has been established for 11B and 10B in the species mentioned, and from the ratios of T1 the quadrupolar origin of the relaxation mechanism has been inferred.

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