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Journal articles on the topic 'Lavrentiev phenomenon'

Author: Grafiati

Published: 9 March 2023

Last updated: 31 July 2025

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1

Loewen, Philip D. "On The Lavrentiev Phenomenon." Canadian Mathematical Bulletin 30, no. 1 (1987): 102–8. http://dx.doi.org/10.4153/cmb-1987-015-7.

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2

Cesari, L., and T. S. Angell. "On the Lavrentiev phenomenon." Calcolo 22, no. 1 (1985): 17–29. http://dx.doi.org/10.1007/bf02576198.

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3

Carstensen, C., and C. Ortner. "Analysis of a Class of Penalty Methods for Computing Singular Minimizers." Computational Methods in Applied Mathematics 10, no. 2 (2010): 137–63. http://dx.doi.org/10.2478/cmam-2010-0008.

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AbstractAmongst the more exciting phenomena in the field of nonlinear partial differential equations is the Lavrentiev phenomenon which occurs in the calculus of variations. We prove that a conforming finite element method fails if and only if the Lavrentiev phenomenon is present. Consequently, nonstandard finite element methods have to be designed for the detection of the Lavrentiev phenomenon in the computational calculus of variations. We formulate and analyze a general strategy for solving variational problems in the presence of the Lavrentiev phenomenon based on a splitting and penalizati

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4

BAI, YU, and ZHI-PING LI. "A TRUNCATION METHOD FOR DETECTING SINGULAR MINIMIZERS INVOLVING THE LAVRENTIEV PHENOMENON." Mathematical Models and Methods in Applied Sciences 16, no. 06 (2006): 847–67. http://dx.doi.org/10.1142/s0218202506001376.

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A numerical method using the truncation technique on the integrand is developed for computing singular minimizers or singular minimizing sequences in variational problems involving the Lavrentiev phenomenon. It is proved that the method can detect absolute minimizers with various singularities whether the Lavrentiev phenomenon is involved or not. It is also proved that, when the absolute infimum is not attainable, the method can produce minimizing sequences. Numerical results on Manià's example and a two-dimensional problem involving the Lavrentiev phenomenon with continuous Sobolev exponent d

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5

Zolezzi, Tullio. "Wellposedness and the Lavrentiev Phenomenon." SIAM Journal on Control and Optimization 30, no. 4 (1992): 787–99. http://dx.doi.org/10.1137/0330043.

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6

BAI, YU, and ZHIPING LI. "NUMERICAL SOLUTION OF NONLINEAR ELASTICITY PROBLEMS WITH LAVRENTIEV PHENOMENON." Mathematical Models and Methods in Applied Sciences 17, no. 10 (2007): 1619–40. http://dx.doi.org/10.1142/s0218202507002406.

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A convergence theory is established for a truncation method in solving polyconvex elasticity problems involving the Lavrentiev phenomenon. Numerical results on a recent example by Foss et al., which has a polyconvex integrand and admits continuous singular minimizers, not only verify our convergence theorems but also provide a sharper estimate on the upper bound of a perturbation parameter for the existence of the Lavrentiev phenomenon in the example.

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7

Foss, M., W. Hrusa, and V. J. Mizel. "The Lavrentiev Phenomenon in Nonlinear Elasticity." Journal of Elasticity 72, no. 1-3 (2003): 173–81. http://dx.doi.org/10.1023/b:elas.0000018778.53392.b7.

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8

Carlson, Dean A. "Property (D) and the Lavrentiev phenomenon." Applicable Analysis 95, no. 6 (2015): 1214–27. http://dx.doi.org/10.1080/00036811.2015.1057703.

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9

Gratwick, Richard. "Singular sets and the Lavrentiev phenomenon." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 145, no. 3 (2015): 513–33. http://dx.doi.org/10.1017/s0308210513001510.

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We show that non-occurrence of the Lavrentiev phenomenon does not imply that the singular set is small. Precisely, given a compact Lebesgue null subsetE⊆ ℝ and an arbitrary superlinearity, there exists a smooth strictly convex Lagrangian with this superlinear growth such that all minimizers of the associated variational problem have singular set exactlyEbut still admit approximation in energy by smooth functions.

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10

Heinricher, Arthur C., and Victor J. Mizel. "A New Example of the Lavrentiev Phenomenon." SIAM Journal on Control and Optimization 26, no. 6 (1988): 1490–503. http://dx.doi.org/10.1137/0326087.

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11

Heinricher, A. C., and V. J. Mizel. "The Lavrentiev phenomenon for invariant variational problems." Archive for Rational Mechanics and Analysis 102, no. 1 (1988): 57–93. http://dx.doi.org/10.1007/bf00250924.

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12

Foss, M., W. J. Hrusa, and V. J. Mizel. "The Lavrentiev Gap Phenomenon in Nonlinear Elasticity." Archive for Rational Mechanics and Analysis 167, no. 4 (2003): 337–65. http://dx.doi.org/10.1007/s00205-003-0249-6.

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13

Buttazzo, Giuseppe, and Victor J. Mizel. "Interpretation of the Lavrentiev phenomenon by relaxation." Journal of Functional Analysis 110, no. 2 (1992): 434–60. http://dx.doi.org/10.1016/0022-1236(92)90038-k.

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14

Braides, A. "The Lavrentiev Phenomenon for Free Discontinuity Problems." Journal of Functional Analysis 127, no. 1 (1995): 1–20. http://dx.doi.org/10.1006/jfan.1995.1001.

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15

Cheng, Chih Wen, and Victor J. Mizel. "On the Lavrentiev phenomenon for autonomous second-order integrands." Archive for Rational Mechanics and Analysis 126, no. 1 (1994): 21–33. http://dx.doi.org/10.1007/bf00375694.

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16

Zaslavski, Alexander J. "Nonoccurrence of the Lavrentiev phenomenon for nonconvex variational problems." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 22, no. 5 (2005): 579–96. http://dx.doi.org/10.1016/j.anihpc.2004.10.004.

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17

Zaslavski, Alexander J. "Nonoccurrence of the Lavrentiev Phenomenon for Many Optimal Control Problems." SIAM Journal on Control and Optimization 45, no. 3 (2006): 1116–46. http://dx.doi.org/10.1137/050640370.

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18

Belloni, Marino. "Interpretation of Lavrentiev phenomenon by relaxation: the higher order case." Transactions of the American Mathematical Society 347, no. 6 (1995): 2011–23. http://dx.doi.org/10.1090/s0002-9947-1995-1290714-9.

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19

Bousquet, Pierre, Carlo Mariconda, and Giulia Treu. "On the Lavrentiev phenomenon for multiple integral scalar variational problems." Journal of Functional Analysis 266, no. 9 (2014): 5921–54. http://dx.doi.org/10.1016/j.jfa.2013.12.020.

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20

Hüsseinov, Farhad. "Relaxation and nonoccurrence of the Lavrentiev phenomenon for nonconvex problems." Acta Mathematica Sinica, English Series 29, no. 6 (2013): 1185–98. http://dx.doi.org/10.1007/s10114-013-1721-3.

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21

Kilpelainen, T., and P. Lindqvist. "The Lavrentiev Phenomenon and the Obstacle Problem for the Dirichlet Integral." Proceedings of the American Mathematical Society 123, no. 8 (1995): 2459. http://dx.doi.org/10.2307/2161275.

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22

Kilpeläinen, T., and P. Lindqvist. "The Lavrentiev phenomenon and the obstacle problem for the Dirichlet integral." Proceedings of the American Mathematical Society 123, no. 8 (1995): 2459. http://dx.doi.org/10.1090/s0002-9939-1995-1264819-8.

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23

Zaslavski, Alexander J. "Nonoccurrence of the Lavrentiev phenomenon for many nonconvex constrained variational problems." Calculus of Variations and Partial Differential Equations 28, no. 3 (2006): 351–81. http://dx.doi.org/10.1007/s00526-006-0047-8.

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24

LI, ZHIPING. "ELEMENT REMOVAL METHOD FOR SINGULAR MINIMIZERS IN PROBLEMS OF HYPERELASTICITY." Mathematical Models and Methods in Applied Sciences 05, no. 03 (1995): 387–99. http://dx.doi.org/10.1142/s0218202595000231.

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A numerical method called element removal method is applied to calculate singular minimizers in problems of hyperelasticity. The method overcomes the difficulty in finite element approximations caused by restrictions, such as det (I+∇u)>0, on admissible functions and can avoid Lavrentiev phenomenon if it does occur in the problem. The convergence of the method is proved.

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25

Cheng, Chih-Wen, and Victor J. Mizel. "On the Lavrentiev Phenomenon for Optimal Control Problems with Second-Order Dynamics." SIAM Journal on Control and Optimization 34, no. 6 (1996): 2172–79. http://dx.doi.org/10.1137/s0363012994261215.

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26

Bonfanti, Giovanni, and Arrigo Cellina. "The Nonoccurrence of the Lavrentiev Phenomenon for a Class of Variational Functionals." SIAM Journal on Control and Optimization 51, no. 2 (2013): 1639–50. http://dx.doi.org/10.1137/12086618x.

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27

Zaslavski, Alexander J. "Generic Nonoccurrence of the Lavrentiev Phenomenon for a Class of Optimal Control Problems." Journal of Dynamical and Control Systems 14, no. 1 (2008): 95–119. http://dx.doi.org/10.1007/s10883-007-9032-6.

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28

Mariconda, Carlo, and Giulia Treu. "Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians." Open Mathematics 18, no. 1 (2020): 1–9. http://dx.doi.org/10.1515/math-2020-0001.

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Abstract We consider the classical functional of the Calculus of Variations of the form $$\begin{array}{} \displaystyle I(u)=\int\limits_{{\it\Omega}}F(x, u(x), \nabla u(x))\,dx, \end{array}$$ where Ω is a bounded open subset of ℝn and F : Ω × ℝ × ℝn → ℝ is a Carathéodory convex function; the admissible functions u coincide with a prescribed Lipschitz function ϕ on ∂Ω. We formulate some conditions under which a given function in ϕ + $\begin{array}{} \displaystyle W^{1,p}_0 \end{array}$(Ω) with I(u) < +∞ can be approximated in the W1,p norm and in energy by a sequence of smooth functions tha

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29

Avci, Mustafa, and Alexander Pankov. "Multivalued elliptic operators with nonstandard growth." Advances in Nonlinear Analysis 7, no. 1 (2018): 35–48. http://dx.doi.org/10.1515/anona-2016-0043.

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AbstractThe paper is devoted to the Dirichlet problem for monotone, in general multivalued, elliptic equations with nonstandard growth condition. The growth conditions are more general than the well-known {p(x)} growth. Moreover, we allow the presence of the so-called Lavrentiev phenomenon. As consequence, at least two types of variational settings of Dirichlet problem are available. We prove results on the existence of solutions in both of these settings. Then we obtain several results on the convergence of certain types of approximate solutions to an exact solution.

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30

Ferriero, Alessandro. "The Approximation of Higher-Order Integrals of the Calculus of Variations and the Lavrentiev Phenomenon." SIAM Journal on Control and Optimization 44, no. 1 (2005): 99–110. http://dx.doi.org/10.1137/s0363012903437721.

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31

Alkhutov, Yu A., and V. V. Zhikov. "Hölder continuity of solutions to a degenerate elliptic equation in the presence of the Lavrentiev phenomenon." Applicable Analysis 93, no. 10 (2014): 2057–75. http://dx.doi.org/10.1080/00036811.2014.936404.

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32

Sarychev, A. V. "First- and second-order integral functionals of the calculus of variations which exhibit the Lavrentiev phenomenon." Journal of Dynamical and Control Systems 3, no. 4 (1997): 565–88. http://dx.doi.org/10.1007/bf02463283.

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33

Mielke, Alexander, and Tomáš Roubíček. "Rate-independent elastoplasticity at finite strains and its numerical approximation." Mathematical Models and Methods in Applied Sciences 26, no. 12 (2016): 2203–36. http://dx.doi.org/10.1142/s0218202516500512.

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Gradient plasticity at large strains with kinematic hardening is analyzed as quasistatic rate-independent evolution. The energy functional with a frame-indifferent polyconvex energy density and the dissipation is approximated numerically by finite elements and implicit time discretization, such that a computationally implementable scheme is obtained. The nonself-penetration as well as a possible frictionless unilateral contact is considered and approximated numerically by a suitable penalization method which keeps polyconvexity and simultaneously bypasses the Lavrentiev phenomenon. The main re

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34

Treu, G., and S. Zagatti. "On the Lavrentiev Phenomenon and the Validity of Euler-Lagrange Equations for a Class of Integral Functionals." Journal of Mathematical Analysis and Applications 184, no. 1 (1994): 56–74. http://dx.doi.org/10.1006/jmaa.1994.1183.

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35

LI, ZHIPING. "NUMERICAL METHODS FOR MINIMIZERS AND MICROSTRUCTURES IN NONLINEAR ELASTICITY." Mathematical Models and Methods in Applied Sciences 06, no. 07 (1996): 957–75. http://dx.doi.org/10.1142/s0218202596000390.

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A standard finite element method and a finite element truncation method are applied to solve the boundary value problems of nonlinear elasticity with certain nonconvex stored energy functions such as those of St. Venant-Kirchhoff materials. Finite element solutions are proved to exist and of the form of minimizers in appropriate sets of admissible finite element functions for both methods. Convergence of the finite element solutions to a solution in the form of a minimizer or microstructure for the boundary value problem is established. It is also shown that in the presence of Lavrentiev pheno

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36

Mazowiecka, Katarzyna, and Paweł Strzelecki. "The Lavrentiev gap phenomenon for harmonic maps into spheres holds on a dense set of zero degree boundary data." Advances in Calculus of Variations 10, no. 3 (2017): 303–14. http://dx.doi.org/10.1515/acv-2015-0040.

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AbstractWe consider the set of smooth zero degree maps {\psi\colon\mathbb{S}^{2}\to\mathbb{S}^{2}} which have the following properties: (i) There is a unique minimizing harmonic map {u\colon\mathbb{B}^{3}\to\mathbb{S}^{2}} which agrees with ψ on the boundary of the unit ball. (ii) The map u has at least N singular points in {\mathbb{B}^{3}}. (iii) The Lavrentiev gap phenomenon holds for ψ, i.e., the infimum of the Dirichlet energies {E(w)} of all smooth extensions {w\colon\mathbb{B}^{3}\to\mathbb{S}^{2}} of ψ is strictly larger than the Dirichlet energy {\int_{\mathbb{B}^{3}}\lvert\nabla u|^{2

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37

De Arcangelis, Riccardo. "The Lavrentieff phenomenon for quadratic functionals." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 125, no. 2 (1995): 329–39. http://dx.doi.org/10.1017/s0308210500028055.

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The paper provides an example of an integral functional in more than two dimensions, with a symmetric and positively defined quadratic integrand q, exhibiting the Lavrentieff phenomenon on a ball B and on a linear boundary datum u0, i.e. for whichThe example is also utilised to discuss nonidentity between some relaxation procedures for a quadratic integral functional and to provide a weighted Sobolev space in the Hilbert case in which smooth functions are not dense.

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38

Antonio Corbo, Esposito, and Riccardo De Arcagelis. "The Lavrentieff phenomenon And Different processes of Homogenization." Communications in Partial Differential Equations 17, no. 9-10 (1992): 1503–38. http://dx.doi.org/10.1080/03605309208820894.

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39

Pastukhova, Svetlana E. "Degenerate equations of monotone type: Lavrent'ev phenomenon and attainability problems." Sbornik: Mathematics 198, no. 10 (2007): 1465–94. http://dx.doi.org/10.1070/sm2007v198n10abeh003892.

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40

Mariconda, Carlo. "Equi-Lipschitz minimizing trajectories for non coercive, discontinuous, non convex Bolza controlled-linear optimal control problems." Transactions of the American Mathematical Society, Series B 8, no. 29 (2021): 899–947. http://dx.doi.org/10.1090/btran/80.

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This article deals with the Lipschitz regularity of the “approximate” minimizers for the Bolza type control functional of the form \[ J t ( y , u ) ≔ ∫ t T Λ ( s , y ( s ) , u ( s ) ) d s + g ( y ( T ) ) J_t(y,u)≔\int _t^T\Lambda (s,y(s), u(s))\,ds+g(y(T)) \] among the pairs ( y , u ) (y,u) satisfying a prescribed initial condition y ( t ) = x y(t)=x , where the state y y is absolutely continuous, the control u u is summable and the dynamic is controlled-linear of the form y ′ = b ( y ) u y’=b(y)u . For b ≡ 1 b\equiv 1 the above becomes a problem of the calculus of variations. The Lagrangian Λ

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41

Esposito, A. Corbo, and T. Durante. "Complete representation of some functionals showing the Lavrentieff phenomenon." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 131, no. 4 (2001): 811–32. http://dx.doi.org/10.1017/s0308210500001128.

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The functional F(u) = ∫Bf(x, Du) is considered, where B is the unit ball in R2, u varies in the set of the locally Lipschitz functions on R2 and f belongs to a family containing, as model case, the following integrand: The computation of the relaxed functional F̄ is provided yielding an explicit representation formula.This formula nevertheless is not integral, because F̄ is not a measure and does not coincide with the obvious extension of F over all W1,p(B).This phenomenon is essentially due to the non-standard growth behaviour of f(x, z) in the variable z.

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42

Dani, Keshav, William J. Hrusa, and Victor J. Mizel. "Lavrentiev's phenomenon for totally unconstrained variational problems in one dimension." Nonlinear Differential Equations and Applications 7, no. 4 (2000): 435–46. http://dx.doi.org/10.1007/pl00001434.

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43

De Arcangelis, Riccardo, Sara Monsurrò, and Elvira Zappale. "On the relaxation and the Lavrentieff phenomenon for variational integrals with pointwise measurable gradient constraints." Calculus of Variations and Partial Differential Equations 21, no. 4 (2004): 357–400. http://dx.doi.org/10.1007/s00526-003-0259-0.

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44

Chlebicka, Iwona, Piotr Gwiazda, and Anna Zatorska–Goldstein. "Parabolic equation in time and space dependent anisotropic Musielak–Orlicz spaces in absence of Lavrentiev's phenomenon." Annales de l'Institut Henri Poincaré C, Analyse non linéaire 36, no. 5 (2019): 1431–65. http://dx.doi.org/10.1016/j.anihpc.2019.01.003.

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45

Chlebicka, Iwona, Piotr Gwiazda, and Anna Zatorska-Goldstein. "Renormalized solutions to parabolic equations in time and space dependent anisotropic Musielak–Orlicz spaces in absence of Lavrentiev's phenomenon." Journal of Differential Equations 267, no. 2 (2019): 1129–66. http://dx.doi.org/10.1016/j.jde.2019.02.005.

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46

Korabel’nikova, E. A. "Anxiety disorders in children with attention deficit/hyperactivity disorder." Russian Journal of Woman and Child Health 3, no. 4 (2020): 302–8. http://dx.doi.org/10.32364/2618-8430-2020-3-4-302-308.

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Aim: to assess the prevalence and presentations of anxiety in preschool children with attention deficit/hyperactivity disorder (ADHD). Patients and Methods: preschoolers (4–6 years) and their parents were enrolled. 30 children with ADHD and their parents were included in the study group and 15 healthy children and their parents were included in the control group. The study was performed in Moscow kindergartens. Clinical anamnestic, experimental psychological (Р. Temple, М. Dorkey, and Е.W. Amen anxiety test adapted by V.M. Astapov; “Fears in Houses” test modified by M.A. Panfilova; questionnai

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47

Karbovska, Karolina. "Personal image: introduction to legal discourse." Law and innovations, no. 4 (36) (December 15, 2021): 117–23. http://dx.doi.org/10.37772/2518-1718-2021-4(36)-17.

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Problem setting. Intensive development of public relations, digitalization and commercialization of new spheres of life, growing popularity of the media sphere necessitates civil protection not only the privacy of individuals who are popular and recognizable, but also their right to publicity, which is directly related to lawful use of intangible benefits that belong to a person and shape his image. Analysis of recent researches and publications. Among the world's most prominent researchers of image as a social phenomenon are R. Burns, I. Hoffman, W. James, J. Rotter, L. Holl, and others. Dome

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48

Parfenova, A. G., and M. V. Safonova. "FEATURES OF THE ACTIVITY OF MODERN ADOLESCENTS AND POSSIBILITY OF ITS DEVELOPMENT WITH THE HELP OF A SUBJECT-SPATIAL COMPONENT OF THE EDUCATIONAL ENVIRONMENT." Bulletin of Krasnoyarsk State Pedagogical University named after V.P. Astafiev 59, no. 1 (2022): 68–84. http://dx.doi.org/10.25146/1995-0861-2022-59-1-318.

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Statement of the problem. The article fixes the relevance of the problem of student activity, since any human manifestations are associated with the phenomenon of activity, emphasizes the importance of studying the nature of activity, the mechanisms of its development and manifestation in order to determine possible ways of its purposeful formation. The purpose of the article is to analyze the results of the study of various manifestations of the activity of the personality of adolescents and to discuss the possibilities of the subject-spatial environment of educational institutions as a means

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49

Burganova, Maria A. "LETTER FROM THE EDITOR." Scientific and analytical journal Burganov House. The space of culture 17, no. 5 (2021): 8–9. http://dx.doi.org/10.36340/2071-6818-2021-17-5-8-9.

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Dear readers, We are pleased to present to you Issue 5, 2021, of the scientific and analytical journal Burganov House. The Space of Culture. Upon the recommendation of the Expert Council of the Higher Attestation Commission, the journal is included in the List of Leading Peer-reviewed Scientific Journals and Publications in which the main scientific results of theses for the academic degrees of doctor and candidate of science must be published. The journal publishes scientific articles by leading specialists in various humanitarian fields, doctoral students, and graduate students. Research are

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50

Cerf, Raphaël, and Carlo Mariconda. "The Lavrentiev Phenomenon." American Mathematical Monthly, October 15, 2024, 1–18. http://dx.doi.org/10.1080/00029890.2024.2397933.

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