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Journal articles on the topic 'M-Integral'

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Author: Grafiati

Published: 14 March 2023

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1

Haddad, Roudy El. "Repeated integration and explicit formula for the \(n\)-th integral of \(x^m(\ln x)^{m'}\)." Open Journal of Mathematical Sciences 6, no. 1 (June 10, 2022): 51–75. http://dx.doi.org/10.30538/oms2022.0178.

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Repeated integration is a major topic of integral calculus. In this article, we study repeated integration. In particular, we study repeated integrals and recurrent integrals. For each of these integrals, we develop reduction formulae for both the definite as well as indefinite form. These reduction formulae express these repetitive integrals in terms of single integrals. We also derive a generalization of the fundamental theorem of calculus that expresses a definite integral in terms of an indefinite integral for repeated and recurrent integrals. From the recurrent integral formulae, we derive some partition identities. Then we provide an explicit formula for the \(n\)-th integral of \(x^m(\ln x)^{m'}\) in terms of a shifted multiple harmonic star sum. Additionally, we use this integral to derive new expressions for the harmonic sum and repeated harmonic sum.

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2

TANIMURA, SHOGO. "PATH INTEGRALS ON RIEMANNIAN MANIFOLDS WITH SYMMETRY AND INDUCED GAUGE STRUCTURE." International Journal of Modern Physics A 16, no. 08 (March 30, 2001): 1443–61. http://dx.doi.org/10.1142/s0217751x01003159.

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We formulate path integrals on any Riemannian manifold which admits the action of a compact Lie group by isometric transformations. We consider a path integral on a Riemannian manifold M on which a Lie group G acts isometrically. Then we show that the path integral on M is reduced to a family of path integrals on a quotient space Q=M/G and that the reduced path integrals are completely classified by irreducible unitary representations of G. It is not necessary to assume that the action of G on M is either free or transitive. Hence our formulation is applicable to a wide class of manifolds, which includes inhomogeneous spaces, and it covers all the inequivalent quantizations. To describe the path integral on inhomogeneous space, stratification geometry, which is a generalization of the concept of principal fiber bundle, is necessarily introduced. Using it we show that the path integral is expressed as a product of three factors; the rotational energy amplitude, the vibrational energy amplitude, and the holonomy factor. When a singular point arises in Q, we determine the boundary condition of the path integral kernel for a path which runs through the singularity.

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3

Zuo, Hong, and Yu-hong Feng. "A new method for M-integral experimental evaluation." International Journal of Damage Mechanics 22, no. 2 (March 27, 2012): 238–46. http://dx.doi.org/10.1177/1056789512442428.

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In this article, the experimental measurement method of M-integral is investigated. Through the detailed analysis to the nondestructive evaluation method of J- and M-integrals suggested by King and Herrmann, it is found that the specimen geometry which they selected and the corresponding clamping mode in their test exists a conflict with the stress distribution assumption on the integral contour. The formulas they proposed cannot represent the selected specimen geometry and the related integral contours. To avoid this conflict, a new experimental measurement method and a simper specimen style is proposed in this study. According to the method, the M-integral is nondestructive evaluated experimentally through the new specimen and the new clamping mode.

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4

Trainin, J. "Integrating expressions of the form and others." Mathematical Gazette 94, no. 530 (July 2010): 216–23. http://dx.doi.org/10.1017/s0025557200006471.

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In an earlier communication to the Gazette [1], the authors in effect showed, in a somewhat complicated manner, how to evaluate the integral One can show in a simpler manner, however, how to evaluate, for integers n and m, a more general integral of the form where n ≥ m, provided that if m = 1, then n is odd.In addition, the final section to this article shows how to extend the procedure to include integrals for which m does not even have to be an integer, and also how to integrate where such an integral converges.

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5

Silva, Bruno Adriano Rodrigues da. "A Concepção Empresarial da Educação Integral e(m) Tempo Integral." Educação & Realidade 43, no. 4 (October 8, 2018): 1613–32. http://dx.doi.org/10.1590/2175-623676399.

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RESUMO O objetivo deste artigo é analisar teoricamente as formulações do Banco Mundial, as prescrições do Centro de Estudos e Pesquisa em Educação, Cultura e Ação Comunitária e as normatizações fixadas no programa Mais Educação sobre o tema da Educação integral e(m) Tempo Integral. Tal análise se faz pertinente em função da legislação educacional brasileira vigente sobre o tema, em especial, a meta 6 do atual Plano Nacional de Educação. Para realizar o estudo utilizou-se pesquisa bibliográfica e análise de documentos que problematizam o tema. Concluímos que existe uma sintonia entre as formulações do Banco Mundial e as prescrições do Cenpec, sendo o programa Mais Educação uma síntese empresarial dessa relação.

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6

González-Morales, M. J., R. Mahillo-Isla, and C. Dehesa-Martínez. "A new integral identity involving the elliptic integral E(m)." Applied Mathematics and Computation 221 (September 2013): 568–70. http://dx.doi.org/10.1016/j.amc.2013.06.081.

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7

Qureshi, Mohammad Idris, and Showkat Ahmad Dar. "Generalizations and applications of Srinivasa Ramanujan’s integral associated with infinite Fourier sine transforms in terms of Meijer’s G-function." Analysis 41, no. 3 (May 19, 2021): 145–53. http://dx.doi.org/10.1515/anly-2018-0067.

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Abstract In this paper, we obtain analytical solutions of an unsolved integral 𝐑 S ⁢ ( m , n ) {\mathbf{R}_{S}(m,n)} of Srinivasa Ramanujan [S. Ramanujan, Some definite integrals connected with Gauss’s sums, Mess. Math. 44 1915, 75–86] with suitable convergence conditions in terms of Meijer’s G-function of one variable, by using Mellin–Barnes type contour integral representations of the sine function, Laplace transform method and some algebraic properties of Pochhammer’s symbol. Also, we have given some generalizations of Ramanujan’s integral 𝐑 S ⁢ ( m , n ) {\mathbf{R}_{S}(m,n)} in the form of integrals ℧ S * ⁢ ( υ , b , c , λ , y ) {\mho_{S}^{*}(\upsilon,b,c,\lambda,y)} , Ξ S ⁢ ( υ , b , c , λ , y ) {\Xi_{S}(\upsilon,b,c,\lambda,y)} , ∇ S ⁡ ( υ , b , c , λ , y ) {\nabla_{S}(\upsilon,b,c,\lambda,y)} and ℧ S ⁢ ( υ , b , λ , y ) {\mho_{S}(\upsilon,b,\lambda,y)} with suitable convergence conditions and solved them in terms of Meijer’s G-functions. Moreover, as applications of Ramanujan’s integral 𝐑 S ⁢ ( m , n ) {\mathbf{R}_{S}(m,n)} , the three new infinite summation formulas associated with Meijer’s G-function are obtained.

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8

Reynolds, Robert, and Allan Stauffer. "Double Integral Involving Logarithmic and Quotient Function with Powers Expressed in terms of the Lerch Function." European Journal of Pure and Applied Mathematics 14, no. 4 (November 10, 2021): 1337–49. http://dx.doi.org/10.29020/nybg.ejpam.v14i4.4085.

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In this work the authors use their contour integral method to derive the double integral given by $\int_{0}^{\infty}\int_{0}^{\infty}\frac{x^{m-1} y^{m+\frac{q}{2}-1} \log ^k(a x y)}{\left(x^q+1\right)^2 \left(y^q+1\right)^2}dxdy$ in terms of the Lerch function. This integral formula is then used to derive closed solutions in terms of fundamental constants and special functions. There are some useful results relating double integrals of certain kinds of functions to ordinary integrals for which we know no general reference. Thus a table of integral pairs is given for interested readers. All the results in this work are new.

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9

Heinzer, William J., James A. Huckaba, and Ira J. Papick. "M-Canonical ideals in integral domains." Communications in Algebra 26, no. 9 (January 1998): 3021–43. http://dx.doi.org/10.1080/00927879808826325.

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10

Glasser, M. L. "Another Definite Integral (M. L. Glasser)." SIAM Review 27, no. 2 (June 1985): 254. http://dx.doi.org/10.1137/1027064.

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11

Goldstein, J. A., and Kosaki H. "An Integral Identity (M. L. Glasser)." SIAM Review 28, no. 4 (December 1986): 570. http://dx.doi.org/10.1137/1028166.

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12

Suo, Z. "Zener’s Crack and the M-Integral." Journal of Applied Mechanics 67, no. 2 (October 12, 1999): 417–18. http://dx.doi.org/10.1115/1.1302302.

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In a pair of bonded solids, the interface may block dislocation gliding. The pileup may cause a crack to nucleate either on the interface, or in one of the solids. The model, proposed by Zener half a century ago, has been analyzed in various forms. This note shows that the energy release rate of the crack can be calculated by an application of the M-integral. Both solids are anisotropic, and the interface is flat. The result leads to a discussion of the crack orientation. [S0021-8936(00)00701-7]

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13

Bogner, Christian, Stefan Müller-Stach, and Stefan Weinzierl. "The unequal mass sunrise integral expressed through iterated integrals on M‾1,3." Nuclear Physics B 954 (May 2020): 114991. http://dx.doi.org/10.1016/j.nuclphysb.2020.114991.

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14

Owa, Shigeyoshi, and H. Özlem Güney. "New Applications of the Bernardi Integral Operator." Mathematics 8, no. 7 (July 17, 2020): 1180. http://dx.doi.org/10.3390/math8071180.

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Let A ( p , n ) be the class of f ( z ) which are analytic p-valent functions in the closed unit disk U ¯ = z ∈ C : z ≤ 1 . The expression B − m − λ f ( z ) is defined by using fractional integrals of order λ for f ( z ) ∈ A ( p , n ) . When m = 1 and λ = 0 , B − 1 f ( z ) becomes Bernardi integral operator. Using the fractional integral B − m − λ f ( z ) , the subclass T p , n α s , β , ρ ; m , λ of A ( p , n ) is introduced. In the present paper, we discuss some interesting properties for f ( z ) concerning with the class T p , n α s , β , ρ ; m , λ . Also, some interesting examples for our results will be considered.

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15

Ohtsuki, Tomotada. "A polynomial invariant of integral homology 3-spheres." Mathematical Proceedings of the Cambridge Philosophical Society 117, no. 1 (January 1995): 83–112. http://dx.doi.org/10.1017/s0305004100072935.

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In 1988 Witten [W] proposed invariants Zk(M) ∈ ℂ (what we call, quantum G invariants) for a 3-manifold M and any integer k associated with a compact simple Lie group G. The invariant Zk(M) is formally expressed by an integral (Feynman path integral) over the (infinite dimensional) quotient space of the all connections in G-bundles on M modulo gauge transformations. If one believes in Feynman path integrals, one can expect the asymptotic formula of Zk(M) for large k predicted by perturbation theory. As in [W], the asymptotic formula (which is a power series in k−1) is given by a sum of contributions from flat connections, since the integral contains an integrand which is wildly oscillatory apart from flat connections for large k. More precise forms of the asymptotic formula are studied in [AS1], [AS2] and [Ko].

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16

Farid, Ghulam, Hafsa Yasmeen, Hijaz Ahmad, and Chahn Yong Jung. "Riemann-Liouville Fractional integral operators with respect to increasing functions and strongly $ (\alpha, m) $-convex functions." AIMS Mathematics 6, no. 10 (2021): 11403–24. http://dx.doi.org/10.3934/math.2021661.

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<abstract><p>In this paper Hadamard type inequalities for strongly $ (\alpha, m) $-convex functions via generalized Riemann-Liouville fractional integrals are studied. These inequalities provide generalizations as well as refinements of several well known inequalities. The established results are further connected with fractional integral inequalities for Riemann-Liouville fractional integrals of convex, strongly convex and strongly $ m $-convex functions. By using two fractional integral identities some more Hadamard type inequalities are proved.</p></abstract>

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17

Jleli, Mohamed, Donal O’Regan, and Bessem Samet. "Some fractional integral inequalities involving $$\varvec{m}$$ m -convex functions." Aequationes mathematicae 91, no. 3 (February 28, 2017): 479–90. http://dx.doi.org/10.1007/s00010-017-0470-2.

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18

SET, ERHAN, AHMET OCAK AKDEMIR, and I. MUMCU. "Hadamard’s inequality and its extensions for conformable fractional integrals of any order α>0." Creative Mathematics and Informatics 27, no. 2 (2018): 197–206. http://dx.doi.org/10.37193/cmi.2018.02.12.

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Recently the authors Abdeljawad [Abdeljawad, T., On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66] and Khalil et al. [Khalil, R., Horani, M. Al., Yousef, A. and Sababheh, M., A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70] introduced a new and simple well-behaved concept of fractional integral called conformable fractional integral. In this article, we establish Hermite-Hadamard’s inequalities for conformable fractional integral. We also give extensions of Hermite-Hadamard type inequalities for conformable fractional integrals.

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19

Kashuri, Artion, and Rozana Liko. "Generalization of different type integral inequalities for generalized (𝑠, 𝑚)-preinvex Godunova–Levin functions." Journal of Applied Analysis 24, no. 2 (December 1, 2018): 211–21. http://dx.doi.org/10.1515/jaa-2018-0020.

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Abstract In the present paper, the notion of generalized {(s,m)} -preinvex Godunova–Levin function of second kind is introduced, and some new integral inequalities involving generalized {(s,m)} -preinvex Godunova–Levin functions of second kind along with beta function are given. By using a new identity for fractional integrals, some new estimates on generalizations of Hermite–Hadamard, Ostrowski and Simpson type inequalities for generalized {(s,m)} -preinvex Godunova–Levin functions of second kind via Riemann–Liouville fractional integral are established.

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20

Jiménez Fernández, E., and E. A. Sánchez Pérez. "Lattice Copies ofℓ2inL1of a Vector Measure and Strongly Orthogonal Sequences." Journal of Function Spaces and Applications 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/357210.

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Letmbe anℓ2-valued (countably additive) vector measure and consider the spaceL2(m) of square integrable functions with respect tom. The integral with respect tomallows to define several notions of orthogonal sequence in these spaces. In this paper, we center our attention in the existence of stronglym-orthonormal sequences. Combining the use of the Kadec-Pelczyński dichotomy in the domain space and the Bessaga-Pelczyński principle in the range space, we construct a two-sided disjointification method that allows to prove several structure theorems for the spacesL1(m) andL2(m). Under certain requirements, our main result establishes that a normalized sequence inL2(m) with a weakly null sequence of integrals has a subsequence that is stronglym-orthonormal inL2(m∗), wherem∗is anotherℓ2-valued vector measure that satisfiesL2(m) = L2(m∗). As an application of our technique, we give a complete characterization of when a space of integrable functions with respect to anℓ2-valued positive vector measure contains a lattice copy ofℓ2.

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21

Kashuri, Artion, and Rozana Liko. "New inequalities for strongly exponentially generalized functions with applications." Proyecciones (Antofagasta) 41, no. 1 (February 1, 2022): 275–300. http://dx.doi.org/10.22199/issn.0717-6279-3641.

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The aim of this paper is to introduce a new class of functions called strongly exponentially generalized (m, ν1, ν2, g1, g2). Some new integral inequalities of trapezium-type for strongly exponentially generalized (m, ν1, ν2, g1, g2) functions with modulus c via Riemann-Liouville fractional integral are established. Also, some new estimates with respect to trapezium-type integral inequalities for strongly exponentially generalized (m, ν1, ν2, g1, g2) functions with modulus c via general fractional integrals are obtained. We show that the strongly exponentially generalized (m, ν1, ν2, g1, g2) functions with modulus c includes several other classes of functions. At the end, some new error estimates for trapezoidal quadrature formula are provided as well. This results may stimulate further research in different areas of pure and applied sciences.

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22

Kim, Young Sik. "Feynman Integral and a Change of Scale Formula about the First Variation and a Fourier–Stieltjes Transform." Mathematics 8, no. 10 (September 28, 2020): 1666. http://dx.doi.org/10.3390/math8101666.

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We prove that the Wiener integral, the analytic Wiener integral and the analytic Feynman integral of the first variation of F(x)=exp{∫0Tθ(t,x(t))dt} successfully exist under the certain condition, where θ(t,u)=∫Rexp{iuv}dσt(v) is a Fourier–Stieltjes transform of a complex Borel measure σt∈M(R) and M(R) is a set of complex Borel measures defined on R. We will find this condition. Moreover, we prove that the change of scale formula for Wiener integrals about the first variation of F(x) sucessfully holds on the Wiener space.

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23

Al-Salam, W. A. "A Weyl Fractional Integral (H. M. Srivastava)." SIAM Review 27, no. 3 (September 1985): 448–49. http://dx.doi.org/10.1137/1027118.

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24

Albalawi, Wedad. "Time-Scale Integral Inequalities of Copson with Steklov Operator in High Dimension." Journal of Mathematics 2022 (October 12, 2022): 1–11. http://dx.doi.org/10.1155/2022/2771854.

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The paper derives some new time-scale (TS) dynamic inequalities for multiple integrals. The obtained inequalities are special cases of Copson integral using Steklov operator in (TS) version with high dimension. We prove the inequalities with several formulas for the operator and in different cases m > μ + 1 and m < μ + 1 for every μ ≥ 1 , using time-scales (TSs) setting for integral properties, chain rules, Fubini’s theorem, and Hölder’s inequality.

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25

Kashuri, Artion, and Themistocles Rassias. "Fractional trapezium-type inequalities for strongly exponentially generalized preinvex functions with applications." Applicable Analysis and Discrete Mathematics, no. 00 (2020): 38. http://dx.doi.org/10.2298/aadm190220038k.

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The aim of this paper is to introduce a new extension of preinvexity called strongly exponentially generalized (m; !1; !2; h1; h2)-preinvexity. Some new integral inequalities of trapezium-type for strongly exponentially generalized (m; !1; !2; h1; h2)-preinvex functions with modulus c via Riemann-Liouville fractional integral are established. Also, some new estimates with respect to trapezium-type integral inequalities for strongly exponentially generalized (m; !1; !2; h1; h2)-preinvex functions with modulus c via general fractional integrals are obtained. We show that the class of strongly exponentially generalized (m; !1; !2; h1; h2)-preinvex functions with modulus c includes several other classes of preinvex functions. At the end, some new error estimates for trapezoidal quadrature formula are provided as well. This results may stimulate further research in different areas of pure and applied sciences.

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26

Bayraktar, Bahtiyar, and Juan Eduardo Nápoles Valdés. "New generalized integral inequalities via $(h,m)$-convex modified functions." Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta 60 (November 2022): 3–15. http://dx.doi.org/10.35634/2226-3594-2022-60-01.

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In this article, we establish several inequalities for $(h,m)$-convex maps, related to weighted integrals, used in previous works. Throughout the work, we show that our results generalize several of the integral inequalities known from the literature.

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27

Anastassiou, George, Artion Kashuri, and Rozana Liko. "Fractional integral inequalities for generalized-$$\mathbf{m }$$-$$((h_{1}^{p},h_{2}^{q});(\eta _{1},\eta _{2}))$$-convex mappings via an extended generalized Mittag–Leffler function." Arabian Journal of Mathematics 9, no. 2 (December 9, 2019): 231–43. http://dx.doi.org/10.1007/s40065-019-00275-9.

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AbstractThe authors discover a new identity concerning differentiable mappings defined on $$\mathbf{m }$$ m -invex set via general fractional integrals. Using the obtained identity as an auxiliary result, some fractional integral inequalities for generalized-$$\mathbf{m }$$ m -$$((h_{1}^{p},h_{2}^{q});(\eta _{1},\eta _{2}))$$ ( ( h 1 p , h 2 q ) ; ( η 1 , η 2 ) ) -convex mappings by involving an extended generalized Mittag–Leffler function are presented. It is pointed out that some new special cases can be deduced from main results. Also these inequalities have some connections with known integral inequalities. At the end, some applications to special means for different positive real numbers are provided as well.

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28

Kashuri, Artion, Muhammad Awan, and Muhammad Noor. "Fractional integral identity, estimation of its bounds and some applications to trapezoidal quadrature rule." Filomat 34, no. 8 (2020): 2629–41. http://dx.doi.org/10.2298/fil2008629k.

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The aim of this paper is to introduce a new extension of preinvexity called exponentially (m,?1,?2, h1,h2)-preinvexity. Some new integral inequalities of Hermite-Hadamard type for exponentially (m,?1,?2,h1,h2)-preinvex functions via Riemann-Liouville fractional integral are established. Also, some new estimates with respect to trapezium-type integral inequalities for exponentially (m,?1,?2,h1,h2)-preinvex functions via general fractional integrals are obtained. We show that the class of exponentially (m,?1,?2, h1,h2)-preinvex functions includes several other classes of preinvex functions. We shown by two basic examples the efficiency of the obtained inequalities on the base of comparing those with the other corresponding existing ones. At the end, some new error estimates for trapezoidal quadrature formula are provided as well. This results may stimulate further research in different areas of pure and applied sciences.

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29

Kashuri, Artion, and Rozana Liko. "Hermite-Hadamard type integral inequalities for products of two generalized (s; m; ξ)-preinvex functions." Moroccan Journal of Pure and Applied Analysis 3, no. 1 (June 27, 2017): 102–15. http://dx.doi.org/10.1515/mjpaa-2017-0009.

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Abstract In this paper, the notion of generalized (s; m; ξ)-preinvex function is introduced and some new integral inequalities for the left-hand side of Gauss-Jacobi type quadrature formula involving generalized (s; m; ξ)-preinvex functions along with beta function are given. Moreover, we establish some new Hermite-Hadamard type integral inequalities for products of two generalized (s; m; ξ)-preinvex functions via classical and Riemann-Liouville fractional integrals. These results not only extend the results appeared in the literature (see [10],[11]), but also provide new estimates on these types. At the end, some conclusions are given.

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Reynolds, Robert, and Allan Stauffer. "DEFINITE INTEGRAL OF LOGARITHMIC FUNCTIONS AND POWERS IN TERMS OF THE LERCH FUNCTION." Ural Mathematical Journal 7, no. 1 (July 30, 2021): 96. http://dx.doi.org/10.15826/umj.2021.1.008.

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A family of generalized definite logarithmic integrals given by $$\int_{0}^{1}\frac{\left(x^{ i m} (\log (a)+i \log (x))^k+x^{-i m} (\log (a)-i \log (x))^k\right)}{(x+1)^2}dx$$built over the Lerch function has its analytic properties and special values listed in explicit detail. We use the general method as given in [6] to derive this integral. We then give a number of examples that can be derived from the general integral in terms of well known functions.

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31

Cavalari Neto, Ranulfo, Thaís Cristina Souza de Oliveira, and Rayanne de Medeiros Gonçalves. "Educação integral (e)m tempo integral e o Programa Mumbuca Futuro em Maricá." INTERFACES DA EDUCAÇÃO 12, no. 35 (November 2, 2021): 719–44. http://dx.doi.org/10.26514/inter.v12i35.6022.

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Este artigo apresenta a interseção da experiência da educação integral em tempo integral com o Programa Mumbuca Futuro (PMF), programa de governo da Secretaria de Economia Solidária de Maricá em parceria com a Secretaria Municipal de Educação. A partir dessa experiência intersetorial entre as duas secretarias, o PMF compôs a Educação Integral em Tempo Integral já em curso no Município em algumas escolas. Temos como premissa que essa articulação potencializa a proposta de formação integral dos estudantes, possibilitando formação crítica e emancipatória. A metodologia utilizada foi o estudo de caso, utilizando-se da análise documental dos relatórios produzidos após as aulas do PMF e conversas com informantes-chave. Tal articulação trouxe contribuições na qualificação do olhar crítico dos estudantes sobre o território e o desenvolvimento econômico local, além de possibilitar aos estudantes um aprendizado pautado na objetividade e subjetividade da práxis, dinamizado pela Educação Popular.

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32

Yan, Tao, Ghulam Farid, Hafsa Yasmeen, and Chahn Yong Jung. "On Hadamard Type Fractional Inequalities for Riemann–Liouville Integrals via a Generalized Convexity." Fractal and Fractional 6, no. 1 (January 3, 2022): 28. http://dx.doi.org/10.3390/fractalfract6010028.

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In the literature of mathematical inequalities, convex functions of different kinds are used for the extension of classical Hadamard inequality. Fractional integral versions of the Hadamard inequality are also studied extensively by applying Riemann–Liouville fractional integrals. In this article, we define (α,h−m)-convex function with respect to a strictly monotone function that unifies several types of convexities defined in recent past. We establish fractional integral inequalities for this generalized convexity via Riemann–Liouville fractional integrals. The outcomes of this work contain compact formulas for fractional integral inequalities which generate results for different kinds of convex functions.

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33

Chen, Y. Z., and Kang Yong Lee. "Analysis of the M-Integral in Plane Elasticity." Journal of Applied Mechanics 71, no. 4 (July 1, 2004): 572–74. http://dx.doi.org/10.1115/1.1748271.

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In this paper, analysis of the M-integral in plane elasticity is carried out. An infinite plate with any number of inclusions and cracks and with any applied forces and remote tractions is considered. To study the problem, the mutual work difference integral (abbreviated as MWDI) is introduced, which is defined by the difference of works done by each other stress field on a large circle. The concept of the derivative stress field is also introduced, which is a real elasticity solution and is derived from the physical stress field. It is found that the M-integral on a large circle is equal to a MWDI from the physical stress field and a derivative stress field. Finally, the expression for M-integral on a large circle is obtained. The variation for the M-integral with respect to the coordinate transformation is addressed. An illustrative example for the use of M-integral is presented.

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34

Kashuri, Artion, and Rozana Liko. "Some new fractional integral inequalities for generalized relative semi-m-(r; h1, h2)-preinvex mappings via generalized Mittag-Leffler function." Arab Journal of Mathematical Sciences 26, no. 1/2 (January 2, 2019): 41–55. http://dx.doi.org/10.1016/j.ajmsc.2018.12.003.

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The authors discover a new identity concerning differentiable mappings defined on m-invex set via fractional integrals. By using the obtained identity as an auxiliary result, some fractional integral inequalities for generalized relative semi- m-(r;h1,h2)-preinvex mappings by involving generalized Mittag-Leffler function are presented. It is pointed out that some new special cases can be deduced from main results of the paper. Also these inequalities have some connections with known integral inequalities. At the end, some applications to special means for different positive real numbers are provided as well.

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35

Iscan, Imdat. "Generalization of different type integral inequalities for (\alpha,m)-convex functions via fractional integrals." Applied Mathematical Sciences 9 (2015): 2925–39. http://dx.doi.org/10.12988/ams.2015.4121023.

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36

Alfaro, J., R. Medina, and L. F. Urrutia. "The itzykson-zuber integral for U(m|n)." Surveys in High Energy Physics 10, no. 1-4 (February 1997): 405–9. http://dx.doi.org/10.1080/01422419708219645.

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37

Banks-Sills, Leslie, and Orly Dolev. "The conservative M-integral for thermal-elastic problems." International Journal of Fracture 125, no. 1 (January 2004): 149–70. http://dx.doi.org/10.1023/b:frac.0000021065.46630.4d.

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38

Chalco-Cano, Y., M. A. Rojas-Medar, and H. Román-Flores. "M-convex fuzzy mappings and fuzzy integral mean." Computers & Mathematics with Applications 40, no. 10-11 (November 2000): 1117–26. http://dx.doi.org/10.1016/s0898-1221(00)00226-1.

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39

Alfaro, Jorge, Ricardo Medina, and Luis F. Urrutia. "The Itzykson–Zuber integral for U(m‖n)." Journal of Mathematical Physics 36, no. 6 (June 1995): 3085–93. http://dx.doi.org/10.1063/1.531014.

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40

HAN, QI, CAISHI WANG, and YULAN ZHOU. "CONVOLUTION OF FUNCTIONALS OF DISCRETE-TIME NORMAL MARTINGALES." Bulletin of the Australian Mathematical Society 86, no. 2 (December 16, 2011): 224–31. http://dx.doi.org/10.1017/s0004972711003091.

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AbstractLet M=(M)n∈ℕ be a discrete-time normal martingale satisfying some mild requirements. In this paper we show that through the full Wiener integral introduced by Wang et al. (‘An alternative approach to Privault’s discrete-time chaotic calculus’, J. Math. Anal. Appl.373 (2011), 643–654), one can define a multiplication-type operation on square integrable functionals of M, which we call the convolution. We examine algebraic and analytical properties of the convolution and, in particular, we prove that the convolution can be used to represent a certain family of conditional expectation operators associated with M. We also present an example of a discrete-time normal martingale to show that the corresponding convolution has an integral representation.

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41

Ahmad, Faiz. "Starlike integral operators." Bulletin of the Australian Mathematical Society 32, no. 2 (October 1985): 217–24. http://dx.doi.org/10.1017/s0004972700009916.

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42

Cohen, Scott M., K. TR Davies, R. W. Davies, and M. Howard Lee. "Principal-value integrals Revisited." Canadian Journal of Physics 83, no. 5 (May 1, 2005): 565–75. http://dx.doi.org/10.1139/p05-025.

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The principal-value (PV) integral has proved a useful tool in many fields of physics. The PV is a specific method for obtaining a finite result for an improper integral. When the integration passes through a simple pole, one speaks of a "first-order" PV. In this paper, we examine first-order PV integrals and analyze several of their important properties. First, we discuss how the PV agrees with one's naïve expectation about these integrals. Next, we show that the basic definition of the first-order PV gives a generalized formula for the complex-variable PV expression. Finally, we show the correspondence between the finite-limit PV integral of x1 along the real axis and the path integral of z1 (where z = x + iy) in the complex plane.PACS Nos.: 02.90.+p, 05.90.+m

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43

Farid, Ghulam, Muhammad Yussouf, and Kamsing Nonlaopon. "Fejér–Hadamard Type Inequalities for (α, h-m)-p-Convex Functions via Extended Generalized Fractional Integrals." Fractal and Fractional 5, no. 4 (December 2, 2021): 253. http://dx.doi.org/10.3390/fractalfract5040253.

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Integral operators of a fractional order containing the Mittag-Leffler function are important generalizations of classical Riemann–Liouville integrals. The inequalities that are extensively studied for fractional integral operators are the Hadamard type inequalities. The aim of this paper is to find new versions of the Fejér–Hadamard (weighted version of the Hadamard inequality) type inequalities for (α, h-m)-p-convex functions via extended generalized fractional integrals containing Mittag-Leffler functions. These inequalities hold simultaneously for different types of well-known convexities as well as for different kinds of fractional integrals. Hence, the presented results provide more generalized forms of the Hadamard type inequalities as compared to the inequalities that already exist in the literature.

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Li, Shasha, Ghulam Farid, Atiq Ur Rehman, and Hafsa Yasmeen. "Fractional Versions of Hadamard-Type Inequalities for Strongly Exponentially α , h − m -Convex Functions." Journal of Mathematics 2021 (August 11, 2021): 1–23. http://dx.doi.org/10.1155/2021/2555974.

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In this article, we prove some fractional versions of Hadamard-type inequalities for strongly exponentially α , h − m -convex functions via generalized Riemann–Liouville fractional integrals. The outcomes of this paper provide inequalities of strongly convex, strongly m -convex, strongly s -convex, strongly α , m -convex, strongly s , m -convex, strongly h − m -convex, strongly α , h − m -convex, strongly exponentially convex, strongly exponentially m -convex, strongly exponentially s -convex, strongly exponentially s , m -convex, strongly exponentially h − m -convex, and exponentially α , h − m -convex functions. The error estimations are also studied by applying two fractional integral identities.

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Chen, Dong, Matloob Anwar, Ghulam Farid, and Waseela Bibi. "Inequalities for q-h-Integrals via ℏ-Convex and m-Convex Functions." Symmetry 15, no. 3 (March 7, 2023): 666. http://dx.doi.org/10.3390/sym15030666.

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This paper investigates several integral inequalities held simultaneously for q and h-integrals in implicit form. These inequalities are established for symmetric functions using certain types of convex functions. Under certain conditions, Hadamard-type inequalities are deducible for q-integrals. All the results are applicable for ℏ-convex, m-convex and convex functions defined on the non-negative part of the real line.

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Pečarić, J. E., and P. R. Beesack. "On Knopp's Inequality for Convex Functions." Canadian Mathematical Bulletin 30, no. 3 (September 1, 1987): 267–72. http://dx.doi.org/10.4153/cmb-1987-038-1.

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AbstractKnopp's inequality for convex functions ϕ on an interval I = [m,M] states thatfor an explicit functional H, and all integrable g: [0, 1] → I. In this paper we give results of this kind in which the integral operator, ∫, is replaced by a general isotonic linear functional.

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Farid, Ghulam, Saira Bano Akbar, Laxmi Rathour, Lakshmi Narayan Mishra, and Vishnu Narayan Mishra. "Riemann-Liouville Fractional Versions of Hadamard inequality for Strongly m-Convex Functions." International Journal of Analysis and Applications 20 (January 18, 2022): 5. http://dx.doi.org/10.28924/2291-8639-20-2022-5.

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This paper deals with Hadamard inequalities for strongly m-convex functions via Riemann-Liouville fractional integrals. These inequalities provide refinements of well known fractional integral inequalities for convex functions. Further, by applying an identity error estimations are obtained and compared with already known error estimations.

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Bayraktar, B., and V. Ch Kudaev. "Some new integral inequalities for (s, m)-convex and (α, m)-convex functions." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 94, no. 2 (June 30, 2019): 15–25. http://dx.doi.org/10.31489/2019m2/15-25.

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Jung, Chahn Yong, Muhammad Yussouf, Yu-Ming Chu, Ghulam Farid, and Shin Min Kang. "Generalized Fractional Hadamard and Fejér–Hadamard Inequalities for Generalized Harmonically Convex Functions." Journal of Mathematics 2020 (November 4, 2020): 1–13. http://dx.doi.org/10.1155/2020/8245324.

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In this paper, we define a new function, namely, harmonically α , h − m -convex function, which unifies various kinds of harmonically convex functions. Generalized versions of the Hadamard and the Fejér–Hadamard fractional integral inequalities for harmonically α , h − m -convex functions via generalized fractional integral operators are proved. From presented results, a series of fractional integral inequalities can be obtained for harmonically convex, harmonically h − m -convex, harmonically α , m -convex, and related functions and for already known fractional integral operators.

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Reynolds, Robert, and Allan Stauffer. "A Definite Integral Involving the Logarithmic Function in Terms of the Lerch Function." Mathematics 7, no. 12 (November 24, 2019): 1148. http://dx.doi.org/10.3390/math7121148.

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We present a method using contour integration to evaluate the definite integral of the form ∫ 0 ∞ log k ( a y ) R ( y ) d y in terms of special functions, where R ( y ) = y m 1 + α y n and k , m , a , α and n are arbitrary complex numbers. We use this method for evaluation as well as to derive some interesting related material and check entries in tables of integrals.

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