Статті в журналах: "Functions of bounded deformation"

1

Dal Maso, Gianni. "Generalised functions of bounded deformation." Journal of the European Mathematical Society 15, no. 5 (2013): 1943–97. http://dx.doi.org/10.4171/jems/410.

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2

Conti, Sergio, Matteo Focardi, and Flaviana Iurlano. "Which special functions of bounded deformation have bounded variation?" Proceedings of the Royal Society of Edinburgh: Section A Mathematics 148, no. 1 (October 17, 2017): 33–50. http://dx.doi.org/10.1017/s030821051700004x.

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Functions of bounded deformation (BD) arise naturally in the study of fracture and damage in a geometrically linear context. They are related to functions of bounded variation (BV), but are less well understood. We discuss here the relation to BV under additional regularity assumptions, which may require the regular part of the strain to have higher integrability or the jump set to have finite area or the Cantor part to vanish. On the positive side, we prove that BD functions that are piecewise affine on a Caccioppoli partition are in GSBV, and we prove that SBDp functions are approximately continuous -almost everywhere away from the jump set. On the negative side, we construct a function that is BD but not in BV and has distributional strain consisting only of a jump part, and one that has a distributional strain consisting of only a Cantor part.

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3

Babadjian, Jean-Francois. "Traces of functions of bounded deformation." Indiana University Mathematics Journal 64, no. 4 (2015): 1271–90. http://dx.doi.org/10.1512/iumj.2015.64.5601.

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4

Ambrosio, Luigi, Alessandra Coscia, and Gianni Dal Maso. "Fine Properties of Functions with Bounded Deformation." Archive for Rational Mechanics and Analysis 139, no. 3 (October 27, 1997): 201–38. http://dx.doi.org/10.1007/s002050050051.

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5

Nie, Ziwei, and Xiaoping Yang. "Deformable Image Registration Using Functions of Bounded Deformation." IEEE Transactions on Medical Imaging 38, no. 6 (June 2019): 1488–500. http://dx.doi.org/10.1109/tmi.2019.2896170.

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6

Hajłasz, Piotr. "On approximate differentiability of functions with bounded deformation." Manuscripta Mathematica 91, no. 1 (December 1996): 61–72. http://dx.doi.org/10.1007/bf02567939.

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7

Chambolle, Antonin. "An approximation result for special functions with bounded deformation." Journal de Mathématiques Pures et Appliquées 83, no. 7 (July 2004): 929–54. http://dx.doi.org/10.1016/j.matpur.2004.02.004.

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8

Nie, Ziwei, Chen Li, Hairong Liu, and Xiaoping Yang. "Deformable Image Registration Based on Functions of Bounded Generalized Deformation." International Journal of Computer Vision 129, no. 5 (February 4, 2021): 1341–58. http://dx.doi.org/10.1007/s11263-021-01439-x.

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9

Ebobisse, François B. "Lusin-type approximation of BD functions." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 129, no. 4 (1999): 697–705. http://dx.doi.org/10.1017/s0308210500013081.

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The purpose of this paper is to establish a Lusin-type approximation of functions with bounded deformation by Lipschitz or C1 functions. The main ingredients inthe proof of our result are the maximal function of the measure Eu, the ‘Poincaré-type’ result by Kohn and the approximate symmetric differentiability of BD functions by Ambrosio and others.

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10

Fuchs, M., and M. Bildhauer. "Compact embeddings of the space of functions with bounded logarithmic deformation." Journal of Mathematical Sciences 172, no. 1 (December 17, 2010): 165–83. http://dx.doi.org/10.1007/s10958-010-0190-9.

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11

Battaglia, Giuseppe, Luigi Gurreri, Andrea Cipollina, Antonina Pirrotta, Svetlozar Velizarov, Michele Ciofalo, and Giorgio Micale. "Fluid–Structure Interaction and Flow Redistribution in Membrane-Bounded Channels." Energies 12, no. 22 (November 8, 2019): 4259. http://dx.doi.org/10.3390/en12224259.

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The hydrodynamics of electrodialysis and reverse electrodialysis is commonly studied by neglecting membrane deformation caused by transmembrane pressure (TMP). However, large frictional pressure drops and differences in fluid velocity or physical properties in adjacent channels may lead to significant TMP values. In previous works, we conducted one-way coupled structural-CFD simulations at the scale of one periodic unit of a profiled membrane/channel assembly and computed its deformation and frictional characteristics as functions of TMP. In this work, a novel fluid–structure interaction model is presented, which predicts, at the channel pair scale, the changes in flow distribution associated with membrane deformations. The continuity and Darcy equations are solved in two adjacent channels by treating them as porous media and using the previous CFD results to express their hydraulic permeability as a function of the local TMP. Results are presented for square stacks of 0.6-m sides in cross and counter flow at superficial velocities of 1 to 10 cm/s. At low velocities, the corresponding low TMP does not significantly affect the flow distribution. As the velocity increases, the larger membrane deformation causes significant fluid redistribution. In the cross flow, the departure of the local superficial velocity from a mean value of 10 cm/s ranges between −27% and +39%.

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12

Friedrich, Manuel. "A Korn-type inequality in SBD for functions with small jump sets." Mathematical Models and Methods in Applied Sciences 27, no. 13 (October 19, 2017): 2461–84. http://dx.doi.org/10.1142/s021820251750049x.

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We present a Korn-type inequality in a planar setting for special functions of bounded deformation. We prove that for each function in [Formula: see text] with a sufficiently small jump set the distance of the function and its derivative from an infinitesimal rigid motion can be controlled in terms of the linearized elastic strain outside of a small exceptional set of finite perimeter. Particularly, the result shows that each function in [Formula: see text] has bounded variation away from an arbitrarily small part of the domain.

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13

De Philippis, Guido, and Filip Rindler. "Fine properties of functions of bounded deformation-an approach via linear PDEs." Mathematics in Engineering 2, no. 3 (2020): 386–422. http://dx.doi.org/10.3934/mine.2020018.

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14

Bellieud, Michel, and Shane Cooper. "Asymptotic Analysis of Stratified Elastic Media in the Space of Functions with Bounded Deformation." SIAM Journal on Mathematical Analysis 49, no. 5 (January 2017): 4275–317. http://dx.doi.org/10.1137/16m107551x.

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15

Dal Maso, Gianni, Gianluca Orlando, and Rodica Toader. "Lower semicontinuity of a class of integral functionals on the space of functions of bounded deformation." Advances in Calculus of Variations 10, no. 2 (April 1, 2017): 183–207. http://dx.doi.org/10.1515/acv-2015-0036.

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AbstractWe study the lower semicontinuity of some free discontinuity functionals with linear growth defined on the space of functions with bounded deformation. The volume term is convex and depends only on the Euclidean norm of the symmetrized gradient. We introduce a suitable class of surface terms, which make the functional lower semicontinuous with respect to ${L^{1}}$ convergence.

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16

DEL PEZZO, LEANDRO M. "OPTIMIZATION PROBLEM FOR EXTREMALS OF THE TRACE INEQUALITY IN DOMAINS WITH HOLES." Communications in Contemporary Mathematics 12, no. 04 (August 2010): 569–86. http://dx.doi.org/10.1142/s0219199710003920.

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We study the Sobolev trace constant for functions defined in a bounded domain Ω that vanish in the subset A. We find a formula for the first variation of the Sobolev trace with respect to the hole. As a consequence of this formula, we prove that when Ω is a centered ball, the symmetric hole is critical when we consider deformation that preserve volume but is not optimal for some case.

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17

Conti, Sergio, Matteo Focardi, and Flaviana Iurlano. "Approximation of fracture energies with p-growth via piecewise affine finite elements." ESAIM: Control, Optimisation and Calculus of Variations 25 (2019): 34. http://dx.doi.org/10.1051/cocv/2018021.

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The modeling of fracture problems within geometrically linear elasticity is often based on the space of generalized functions of bounded deformation GSBDp(Ω), p ∈ (1, ∞), their treatment is however hindered by the very low regularity of those functions and by the lack of appropriate density results. We construct here an approximation of GSBDp functions, for p ∈ (1, ∞), with functions which are Lipschitz continuous away from a jump set which is a finite union of closed subsets of C1 hypersurfaces. The strains of the approximating functions converge strongly in Lp to the strain of the target, and the area of their jump sets converge to the area of the target. The key idea is to use piecewise affine functions on a suitable grid, which is obtained via the Freudenthal partition of a cubic grid.

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18

Qin, Long, Yabing Zha, Quanjun Yin, and Yong Peng. "Formation Control of Robotic Swarm Using Bounded Artificial Forces." Scientific World Journal 2013 (2013): 1–15. http://dx.doi.org/10.1155/2013/194280.

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Formation control of multirobot systems has drawn significant attention in the recent years. This paper presents a potential field control algorithm, navigating a swarm of robots into a predefined 2D shape while avoiding intermember collisions. The algorithm applies in both stationary and moving targets formation. We define the bounded artificial forces in the form of exponential functions, so that the behavior of the swarm drove by the forces can be adjusted via selecting proper control parameters. The theoretical analysis of the swarm behavior proves the stability and convergence properties of the algorithm. We further make certain modifications upon the forces to improve the robustness of the swarm behavior in the presence of realistic implementation considerations. The considerations include obstacle avoidance, local minima, and deformation of the shape. Finally, detailed simulation results validate the efficiency of the proposed algorithm, and the direction of possible futrue work is discussed in the conclusions.

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19

BĚLÍK, PAVEL, and MITCHELL LUSKIN. "APPROXIMATION BY PIECEWISE CONSTANT FUNCTIONS IN A BV METRIC." Mathematical Models and Methods in Applied Sciences 13, no. 03 (March 2003): 373–93. http://dx.doi.org/10.1142/s0218202503002556.

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We study the approximation properties of piecewise constant functions with respect to triangular and rectangular finite elements in a metric defined on functions of bounded variation. We apply our results to a thin film model for martensitic crystals and to the approximation of deformations with microstructure.

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20

Eziefula, Uchechi G. "Analysis of inelastic buckling of rectangular plates with a free edge using polynomial deflection functions." International Review of Applied Sciences and Engineering 11, no. 1 (April 2020): 15–21. http://dx.doi.org/10.1556/1848.2020.00003.

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AbstractThe inelastic buckling behaviour of different rectangular thin isotropic plates having a free edge is studied. Various combinations of boundary conditions are subject to in-plane uniaxial compression and each rectangular plate is bounded by an unloaded free edge. The characteristic deflection function of each plate is formulated using a polynomial function in form of Taylor–Maclaurin series. A deformation plasticity approach is adopted and the buckling load equation is modified using a work principle technique. Buckling coefficients of the plates are calculated for various aspect ratios and moduli ratios. Findings obtained from the investigation are found to reasonably agree with data published in the literature.

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21

Fuchs, M., and S. Repin. "Some Poincaré-type inequalities for functions of bounded deformation involving the deviatoric part of the symmetric gradient." Journal of Mathematical Sciences 178, no. 3 (September 24, 2011): 367–72. http://dx.doi.org/10.1007/s10958-011-0554-9.

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22

Rindler, Filip. "Lower Semicontinuity for Integral Functionals in the Space of Functions of Bounded Deformation Via Rigidity and Young Measures." Archive for Rational Mechanics and Analysis 202, no. 1 (March 29, 2011): 63–113. http://dx.doi.org/10.1007/s00205-011-0408-0.

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23

Fonseca, Irene, Adrian Hagerty, and Roberto Paroni. "Second-Order Structured Deformations in the Space of Functions of Bounded Hessian." Journal of Nonlinear Science 29, no. 6 (June 13, 2019): 2699–734. http://dx.doi.org/10.1007/s00332-019-09556-1.

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24

Carrillo Rouse, Paulo. "Compactly supported analytic indices for Lie groupoids." Journal of K-Theory 4, no. 2 (October 2009): 223–62. http://dx.doi.org/10.1017/is008003015jkt069.

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AbstractFor any Lie groupoid we construct an analytic index morphism taking values in a modified K-theory group which involves the convolution algebra of compactly supported smooth functions over the groupoid. The construction is performed by using the deformation algebra of smooth functions over the tangent groupoid constructed in [CR06]. This allows us in particular to prove a more primitive version of the Connes-Skandalis longitudinal index theorem for foliations, that is, an index theorem taking values in a group which pairs with cyclic cocycles. As another application, for D a -PDO elliptic operator with associated index ind we prove that the pairingwith τ a bounded continuous cyclic cocycle, only depends on the principal symbol class [σ(D)]∈K0. The result is completely general for étale groupoids. We discuss some potential applications to the Novikov conjecture.

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25

Baía, Margarida, José Matias, and Pedro Miguel Santos. "A relaxation result in the framework of structured deformations in a bounded variation setting." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 142, no. 2 (March 21, 2012): 239–71. http://dx.doi.org/10.1017/s0308210510001460.

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We obtain an integral representation of an energy for structured deformations of continua in the space of functions of bounded variation, as a first step to the study of asymptotic models for thin defective crystalline structures, where phenomena as slips, vacancies and dislocations prevent the effectiveness of classical theories.

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26

Kabanova, L. A., and A. V. Khokhlov. "CREEP OF A THICKWALLED QUASILINEAR VISCOELASTIC TUBE UNDER A CONSTANT EXTERNAL AND INTERNAL PRESSURE." Problems of strenght and plasticity 83, no. 2 (2021): 170–87. http://dx.doi.org/10.32326/1814-9146-2021-83-2-170-187.

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We consider the creep problem for a quasilinear viscoelastic model of a thickwalled tube, loaded with constant internal and external pressure; the material is supposed to be incompressible. An exact solution to this problem was received by one of the authors in previous papers, assuming the state of a tube to be plain deformation; hereby we study properties of this solution for arbitrary material functions of quasilinear viscoelasticity constitutive relation. A criterion of stress stationarity is derived; the stress field of a thickwalled tube under a constant pressure evolves in time in the case of unbounded creep function and arbitrary nonlinearity function, except some particular types. The monotonicity of stress field components is studied: the radial stress monotonicity depends only on internal and external pressure values (for internal pressure, greater than an external one, it is negative and increases in radii). For other stress components, there are derived sufficient conditions of monotonicity. For an exponential nonlinearity function and unbounded creep function, a creep curve is determined to be concave up at the initial moment, and concave down during prolonged observation; the creep curve of a bipower nonlinearity function model may change its convexity. The stressstrain state of a model with a bounded creep function is proved to be bounded.

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27

Giacomini, Alessandro, and Marcello Ponsiglione. "Non-interpenetration of matter for SBV deformations of hyperelastic brittle materials." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 138, no. 5 (October 2008): 1019–41. http://dx.doi.org/10.1017/s0308210507000121.

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We prove that the Ciarlet–Nečas non-interpenetration of matter condition can be extended to the case of deformations of hyperelastic brittle materials belonging to the class of special functions of bounded variation (SBV), and can be taken into account for some variational models in fracture mechanics. In order to formulate such a condition, we define the deformed configuration under an SBV map by means of the approximately differentiable representative, and we prove some connected stability results under weak convergence. We provide an application to the case of brittle Ogden materials.

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28

Chambolle, Antonin. "Addendum to “An approximation result for special functions with bounded deformation” [J. Math. Pures Appl. (9) 83 (7) (2004) 929–954]: the N-dimensional case." Journal de Mathématiques Pures et Appliquées 84, no. 1 (January 2005): 137–45. http://dx.doi.org/10.1016/j.matpur.2004.11.001.

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29

Gryshchuk, Serhii. "Monogenic functions in two dimensional commutative algebras to equations of plane orthotropy." Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine 32 (December 28, 2018): 18–29. http://dx.doi.org/10.37069/1683-4720-2018-32-3.

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Among all two-dimensional commutative and assosiative algebras of the second rank with the unity $e$ over the field of complex numbers $\mathbb{C}$ we find a semi-simple algebra $\mathbb{B}_{0} := \{c_1 e+c_2 \omega: c_k\in\mathbb{C}, k=1,2\}$, $\omega^2=e$, containing a basis $(e_1,e_2)$, such that $ e_1^4 + 2p e_1^2 e_2^2 + e_2^4 = 0 $ for any fixed $ p $ such that $-1 \lt p \gt 1 $. A domain $\mathcal{B}_{1}=\{(e_1,e_2)\}$, $e_1=e$, is discribed in an explicit form. We consider an approach of $\mathbb{B}_{0}$-valued ''analytic'' functions $\Phi(xe_1+ye_2) = U_{1}(x,y)e_1 + U_{2}(x,y)ie_1+ U_{3}(x,y)e_2 + U_{4}(x,y)ie_2$ ($(e_1,e_2)\in \mathcal{B}$, $x$ and $y$ are real variables) such that their real-valued components $U_{k}$, $k=\overline{1,4}$, satisfy the equation on finding the stress function $u$ in the case of orthotropic plane deformations (with absence of body forses): $ \left(\frac{\partial^4}{\partial x^4} +2p\frac{\partial^4 }{\partial x^2\partial y^2}+ \frac{\partial^4 }{\partial y^4} \right) u(x,y)=0$ for every $(x,y)\in D$, where $D$ is a domain of the Cartesian plane $xOy$. A characterization of solutions $u$ for this equation in a bounded simply-connected domain via real components $U_{k}$, $k=\overline{1,4}$, of the function $\Phi$ is done in the following sense: let $D$ be a bounded and simply-connected domain, a solution $u$ is fixed, then $u$ is a first component of monogenic function $\Phi_{u}$. The variety of such $\Phi_{u}$ is found in a complete form. We consider a particular case of $(e,e_2)\in \mathcal{B}_{1}$ for which $\Phi_{u}$ can be found in an explicit form. For this case a function $\Phi_{u}$ is obtained in an explicit form. Note, that in case of orthotropic plane deformations, when Eqs. of the stress function is of the form: $ \left(\frac{\partial^4}{\partial x^4} +2p\frac{\partial^4}{\partial x^2\partial y^2}+\frac{\partial^4 } {\partial y^4} \right) u(x,y)=0$, here $p$ is a fixed number such that $p>1$, a similar research is done in [Gryshchuk S. V. Сommutative сomplex algebras of the second rank with unity and some cases of plane orthotropy. I. Ukr. Mat. Zh. 2018. 70, No. 8. pp. 1058-1071 (Ukrainian); Gryshchuk S. V. Сommutative сomplex algebras of the second rank with unity and some cases of plane orthotropy. II. Ukr. Mat. Zh. 2018. 70, No. 10. pp. 1382-1389 (Ukrainian)].

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30

Braides, Andrea, and Nicola A. Nodargi. "Homogenization of cohesive fracture in masonry structures." Mathematics and Mechanics of Solids 25, no. 2 (August 30, 2019): 181–200. http://dx.doi.org/10.1177/1081286519870222.

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We derive a homogenized mechanical model of a masonry-type structure constituted by a periodic assemblage of blocks with interposed mortar joints. The energy functionals in the model under investigation consist of (i) a linear elastic contribution within the blocks, (ii) a Barenblatt’s cohesive contribution at contact surfaces between blocks, and (iii) a suitable unilateral condition on the strain across contact surfaces, and are governed by a small parameter representing the typical ratio between the length of the blocks and the dimension of the structure. Using the terminology of [Formula: see text]-convergence and within the functional setting supplied by the functions of bounded deformation, we analyze the asymptotic behavior of such energy functionals when the parameter tends to zero, and derive a simple homogenization formula for the limit energy. Furthermore, we highlight the main mathematical and mechanical properties of the homogenized energy, including its non-standard growth conditions under tension or compression. The key point in the limit process is the definition of macroscopic tensile and compressive stresses, which are determined by the unilateral conditions on contact surfaces and the geometry of the blocks.

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31

Salesi, G., M. Greselin, L. Deleidi, and R. A. Peruzza. "Modified Lorentz transformations in deformed special relativity." International Journal of Modern Physics A 32, no. 15 (May 26, 2017): 1750086. http://dx.doi.org/10.1142/s0217751x17500865.

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We have extended a recent approach to Deformed Special Relativity based on deformed dispersion laws, entailing modified Lorentz transformations and, at the same time, noncommutative geometry and intrinsically discrete space–time. In so doing we have obtained the explicit form of the modified Lorentz transformations for a special class of modified momentum-energy relations often found in literature and arising from quantum gravity and elementary particle physics. Actually, our theory looks as a very simple and natural extension of special relativity to include a momentum cutoff at the Planck scale. In particular, the new Lorentz transformations do imply that for high boost speed [Formula: see text] the deformed Lorentz factor does not diverge as in ordinary relativity, but results to be upper bounded by a large finite value of the order of the ratio between the Planck mass and the particle mass. We have also predicted that a generic boost leaves unchanged Planck energy and momentum, which result invariant with respect to any reference frame. Finally, through matrix deformation functions, we have extended our theory to more general cases with dispersion laws containing momentum-energy mixed terms.

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32

Acerbi, Emilio, Irene Fonseca, and Nicola Fusco. "Regularity results for equilibria in a variational model for fracture." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 127, no. 5 (1997): 889–902. http://dx.doi.org/10.1017/s0308210500026780.

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SynopsisIn recent years models describing interactions between fracture and damage have been proposed in which the relaxed energy of the material is given by a functional involving bulk and interfacial terms, of the formwhere Ω is an open, bounded subset of ℝN, q ≧1, g ∈ L∞ (Ω ℝN), λ, β > 0, the bulk energy density F is quasiconvex, K⊂ℝN is closed, and the admissible deformation u:Ω→ ℝN is C1 in Ω\K One of the main issues has to do with regularity properties of the ‘crack site’ K for a minimising pair (K, u). In the scalar case, i.e. when uΩ→ ℝ, similar models were adopted to image segmentation problems, and the regularity of the ‘edge’ set K has been successfully resolved for a quite broad class of convex functions F with growth p > 1 at infinity. In turn, this regularity entails the existence of classical solutions. The methods thus used cannot be carried out to the vectorial case, except for a very restrictive class of integrands. In this paper we deal with a vector-valued case on the plane, obtaining regularity for minimisers of corresponding to polyconvex bulk energy densities of the formwhere the convex function h grows linearly at infinity.

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33

Feng, Mei, Meijian An, James Mechie, Wenjin Zhao, Guangqi Xue, and Heping Su. "Lithospheric structures of and tectonic implications for the central–east Tibetan plateau inferred from joint tomography of receiver functions and surface waves." Geophysical Journal International 223, no. 3 (August 26, 2020): 1688–707. http://dx.doi.org/10.1093/gji/ggaa403.

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SUMMARY We present an updated joint tomographic method to simultaneously invert receiver function waveforms and surface wave dispersions for a 3-D S-wave velocity (Vs) model. By applying this method to observations from ∼900 seismic stations and with a priori Moho constraints from previous studies, we construct a 3-D lithospheric S-wave velocity model and crustal-thickness map for the central–east Tibetan plateau. Data misfit/fitting shows that the inverted model can fit the receiver functions and surface wave dispersions reasonably well, and checkerboard tests show the model can retrieve major structural information. The results highlight several features. Within the plateau crustal thickness is >60 km and outwith the plateau it is ∼40 km. Obvious Moho offsets and lateral variations of crustal velocities exist beneath the eastern (Longmen Shan Fault), northern (central–east Kunlun Fault) and northeastern (east Kunlun Fault) boundaries of the plateau, but with decreasing intensity. Segmented high upper-mantle velocities have varied occurrences and depth extents from south/southwest to north/northeast in the plateau. A Z-shaped upper-mantle low-velocity channel, which was taken as Tibetan lithospheric mantle, reflecting deformable material lies along the northern and eastern periphery of the Tibetan plateau, seemingly separating two large high-velocity mantle areas that, respectively, correspond to the Indian and Asian lithospheres. Other small high-velocity mantle segments overlain by the Z-shaped channel are possibly remnants of cold microplates/slabs associated with subductions/collisions prior to the Indian–Eurasian collision during the accretion of the Tibetan region. By integrating the Vs structures with known tectonic information, we derive that the Indian slab generally underlies the plateau south of the Bangong–Nujiang suture in central Tibet and the Jinsha River suture in eastern Tibet and west of the Lanchangjiang suture in southeastern Tibet. The eastern, northern, northeastern and southeastern boundaries of the Tibetan plateau have undergone deformation with decreasing intensity. The weakly resisting northeast and southeast margins, bounded by a wider softer channel of uppermost mantle material, are two potential regions for plateau expansion in the future.

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34

Pham, Duc Thong, and Dmitry V. Tarlakovskii. "Dynamic Bending of an Infinite Electromagnetoelastic Rod." Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics 20, no. 4 (2020): 493–501. http://dx.doi.org/10.18500/1816-9791-2020-20-4-493-501.

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The problem of non-stationary bending of an infinite electromagnetoelastic rod is considered. It is assumed that the material of the rod is a homogeneous isotropic conductor. The closed-form system of process equations is constructed under the assumption that the desired functions depend only on the longitudinal coordinate and time using the corresponding relations for shells which take into account the initial electromagnetic field, the Lorentz force, Maxwell’s equations, and the generalized Ohm’s law. The desired functions are assumed to be bounded, and the initial conditions are assumed to be null. The solution of the problem is constructed in an integral form with kernels in the form of influence functions. Images of kernel are found in the space of Laplace transformations in time and Fourier transformations in spatial coordinates. It is noted that the images are rational functions of the Laplace transform parameter, which makes it quite easy to find the originals. However, for a general model that takes into account shear deformations, the subsequent inversion of the Fourier transform can be carried out only numerically, which leads to computational problems associated with the presence of rapidly oscillating integrals. Therefore, the transition to simplified equations corresponding to the Bernoulli – Euler rod and the quasistationary electromagnetic field is carried out. The method of a small parameter is used for which a coefficient is selected that relates the mechanical and electromagnetic fields. In the linear approximation, influence functions are found for which images and originals are constructed. In this case, the zeroth approximation corresponds to a purely elastic solution. Originals are found explicitly using transform properties and tables. Examples of calculations are given for an aluminum rod with a square cross section. It is shown that for the selected material the quantitative difference from the elastic solution is insignificant. At the same time, taking into account the connectedness of the process leads to additional significant qualitative effects.

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35

Berdnyk, M. H. "Mathematical modeling of temperature fields of spacecraft antenna reflectors with a new inte-gral transformation." Mathematical machines and systems 3 (2022): 158–65. http://dx.doi.org/10.34121/1028-9763-2022-3-158-165.

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Prediction of the mechanical properties of the reflector, and above all, the deviation of the highly accurate shape of the reflecting surface (RSS) from the given one is the main goal of designing spacecraft antennas. Distortion of the RSS is determined by the stress-deformed state of the elements of the reflector structures under the conditions of orbital operation. At the same time, the main factor determining the distortion of reflectors with RSS in open space is temperature deformation due to the uneven distribution of solar heat fluxes among structural elements. Therefore, the development of methods and models for calculating temperature fields in reflectors during heat flows on the surface is relevant. In the article, for the first time, a new finite integral transformation for the Laplace equation in a cylindrical coordinate system is constructed for a region bounded by several closed piecewise smooth contours. The inverse transformation formula is given. In the article, for the first time, a mathematical model for the calculation of temperature fields in a paraboloid rotating with a constant angular velocity is constructed, taking into account the finite speed of heat propagation in the form of a boundary value problem of mathematical physics for the hyperbolic equation of heat conduction with Dirichlet boundary conditions. With the help of the developed integral transformation, the temperature fields in the paraboloid were found in the form of convergent series according to the Fourier functions. The found solution to the generalized boundary value problem of the heat transfer of the paraboloid of rotation can find application in modulating the temperature fields that arise in the antenna reflectors of space vehicles. The developed integral transformation makes it possible to obtain solutions to complex boundary value problems of mathematical physics.

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36

Barakat, Joseph M., and Eric S. G. Shaqfeh. "Stokes flow of vesicles in a circular tube." Journal of Fluid Mechanics 851 (July 30, 2018): 606–35. http://dx.doi.org/10.1017/jfm.2018.533.

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The inertialess motion of lipid-bilayer vesicles flowing through a circular tube is investigated via direct numerical simulation and lubrication theory. A fully three-dimensional boundary integral equation method, previously used to study unbounded and wall-bounded Stokes flows around freely suspended vesicles, is extended to study the hindered mobility of vesicles through conduits of arbitrary cross-section. This study focuses on the motion of a periodic train of vesicles positioned concentrically inside a circular tube, with particular attention given to the effects of tube confinement, vesicle deformation and membrane bending elasticity. When the tube diameter is comparable to the transverse dimension of the vesicle, axisymmetric lubrication theory provides an approximate solution to the full Stokes-flow problem. By combining the present numerical results with a previously reported asymptotic theory (Barakat & Shaqfeh, J. Fluid Mech., vol. 835, 2018, pp. 721–761), useful correlations are developed for the vesicle velocity $U$ and extra pressure drop $\unicode[STIX]{x0394}p^{+}$. When bending elasticity is relatively weak, these correlations are solely functions of the geometry of the system (independent of the imposed flow rate). The prediction of Barakat & Shaqfeh (2018) supplies the correct limiting behaviour of $U$ and $\unicode[STIX]{x0394}p^{+}$ near maximal confinement, whereas the present study extends this result to all regimes of confinement. Vesicle–vesicle interactions, shape transitions induced by symmetry breaking, and unsteadiness introduce quantitative changes to $U$ and $\unicode[STIX]{x0394}p^{+}$. By contrast, membrane bending elasticity can qualitatively affect the hydrodynamics at sufficiently low flow rates. The dependence of $U$ and $\unicode[STIX]{x0394}p^{+}$ on the membrane bending stiffness (relative to a characteristic viscous stress scale) is found to be rather complex. In particular, the competition between viscous forces and bending forces can hinder or enhance the vesicle’s mobility, depending on the geometry and flow conditions.

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37

Mimna and Wingler. "LOCALLY BOUNDED FUNCTIONS." Real Analysis Exchange 23, no. 1 (1997): 251. http://dx.doi.org/10.2307/44152849.

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38

Esterle, J. "Bounded Cosine Functions Close to Continuous Scalar Bounded Cosine Functions." Integral Equations and Operator Theory 85, no. 3 (June 7, 2016): 347–57. http://dx.doi.org/10.1007/s00020-016-2304-3.

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39

Cohen, Joel M., and Flavia Colonna. "Bounded holomorphic functions on bounded symmetric domains." Transactions of the American Mathematical Society 343, no. 1 (January 1, 1994): 135–56. http://dx.doi.org/10.1090/s0002-9947-1994-1176085-6.

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40

Koepf, Wolfram, and Dieter Schmersau. "Bounded nonvanishing functions and bateman functions." Complex Variables, Theory and Application: An International Journal 25, no. 3 (August 1994): 237–59. http://dx.doi.org/10.1080/17476939408814746.

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41

Cadena, M., M. Kratz, and E. Omey. "On Functions Bounded by Karamata Functions." Journal of Mathematical Sciences 237, no. 5 (February 19, 2019): 621–30. http://dx.doi.org/10.1007/s10958-019-04187-z.

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42

Libera, Richard J., and Eligiusz Złotkiewicz. "Bounded Montel univalent functions." Colloquium Mathematicum 56, no. 1 (1988): 169–77. http://dx.doi.org/10.4064/cm-56-1-169-177.

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43

Cang, Dingbang. "Polynomially Bounded Cosine Functions." Analysis in Theory and Applications 28, no. 1 (June 2012): 13–18. http://dx.doi.org/10.4208/ata.2012.v28.n1.2.

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44

., Jyoti. "Functions of Bounded Variation." Journal of Advances and Scholarly Researches in Allied Education 15, no. 4 (June 1, 2018): 250–52. http://dx.doi.org/10.29070/15/57855.

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45

Bang, Ha Huy. "Functions with bounded spectrum." Transactions of the American Mathematical Society 347, no. 3 (March 1, 1995): 1067–80. http://dx.doi.org/10.1090/s0002-9947-1995-1283539-1.

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46

Vogt, Andreas. "On Bounded Universal Functions." Computational Methods and Function Theory 12, no. 1 (January 21, 2012): 213–19. http://dx.doi.org/10.1007/bf03321823.

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47

Thill, Marco. "Exponentially bounded indefinite functions." Mathematische Annalen 285, no. 2 (October 1989): 297–307. http://dx.doi.org/10.1007/bf01443520.

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48

Pipher, Jill. "Bounded double square functions." Annales de l’institut Fourier 36, no. 2 (1986): 69–82. http://dx.doi.org/10.5802/aif.1048.

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49

Baldi, Paolo, and Hykel Hosni. "Depth-bounded Belief functions." International Journal of Approximate Reasoning 123 (August 2020): 26–40. http://dx.doi.org/10.1016/j.ijar.2020.05.001.

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50

Roy, Ranjan, and S. M. Shah. "Functions of bounded index, bounded value distribution and v-bounded index." Nonlinear Analysis: Theory, Methods & Applications 11, no. 12 (December 1987): 1383–90. http://dx.doi.org/10.1016/0362-546x(87)90090-3.

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