Увійти

Готові списки джерел за темами / Bounded measurable coefficients / Статті в журналах

Статті в журналах з теми "Bounded measurable coefficients"

Щоб переглянути інші типи публікацій з цієї теми, перейдіть за посиланням: Bounded measurable coefficients.

Автор: Grafiati

Опубліковано: 10 грудня 2022

Оновлено: 28 січня 2023

Оформте джерело за APA, MLA, Chicago, Harvard та іншими стилями

Оберіть тип джерела:

Ознайомтеся з топ-50 статей у журналах для дослідження на тему "Bounded measurable coefficients".

Біля кожної праці в переліку літератури доступна кнопка «Додати до бібліографії». Скористайтеся нею – і ми автоматично оформимо бібліографічне посилання на обрану працю в потрібному вам стилі цитування: APA, MLA, «Гарвард», «Чикаго», «Ванкувер» тощо.

Також ви можете завантажити повний текст наукової публікації у форматі «.pdf» та прочитати онлайн анотацію до роботи, якщо відповідні параметри наявні в метаданих.

Переглядайте статті в журналах для різних дисциплін та оформлюйте правильно вашу бібліографію.

1

Jensen, Robert R. "Uniformly Elliptic PDEs with Bounded, Measurable Coefficients." Journal of Fourier Analysis and Applications 2, no. 3 (June 1995): 237–59. http://dx.doi.org/10.1007/s00041-001-4031-6.

Повний текст джерела

Стилі APA, Harvard, Vancouver, ISO та ін.

2

Andrew, Paul. "On Singular Integral Operators with Bounded Measurable Coefficients." Mathematische Nachrichten 165, no. 1 (1994): 183–89. http://dx.doi.org/10.1002/mana.19941650112.

Повний текст джерела

Стилі APA, Harvard, Vancouver, ISO та ін.

3

Anceschi, Francesca, Michela Eleuteri, and Sergio Polidoro. "A geometric statement of the Harnack inequality for a degenerate Kolmogorov equation with rough coefficients." Communications in Contemporary Mathematics 21, no. 07 (October 10, 2019): 1850057. http://dx.doi.org/10.1142/s0219199718500578.

Повний текст джерела

Анотація:

We consider weak solutions of second-order partial differential equations of Kolmogorov–Fokker–Planck-type with measurable coefficients in the form [Formula: see text] where [Formula: see text] is a symmetric uniformly positive definite matrix with bounded measurable coefficients; [Formula: see text] and the components of the vector [Formula: see text] are bounded and measurable functions. We give a geometric statement of the Harnack inequality recently proved by Golse et al. As a corollary, we obtain a strong maximum principle.

Стилі APA, Harvard, Vancouver, ISO та ін.

4

Dong, Hongjie, and Doyoon Kim. "Lq-Estimates for stationary Stokes system with coefficients measurable in one direction." Bulletin of Mathematical Sciences 09, no. 01 (April 2019): 1950004. http://dx.doi.org/10.1142/s1664360719500048.

Повний текст джерела

Анотація:

We study the stationary Stokes system with variable coefficients in the whole space, a half space, and on bounded Lipschitz domains. In the whole and half spaces, we obtain a priori [Formula: see text]-estimates for any [Formula: see text] when the coefficients are merely measurable functions in one fixed direction. For the system on bounded Lipschitz domains with a small Lipschitz constant, we obtain a [Formula: see text]-estimate and prove the solvability for any [Formula: see text] when the coefficients are merely measurable functions in one direction and have locally small mean oscillations in the orthogonal directions in each small ball, where the direction is allowed to depend on the ball.

Стилі APA, Harvard, Vancouver, ISO та ін.

5

Lyons, T. J., and W. A. Zheng. "Diffusion processes with non-smooth diffusion coefficients and their density functions." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 115, no. 3-4 (1990): 231–42. http://dx.doi.org/10.1017/s0308210500020618.

Повний текст джерела

Анотація:

SynopsisDenote by Xt an n-dimensional symmetric Markov process associated with an elliptic operatorwhere (aij) is a bounded measurable uniformly positive definite matrix-valued function of x. Let f(x, t) be a measurable function defined on Rn × [0, 1]. In this paper, we prove that f(Xt, t) is a regular Dirichlet process if and only if the following two conditions are satisfied:(i) For almost every and (ii) Let be a sequence of subdivisions of [0,1] so thatThenAs an application of the above result, we prove the following fact: Let p(y, t) be the probability density of the diffusion process Yt, associated with the elliptic operatorwhere (bi) are bounded measurable functions of x and we suppose that . Then, p(Yt, t) is a regular Dirichlet process and therefore p(.,.) satisfies (i) and (ii).

Стилі APA, Harvard, Vancouver, ISO та ін.

6

Yan, Lixin. "A remark on Littlewood–Paley g-function." Bulletin of the Australian Mathematical Society 66, no. 1 (August 2002): 33–41. http://dx.doi.org/10.1017/s0004972700020657.

Повний текст джерела

Стилі APA, Harvard, Vancouver, ISO та ін.

7

Chen, Yanping, Qingquan Deng та Yong Ding. "Commutators with fractional differentiation for second-order elliptic operators on ℝn". Communications in Contemporary Mathematics 22, № 02 (28 лютого 2019): 1950010. http://dx.doi.org/10.1142/s021919971950010x.

Повний текст джерела

Анотація:

Let [Formula: see text] be a second-order divergence form elliptic operator and [Formula: see text] an accretive, [Formula: see text] matrix with bounded measurable complex coefficients in [Formula: see text] In this paper, we establish [Formula: see text] theory for the commutators generated by the fractional differential operators related to [Formula: see text] and bounded mean oscillation (BMO)–Sobolev functions.

Стилі APA, Harvard, Vancouver, ISO та ін.

8

Shargorodsky, Eugene. "A Remark on the Essential Spectra of Toeplitz Operators with Bounded Measurable Coefficients." Integral Equations and Operator Theory 57, no. 1 (December 22, 2006): 127–32. http://dx.doi.org/10.1007/s00020-006-1470-0.

Повний текст джерела

Стилі APA, Harvard, Vancouver, ISO та ін.

9

Duong, Xuan Thinh, and Alan McIntosh. "Functional calculi of second-order elliptic partial differential operators with bounded measurable coefficients." Journal of Geometric Analysis 6, no. 2 (June 1996): 181–205. http://dx.doi.org/10.1007/bf02921599.

Повний текст джерела

Стилі APA, Harvard, Vancouver, ISO та ін.

10

Healey, Timothy J., Hansjörg Kielhöfer, and Charles A. Stuart. "Global branches of positive weak solutions of semilinear elliptic problems over nonsmooth domains." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 124, no. 2 (1994): 371–88. http://dx.doi.org/10.1017/s0308210500028535.

Повний текст джерела

Анотація:

We consider the nonlinear eigenvalue problem posed by a parameter-dependent semilinear second-order elliptic equation on a bounded domain with the Dirichlet boundary condition. The coefficients of the elliptic operator are bounded measurable functions and the boundary of the domain is only required to be regular in the sense of Wiener. The main results establish the existence of an unbounded branch of positive weak solutions.

Стилі APA, Harvard, Vancouver, ISO та ін.

11

Kunstmann, Peer Christian, and Matthias Uhl. "Lp-Spectral Multipliers for some Elliptic Systems." Proceedings of the Edinburgh Mathematical Society 58, no. 1 (October 10, 2014): 231–53. http://dx.doi.org/10.1017/s001309151400008x.

Повний текст джерела

Анотація:

AbstractWe show results on Lp-spectral multipliers for Maxwell operators with bounded measurable coefficients. We also present similar results for the Stokes operator with Hodge boundary conditions and the Lamé system. Here, we rely on resolvent estimates recently established by Mitrea and Monniaux.

Стилі APA, Harvard, Vancouver, ISO та ін.

12

Hofmann, Steve, Carlos Kenig, Svitlana Mayboroda, and Jill Pipher. "The regularity problem for second order elliptic operators with complex-valued bounded measurable coefficients." Mathematische Annalen 361, no. 3-4 (September 4, 2014): 863–907. http://dx.doi.org/10.1007/s00208-014-1087-6.

Повний текст джерела

Стилі APA, Harvard, Vancouver, ISO та ін.

13

Cho, Chung-Ki, and Hi Jun Choe. "The Asymptotic Behaviour of Solutions of a Porous Medium Equation with Bounded Measurable Coefficients." Journal of Mathematical Analysis and Applications 210, no. 1 (June 1997): 241–56. http://dx.doi.org/10.1006/jmaa.1997.5400.

Повний текст джерела

Стилі APA, Harvard, Vancouver, ISO та ін.

14

PASCUCCI, ANDREA, and SERGIO POLIDORO. "THE MOSER'S ITERATIVE METHOD FOR A CLASS OF ULTRAPARABOLIC EQUATIONS." Communications in Contemporary Mathematics 06, no. 03 (June 2004): 395–417. http://dx.doi.org/10.1142/s0219199704001355.

Повний текст джерела

Анотація:

We adapt the iterative scheme by Moser, to prove that the weak solutions to an ultraparabolic equation, with measurable coefficients, are locally bounded functions. Due to the strong degeneracy of the equation, our method differs from the classical one in that it is based on some ad hoc Sobolev type inequalities for solutions.

Стилі APA, Harvard, Vancouver, ISO та ін.

15

Zecca, Gabriella. "Lp and BMO-solvability of the Dirichlet Problem for Elliptic Operators." International Journal of Pure Mathematics 9 (March 12, 2022): 31–34. http://dx.doi.org/10.46300/91019.2022.9.8.

Повний текст джерела

Анотація:

We establish a connection between the solvability of end-point BMO and Lp Dirichlet problems for a second order divergence form elliptic operator (not necessarily symmetric) with bounded measurable coefficients. In particular, we give a lower bound for the exponent p > 1 in terms of the BMO-constant of L (see Definition 8).

Стилі APA, Harvard, Vancouver, ISO та ін.

16

Wu, Xiongtao, Wenyu Tao, Yanping Chen, and Kai Zhu. "Commutators of Square Functions Related to Fractional Differentiation for Second-Order Elliptic Operators." Journal of Function Spaces 2018 (December 2, 2018): 1–10. http://dx.doi.org/10.1155/2018/2624057.

Повний текст джерела

Анотація:

Let L=-div(A∇) be a second-order divergence form elliptic operator, where A is an accretive n×n matrix with bounded measurable complex coefficients in Rn. In this paper, we mainly establish the Lp boundedness for the commutators generated by b∈Iα(BMO) and the square function related to fractional differentiation for second-order elliptic operators.

Стилі APA, Harvard, Vancouver, ISO та ін.

17

Ryazanov, Vladimir. "On the Riemann-Hilbert problem in multiply connected domains." Open Mathematics 14, no. 1 (January 1, 2016): 13–18. http://dx.doi.org/10.1515/math-2016-0002.

Повний текст джерела

Анотація:

AbstractWe proved the existence of multivalent solutions with the infinite number of branches for the Riemann-Hilbert problem in the general settings of finitely connected domains bounded by mutually disjoint Jordan curves, measurable coefficients and measurable boundary data. The theorem is formulated in terms of harmonic measure and principal asymptotic values. It is also given the corresponding reinforced criterion for domains with rectifiable boundaries stated in terms of the natural parameter and nontangential limits. Furthermore, it is shown that the dimension of the spaces of these solutions is infinite.

Стилі APA, Harvard, Vancouver, ISO та ін.

18

Cho, C. K., and H. J. Choe. "The initial trace of a solution of a porous medium equation with bounded measurable coefficients." Nonlinear Analysis: Theory, Methods & Applications 33, no. 6 (September 1998): 657–73. http://dx.doi.org/10.1016/s0362-546x(97)00660-3.

Повний текст джерела

Стилі APA, Harvard, Vancouver, ISO та ін.

19

Chen, Yanping, and Yong Ding. "Gradient Estimates for Commutators of Square Roots of Elliptic Operators with Complex Bounded Measurable Coefficients." Journal of Geometric Analysis 27, no. 1 (February 19, 2016): 466–91. http://dx.doi.org/10.1007/s12220-016-9687-x.

Повний текст джерела

Стилі APA, Harvard, Vancouver, ISO та ін.

20

Consiglieri, Luisa. "Explicit Estimates for Solutions of Mixed Elliptic Problems." International Journal of Partial Differential Equations 2014 (March 31, 2014): 1–16. http://dx.doi.org/10.1155/2014/845760.

Повний текст джерела

Анотація:

We deal with the existence of quantitative estimates for solutions of mixed problems to an elliptic second-order equation in divergence form with discontinuous coefficients. Our concern is to estimate the solutions with explicit constants, for domains in ℝn (n≥2) of class C0,1. The existence of L∞ and W1,q estimates is assured for q=2 and any q<n/(n-1) (depending on the data), whenever the coefficient is only measurable and bounded. The proof method of the quantitative L∞ estimates is based on the De Giorgi technique developed by Stampacchia. By using the potential theory, we derive W1,p estimates for different ranges of the exponent p depending on the fact that the coefficient is either Dini-continuous or only measurable and bounded. In this process, we establish new existences of Green functions on such domains. The last but not least concern is to unify (whenever possible) the proofs of the estimates to the extreme Dirichlet and Neumann cases of the mixed problem.

Стилі APA, Harvard, Vancouver, ISO та ін.

21

Hofmann, Steve, Svitlana Mayboroda, and Alan McIntosh. "Second order elliptic operators with complex bounded measurable coefficients in $L^p$, Sobolev and Hardy spaces." Annales scientifiques de l'École normale supérieure 44, no. 5 (2011): 723–800. http://dx.doi.org/10.24033/asens.2154.

Повний текст джерела

Стилі APA, Harvard, Vancouver, ISO та ін.

22

Kravchenko, V. G., A. B. Lebre, and J. S. Rodríguez. "Factorization of singular integral operators with a Carleman backward shift: The case of bounded measurable coefficients." Journal d'Analyse Mathématique 107, no. 1 (January 2009): 1–37. http://dx.doi.org/10.1007/s11854-009-0001-8.

Повний текст джерела

Стилі APA, Harvard, Vancouver, ISO та ін.

23

Kaplický, P., and J. Wolf. "On the higher integrability of weak solutions to the generalized Stokes system with bounded measurable coefficients." Dynamics of Partial Differential Equations 15, no. 2 (2018): 127–46. http://dx.doi.org/10.4310/dpde.2018.v15.n2.a3.

Повний текст джерела

Стилі APA, Harvard, Vancouver, ISO та ін.

24

Mengütürk, Levent Ali, and Murat Cahit Mengütürk. "Captive diffusions and their applications to order-preserving dynamics." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2241 (September 2020): 20200294. http://dx.doi.org/10.1098/rspa.2020.0294.

Повний текст джерела

Анотація:

We propose a class of stochastic processes that we call captive diffusions, which evolve within measurable pairs of càdlàg bounded functions that admit bounded right-derivatives at points where they are continuous. In full generality, such processes allow reflection and absorption dynamics at their boundaries—possibly in a hybrid manner over non-overlapping time periods—and if they are martingales, continuous boundaries are necessarily monotonic. We employ multi-dimensional captive diffusions equipped with a totally ordered set of boundaries to model random processes that preserve an initially determined rank. We run numerical simulations on several examples governed by different drift and diffusion coefficients. Applications include interacting particle systems, random matrix theory, epidemic modelling and stochastic control.

Стилі APA, Harvard, Vancouver, ISO та ін.

25

Cao, Jun, Svitlana Mayboroda, and Dachun Yang. "Maximal function characterizations of Hardy spaces associated to homogeneous higher order elliptic operators." Forum Mathematicum 28, no. 5 (September 1, 2016): 823–56. http://dx.doi.org/10.1515/forum-2014-0127.

Повний текст джерела

Анотація:

AbstractLet L be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients and ${(p_{-}(L),p_{+}(L))}$ be the maximal interval of exponents ${q\in[1,\infty]}$ such that the semigroup ${\{e^{-tL}\}_{t>0}}$ is bounded on ${L^{q}(\mathbb{R}^{n})}$. In this article, the authors establish the non-tangential maximal function characterizations of the associated Hardy spaces ${H_{L}^{p}(\mathbb{R}^{n})}$ for all ${p\in(0,p_{+}(L))}$, which when ${p=1}$, answers a question asked by Deng, Ding and Yao in [21]. Moreover, the authors characterize ${H_{L}^{p}(\mathbb{R}^{n})}$ via various versions of square functions and Lusin-area functions associated to the operator L.

Стилі APA, Harvard, Vancouver, ISO та ін.

26

Gröchenig, Karlheinz, and Philippe Jaming. "THE CRAMÉR–WOLD THEOREM ON QUADRATIC SURFACES AND HEISENBERG UNIQUENESS PAIRS." Journal of the Institute of Mathematics of Jussieu 19, no. 1 (November 7, 2017): 117–35. http://dx.doi.org/10.1017/s1474748017000457.

Повний текст джерела

Анотація:

Two measurable sets $S,\unicode[STIX]{x1D6EC}\subseteq \mathbb{R}^{d}$ form a Heisenberg uniqueness pair, if every bounded measure $\unicode[STIX]{x1D707}$ with support in $S$ whose Fourier transform vanishes on $\unicode[STIX]{x1D6EC}$ must be zero. We show that a quadratic hypersurface and the union of two hyperplanes in general position form a Heisenberg uniqueness pair in $\mathbb{R}^{d}$. As a corollary we obtain a new, surprising version of the classical Cramér–Wold theorem: a bounded measure supported on a quadratic hypersurface is uniquely determined by its projections onto two generic hyperplanes (whereas an arbitrary measure requires the knowledge of a dense set of projections). We also give an application to the unique continuation of eigenfunctions of second-order PDEs with constant coefficients.

Стилі APA, Harvard, Vancouver, ISO та ін.

27

Vas’kovskii, M. M. "Existence of β-martingale solutions of stochastic evolution functional equations of parabolic type with measurable locally bounded coefficients". Differential Equations 48, № 8 (серпень 2012): 1065–80. http://dx.doi.org/10.1134/s0012266112080022.

Повний текст джерела

Стилі APA, Harvard, Vancouver, ISO та ін.

28

Mayboroda, Svitlana. "The connections between Dirichlet, regularity and Neumann problems for second order elliptic operators with complex bounded measurable coefficients." Advances in Mathematics 225, no. 4 (November 2010): 1786–819. http://dx.doi.org/10.1016/j.aim.2010.04.019.

Повний текст джерела

Стилі APA, Harvard, Vancouver, ISO та ін.

29

Yang, Dachun, Junqiang Zhang, and Ciqiang Zhuo. "Variable Hardy Spaces Associated with Operators Satisfying Davies–Gaffney Estimates." Proceedings of the Edinburgh Mathematical Society 61, no. 3 (May 21, 2018): 759–810. http://dx.doi.org/10.1017/s0013091517000414.

Повний текст джерела

Анотація:

AbstractLetLbe a one-to-one operator of type ω inL2(ℝn), with ω∈[0, π/2), which has a bounded holomorphic functional calculus and satisfies the Davies–Gaffney estimates. Letp(·): ℝn→(0, 1] be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors introduce the variable Hardy space$H_L^{p(\cdot )} ({\open R}^n)$associated withL. By means of variable tent spaces, the authors establish the molecular characterization of$H_L^{p(\cdot )} ({\open R}^n)$. Then the authors show that the dual space of$H_L^{p(\cdot )} ({\open R}^n)$is the bounded mean oscillation (BMO)-type space${\rm BM}{\rm O}_{p(\cdot ),{\kern 1pt} L^ * }({\open R}^n)$, whereL* denotes the adjoint operator ofL. In particular, whenLis the second-order divergence form elliptic operator with complex bounded measurable coefficients, the authors obtain the non-tangential maximal function characterization of$H_L^{p(\cdot )} ({\open R}^n)$and show that the fractional integralL−αfor α∈(0, (1/2)] is bounded from$H_L^{p(\cdot )} ({\open R}^n)$to$H_L^{q(\cdot )} ({\open R}^n)$with (1/p(·))−(1/q(·))=2α/n, and the Riesz transform ∇L−1/2is bounded from$H_L^{p(\cdot )} ({\open R}^n)$to the variable Hardy spaceHp(·)(ℝn).

Стилі APA, Harvard, Vancouver, ISO та ін.

30

STEFANOPOULOS, VANGELIS. "Uniform approximation by universal series on arbitrary sets." Mathematical Proceedings of the Cambridge Philosophical Society 144, no. 1 (January 2008): 207–16. http://dx.doi.org/10.1017/s0305004107000515.

Повний текст джерела

Анотація:

AbstractBy considering a tree-like decomposition of an arbitrary set we prove the existence of an associated series with the property that its partial sums approximate uniformly all elements in a relevant space of bounded functions. In a topological setting we show that these sums are dense in the space of continuous functions, hence in turn any Borel measurable function is the almost everywhere limit of an appropriate sequence of partial sums of the same series. The coefficients of the series may be chosen in c0, or in a weighted ℓp with 1 < p < ∞, but not in the corresponding weighted ℓ1.

Стилі APA, Harvard, Vancouver, ISO та ін.

31

Hou, Yuexia. "Weighted Sobolev–Morrey Estimates for Nondivergence Degenerate Operators with Drift on Homogeneous Groups." Symmetry 13, no. 11 (November 1, 2021): 2061. http://dx.doi.org/10.3390/sym13112061.

Повний текст джерела

Анотація:

Let X0,X1,…,Xq(q<N) be real vector fields, which are left invariant on homogeneous group G, provided that X0 is homogeneous of degree two and X1,…,Xq are homogeneous of degree one. We consider the following nondivergence degenerate operator with drift L=∑i,j=1qaij(x)XiXj+a0(x)X0, where the coefficients aij(x), a0(x) belonging to vanishing mean oscillation space are bounded measurable functions. Furthermore, aij(x) satisfies the uniform ellipticity condition on Rq and a0(x)≠0. We obtain the local weighted Sobolev–Morrey estimates by applying the boundedness of commutators and interpolation inequalities on weighted Morrey spaces.

Стилі APA, Harvard, Vancouver, ISO та ін.

32

Arroyo, Ángel, and Mikko Parviainen. "Asymptotic Hölder regularity for the ellipsoid process." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 112. http://dx.doi.org/10.1051/cocv/2020034.

Повний текст джерела

Анотація:

We obtain an asymptotic Hölder estimate for functions satisfying a dynamic programming principle arising from a so-called ellipsoid process. By the ellipsoid process we mean a generalization of the random walk where the next step in the process is taken inside a given space dependent ellipsoid. This stochastic process is related to elliptic equations in non-divergence form with bounded and measurable coefficients, and the regularity estimate is stable as the step size of the process converges to zero. The proof, which requires certain control on the distortion and the measure of the ellipsoids but not continuity assumption, is based on the coupling method.

Стилі APA, Harvard, Vancouver, ISO та ін.

33

ARENA, ORAZIO, and CRISTINA GIANNOTTI. "ELLIPTIC EXTENSIONS IN THE DISK WITH OPERATORS IN DIVERGENCE FORM." Bulletin of the Australian Mathematical Society 88, no. 1 (August 20, 2012): 51–55. http://dx.doi.org/10.1017/s000497271200069x.

Повний текст джерела

Анотація:

AbstractLet $\varphi _0$ and $\varphi _1$ be regular functions on the boundary $\partial D$ of the unit disk $D$ in $\mathbb {R}^2$, such that $\int _{0}^{2\pi }\varphi _1\,d\theta =0$ and $\int _{0}^{2\pi }\sin \theta (\varphi _1-\varphi _0)\,d\theta =0$. It is proved that there exist a linear second-order uniformly elliptic operator $L$ in divergence form with bounded measurable coefficients and a function $u$ in $W^{1,p}(D)$, $1 \lt p \lt 2$, such that $Lu=0$ in $D$ and with $u|_{\partial D}= \varphi _0$ and the conormal derivative $\partial u/\partial N|_{\partial D}=\varphi _1$.

Стилі APA, Harvard, Vancouver, ISO та ін.

34

Tran, Tri Dung. "Musielak–Orlicz Hardy Spaces Associated with Divergence Form Elliptic Operators Without Weight Assumptions." Nagoya Mathematical Journal 216 (2014): 71–110. http://dx.doi.org/10.1017/s0027763000022443.

Повний текст джерела

Анотація:

AbstractLetLbe a divergence form elliptic operator with complex bounded measurable coefficients, letωbe a positive Musielak-Orlicz function on (0, ∞) of uniformly strictly critical lower-typepω∈ (0, 1], and letρ(x,t) = t−1/ω−1(x,t−1) forx∈ ℝn, t∊ (0, ∞). In this paper, we study the Musielak-Orlicz Hardy spaceHω,L(ℝn) and its dual space BMOρ,L* (ℝn), whereL*denotes the adjoint operator ofLinL2(ℝn). Theρ-Carleson measure characterization and the John-Nirenberg inequality for the space BMOρ,L(ℝn) are also established. Finally, as applications, we show that the Riesz transform ∇L−1/2and the Littlewood–Paleyg-functiongLmapHω,L(ℝn) continuously intoL(ω).

Стилі APA, Harvard, Vancouver, ISO та ін.

35

MASO, GIANNI DAL, and ADRIANA GARRONI. "NEW RESULTS ON THE ASYMPTOTIC BEHAVIOR OF DIRICHLET PROBLEMS IN PERFORATED DOMAINSDIRICHLET PROBLEMS IN PERFORATED DOMAINS." Mathematical Models and Methods in Applied Sciences 04, no. 03 (June 1994): 373–407. http://dx.doi.org/10.1142/s0218202594000224.

Повний текст джерела

Анотація:

Let A be a linear elliptic operator of the second order with bounded measurable coefficients on a bounded open set Ω of Rn and let (Ωh) be an arbitrary sequence of open subsets of Ω. We prove the following compactness result: there exist a subsequence, still denoted by (Ωh), and a positive Borel measure μ on Ω, not charging polar sets, such that, for every f∈H−1(Ω) the solutions [Formula: see text] of the equations Auh=f in Ωh, extended to 0 on Ω\Ωh, converge weakly in [Formula: see text] to the unique solution [Formula: see text] of the problem [Formula: see text] When A is symmetric, this compactness result is already known and was obtained by Γ-convergence techniques. Our new proof, based on the method of oscillating test functions, extends the result to the non-symmetric case. The new technique, which is completely independent of Γ-convergence, relies on the study of the behavior of the solutions [Formula: see text] of the equations [Formula: see text] where A* is the adjoint operator. We prove also that the limit measure μ does not change if A is replaced by A*. Moreover, we prove that µ depends only on the symmetric part of the operator A, if the coefficients of the skew-symmetric part are continuous, while an explicit example shows that μ may depend also on the skew-symmetric part of A, when the coefficients are discontinuous.

Стилі APA, Harvard, Vancouver, ISO та ін.

36

Chaudhuri, Nirmalendu, and Mythily Ramaswamy. "Existence of positive solutions of some semilinear elliptic equations with singular coefficients." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 131, no. 6 (December 2001): 1275–95. http://dx.doi.org/10.1017/s0308210500001396.

Повний текст джерела

Анотація:

In this paper we consider the semilinear elliptic problem in a bounded domain Ω ⊆ Rn, where μ ≥ 0, 0 ≤ α ≤ 2, 2α* := 2(n − α)/(n − 2), f : Ω → R+ is measurable, f > 0 a.e, having a lower-order singularity than |x|-2 at the origin, and g : R → R is either linear or superlinear. For 1 < p < n, we characterize a class of singular functions Ip for which the embedding is compact. When p = 2, α = 2, f ∈ I2 and 0 ≤ μ < (½(n − 2))2, we prove that the linear problem has -discrete spectrum. By improving the Hardy inequality we show that for f belonging to a certain subclass of I2, the first eigenvalue goes to a positive number as μ approaches (½(n − 2))2. Furthermore, when g is superlinear, we show that for the same subclass of I2, the functional corresponding to the differential equation satisfies the Palais-Smale condition if α = 2 and a Brezis-Nirenberg type of phenomenon occurs for the case 0 ≤ α < 2.

Стилі APA, Harvard, Vancouver, ISO та ін.

37

Arendt, Wolfgang, A. F. M. ter Elst, and Jochen Glück. "Strict Positivity for the Principal Eigenfunction of Elliptic Operators with Various Boundary Conditions." Advanced Nonlinear Studies 20, no. 3 (August 1, 2020): 633–50. http://dx.doi.org/10.1515/ans-2020-2091.

Повний текст джерела

Анотація:

AbstractWe consider elliptic operators with measurable coefficients and Robin boundary conditions on a bounded domain {\Omega\subset\mathbb{R}^{d}} and show that the first eigenfunction v satisfies {v(x)\geq\delta>0} for all {x\in\overline{\Omega}}, even if the boundary {\partial\Omega} is only Lipschitz continuous. Under such weak regularity assumptions the Hopf–Oleĭnik boundary lemma is not available; instead we use a new approach based on an abstract positivity improving condition for semigroups that map {L_{p}(\Omega)} into {C(\overline{\Omega})}. The same tool also yields corresponding results for Dirichlet or mixed boundary conditions. Finally, we show that our results can be used to derive strong minimum and maximum principles for parabolic and elliptic equations.

Стилі APA, Harvard, Vancouver, ISO та ін.

38

Tran, Tri Dung. "Musielak–Orlicz Hardy Spaces Associated with Divergence Form Elliptic Operators Without Weight Assumptions." Nagoya Mathematical Journal 216 (2014): 71–110. http://dx.doi.org/10.1215/00277630-2817420.

Повний текст джерела

Анотація:

AbstractLet L be a divergence form elliptic operator with complex bounded measurable coefficients, let ω be a positive Musielak-Orlicz function on (0, ∞) of uniformly strictly critical lower-type pω ∈ (0, 1], and let ρ(x,t) = t−1/ω−1 (x,t−1) for x ∈ ℝn, t ∊ (0, ∞). In this paper, we study the Musielak-Orlicz Hardy space Hω,L(ℝn) and its dual space BMOρ,L* (ℝ n), where L* denotes the adjoint operator of L in L2 (ℝ n). The ρ-Carleson measure characterization and the John-Nirenberg inequality for the space BMOρ,L (ℝn) are also established. Finally, as applications, we show that the Riesz transform ∇L−1/2 and the Littlewood–Paley g-function gL map Hω,L(ℝn) continuously into L(ω).

Стилі APA, Harvard, Vancouver, ISO та ін.

39

Garain, Prashanta, and Kaj Nyström. "On regularity and existence of weak solutions to nonlinear Kolmogorov-Fokker-Planck type equations with rough coefficients." Mathematics in Engineering 5, no. 2 (2022): 1–37. http://dx.doi.org/10.3934/mine.2023043.

Повний текст джерела

Анотація:

<abstract><p>We consider nonlinear Kolmogorov-Fokker-Planck type equations of the form</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} (\partial_t+X\cdot\nabla_Y)u = \nabla_X\cdot(A(\nabla_X u, X, Y, t)). \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>The function $ A = A(\xi, X, Y, t): \mathbb R^m\times \mathbb R^m\times \mathbb R^m\times \mathbb R\to \mathbb R^m $ is assumed to be continuous with respect to $ \xi $, and measurable with respect to $ X, Y $ and $ t $. $ A = A(\xi, X, Y, t) $ is allowed to be nonlinear but with linear growth. We establish higher integrability and local boundedness of weak sub-solutions, weak Harnack and Harnack inequalities, and Hölder continuity with quantitative estimates. In addition we establish existence and uniqueness of weak solutions to a Dirichlet problem in certain bounded $ X $, $ Y $ and $ t $ dependent domains.</p></abstract>

Стилі APA, Harvard, Vancouver, ISO та ін.

40

Zhuo, Ciqiang, та Dachun Yang. "Variable weak Hardy spaces WH L p(·)(ℝ n ) associated with operators satisfying Davies–Gaffney estimates". Forum Mathematicum 31, № 3 (1 травня 2019): 579–605. http://dx.doi.org/10.1515/forum-2018-0125.

Повний текст джерела

Анотація:

Abstract Let {p(\,\cdot\,)\colon\mathbb{R}^{n}\to[0,1]} be a variable exponent function satisfying the globally log-Hölder continuous condition, and L a one-to-one operator of type ω in {L^{2}({\mathbb{R}}^{n})} , with {\omega\in[0,\pi/2)} , which has a bounded holomorphic functional calculus and satisfies the Davies–Gaffney estimates. In this article, we introduce the variable weak Hardy space {\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})} , associated with L via the corresponding square function. Its molecular characterization is then established by means of the atomic decomposition of the variable weak tent space {\mathrm{WT}^{p(\,\cdot\,)}(\mathbb{R}_{+}^{n+1})} , which is also obtained in this article. In particular, when L is non-negative and self-adjoint, we obtain the atomic characterization of {\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})} . As an application of the molecular characterization, when L is the second-order divergence form elliptic operator with complex bounded measurable coefficients, we prove that the associated Riesz transform {\nabla L^{-1/2}} is bounded from {\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})} to the variable weak Hardy space {\mathrm{WH}^{p(\,\cdot\,)}(\mathbb{R}^{n})} . Moreover, when L is non-negative and self-adjoint with the kernels of {\{e^{-tL}\}_{t>0}} satisfying the Gaussian upper bound estimates, the atomic characterization of {\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})} is further used to characterize this space via non-tangential maximal functions.

Стилі APA, Harvard, Vancouver, ISO та ін.

41

Hofmann, Steve, and Guoming Zhang. "L2 estimates for commutators of the Dirichlet-to-Neumann Map associated to elliptic operators with complex-valued bounded measurable coefficients on R+n+1." Journal of Mathematical Analysis and Applications 504, no. 2 (December 2021): 125408. http://dx.doi.org/10.1016/j.jmaa.2021.125408.

Повний текст джерела

Стилі APA, Harvard, Vancouver, ISO та ін.

42

Arguchintsev, Alexander, Vasilisa Poplevko, and Alexander Sinitsyn. "Variational optimality condition in control of hyperbolic systems with boundary delay parameters." Cybernetics and Physics, Volume 11, 2022, Number 2 (September 30, 2022): 61–66. http://dx.doi.org/10.35470/2226-4116-2022-11-2-61-66.

Повний текст джерела

Анотація:

An optimal control problem of a first-order hyperbolic system is studied, in which a boundary condition at one of the ends is determined from a controlled system of ordinary differential equations with constant state lag. Control functions are bounded and measurable functions. The system of ordinary differential equations at the boundary is linear in state. However the matrix of coefficients depends on control functions. Therefore, the optimality condition of Pontryagin’s maximum principle type in this problem is a necessary, but not a sufficient optimality condition. In this paper, the problem is reduced to an optimal control problem of a special system of ordinary differential equations. The proposed approach is based on the use of an exact formula of the cost functional increment. The reduced problem can be solved using a wide range of effective methods used for optimization problems in systems of ordinary differential equations. Problems of this kind arise when modeling thermal separation processes, suppression of mechanical vibrations in drilling, wave processes and population dynamics.

Стилі APA, Harvard, Vancouver, ISO та ін.

43

Slavnov, Sergey. "Linear logic in normed cones: probabilistic coherence spaces and beyond." Mathematical Structures in Computer Science 31, no. 5 (May 2021): 495–534. http://dx.doi.org/10.1017/s0960129521000177.

Повний текст джерела

Анотація:

Abstract Ehrhard et al. (2018. Proceedings of the ACM on Programming Languages, POPL 2, Article 59.) proposed a model of probabilistic functional programming in a category of normed positive cones and stable measurable cone maps, which can be seen as a coordinate-free generalization of probabilistic coherence spaces (PCSs). However, unlike the case of PCSs, it remained unclear if the model could be refined to a model of classical linear logic. In this work, we consider a somewhat similar category which gives indeed a coordinate-free model of full propositional linear logic with nondegenerate interpretation of additives and sound interpretation of exponentials. Objects are dual pairs of normed cones satisfying certain specific completeness properties, such as existence of norm-bounded monotone weak limits, and morphisms are bounded (adjointable) positive maps. Norms allow us a distinct interpretation of dual additive connectives as product and coproduct. Exponential connectives are modeled using real analytic functions and distributions that have representations as power series with positive coefficients. Unlike the familiar case of PCSs, there is no reference or need for a preferred basis; in this sense the model is invariant. PCSs form a full subcategory, whose objects, seen as posets, are lattices. Thus, we get a model fitting in the tradition of interpreting linear logic in a linear algebraic setting, which arguably is free from the drawbacks of its predecessors.

Стилі APA, Harvard, Vancouver, ISO та ін.

44

Gutlyanskii, Vladimir, Vladimir Ryazanov, Eduard Yakubov, and Artyem Yefimushkin. "On the Hilbert boundary-value problem for Beltrami equations with singularities." Ukrainian Mathematical Bulletin 17, no. 4 (December 14, 2020): 484–508. http://dx.doi.org/10.37069/1810-3200-2020-17-4-2.

Повний текст джерела

Анотація:

We investigate the Hilbert boundary-value problem for Beltrami equations $\overline\partial f=\mu\partial f$ with singularities in generalized quasidisks $D$ whose Jordan boundary $\partial D$ consists of a countable collection of open quasiconformal arcs and, maybe, a countable collection of points. Such generalized quasicircles can be nowhere even locally rectifiable but include, for instance, all piecewise smooth curves, as well as all piecewise Lipschitz Jordan curves. Generally speaking, generalized quasidisks do not satisfy the standard $(A)-$condition in PDE by Ladyzhenskaya-Ural'tseva, in particular, the outer cone touching condition, as well as the quasihyperbolic boundary condition by Gehring-Martio that we assumed in our last paper for the uniformly elliptic Beltrami equations. In essence, here, we admit any countable collection of singularities of the Beltrami equations on the boundary and arbitrary singularities inside the domain $D$ of a general nature. As usual, a point in $\overline D$ is called a singularity of the Beltrami equation, if the dilatation quotient $K_{\mu}:=(1+|\mu|)/(1-|\mu|)$ is not essentially bounded in all its neighborhoods. Presupposing that the coefficients of the problem are arbitrary functions of countable bounded variation and the boundary data are arbitrary measurable with respect to the logarithmic capacity, we prove the existence of regular solutions of the Hilbert boundary-value problem. As a consequence, we derive the existence of nonclassical solutions of the Dirichlet, Neumann, and Poincar\'{e} boundary-value problems for equations of mathematical physics with singularities in anisotropic and inhomogeneous media.

Стилі APA, Harvard, Vancouver, ISO та ін.

45

Chen, Yanping, and Yong Ding. "Commutators of the fractional integrals for second-order elliptic operators on Morrey spaces." Forum Mathematicum 30, no. 3 (May 1, 2018): 617–29. http://dx.doi.org/10.1515/forum-2017-0062.

Повний текст джерела

Анотація:

AbstractLet {L=-\operatorname{div}(A\nabla)} be a second-order divergence form elliptic operator and let A be an accretive, {n\times n} matrix with bounded measurable complex coefficients in {{\mathbb{R}}^{n}}. Let {L^{-\frac{\alpha}{2}}} be the fractional integral associated to L for {0<\alpha<n}. For {b\in L_{\mathrm{loc}}({\mathbb{R}}^{n})} and {k\in{\mathbb{N}}}, the k-th order commutator of b and {L^{-\frac{\alpha}{2}}} is given by(L^{-\frac{\alpha}{2}})_{b,k}f(x)=L^{-\frac{\alpha}{2}}((b(x)-b)^{k}f)(x).In the paper, we mainly show that if {b\in\mathrm{BMO}({\mathbb{R}}^{n})}, {0<\lambda<n} and {0<\alpha<n-\lambda}, then {(L^{-\frac{\alpha}{2}})_{b,k}} is bounded from {L^{p,\lambda}} to {L^{q,\lambda}} for {p_{-}(L)<p<q<p_{+}(L)\frac{n-\lambda}{n}} and {\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n-\lambda}}, where {p_{-}(L)} and {p_{+}(L)} are the two critical exponents for the {L^{p}} uniform boundedness of the semigroup {\{e^{-tL}\}_{t>0}}. Also, we establish the boundedness of the commutator of the fractional integral with Lipschitz function on Morrey spaces. The results encompass what is known for the classical Riesz potentials and elliptic operators with Gaussian domination by the classical heat operator.

Стилі APA, Harvard, Vancouver, ISO та ін.

46

Sini, Mourad. "On the Uniqueness and Reconstruction of Rough and Complex Obstacles from Acoustic Scattering Data." Computational Methods in Applied Mathematics 11, no. 1 (2011): 83–104. http://dx.doi.org/10.2478/cmam-2011-0005.

Повний текст джерела

Анотація:

Abstract We deal with the inverse scattering problem by an obstacle at a fixed frequency. The obstacle is characterized by its shape, the type of boundary conditions on its surface and the eventual coefficients distributed on this surface. In this paper, we assume that the surface ∂D of the obstacle D is Lipschitz and the surface impedance, λ, is given by a complex valued, measurable and bounded function. We prove uniqueness of (∂D,λ) from the far field map under these regularity conditions. The usual proof of uniqueness for obstacles, based on the use of singular solutions, is divided into two steps. The first one consists of the use of Rellich type lemma to go from the far fields to the near fields and then use the singularities of the singular solutions, via orthogonality relations, to show uniqueness of ∂D. The second step is to use the boundary conditions to prove uniqueness of λ on ∂D via the unique continuation property. This last step requires the surface impedance to be continuous. We propose an approach using layer potentials to transform the inverse problem to the invertibility of integral equations of second kind involving the unknowns ∂D and λ. This enables us to weaken the required regularity conditions by assuming ∂D to be Lipschitz and λ to be only bounded. The procedure of the proof is reconstructive and provides a method to compute the complex valued and bounded surface impedance λ on ∂D by inverting an invertible integral equation. In addition, assuming ∂D to be C^2 regular and λ to be of class C^{0,α}, with α>0, we give a direct and stable formula as another method to reconstruct the surface impedance on ∂D.

Стилі APA, Harvard, Vancouver, ISO та ін.

47

Mansimov, K. B., and R. O. Mastaliyev. "Representation of the Solution of Goursat Problem for Second Order Linear Stochastic Hyperbolic Differential Equations." Bulletin of Irkutsk State University. Series Mathematics 36 (2021): 29–43. http://dx.doi.org/10.26516/1997-7670.2021.36.29.

Повний текст джерела

Анотація:

The article considers second-order system of linear stochastic partial differential equations of hyperbolic type with Goursat boundary conditions. Earlier, in a number of papers, representations of the solution Goursat problem for linear stochastic equations of hyperbolic type in the classical way under the assumption of sufficient smoothness of the coefficients of the terms included in the right-hand side of the equation were obtained. Meanwhile, study of many stochastic applied optimal control problems described by linear or nonlinear second-order stochastic differential equations, in partial derivatives hyperbolic type, the assumptions of sufficient smoothness of these equations are not natural. Proceeding from this, in the considered Goursat problem, in contrast to the known works, the smoothness of the coefficients of the terms in the right-hand side of the equation is not assumed. They are considered only measurable and bounded matrix functions. These assumptions, being natural, allow us to further investigate a wide class of optimal control problems described by systems of second-order stochastic hyperbolic equations. In this work, a stochastic analogue of the Riemann matrix is introduced, an integral representation of the solution of considered boundary value problem in explicit form through the boundary conditions is obtained. An analogue of the Riemann matrix was introduced as a solution of a two-dimensional matrix integral equation of the Volterra type with one-dimensional terms, a number of properties of an analogue of the Riemann matrix were studied.

Стилі APA, Harvard, Vancouver, ISO та ін.

48

Domoshnitsky, Alexander, Irina Volinsky, and Anatoly Polonsky. "Stabilization of third order differential equation by delay distributed feedback control with unbounded memory." Mathematica Slovaca 69, no. 5 (October 25, 2019): 1165–76. http://dx.doi.org/10.1515/ms-2017-0298.

Повний текст джерела

Анотація:

Abstract There are almost no results on the exponential stability of differential equations with unbounded memory in mathematical literature. This article aimes to partially fill this gap. We propose a new approach to the study of stability of integro-differential equations with unbounded memory of the following forms $$\begin{array}{} \begin{split} \displaystyle x'''(t)+\sum_{i=1}^{m}\int\limits_{t-\tau_{i}(t)}^{t}b_{i}(t)\text{e}^{-\alpha _{i}(t-s) }x(s)\text{d} s &=0, \\ x'''(t)+\sum_{i=1}^{m}\int\limits_{0}^{t-\tau _{i}(t)}b_{i}(t)\text{e}^{-\alpha _{i}(t-s) }x(s)\text{d} s &= 0, \end{split} \end{array}$$ with measurable essentially bounded bi(t) and τi(t), i = 1, …, m. We demonstrate that, under certain conditions on the coefficients, integro-differential equations of these forms are exponentially stable if the delays τi(t), i = 1, …, m, are small enough. This opens new possibilities for stabilization by distributed input control. According to common belief this sort of stabilization requires first and second derivatives of x. Results obtained in this paper prove that this is not the case.

Стилі APA, Harvard, Vancouver, ISO та ін.

49

Cao, Jun, Svitlana Mayboroda, and Dachun Yang. "Local Hardy spaces associated with inhomogeneous higher order elliptic operators." Analysis and Applications 15, no. 02 (January 25, 2017): 137–224. http://dx.doi.org/10.1142/s0219530515500189.

Повний текст джерела

Анотація:

Let [Formula: see text] be a divergence form inhomogeneous higher order elliptic operator with complex bounded measurable coefficients. In this paper, for all [Formula: see text] and [Formula: see text] satisfying a weak ellipticity condition, the authors introduce the local Hardy spaces [Formula: see text] associated with [Formula: see text], which coincide with Goldberg’s local Hardy spaces [Formula: see text] for all [Formula: see text] when [Formula: see text] (the Laplace operator). The authors also establish a real-variable theory of [Formula: see text], which includes their characterizations in terms of the local molecules, the square functions or the maximal functions, the complex interpolation and dual spaces. These real-variable characterizations on the local Hardy spaces are new even when [Formula: see text] (the divergence form homogeneous second-order elliptic operator). Moreover, the authors show that [Formula: see text] coincides with the Hardy space [Formula: see text] associated with the operator [Formula: see text] for all [Formula: see text], where [Formula: see text] is some positive constant depending on the ellipticity and the off-diagonal estimates of [Formula: see text]. As an application, the authors establish some mapping properties for the local Riesz transforms [Formula: see text] on [Formula: see text], where [Formula: see text] and [Formula: see text].

Стилі APA, Harvard, Vancouver, ISO та ін.

50

Makhnei, O. V. "Mixed problem for the singular partial differential equation of parabolic type." Carpathian Mathematical Publications 10, no. 1 (July 3, 2018): 165–71. http://dx.doi.org/10.15330/cmp.10.1.165-171.

Повний текст джерела

Анотація:

The scheme for solving of a mixed problem is proposed for a differential equation \[a(x)\frac{\partial T}{\partial \tau}= \frac{\partial}{\partial x} \left(c(x)\frac{\partial T}{\partial x}\right) -g(x)\, T\] with coefficients $a(x)$, $g(x)$ that are the generalized derivatives of functions of bounded variation, $c(x)>0$, $c^{-1}(x)$ is a bounded and measurable function. The boundary and initial conditions have the form $$p_{1}T(0,\tau)+p_{2}T^{[1]}_x (0,\tau)= \psi_1(\tau), q_{1}T(l,\tau)+q_{2}T^{[1]}_x (l,\tau)= \psi_2(\tau), $$ $$T(x,0)=\varphi(x), $$ where $p_1 p_2\leq 0$, $q_1 q_2\geq 0$ and by $T^{[1]}_x (x,\tau)$ we denote the quasiderivative $c(x)\frac{\partial T}{\partial x}$. A solution of this problem seek by the reduction method in the form of sum of two functions $T(x,\tau)=u(x,\tau)+v(x,\tau)$. This method allows to reduce solving of proposed problem to solving of two problems: a quasistationary boundary problem with initial and boundary conditions for the search of the function $u(x,\tau)$ and a mixed problem with zero boundary conditions for some inhomogeneous equation with an unknown function $v(x,\tau)$. The first of these problems is solved through the introduction of the quasiderivative. Fourier method and expansions in eigenfunctions of some boundary value problem for the second-order quasidifferential equation $\big(c(x)X'(x)\big)' -g(x)X(x)+ \omega a(x)X(x)=0$ are used for solving of the second problem. The function $v(x,\tau)$ is represented as a series in eigenfunctions of this boundary value problem. The results can be used in the investigation process of heat transfer in a multilayer plate.

Стилі APA, Harvard, Vancouver, ISO та ін.

Ми пропонуємо знижки на всі преміум-плани для авторів, чиї праці увійшли до тематичних добірок літератури. Зв'яжіться з нами, щоб отримати унікальний промокод!