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Published: 17 August 2022
Last updated: 10 September 2022

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1

Pavlovsky, Vladislav A., and Igor L. Vasiliev. "On properties of h-differentiable functions." Journal of the Belarusian State University. Mathematics and Informatics, no. 2 (August 5, 2021): 29–37. http://dx.doi.org/10.33581/2520-6508-2021-2-29-37.

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Research in the theory of functions of an h-complex variable is of interest in connection with existing applications in non-Euclidean geometry, theoretical mechanics, etc. This article is devoted to the study of the properties of h-differentiable functions. Criteria for h-differentiability and h-holomorphy are found, formulated and proved a theorem on finite increments for an h-holomorphic function. Sufficient conditions for h-analyticity are given, formulated and proved a uniqueness theorem for h-analytic functions.
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2

Gaydu, Michaël. "An iterative method for solving $$H$$ H -differentiable inclusions." Rendiconti del Circolo Matematico di Palermo (1952 -) 63, no. 3 (June 27, 2014): 389–97. http://dx.doi.org/10.1007/s12215-014-0161-y.

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3

Romanov, V. A. "Asymptotics of H-continuous and H-differentiable measures in Hilbert space." Mathematical Notes of the Academy of Sciences of the USSR 37, no. 1 (January 1985): 49–53. http://dx.doi.org/10.1007/bf01652514.

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4

Buczolich, Zoltán. "Functions with finite intersections with analytic functions." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 112, no. 3-4 (1989): 271–75. http://dx.doi.org/10.1017/s0308210500018746.

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SynopsisWe prove that for every dense Gδ set H, there exists a continuous function f, such that f intersects every analytic function in finitely many points and f is infinitely differentiable exactly at the points of H. This answers a problem of S. Agronsky, A. M. Bruckner, M. Laczkovich and D. Preiss. They proved a result which implies that every continuous function with finite intersections with analytic functions is infinitely differentiable at the points of a dense Gδ set.
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5

Latif, Muhammad Amer, Sever Silvestru Dragomir, and Ebrahim Momoniat. "Some weighted integral inequalities for differentiable h-preinvex functions." Georgian Mathematical Journal 25, no. 3 (September 1, 2018): 441–50. http://dx.doi.org/10.1515/gmj-2016-0081.

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AbstractIn this paper, by using a weighted identity for functions defined on an open invex subset of the set of real numbers, by using the Hölder integral inequality and by using the notion of h-preinvexity, we present weighted integral inequalities of Hermite–Hadamard-type for functions whose derivatives in absolute value raised to certain powers are h-preinvex functions. Some new Hermite–Hadamard-type integral inequalities are obtained when h is super-additive. Inequalities of Hermite–Hadamard-type for s-preinvex functions are given as well as a special case of our results.
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6

Hernández-Verón, Miguel A., Sonia Yadav, Ángel Alberto Magreñán, Eulalia Martínez, and Sukhjit Singh. "An Algorithm Derivative-Free to Improve the Steffensen-Type Methods." Symmetry 14, no. 1 (December 21, 2021): 4. http://dx.doi.org/10.3390/sym14010004.

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Solving equations of the form H(x)=0 is one of the most faced problem in mathematics and in other science fields such as chemistry or physics. This kind of equations cannot be solved without the use of iterative methods. The Steffensen-type methods, defined using divided differences are derivative free, are usually considered to solve these problems when H is a non-differentiable operator due to its accuracy and efficiency. However, in general, the accessibility of these iterative methods is small. The main interest of this paper is to improve the accessibility of Steffensen-type methods, this is the set of starting points that converge to the roots applying those methods. So, by means of using a predictor–corrector iterative process we can improve this accessibility. For this, we use a predictor iterative process, using symmetric divided differences, with good accessibility and then, as corrector method, we consider the Center-Steffensen method with quadratic convergence. In addition, the dynamical studies presented show, in an experimental way, that this iterative process also improves the region of accessibility of Steffensen-type methods. Moreover, we analyze the semilocal convergence of the predictor–corrector iterative process proposed in two cases: when H is differentiable and H is non-differentiable. Summing up, we present an effective alternative for Newton’s method to non-differentiable operators, where this method cannot be applied. The theoretical results are illustrated with numerical experiments.
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7

Walther, Hans-Otto. "Delay Differential Equations with Differentiable Solution Operators on Open Domains in C((-∞, 0], Rn) and Processes for Volterra Integro-Differential Equations." Contemporary Mathematics. Fundamental Directions 67, no. 3 (December 15, 2021): 483–506. http://dx.doi.org/10.22363/2413-3639-2021-67-3-483-506.

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For autonomous delay differential equations x'(t)=f(xt){x'(t)=f(x_t)} we construct a continuous semiflow of continuously differentiable solution operators x0xt{x_0 \to x_t}, t0{t \le 0}, on open subsets of the Frechet space C((-,0],Rn){C((-\infty, 0], R^n)}. For nonautonomous equations this yields a continuous process of differentiable solution operators. As an application, we obtain processes which incorporate all solutions of Volterra integro-differential equations x'(t)=∫0tk(t,s)h(x(s))ds{x'(t)={\int_0}^t k(t,s) h(x(s)) ds}.
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8

Awan, Muhammad, Muhammad Noor, Marcela Mihai, and Khalida Noor. "Two point trapezoidal like inequalities involving hypergeometric functions." Filomat 31, no. 8 (2017): 2281–92. http://dx.doi.org/10.2298/fil1708281a.

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In this paper, we derive a new integral identity for differentiable function. Using this new integral identity as an auxiliary result, we derive some new two point trapezoidal like inequalities for differentiable harmonic h-convex functions. These inequalities can also be viewed as Hermite-Hadamard type inequalities. We also discuss some new special cases which can be deduced from our main results.
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9

Latif, Muhammad Amer. "New Weighted Hermite–Hadamard Type Inequalities for Differentiable h -Convex and Quasi h -Convex Mappings." Journal of Mathematics 2021 (July 5, 2021): 1–14. http://dx.doi.org/10.1155/2021/4495588.

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In this paper, new weighted Hermite–Hadamard type inequalities for differentiable h -convex and quasi h -convex functions are proved. These results generalize many results proved in earlier works for these classes of functions. Applications of some of our results to s ˘ -divergence and to statistics are given.
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10

Gaydu, M., M. H. Geoffroy, and C. Jean-Alexis. "An inverse mapping theorem for H-differentiable set-valued maps." Journal of Mathematical Analysis and Applications 421, no. 1 (January 2015): 298–313. http://dx.doi.org/10.1016/j.jmaa.2014.07.006.

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11

Cakmak, Musa, Mevlüt Tunc, and Aysegül Acem. "Some new inequalities for differentiable h-convex functions and applications." Miskolc Mathematical Notes 22, no. 1 (2021): 107. http://dx.doi.org/10.18514/mmn.2021.2444.

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12

Mahreen, Kahkashan, and Hüseyin Budak. "Generalized midpoint fractional integral inequalities via h-convexity." Filomat 35, no. 11 (2021): 3821–32. http://dx.doi.org/10.2298/fil2111821m.

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In this research, generalizations of midpoint type inequalities are established. h-convexity is used as a tool. These inequalities are for differentiable functions which involve Riemann-Liouville fractional integrals. Also, some consequences of these established inequalities are obtained.
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13

Morozov, Anatoly N. "On the Taylor Differentiability in Spaces Lp, 0 < p ≤ ∞." Modeling and Analysis of Information Systems 25, no. 3 (June 30, 2018): 323–30. http://dx.doi.org/10.18255/1818-1015-2018-3-323-330.

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The function \(f\in L_p[I], \;p>0,\) is called \((k,p)\)-differentiable at a point \(x_0\in I\) if there exists an algebraic polynomial of \(\pi\) of degree no more than \(k\) for which holds \( \Vert f-\pi \Vert_{L_p[J_h]} = o(h^{k+\frac{1}{p}}), \) where \(\;J_h=[x_0-h; x_0+h]\cap I.\) At an internal point for \(k=1\) and \(p=\infty\) this is equivalent to the usual definition of the function differentiability. At an interior point for \(k=1\) and \(p=\infty\), the definition is equivalent to the usual differentiability of the function. There is a standard "hierarchy" for the existence of differentials(if \(p_1<p_2,\) then \((k,p_2)\)-differentiability should be \((k,p_1)\)-differentiability. In the works of S.N. Bernstein, A.P. Calderon and A. Zygmund were given applications of such a construction to build a description of functional spaces (\(p=\infty\)) and the study of local properties of solutions of differential equations \((1\le p\le\infty)\), respectively. This article is related to the first mentioned work. The article introduces the concept of uniform differentiability. We say that a function \(f\), \((k,p)\)-differentiable at all points of the segment \(I\), is uniformly \((k,p)\)-differentiable on \(I\) if for any number \(\varepsilon>0\) there is a number \(\delta>0\) such that for each point \(x\in I\) runs \( \Vert f-\pi\Vert_{L_p[J_h]}<\varepsilon\cdot h^{k+\frac{1}{p}} \; \) for \(0<h<\delta, \; J_h = [x\!-\!H; x\!+\!h]\cap I,\) where \(\pi\) is the polynomial of the terms of the \((k, p)\)-differentiability at the point \(x\). Based on the methods of local approximations of functions by algebraic polynomials it is shown that a uniform \((k,p)\)-differentiability of the function \(f\) at some \(1\le p\le\infty\) implies \(f\in C^k[I].\) Therefore, in this case the differentials are "equivalent". Since every function from \(C^k[I]\) is uniformly \((k,p)\)-differentiable on the interval \(I\) at \(1\le p\le\infty,\) we obtain a certain criterion of belonging to this space. The range \(0<p<1,\) obviously, can be included into the necessary condition the membership of the function \(C^k[I]\), but the sufficiency of Taylor differentiability in this range has not yet been fully proven.
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14

Dragomir, Sever. "General Lebesgue integral inequalities of Jensen and Ostrowski type for differentiable functions whose derivatives in absolute value are h-convex and applications." Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica 69, no. 2 (December 30, 2015): 17. http://dx.doi.org/10.17951/a.2015.69.2.17-45.

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Some inequalities related to Jensen and Ostrowski inequalities for general Lebesgue integral of differentiable functions whose derivatives in absolute value are h-convex are obtained. Applications for f-divergence measure are provided as well.
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15

Ivankovic, Bovzidar. "Quasi-arithmetic Means Inequalities Criteria for Differentiable Functions." Journal of Mathematics Research 7, no. 4 (November 6, 2015): 130. http://dx.doi.org/10.5539/jmr.v7n4p130.

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Quasi-arithmetic means are defined for continuous, strictly monotone functions. In the case that functions are twice differentiable, we obtained criteria for inequalities between finite number of quasi-arithmetic means in additional and multiplicative case. Applications for H\"older and Minkowski type inequalities are given.
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16

Shpakivskyi, Vitalii, and Tetiana Kuzmenko. "Hausdorff analytic functions in a three-dimensional associative noncommutative algebra." Ukrainian Mathematical Bulletin 19, no. 1 (January 28, 2022): 103–20. http://dx.doi.org/10.37069/1810-3200-2022-19-1-7.

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A class of H-analytic (differentiable by Hausdorff) functions in a three-dimensional noncommutative algebra $\mathbb{\widetilde{A}}_{2}$ over the field $\mathbb{C}$ is considered. All $H$-analytic functions are described in the explicit form. The obtained description is applied to the integral representation of these functions, and the mentioned functions are also applied when solving some PDEs.
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17

Pirzada, U. M., and D. C. Vakaskar. "Existence of Hukuhara Differentiability of Fuzzy-Valued Functions." Journal of the Indian Mathematical Society 84, no. 3-4 (July 1, 2017): 239. http://dx.doi.org/10.18311/jims/2017/5824.

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In this paper, we discuss existence of Hukuhara differentiability of fuzzy-valued functions. Several examples are worked out to check that fuzzy-valued functions are one time, two times and n-times H-differentiable. We study the effects of fuzzy modelling on existence of Hukuhara differentiability of fuzzy-valued functions. We discuss existence of gH-differentiability and its comparison with H-differentiability.
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18

Thanh, N. T., and V. N. Phat. "H∞ control for nonlinear systems with interval non-differentiable time-varying delay." European Journal of Control 19, no. 3 (May 2013): 190–98. http://dx.doi.org/10.1016/j.ejcon.2013.05.002.

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19

Noor, Muhammad Aslam, Khalida Inayat Noor, Muhammad Uzair Awan, and Sundas Khan. "Hermite–Hadamard type inequalities for differentiable $${h_{\varphi}}$$ h φ -preinvex functions." Arabian Journal of Mathematics 4, no. 1 (January 8, 2015): 63–76. http://dx.doi.org/10.1007/s40065-014-0124-3.

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20

Alcorta-García, María Aracelia, Martín Eduardo Frías-Armenta, María Esther Grimaldo-Reyna, and Elifalet López-González. "Algebrization of Nonautonomous Differential Equations." Journal of Applied Mathematics 2015 (2015): 1–10. http://dx.doi.org/10.1155/2015/632150.

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Given a planar system of nonautonomous ordinary differential equations,dw/dt=F(t,w), conditions are given for the existence of an associative commutative unital algebraAwith uniteand a functionH:Ω⊂R2×R2→R2on an open setΩsuch thatF(t,w)=H(te,w)and the mapsH1(τ)=H(τ,ξ)andH2(ξ)=H(τ,ξ)are Lorch differentiable with respect toAfor all(τ,ξ)∈Ω, whereτandξrepresent variables inA. Under these conditions the solutionsξ(τ)of the differential equationdξ/dτ=H(τ,ξ)overAdefine solutions(x(t),y(t))=ξ(te)of the planar system.
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21

Almutairi, Ohud Bulayhan. "Quantum Estimates for Different Type Intequalities through Generalized Convexity." Entropy 24, no. 5 (May 20, 2022): 728. http://dx.doi.org/10.3390/e24050728.

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This article estimates several integral inequalities involving (h−m)-convexity via the quantum calculus, through which Important integral inequalities including Simpson-like, midpoint-like, averaged midpoint-trapezoid-like and trapezoid-like are extended. We generalized some quantum integral inequalities for q-differentiable (h−m)-convexity. Our results could serve as the refinement and the unification of some classical results existing in the literature by taking the limit q→1−.
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22

Verma, Ram U. "Partially relaxed cocoercive variational inequalities and auxiliary problem principle." Journal of Applied Mathematics and Stochastic Analysis 2004, no. 2 (January 1, 2004): 143–48. http://dx.doi.org/10.1155/s1048953304305010.

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Let T:K→H be a mapping from a nonempty closed convex subset K of a finite-dimensional Hilbert space H into H. Let f:K→ℝ be proper, convex, and lower semicontinuous on K and let h:K→ℝ be continuously Frećhet-differentiable on K with h′ (gradient of h), α-strongly monotone, and β-Lipschitz continuous on K. Then the sequence {xk} generated by the general auxiliary problem principle converges to a solution x* of the variational inequality problem (VIP) described as follows: find an element x*∈K such that 〈T(x*),x−x*〉+f(x)−f(x*)≥0 for all x∈K.
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23

Cianciaruso, Filomena, Giuseppe Marino, Luigi Muglia, and Haiyun Zhou. "Strong Convergence of Viscosity Methods for Continuous Pseudocontractions in Banach Spaces." Journal of Applied Mathematics and Stochastic Analysis 2008 (December 31, 2008): 1–11. http://dx.doi.org/10.1155/2008/149483.

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We define a viscosity method for continuous pseudocontractive mappings defined on closed and convex subsets of reflexive Banach spaces with a uniformly Gâteaux differentiable norm. We prove the convergence of these schemes improving the main theorems in the work by Y. Yao et al. (2007) and H. Zhou (2008).
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24

Afuwape, Anthony Uyi. "Convergence of the solutions for the equationx(iv)+ax⃛+bx¨+g(x˙)+h(x)=p(t,x,x˙,x¨,x⃛)." International Journal of Mathematics and Mathematical Sciences 11, no. 4 (1988): 727–33. http://dx.doi.org/10.1155/s0161171288000882.

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This paper is concerned with differential equations of the formx(iv)+ax⃛+bx¨+g(x˙)+h(x)=p(t,x,x˙,x¨,x⃛)wherea,bare positive constants and the functionsg,handpare continuous in their respective arguments, with the functionhnot necessarily differentiable. By introducing a Lyapunov function, as well as restricting the incrementary ratioη−1{h(ζ+η)−h(ζ)},(η≠0), ofhto a closed sub-interval of the Routh-Hurwitz interval, we prove the convergence of solutions for this equation. This generalizes earlier results.
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25

Yang, Yingxia, Muhammad Shoaib Saleem, Mamoona Ghafoor, and Muhammad Imran Qureshi. "Fractional Integral Inequalities of Hermite–Hadamard Type for Differentiable Generalized h-Convex Functions." Journal of Mathematics 2020 (June 25, 2020): 1–13. http://dx.doi.org/10.1155/2020/2301606.

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In the present paper, some fractional integral inequalities of Hermite–Hadamard type for functions whose derivatives are generalized h-convex are established. Moreover, several particular cases are also discussed which can be deduced from our results. As special cases, one can obtain several new versions of the results of generalized h-convexity for other various generalizations of convex functions.
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26

Almutairi, Ohud, and Adem Kılıçman. "A Review of Hermite–Hadamard Inequality for α-Type Real-Valued Convex Functions." Symmetry 14, no. 5 (April 19, 2022): 840. http://dx.doi.org/10.3390/sym14050840.

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Inequalities play important roles not only in mathematics but also in other fields, such as economics and engineering. Even though many results are published as Hermite–Hadamard (H-H)-type inequalities, new researchers to these fields often find it difficult to understand them. Thus, some important discoverers, such as the formulations of H-H-type inequalities of α-type real-valued convex functions, along with various classes of convexity through differentiable mappings and for fractional integrals, are presented. Some well-known examples from the previous literature are used as illustrations. In the many above-mentioned inequalities, the symmetrical behavior arises spontaneously.
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27

Terrang, Abubakar, Awumtiya Isa, Felix Bakare, and Patience Iliya. "On Solution of First Order Initial Value Problems using Laplace Transform in Fuzzy Environment." International Journal of Mathematical Analysis and Optimization: Theory and Applications 7, no. 2 (November 16, 2021): 21–29. http://dx.doi.org/10.52968/28305366.

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This study aimed at solving a nonhomogeneous linear first order initial value problem by means of Laplace transform method in fuzzy environment. The conditions for a fuzzy function to be H−differentiable and gH−differentiability are well established. Finally, example is constructed to test the applicability or otherwise of the established results.
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Wu, Huiling, and Yongqing Li. "Ground State for a Coupled Elliptic System with Critical Growth." Advanced Nonlinear Studies 18, no. 1 (February 1, 2018): 1–15. http://dx.doi.org/10.1515/ans-2017-6019.

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Abstract We study the following coupled elliptic system with critical nonlinearities: \left\{\begin{aligned} &\displaystyle-\triangle{u}+u=f(u)+\beta h(u)K(v),&&% \displaystyle x\in{\mathbb{R}}^{N},\\ &\displaystyle-\triangle{v}+v=g(v)+\beta H(u)k(v),&&\displaystyle x\in{\mathbb% {R}}^{N},\\ &\displaystyle u,v\in H^{1}({\mathbb{R}}^{N}),\end{aligned}\right. where {\beta>0} ; f, g are differentiable functions with critical growth; and {H,K} are primitive functions of h and k, respectively. Under some assumptions on f, g, h and k, we obtain the existence of a positive ground state solution of this system for {N\geq 2} .
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29

Sergeev, Armen. "The Sobolev space of half-differentiable functions and quasisymmetric homeomorphisms." Georgian Mathematical Journal 23, no. 4 (December 1, 2016): 615–22. http://dx.doi.org/10.1515/gmj-2016-0047.

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AbstractIn this paper, we give an interpretation of some classical objects of function theory in terms of Banach algebras of linear operators in a Hilbert space. We are especially interested in quasisymmetric homeomorphisms of the circle. They are boundary values of quasiconformal homeomorphisms of the disk and form a group ${\operatorname{QS}(S^{1})}$ with respect to composition. This group acts on the Sobolev space ${H^{1/2}_{0}(S^{1},\mathbb{R})}$ of half-differentiable functions on the circle by reparameterization. We give an interpretation of the group ${\operatorname{QS}(S^{1})}$ and the space ${H^{1/2}_{0}(S^{1},\mathbb{R})}$ in terms of noncommutative geometry.
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30

Cannarsa, Piermarco, Wei Cheng, and Albert Fathi. "Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry." Publications mathématiques de l'IHÉS 133, no. 1 (June 2021): 327–66. http://dx.doi.org/10.1007/s10240-021-00125-5.

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AbstractIf $U:[0,+\infty [\times M$ U : [ 0 , + ∞ [ × M is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation $$ \partial _{t}U+ H(x,\partial _{x}U)=0, $$ ∂ t U + H ( x , ∂ x U ) = 0 , where $M$ M is a not necessarily compact manifold, and $H$ H is a Tonelli Hamiltonian, we prove the set $\Sigma (U)$ Σ ( U ) , of points in $]0,+\infty [\times M$ ] 0 , + ∞ [ × M where $U$ U is not differentiable, is locally contractible. Moreover, we study the homotopy type of $\Sigma (U)$ Σ ( U ) . We also give an application to the singularities of the distance function to a closed subset of a complete Riemannian manifold.
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31

Liu, Guanting, Yujie Zhong, Sheng Guo, Matthew R. Scott, and Weilin Huang. "Unchain the Search Space with Hierarchical Differentiable Architecture Search." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 10 (May 18, 2021): 8644–52. http://dx.doi.org/10.1609/aaai.v35i10.17048.

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Differentiable architecture search (DAS) has made great progress in searching for high-performance architectures with reduced computational cost. However, DAS-based methods mainly focus on searching for a repeatable cell structure, which is then stacked sequentially in multiple stages to form the networks. This configuration significantly reduces the search space, and ignores the importance of connections between the cells. To overcome this limitation, in this paper, we propose a Hierarchical Differentiable Architecture Search (H-DAS) that performs architecture search both at the cell level and at the stage level. Specifically, the cell-level search space is relaxed so that the networks can learn stage-specific cell structures. For the stage-level search, we systematically study the architectures of stages, including the number of cells in each stage and the connections between the cells. Based on insightful observations, we design several search rules and losses, and mange to search for better stage-level architectures. Such hierarchical search space greatly improves the performance of the networks without introducing expensive search cost. Extensive experiments on CIFAR10 and ImageNet demonstrate the effectiveness of the proposed H-DAS. Moreover, the searched stage-level architectures can be combined with the cell structures searched by existing DAS methods to further boost the performance. Code is available at: https://github.com/msight-tech/research-HDAS
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32

Wu, Yuhu, and Xiaoping Xue. "ASYMPTOTICS FOR A DISSIPATIVE DYNAMICAL SYSTEM WITH LINEAR AND GRADIENT-DRIVEN DAMPING." Mathematical Modelling and Analysis 18, no. 5 (December 1, 2013): 654–74. http://dx.doi.org/10.3846/13926292.2013.868842.

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We study, in the setting of a real Hilbert space H, the asymptotic behavior of trajectories of the second-order dissipative dynamical system with linear and gradient-driven nonlinear damping where λ > 0 and f, Φ: H → R are two convex differentiable functions. It is proved that if Φ is coercive and bounded from below, then the trajectory converges weakly towards a minimizer of Φ. In particular, we state that under suitable conditions, the trajectory strongly converges to the minimizer of Φ exponentially or polynomially.
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33

Rashid, Saima, Muhammad Amer Latif, Zakia Hammouch, and Yu-Ming Chu. "Fractional Integral Inequalities for Strongly h -Preinvex Functions for a kth Order Differentiable Functions." Symmetry 11, no. 12 (November 25, 2019): 1448. http://dx.doi.org/10.3390/sym11121448.

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The objective of this paper is to derive Hermite-Hadamard type inequalities for several higher order strongly h -preinvex functions via Riemann-Liouville fractional integrals. These results are the generalizations of the several known classes of preinvex functions. An identity associated with k-times differentiable function has been established involving Riemann-Liouville fractional integral operator. A number of new results can be deduced as consequences for the suitable choices of the parameters h and σ . Our outcomes with these new generalizations have the abilities to be implemented for the evaluation of many mathematical problems related to real world applications.
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Matsugu, Yasuo, and Jun Miyazawa. "A characterization of weighted Bergman-Orlicz spaces on the unit ball in Cn." Journal of the Australian Mathematical Society 74, no. 1 (February 2003): 5–18. http://dx.doi.org/10.1017/s1446788700003074.

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AbstractLet B denote the unit ball in Cn, and ν the normalized Lebesgue measure on B. For α > −1, define Here cα is a positive constant such that να(B) = 1. Let H(B) denote the space of all holomorphic functions in B. For a twice differentiable, nondecreasing, nonnegative strongly convex function ϕ on the real line R, define the Bergman-Orlicz space Aϕ(να) by In this paper we prove that a function f ∈ H(B) is in Aϕ(να) if and only if where is the radial derivative of f.
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35

Afanas'eva, Elena, Anatoly Golberg, and Ruslan Salimov. "Distortion theorems for homeomorphic Sobolev mappings of integrable p-dilatations." Studia Universitatis Babes-Bolyai Matematica 67, no. 2 (June 8, 2022): 403–20. http://dx.doi.org/10.24193/subbmath.2022.2.15.

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"We study the distortion features of homeomorphisms of Sobolev class $W^{1,1}_{\rm loc}$ admitting integrability for $p$-outer dilatation. We show that such mappings belong to $W^{1,n-1}_{\rm loc},$ are differentiable almost everywhere and possess absolute continuity in measure. In addition, such mappings are both ring and lower $Q$-homeomorphisms with appropriate measurable functions $Q.$ This allows us to derive various distortion results like Lipschitz, H\""older, logarithmic H\""older continuity, etc. We also establish a weak bounded variation property for such class of homeomorphisms."
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36

Motornyi, V. P., and D. A. Ovsyannikov. "Estimates of the error of interval quadrature formulas on some classes of differentiable functions." Researches in Mathematics 28, no. 1 (August 19, 2020): 12. http://dx.doi.org/10.15421/242002.

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The exact value of error of interval quadrature formulas$$\int_0^{2\pi}f(t)dt -\frac{\pi}{nh}\sum_{k=0}^{n-1}\int_{-h}^hf(t+\frac {2k\pi}{n})dt = R_n(f;\vec{c_0};\vec{x_0};h)$$obtained for the classes $W^rH^{\omega} (r=1,2,...)$ of differentiable periodic functions for which the modulus of continuity of the $r -$th derivative is majorized by the given modulus of continuity $\omega(t)$. This interval quadrature formula coincides with the rectangles formula for the Steklov functions $f_h(t)$ and is optimal for some important classes of functions.
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37

Sekigawa, Kouei, and Takashi Koda. "Compact Hermitian surfaces of pointwise constant holomorphic sectional curvature." Glasgow Mathematical Journal 37, no. 3 (September 1995): 343–49. http://dx.doi.org/10.1017/s0017089500031621.

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Let M = (M, J, g) be an almost Hermitian manifold and U(M)the unit tangent bundle of M. Then the holomorphic sectional curvature H = H(x) can be regarded as a differentiable function on U(M). If the function H is constant along each fibre, then M is called a space of pointwise constant holomorphic sectional curvature. Especially, if H is constant on the whole U(M), then M is called a space of constant holomorphic sectional curvature. An almost Hermitian manifold with an integrable almost complex structure is called a Hermitian manifold. A real 4-dimensional Hermitian manifold is called a Hermitian surface. Hermitian surfaces of pointwise constant holomorphic sectional curvature have been studied by several authors (cf. [2], [3], [5], [6] and so on).
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38

Dimitrijevic, Ivan, Branko Dragovich, Jelena Grujic, and Zoran Rakic. "Some cosmological solutions of a nonlocal modified gravity." Filomat 29, no. 3 (2015): 619–28. http://dx.doi.org/10.2298/fil1503619d.

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We consider nonlocal modification of the Einstein theory of gravity in framework of the pseudo- Riemannian geometry. The nonlocal term has the form H(R)F(?)G(R), where H and G are differentiable functions of the scalar curvature R, and F(?) = ??n=0 fn?n is an analytic function of the d?Alambert operator ?. Using calculus of variations of the action functional, we derived the corresponding equations of motion. The variation of action is induced by variation of the gravitational field, which is the metric tensor g?v. Cosmological solutions are found for the case when the Ricci scalar R is constant.
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39

Matłoka, Marian. "On Some New Inequalities for Differentiable (h1,h2)– Preinvex Functions on the Co-Ordinates." Mathematics and Statistics 2, no. 1 (January 2014): 6–14. http://dx.doi.org/10.13189/ms.2014.020102.

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40

Vivas-Cortez, Miguel, Muhammad Shoaib Saleem, Sana Sajid, Muhammad Sajid Zahoor, and Artion Kashuri. "Hermite–Jensen–Mercer-Type Inequalities via Caputo–Fabrizio Fractional Integral for h-Convex Function." Fractal and Fractional 5, no. 4 (December 10, 2021): 269. http://dx.doi.org/10.3390/fractalfract5040269.

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Integral inequalities involving many fractional integral operators are used to solve various fractional differential equations. In the present paper, we will generalize the Hermite–Jensen–Mercer-type inequalities for an h-convex function via a Caputo–Fabrizio fractional integral. We develop some novel Caputo–Fabrizio fractional integral inequalities. We also present Caputo–Fabrizio fractional integral identities for differentiable mapping, and these will be used to give estimates for some fractional Hermite–Jensen–Mercer-type inequalities. Some familiar results are recaptured as special cases of our results.
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41

Kashuri, Artion, Tingsong Du, and Rozana Liko. "On some new integral inequalities concerning twice differentiable generalized relative semi-(m,h)-preinvex mappins." Studia Universitatis Babes-Bolyai Matematica 64, no. 1 (March 14, 2019): 43–61. http://dx.doi.org/10.24193/subbmath.2019.1.05.

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42

sci, global. "H\"{o}lder Continuity of Spectral Measures for the Finitely Differentiable Quasi-Periodic Schrodinger Operators." Analysis in Theory and Applications 36, no. 1 (June 2020): 33–51. http://dx.doi.org/10.4208/ata.oa-0019.

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43

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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44

Sahoo, Soubhagya Kumar, Pshtiwan Othman Mohammed, Bibhakar Kodamasingh, Muhammad Tariq, and Y. S. Hamed. "New Fractional Integral Inequalities for Convex Functions Pertaining to Caputo–Fabrizio Operator." Fractal and Fractional 6, no. 3 (March 19, 2022): 171. http://dx.doi.org/10.3390/fractalfract6030171.

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In this article, a generalized midpoint-type Hermite–Hadamard inequality and Pachpatte-type inequality via a new fractional integral operator associated with the Caputo–Fabrizio derivative are presented. Furthermore, a new fractional identity for differentiable convex functions of first order is proved. Then, taking this identity into account as an auxiliary result and with the assistance of Hölder, power-mean, Young, and Jensen inequality, some new estimations of the Hermite-Hadamard (H-H) type inequality as refinements are presented. Applications to special means and trapezoidal quadrature formula are presented to verify the accuracy of the results. Finally, a brief conclusion and future scopes are discussed.
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45

Chen, Bang-Yen. "Classification of $h$-homogeneous production functions with constant elasticity of substitution." Tamkang Journal of Mathematics 43, no. 2 (June 25, 2012): 321–28. http://dx.doi.org/10.5556/j.tkjm.43.2012.1145.

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Almost all economic theories presuppose a production function, either on the firm level or the aggregate level. In this sense the production function is one of the key concepts of mainstream neoclassical theories. There is a very important class of production functions that are often analyzed in both microeconomics and macroeonomics; namely, $h$-homogeneous production functions. This class of production functions includes two important production functions; namely, the generalized Cobb-Douglas production functions and ACMS production functions. It was proved in 2010 by L. Losonczi \cite{L} that twice differentiable two-inputs $h$-homogeneous production functions with constant elasticity of substitution (CES) property are Cobb-Douglas' and ACMS production functions. Lozonczi also pointed out in \cite{L} that his proof does not work for production functions of $n$-inputs with $n>2$
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46

Kashuri, Artion, and Rozana Liko. "Some different type integral inequalities concerning twice differentiable generalized relative semi-$(r; m, h)$-preinvex mappings." Tbilisi Mathematical Journal 11, no. 1 (January 2018): 79–97. http://dx.doi.org/10.2478/tmj-2018-0006.

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47

Obeidat, Sofian, M. A. Latif, and S. S. Dragomir. "Fejér and Hermite-Hadamard type inequalities for differentiable h-convex and quasi convex functions with applications." Miskolc Mathematical Notes 23, no. 1 (2022): 401. http://dx.doi.org/10.18514/mmn.2022.3065.

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48

Morozov, Anatoly Nikolaevich. "Numerical Modeling Tools and S-derivatives." Modeling and Analysis of Information Systems 29, no. 1 (March 17, 2022): 20–29. http://dx.doi.org/10.18255/1818-1015-2022-1-20-29.

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Numerical study of various processes leads to the need of clarification (extensions) of the limits of applicability of computational constructs and modeling tools. For dynamical systems, this question may be related with a generalization of the concept of a derivative, which keeps the used constructions relevant. In this article we introduce the concept of weak local differentiability in a space of Lebesgue integrable functions and consider the consistency of this concept with such fundamental computational constructions as the Taylor expansion and finite differences, as well as properties of functions with a given type of differentiability on a segment. The function f from L₁[a; b] is called S-differentiable at the point x₀ from (a; b), if there are coefficients c and q, for which fx₀x₀+h (f (x) - c - q·(x-x₀)) dx = o(h²). Formulas are found for calculating the coefficients c and q, coefficients c and q, which are conveniently denoted fₛ(x₀) and fₛ ˊ(x₀) respectively. It is shown that if the function f belongs to W₁ⁿ⁻¹[a; b], n is greater than 1, and the function f⁽ⁿ⁻¹⁾ is S-differentiable at the point xₒ from (a; b), then f is approximated by a Taylor polynomial with accuracy o((x-xₒ)ⁿ), and the ratio of Δⁿₕ(f, xₒ) to hⁿ tends to fₛ⁽ⁿ⁾(xₒ) as h tends to 0. Based on the quotient Δⁿₕ (f, ·) and hⁿ, a sequence is built {Ʌₘⁿ [f]} piecewise constant functions subordinate to partitions segment [a; b] into m equal parts. It is shown that for the function f from W₁ⁿ⁻¹[a; b], for which the value is defined f ₛ⁽ⁿ⁾(xₒ), { Ʌₘⁿ [f] (xₒ)} converges to f ⁽ⁿ⁾(xₒ) as m tends to infinity, and for f from Wₚⁿ[a; b] the sequence { Ʌₘⁿ [f] } converges to f⁽ⁿ⁾ in the norm of the space Lₚ [I]. The place of S-differentiability in practical and theoretical terms is determined by its bilateral relations with ordinary differentiability. It is proved that if f belongs to W₁ⁿ⁻¹[I] and the function f⁽ⁿ⁻¹⁾ is uniformly S-differentiable on I, then f belongs to Cⁿ[f]. The constructions are algorithmic in nature and can be applied in numerically computer research of various relevant models.
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49

Rashid, Saima, Saad Ihsan Butt, Shazia Kanwal, Hijaz Ahmad, and Miao-Kun Wang. "Quantum Integral Inequalities with Respect to Raina’s Function via Coordinated Generalized Ψ -Convex Functions with Applications." Journal of Function Spaces 2021 (January 26, 2021): 1–16. http://dx.doi.org/10.1155/2021/6631474.

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In accordance with the quantum calculus, we introduced the two variable forms of Hermite-Hadamard- ( H H -) type inequality over finite rectangles for generalized Ψ -convex functions. This novel framework is the convolution of quantum calculus, convexity, and special functions. Taking into account the q ^ 1 q ^ 2 -integral identity, we demonstrate the novel generalizations of the H H -type inequality for q ^ 1 q ^ 2 -differentiable function by acquainting Raina’s functions. Additionally, we present a different approach that can be used to characterize H H -type variants with respect to Raina’s function of coordinated generalized Ψ -convex functions within the quantum techniques. This new study has the ability to generate certain novel bounds and some well-known consequences in the relative literature. As application viewpoint, the proposed study for changing parametric values associated with Raina’s functions exhibits interesting results in order to show the applicability and supremacy of the obtained results. It is expected that this method which is very useful, accurate, and versatile will open a new venue for the real-world phenomena of special relativity and quantum theory.
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50

Rashid, Saima, Saima Parveen, Hijaz Ahmad, and Yu-Ming Chu. "New quantum integral inequalities for some new classes of generalized ψ-convex functions and their scope in physical systems." Open Physics 19, no. 1 (January 1, 2021): 35–50. http://dx.doi.org/10.1515/phys-2021-0001.

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Abstract In the present study, two new classes of convex functions are established with the aid of Raina’s function, which is known as the ψ-s-convex and ψ-quasi-convex functions. As a result, some refinements of the Hermite–Hadamard ( {\mathcal{ {\mathcal H} {\mathcal H} }} )-type inequalities regarding our proposed technique are derived via generalized ψ-quasi-convex and generalized ψ-s-convex functions. Considering an identity, several new inequalities connected to the {\mathcal{ {\mathcal H} {\mathcal H} }} type for twice differentiable functions for the aforesaid classes are derived. The consequences elaborated here, being very broad, are figured out to be dedicated to recapturing some known results. Appropriate links of the numerous outcomes apprehended here with those connecting comparatively with classical quasi-convex functions are also specified. Finally, the proposed study also allows the description of a process analogous to the initial and final condition description used by quantum mechanics and special relativity theory.
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