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Journal articles on the topic 'Beckner inequality'

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Published: 1 April 2023

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1

Kondratyev, Stanislav, Léonard Monsaingeon, and Dmitry Vorotnikov. "A new multicomponent Poincaré–Beckner inequality." Journal of Functional Analysis 272, no. 8 (April 2017): 3281–310. http://dx.doi.org/10.1016/j.jfa.2016.12.018.

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2

Deng, Pingji, and Fengyu Wang. "Beckner Inequality on Finite- and Infinite-Dimensional Manifolds*." Chinese Annals of Mathematics, Series B 27, no. 5 (October 2006): 581–94. http://dx.doi.org/10.1007/s11401-004-0317-8.

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3

AMRI, BESMA, and LAKHDAR T. RACHDI. "BECKNER LOGARITHMIC UNCERTAINTY PRINCIPLE FOR THE RIEMANN–LIOUVILLE OPERATOR." International Journal of Mathematics 24, no. 09 (August 2013): 1350070. http://dx.doi.org/10.1142/s0129167x13500705.

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First, we establish the Stein–Weiss inequality for the B-Riesz potential generated by the Riemann–Liouville operator. Next, we prove the Pitt's and Beckner logarithmic inequalities related to the connected Fourier transform.

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4

Kim, Meelae. "Linearised Moser-Trudinger inequality." Bulletin of the Australian Mathematical Society 62, no. 3 (December 2000): 445–57. http://dx.doi.org/10.1017/s0004972700018967.

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As a limiting case of the Sobolev imbedding theorem, the Moser-Trudinger inequality was obtained for functions in with resulting exponential class integrability. Here we prove this inequality again and at the same time get sharper information for the bound. We also generalise the Linearised Moser inequality to higher dimensions, which was first introduced by Beckner for functions on the unit disc. Both of our results are obtained by using the method of Carleson and Chang. The last section introduces an analogue of each inequality for the Laplacian instead of the gradient under some restricted conditions.

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5

Yao, Nian, and Zhengliang Zhang. "Beckner inequalities for Moebius measures on spheres." ESAIM: Probability and Statistics 23 (2019): 552–66. http://dx.doi.org/10.1051/ps/2018025.

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In this paper, we consider the Moebius measures μxn indexed by dimension n and |x| < 1 on the unit sphere Sn−1 in ℝn (n ≥ 3), and provide a precise two-sided estimate on the order of the Beckner inequality constant with exponent p ∈ [1, 2) in the three parameters. As special cases for p = 1 and p tending to 2, our results cover those in Barthe et al. [Forum Math. (submitted for publication)] for n ≥ 3 and explore an interesting phenomenon.

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6

Da Pelo, Paolo, Alberto Lanconelli, and Aurel I. Stan. "A sharp interpolation between the Hölder and Gaussian Young inequalities." Infinite Dimensional Analysis, Quantum Probability and Related Topics 19, no. 01 (March 2016): 1650001. http://dx.doi.org/10.1142/s0219025716500016.

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We prove a very general sharp inequality of the Hölder–Young-type for functions defined on infinite dimensional Gaussian spaces. We begin by considering a family of commutative products for functions which interpolates between the pointwise and Wick products; this family arises naturally in the context of stochastic differential equations, through Wong–Zakai-type approximation theorems, and plays a key role in some generalizations of the Beckner-type Poincaré inequality. We then obtain a crucial integral representation for that family of products which is employed, together with a generalization of the classic Young inequality due to Lieb, to prove our main theorem. We stress that our main inequality contains as particular cases the Hölder inequality and Nelson’s hyper-contractive estimate, thus providing a unified framework for two fundamental results of the Gaussian analysis.

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7

Cazacu, Cristian. "A new proof of the Hardy–Rellich inequality in any dimension." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 6 (August 19, 2019): 2894–904. http://dx.doi.org/10.1017/prm.2019.50.

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The Hardy-Rellich inequality in the whole space with the best constant was firstly proved by Tertikas and Zographopoulos in Adv. Math. (2007) in higher dimensions N ⩾ 5. Then it was extended to lower dimensions N ∈ {3, 4} by Beckner in Forum Math. (2008) and Ghoussoub-Moradifam in Math. Ann. (2011) by applying totally different techniques.In this note, we refine the method implemented by Tertikas and Zographopoulos, based on spherical harmonics decomposition, to give an easy and compact proof of the optimal Hardy–Rellich inequality in any dimension N ⩾ 3. In addition, we provide minimizing sequences which were not explicitly mentioned in the quoted papers in lower dimensions N ∈ {3, 4}, emphasizing their symmetry breaking. We also show that the best constant is not attained in the proper functional space.

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8

Santana-Carrillo, R., Jesus S. González-Flores, Emilio Magaña-Espinal, Luis F. Quezada, Guo-Hua Sun, and Shi-Hai Dong. "Quantum Information Entropy of Hyperbolic Potentials in Fractional Schrödinger Equation." Entropy 24, no. 11 (October 24, 2022): 1516. http://dx.doi.org/10.3390/e24111516.

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In this work we have studied the Shannon information entropy for two hyperbolic single-well potentials in the fractional Schrödinger equation (the fractional derivative number (0<n≤2) by calculating position and momentum entropy. We find that the wave function will move towards the origin as the fractional derivative number n decreases and the position entropy density becomes more severely localized in more fractional system, i.e., for smaller values of n, but the momentum probability density becomes more delocalized. And then we study the Beckner Bialynicki-Birula–Mycieslki (BBM) inequality and notice that the Shannon entropies still satisfy this inequality for different depth u even though this inequality decreases (or increases) gradually as the depth u of the hyperbolic potential U1 (or U2) increases. Finally, we also carry out the Fisher entropy and observe that the Fisher entropy increases as the depth u of the potential wells increases, while the fractional derivative number n decreases.

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9

Mossel, Elchanan, Ryan O'Donnell, Oded Regev, Jeffrey E. Steif, and Benny Sudakov. "Non-interactive correlation distillation, inhomogeneous Markov chains, and the reverse Bonami-Beckner inequality." Israel Journal of Mathematics 154, no. 1 (December 2006): 299–336. http://dx.doi.org/10.1007/bf02773611.

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10

Pedram, Pouria. "The Minimal Length and the Shannon Entropic Uncertainty Relation." Advances in High Energy Physics 2016 (2016): 1–8. http://dx.doi.org/10.1155/2016/5101389.

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In the framework of the generalized uncertainty principle, the position and momentum operators obey the modified commutation relationX,P=iħ1+βP2, whereβis the deformation parameter. Since the validity of the uncertainty relation for the Shannon entropies proposed by Beckner, Bialynicki-Birula, and Mycielski (BBM) depends on both the algebra and the used representation, we show that using the formally self-adjoint representation, that is,X=xandP=tan⁡βp/β, where[x,p]=iħ, the BBM inequality is still valid in the formSx+Sp≥1+ln⁡πas well as in ordinary quantum mechanics. We explicitly indicate this result for the harmonic oscillator in the presence of the minimal length.

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11

Saito, Kichi-Suke, and Ryotaro Tanaka. "On generalized Beckner's inequality." Annals of Functional Analysis 6, no. 1 (2015): 267–78. http://dx.doi.org/10.15352/afa/06-1-20.

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12

Zhang, Lina, and Xuesi Ma. "New Refinements and Improvements of Some Trigonometric Inequalities Based on Padé Approximant." Journal of Mathematics 2020 (May 19, 2020): 1–7. http://dx.doi.org/10.1155/2020/2753691.

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A multiple-point Padé approximant method is presented for approximating and bounding some trigonometric functions in this paper. We give new refinements and improvements of some trigonometric inequalities including Jordan’s inequality, Kober’s inequality, and Becker-Stark’s inequality. The analysis results show that our conclusions are better than the previous conclusions.

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13

Tanaka, Ryotaro, Kichi-Suke Saito, and Naoto Komuro. "Another approach to Beckner's inequality." Journal of Mathematical Inequalities, no. 3 (2013): 543–49. http://dx.doi.org/10.7153/jmi-07-50.

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14

Nishizawa, Yusuke. "Extended constant parts of Becker-Stark's and Shafer-Fink's inequalities." Tamkang Journal of Mathematics 47, no. 4 (December 30, 2016): 385–91. http://dx.doi.org/10.5556/j.tkjm.47.2016.2024.

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15

Zhu, Ling. "New Sharp Double Inequality of Becker–Stark Type." Mathematics 10, no. 4 (February 11, 2022): 558. http://dx.doi.org/10.3390/math10040558.

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In this paper, we establish new sharp double inequality of Becker–Stark type by using a role of the monotonicity criterion for the quotient of power series and the estimation of the ratio of two adjacent even-indexed Bernoulli numbers. The inequality results are better than those in the existing literature.

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16

Gwynne, Ewain, and Elton Hsu. "On Beckner’s inequality for Gaussian measures." Elemente der Mathematik 70, no. 1 (2015): 33–38. http://dx.doi.org/10.4171/em/270.

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17

Wu, Shengjian, and Shanshuang Yang. "On Symmetric Quasicircles." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 68, no. 1 (February 2000): 131–44. http://dx.doi.org/10.1017/s1446788700001622.

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AbstractWe study an important subclass of quasicircles, namely, symmetric quasicircles. Several characterizations for quasicircles, such as the reverse triangle inequality, the M -condition and the quasiconformal extension property, have been extended to symmetric quasicircles by Becker and Pommerenke and by Gardiner and Sullivan. In this paper we establish several relations among various domain constants such as quasiextremal distance constants, (local) reflection constants and (local) extension constants for this class. We also give several characterizations for symmetric quasicircles such as the strong quadrilateral inequality and the strong extremal distance property. They correspond to the quadrilateral inequality and the extremal distance property for quasicircles.

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18

Ge, Hua-feng. "New Sharp Bounds for the Bernoulli Numbers and Refinement of Becker-Stark Inequalities." Journal of Applied Mathematics 2012 (2012): 1–7. http://dx.doi.org/10.1155/2012/137507.

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We obtain new sharp bounds for the Bernoulli numbers:2(2n)!/(π2n(22n−1))<|B2n|≤(2(22k−1)/22k)ζ(2k)(2n)!/(π2n(22n−1)),n=k,k+1,…, k∈N+, and establish sharpening of Papenfuss's inequalities, the refinements of Becker-Stark, and Steckin's inequalities. Finally, we show a new simple proof of Ruehr-Shafer inequality.

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19

Gui, Changfeng, Yeyao Hu, and Weihong Xie. "Improved Beckner's inequality for axially symmetric functions on Sn." Journal of Functional Analysis 282, no. 5 (March 2022): 109335. http://dx.doi.org/10.1016/j.jfa.2021.109335.

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20

Hleili, Khaled. "A variety of uncertainty principles for the Hankel-Stockwell transform." Open Journal of Mathematical Analysis 5, no. 1 (January 29, 2021): 22–34. http://dx.doi.org/10.30538/psrp-oma2021.0079.

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In this work, we establish \(L^p\) local uncertainty principle for the Hankel-Stockwell transform and we deduce \(L^p\) version of Heisenberg-Pauli-Weyl uncertainty principle. Next, By combining these principles and the techniques of Donoho-Stark we present uncertainty principles of concentration type in the \(L^p\) theory, when \(1< p\leqslant2\). Finally, Pitt's inequality and Beckner's uncertainty principle are proved for this transform.

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21

Nielsen, Ole A. "Sharpness in Young's Inequality for Convolution Products." Canadian Journal of Mathematics 46, no. 06 (December 1994): 1287–98. http://dx.doi.org/10.4153/cjm-1994-073-7.

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AbstractSuppose that Gis a locally compact group with modular function Δ and that p, q, r are three numbers in the interval (l,∞) satisfying. If cp,q(G) is the smallest constant c such thatfor all functions f, g ∈ Cc(G) (here the convolution product is with respect to left Haar measure andis the exponent which is conjugate to p) then Young's inequality asserts that cp,q(G) ≤ 1. This paper contains three results about these constants. Firstly, if G contains a compact open subgroup then cp,q(G) = 1 and, as an extension of an earlier result of J. J. F. Fournier, it is shown that there is a constant cp,q< 1 such that if G does not contain a compact open subgroup then c<(G) ≤ c≤. Secondly, Beckner's calculation ofis used to obtain the value of cp,q(G) for all simply-connected solvable Lie groups and all nilpotent Lie groups. And thirdly, it is shown that for a nilpotent Lie group the setis not contained in the union of the spaces Ls(G),.

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22

Ziliak, Stephen T. "Urban Inequality: Evidence from Four Cities. Edited by Alice O'Connor, Chris Tilly, and Lawrence D. Bobo. New York: Russell Sage Foundation, 2001. Pp. xii, 549. $45.00." Journal of Economic History 61, no. 4 (December 2001): 1145–46. http://dx.doi.org/10.1017/s0022050701005812.

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In some inquiries, racism is a given, a fundamental assumption and fact that is known to shape economic and social and spatial inequalities. In such inquiries, from the empirical studies of W. E. B. Du Bois to the social thought of Patricia Williams, the world is believed to be racialized “always already”: the task of the researcher, then, is to probe the roots of racism in particular cases, and to liberate (or compensate) the people who are oppressed by its institutional manifestations. Other inquiries—guided by a methodology of rational choice—do not assume racism to be a given. For example, in mainstream economics after Gary Becker, race is a possible “factor,” one vector of a matrix, whose strength and direction of force on earnings or housing differences is to be examined in a regression equation once all other factors have been held constant. Indeed, some economists after Becker argue that racism cannot explain differences in earnings or in opportunities because if the marginal product of labor (or the bid on a house) of a person of color exceeds that of the white competition, the argument goes, then no profit-maximizing employer or seller would deny the offer of the person of color. And since all employers and households are assumed to be profit-maximizers, the color of one's skin will have no economic significance at all.

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23

Da Pelo, Paolo, Alberto Lanconelli, and Aurel I. Stan. "An extension of the Beckner’s type Poincaré inequality to convolution measures on abstract Wiener spaces." Stochastic Analysis and Applications 34, no. 1 (December 23, 2015): 47–64. http://dx.doi.org/10.1080/07362994.2015.1099443.

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24

Bosi, Stefano, Raouf Boucekkine, and Thomas Seegmuller. "The Dynamics of Wealth Inequality under Endogenous Fertility: A Remark on the Barro-Becker Model with Heterogenous Endowments." Theoretical Economics Letters 01, no. 01 (2011): 3–7. http://dx.doi.org/10.4236/tel.2011.11002.

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25

Menezes, Eliomar Cruz, Alberto Moreira Silva-Neto, Francisco Eriberto Lima Nascimento, and Freddy Ruben Bravo. "Lista dos Cerambycidae, incluindo 12 Holótipos, Presentes no Museu de Zoologia da Universidade Estadual de Feira de Santana e o Primeiro Registro da espécie Chrysoprasis airi Napp & Martins para o Brasil." EntomoBrasilis 5, no. 1 (March 25, 2012): 49–58. http://dx.doi.org/10.12741/ebrasilis.v5i1.191.

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O Brasil é considerado um das nações mais ricas em biodiversidade de insetos, porém possui uma grande desigualdade de estudos na área de zoologia ao longo de suas regiões geográficas, devido a desigual divisão de recursos e de mão de obra especializada. A falta de coleções zoológicas importantes na região nordeste, as dificuldades em obter os recursos necessários para arcar com os altos custos de manutenção destas coleções e a falta de divulgação de suas bases de dados são pontos chaves das causas dessa desigualdade. O objetivo deste trabalho é divulgar a lista das espécies da família Cerambicydae, incluindo 12 holótipos, presentes na coleção entomológica Professor Johann Becker do Museu de Zoologia da Universidade Estadual de Feira de Santana (MZFS) e fazer o primeiro registro da espécie Chrysoprasis airi Napp & Martins para o Brasil. Foram contabilizados mil e sessenta e quatro espécimes de Cerambycidae distribuídas em cento e cinqüenta e quatro espécies, cento e nove gêneros, quarenta e cinco tribos e três subfamílias, depositadas no MZFS. Do total dessas espécies 83,3% foram coletadas no estado da Bahia, sendo ainda 90,4% pertencentes a municípios da região do Semi-árido, que possui sua fauna de insetos pouco estudada. Com base nos resultados pode-se concluir que as publicações dos dados armazenados nas coleções entomológicas podem contribuir em muito para um melhor entendimento da real diversidade e distribuição da entomofauna brasileira.List of Cerambycidae, Including 12 Holotypes, Present in the Museum of Zoology, State University of Feira de Santana and the First Record of the Species Chrysoprasis airi Napp & Martins to Brazil.Abstract. Brazil is considered one of the richest nations on the biodiversity of insects, but has a great inequality in the studies of zoology throughout their geographic regions due to unequal division of resources and skilled labor. The lack of major zoological collections in the Northeast, the difficulties in obtaining the resources needed to afford the high costs of maintaining these collections and the lack of disclosure of their databases are key points of the causes of this inequality. The objective of this work is to make list of species of the family Cerambycidae, including 12 holotypes, present in the entomological collection of Prof. Johann Becker Museum of Zoology, State University of Feira de Santana (MZFS) and make the first record of the species Chrysoprasis airi Napp & Martins to Brazil. We counted one thousand and sixty-four specimens of Cerambycidae distributed in one hundred and fifty-four species, one hundred and nine genera, forty-five tribes and three subfamilies, deposited in MZUEFS. Of the total 83.3% of these species were collected in the state of Bahia, is still 90.4% owned by municipalities in the semi-arid region, which has its insect fauna of poorly studied. Based on the results we can conclude that the publications of the data stored in the entomological collections can contribute greatly to a better understanding of the actual diversity and distribution of Brazilian insect fauna.

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26

Edward Curtis Jr., James. "Wealth Discrimination Theory." International Research in Economics and Finance 2, no. 2 (August 8, 2018): 1. http://dx.doi.org/10.20849/iref.v2i2.443.

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One approach to analyzing inequality is to compare average economic choices from a classical theoretical framework. Another approach considers the impact of the formation of society, through statutes and institutions, on average economic outcomes. This paper studies the effects of slavery on black-white wealth inequality upon the emancipation of slaves in the US using historical data. The purpose of wealth has varied from over time. From an economics perspective, wealth is the accumulation of resources that have market value and can be liquidated for present and future consumption. This study proceeds based on the most measurable assumption: households reside in a country with a mixed economy of markets and social planning, such that they have an incentive to accumulate material wealth for intertemporal household consumption and social influence. Becker (1957) and Arrow (1972) developed the most general theories of wage discrimination and favoritism. Oaxaca (1973) and Blinder (1973) have mechanized their theories for empirical analysis. While their findings are insightful, they cannot be directly applied to studying wealth differences since wealth is a complex combination of wages and other variables. Finally, since unexplained differences in states that abolished slavery after the Civil War were 10 percent higher than unexplained effects in states that abolished slavery well before the Civil War and the magnitudes of the unexplained effects were similar over the long-run, we cannot reject the existence of a negatively bounded correlation between the duration of time from enslavement and the magnitude of unexplained differences in wealth. This research was funded in part by the National Science Foundation under Grant SES 0096414. I would like to thank John Ham, Richard Steckel, Randall Olsen, Bruce Weinberg, Audrey Light, Nori Hashimoto, James Peck, Patricia Reagan, Charles Kirwin, Rebecca Blank, Charles Betsey, Alvin Thornton, Leibert Morris, Maude Toussaint-Comeau, Simone Wegge, James Wilbanks, Thomas Maloney, and William Collins for their insightful comments. I would also like to thank participants in workshops and seminars at the Ohio State University, Howard University, University of Michigan, American Economic Association Summer Program and Pipeline Conferences, Western Economics Association International meetings, and Social Science History Association meetings. I would also like to thank James Curtis Sr, K D Curtis, Karen Curtis (deceased), Lariece Grant-Brown, Barbara Broadnax, Dwayne Broadnax, Rudy Broadnax, Zee Curtis-Grant, Raymond Tillery, Chris Cooper, Dr. K A Troy, Dr. H. Beecher Hicks, Reverend Charles Lewis, Reverend Cornelius Wheeler, Reverend James Lewis, Elder David Treadwell, Dr. Stephen Tucker and Roberta Tucker, Minister Charles Webb, Minister David Surles, and Elder Gregory Strong for their support. This draft is a revision of a November 2010 paper and August 2001 paper.

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27

Shareef, Fareed, Muhammad Junaid Khawaja, and Toseef Azid. "Does parents’ income matter in intergenerational transmission of human capital? A decomposition analysis." International Journal of Social Economics 44, no. 2 (February 13, 2017): 267–83. http://dx.doi.org/10.1108/ijse-11-2014-0235.

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Purpose Since the pristine works of Schultz (1961) and Becker (1964, 1975), the concept of intergenerational transmission has constantly been in the front line of discussion among the social scientists to divulge the sources and channels through which diffusion of socio-economic status can take place across the generations. The purpose of this paper is to explore the intergeneration links via monetary channels through decomposition technique. Design/methodology/approach Using a sample of 613 households selecting through systematic sampling from Multan district (Pakistan). Making a three tier analyses, i.e. simple, sequential and double decomposition, the findings of the models support the hypothesis of the study that children of high-income parents also fall in high-income groups. Findings The simple decomposition analysis using education as the pathway factor reveals that parental income is pivotal in determining the education and ultimately the level of their child’s income. The sequential analysis incorporates occupation and depicts a positive association between the offspring education and occupation. In the double decomposition analysis, the direct component reveals that even among those children with the same level of education, higher parental income is linked with the better occupational achievements, whereas indirect component explains the impact of parental income on occupation via education of the children. In other words, it explains the degree to which children with higher family income acquire more education and consequently get better jobs. Research limitations/implications In Pakistan like the other developing countries nationwide surveys are not conducted at the government level. Practical implications This study is providing the guideline to the policy makers for the formulating their policies for developing and managing the human capital. Social implications The findings of this study are useful for reducing the inequality in the society. Originality/value This is an original and first time it is going to be conducted in a country like Pakistan

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28

Liadova, Anna V. "The peculiarities of the development of sociology of medicine as the interdisciplinary scientific direction." Moscow State University Bulletin. Series 18. Sociology and Political Science 25, no. 1 (April 18, 2019): 42–61. http://dx.doi.org/10.24290/1029-3736-2019-25-1-42-61.

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The article examines the main trends in the development of sociology of medicine as a branch of sociology. As it is pointed out by the author, nowadays sociology of medicine is an actual scientific course about social aspects of health and illness, medicine as a social structure. All these items are important under the modern social development. Based on the analysis of the main stages in the history of sociology of medicine, the author notes the contribution of physicians to this process and practical meaning of early studies. The accumulation of empirical knowledge on health led to the theoretical comprehension, that was done by American sociologists, such as T. Parsons, E. Freidson, E. Goffman, H. Becker, R. Merton, and others. The author pays special attention to the study of T. Parsons “The Social System”, in which a structural and functional analysis of medicine as a social institution was presented. After it became obvious that the sociology of medicine can develop not only as an applied field of scientific research, but also a theoretical one. As A.V. Reshetnikov, W. Cockerheim, S. Bloom note, the process of institutionalization of the sociology of medicine ends in the second half of the 20th century in Europe and America, at the end of the 20th and beginning of the 21st centuries in Russia. The main trends in studies around sociology of medicine are based on such theoretical approaches of sociology as symbolic interactionism, structural functionalism, social constructionism. In conclusion, the author underlines that, the sociology of medicine, like any field of scientific knowledge, reacts dynamically to the ongoing changes in social reality, its structure, social institutions. As a result of this influence, the research field of the sociology of medicine is expanding. Among the topical issues are social inequality in health, the features of the development of digital health, new practices in health, transformation of the value of health in the post-industrial society, pharmaceticisation, biomedicalization, etc.

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29

DEACON, ALAN. "Saul Becker, Responding to Poverty, Longman, London, 1997, xii + 223 pp., £12.99 pbk. Alissa Goodman, Paul Johnson and Steven Webb, Inequality in the UK, Oxford University Press, Oxford, 1997, xvi + 297 pp., £35.00, £15.99 pbk. David Gordon and Christine Pantazis (eds.), Breadline Britain in the 1990s, Ashgate, Aldershot, 1997, xxiv + 309 pp., £42.00. David G. Green, Benefit Dependency, Institute of Economic Affairs, London, 1998, xi + 49 pp., £7.00 pbk. John Hills, Income and Wealth, Joseph Rowntree Foundation, York, 1998, 80 pp., £13.95 pbk." Journal of Social Policy 28, no. 1 (January 1999): 139–73. http://dx.doi.org/10.1017/s0047279499375504.

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30

Kim, Daesung. "Instability results for the logarithmic Sobolev inequality and its application to related inequalities." Discrete and Continuous Dynamical Systems, 2022, 0. http://dx.doi.org/10.3934/dcds.2022053.

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<p style='text-indent:20px;'>We show that there are no general stability results for the logarithmic Sobolev inequality in terms of the Wasserstein distances and <inline-formula><tex-math id="M1">\begin{document}$ L^{p}(d\gamma) $\end{document}</tex-math></inline-formula> distance for <inline-formula><tex-math id="M2">\begin{document}$ p>1 $\end{document}</tex-math></inline-formula>. To this end, we construct a sequence of centered probability measures such that the deficit of the logarithmic Sobolev inequality converges to zero but the relative entropy and the moments do not, which leads to instability for the logarithmic Sobolev inequality. As an application, we prove instability results for Talagrand's transportation inequality and the Beckner–Hirschman inequality.</p>

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31

Singh, Sudin. "A Theoretical Study on the Information Theoretic Inequalities and Fisher-Shannon Product of a Free Particle." Asian Journal of Research and Reviews in Physics, October 17, 2022, 25–32. http://dx.doi.org/10.9734/ajr2p/2022/v6i3119.

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In this article, the plane wave solution for a free particle in three dimensions is considered and the wave function is normalized in an arbitrarily large but finite cube. The momentum space wave function is obtained by taking the Fourier transform of the coordinate space wave function. The probability densities are employed to compute the numerical values of the information theoretic quantities such as Shannon information entropy (S), Fisher information entropy (I), Shannon power (J) and the Fisher–Shannon product (P) both in coordinate and momentum spaces for different values of the length (L) of the cubical box. Numerical values so found satisfy the Beckner, Bialynicki-Birula and Myceilski (BBM) inequality relation; Stam-Cramer-Rao inequalities (better known as the Fisher based uncertainty relation) and Fisher-Shannon product relation. This establishes the validity of the information theoretic inequalities in respect of the motion of a free particle.

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32

Zhou, Huaiyu, and Jingbo Dou. "Classifications of positive solutions to an integral system involving the multilinear fractional integral inequality." Discrete and Continuous Dynamical Systems, 2022, 0. http://dx.doi.org/10.3934/dcds.2022070.

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<p style='text-indent:20px;'>In this paper, we classify the positive solutions to the following integral system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \begin{cases} u_s(x_s) = \int_{\mathbb{R}^{N(k-1)}}\frac{\prod\limits_{j\neq s}u^{p_j}_j(x_j)}{\prod_{1\leq i<j\leq k}|x_i-x_j|^{N-h_{ij}}}dX_{\widehat{s}}, \\ u_s\geq0, \; \rm{in}\quad\mathbb{R}^N, \quad s = 1, 2, \cdots, k, \end{cases} \end{eqnarray*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ N\ge1, p_j>1, 0<h_{ij}<N $\end{document}</tex-math></inline-formula> for all <inline-formula><tex-math id="M2">\begin{document}$ i, j\in\{1, 2, \cdots, k\} $\end{document}</tex-math></inline-formula>. Up to a positive constant multiplier, this system is the Euler-Lagrangian equations associated to the multilinear fractional integral inequality established by Beckner. Employing the method of moving spheres, we give the explicit form of positive solutions to the above system with <inline-formula><tex-math id="M3">\begin{document}$ p_j = \frac{\sum_{1\leq i<j\leq k} \ \ h_{ij}-(k-3)N}{(k-1)N-\sum_{1\leq i<j\leq k} \ \ h_{ij}} $\end{document}</tex-math></inline-formula> satisfying</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \sum\limits^k_{j = 1}\frac{1}{p_j+1} = \sum\limits_{1\leq i<j\leq k}\frac{N-h_{ij}}{N}, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and show the nonexistence of positive solutions for <inline-formula><tex-math id="M4">\begin{document}$ p_j>1 $\end{document}</tex-math></inline-formula> with</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ \sum\limits^k_{j = 1}\frac{1}{p_j+1}>\sum\limits_{1\leq i<j\leq k}\frac{N-h_{ij}}{N}. $\end{document} </tex-math></disp-formula></p>

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"Inequality and economic policy: essays in memory of Gary Becker." Choice Reviews Online 53, no. 11 (June 21, 2016): 53–4881. http://dx.doi.org/10.5860/choice.196565.

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OTINASHVILI, RAMAZ. "HUMAN RESOURCE AND PECULIARITIES OF INCOME DISTRIBUTION." Globalization and Business, July 5, 2021, 101–7. http://dx.doi.org/10.35945/gb.2021.11.014.

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The country's leading factor in human development that is a human resource. It is the creator of everything and it is necessary to create a law and protect human resource. People need to implement, pursue the economic policy and ensure business efficiency. That functions, to prevent crime and to restore order. Man builds the country, but to build well, knowledge is needed and knowledge needs education. Humans capital is a body of knowledge and includes philosophical, social, economic and psychological aspects. This term dates from the eighty-seventh year of the twentieth century to its end. It is founder is Nobel Prize Laureate G. Becker. Human capital consists of: educational, professional-quality , information-cultural and social connections. Among them, the capital of the Social Union dominates in the conditions of Georgia. According to the GAERO, Georgia ranks 61st out of 189 countries in the Human Development Index. Human resources determine a person’s income. It is important to note that human capital is one of the main causes of income inequality. We differ from each other in terms of data and capabilities. It’s important that state should have a fair social-economic environment, which gives all possible conditions to citizens for the realization of opportunity

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Папітченко, Лінда. "ТЕОРЕТИЧНИЙ АНАЛІЗ ПРОБЛЕМИ ЗНАЧУЩОСТІ САМООЦІНКИ В СТРУКТУРІ ГІДНОСТІ ОСОБИСТОСТІ." Науковий часопис НПУ імені М. П. Драгоманова. Серія 12. Психологічні науки, December 29, 2020, 79–89. http://dx.doi.org/10.31392/npu-nc.series12.2020.12(57).07.

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У статті презентовано результати теоретичного аналізу позицій науковців щодо розуміння феномена «самооцінки» від початку зародження цього поняття в кінці дев’ятнадцятого століття до сьогодення. Мета дослідження полягала в здійсненні теоретичного аналізу проблеми формування самооцінки і визначення її значущості в структурі гідності особистості. Завдання дослідження: простежити ключові позиції науковців у розумінні феномена самооцінки та розглянути її місце в структурній моделі гідності особисті, а також теоретично обґрунтувати наявність впливу соціокультурного середовища на формування самооцінки особистості.Теоретично проаналізовано і схематично подано взаємозв’язок структурних компонентів самооцінки у моделі особистісної гідності. Звернено увагу на наявність експліцитної та імпліцитної самооцінки особистості, відмінності між ними й рівень її усвідомленості: когнітивно доступний і підсвідомий. Зазначено, що структура самооцінки презентована трьома компонентами: 1) емоційним, що засвідчує ставлення до самого себе і міру задоволеності собою; 2) когнітивним, що відбиває знання людини про себе; 3) поведінковим, який окреслює ставлення до себе на рівні дії. Окреслено важливість самооцінки в життєдіяльності особистості, адже вона впливає на рівень домагань і прагнень людини в різних сферах життя, а також демонструє зіставлення між «Я-реальним» і «Я-ідеальним». Акцентовано увагу на тому, що особистість отримує (надає собі) найвищу самооцінку із аспектів самоідентичності, які найкращим чином відповідають цінностям культури, що її оточує. Система самооцінки є важливим способом, за допомогою якого люди засвоюють цінності своєї культури на імпліцитному рівні, навіть якщо вони не усвідомлюють цього. У результаті теоретичного аналізу виявлено, що люди з неконгруентною самооцінкою більш непередбачувані та реакційні. 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