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Journal articles on the topic 'Theorems'

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Published: 4 June 2021

Last updated: 6 September 2023

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1

KIKUCHI, MAKOTO, and TAISHI KURAHASHI. "GENERALIZATIONS OF GÖDEL’S INCOMPLETENESS THEOREMS FOR ∑n-DEFINABLE THEORIES OF ARITHMETIC." Review of Symbolic Logic 10, no. 4 (November 7, 2017): 603–16. http://dx.doi.org/10.1017/s1755020317000235.

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AbstractIt is well known that Gödel’s incompleteness theorems hold for ∑1-definable theories containing Peano arithmetic. We generalize Gödel’s incompleteness theorems for arithmetically definable theories. First, we prove that every ∑n+1-definable ∑n-sound theory is incomplete. Secondly, we generalize and improve Jeroslow and Hájek’s results. That is, we prove that every consistent theory having ∏n+1 set of theorems has a true but unprovable ∏n sentence. Lastly, we prove that no ∑n+1-definable ∑n -sound theory can prove its own ∑n-soundness. These three results are generalizations of Rosser’s improvement of the first incompleteness theorem, Gödel’s first incompleteness theorem, and the second incompleteness theorem, respectively.

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2

LIEBSCHER, VOLKMAR. "NOTE ON ENTANGLED ERGODIC THEOREMS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 02, no. 02 (June 1999): 301–4. http://dx.doi.org/10.1142/s0219025799000175.

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We prove the entangled ergodic theorem, a notion recently proposed by Accardi, Hashimoto and Obata in connection with central limit theorems1 provided a multidimensional noncommutative analogue of the spectral theorem is valid. This shows at least the possible structure of limit states in such central limit theorems.

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3

Maran, A. K. "Maran's theorem (New theorem) on Right-angled triangle." Mapana - Journal of Sciences 3, no. 1 (October 6, 2004): 7–10. http://dx.doi.org/10.12723/mjs.5.2.

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In Geometric, right-angled triangle is One Of the two—dimensional plane having three sides With one Of its angle is 9C and whk-h is important to solve problems related to Geometry and sometimes in Other subiect os well. Some fundamental concept 'theorems Of triangles are required to solve such problems and such theorems cre Pythagoras theorern' Pythagoras theorern (ii0 Appollonius theorem Euclids theorem' and (v) Eucli&s 20 theorem (Altitude theorem).3 I n addition to these, the author attempted to develop a new theorem related to right-angled triangle (Maran's theorem of right-angled triangle). The new theorem have been discussed and proved With relevant examples.

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4

Fu, Yaoshun, and Wensheng Yu. "Formalization of the Equivalence among Completeness Theorems of Real Number in Coq." Mathematics 9, no. 1 (December 25, 2020): 38. http://dx.doi.org/10.3390/math9010038.

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The formalization of mathematics based on theorem prover becomes increasingly important in mathematics and computer science, and, particularly, formalizing fundamental mathematical theories becomes especially essential. In this paper, we describe the formalization in Coq of eight very representative completeness theorems of real numbers. These theorems include the Dedekind fundamental theorem, Supremum theorem, Monotone convergence theorem, Nested interval theorem, Finite cover theorem, Accumulation point theorem, Sequential compactness theorem, and Cauchy completeness theorem. We formalize the real number theory strictly following Landau’s Foundations of Analysis where the Dedekind fundamental theorem can be proved. We extend this system and complete the related notions and properties for finiteness and sequence. We prove these theorems in turn from Dedekind fundamental theorem, and finally prove the Dedekind fundamental theorem by the Cauchy completeness theorem. The full details of formal proof are checked by the proof assistant Coq, which embodies the characteristics of reliability and interactivity. This work can lay the foundation for many applications, especially in calculus and topology.

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5

Yi, Piljin. "Index Theorems for Gauge Theories." Journal of the Korean Physical Society 73, no. 4 (August 2018): 436–48. http://dx.doi.org/10.3938/jkps.73.436.

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6

Weinberg, Steven. "Nonrenormalization Theorems in Nonrenormalizable Theories." Physical Review Letters 80, no. 17 (April 27, 1998): 3702–5. http://dx.doi.org/10.1103/physrevlett.80.3702.

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7

Graczyk, P., and C. R. E. Raja. "Classical theorems of probability on Gelfand pairs—Khinchin’s theorems and Cramér’s theorem." Israel Journal of Mathematics 132, no. 1 (December 2002): 61–107. http://dx.doi.org/10.1007/bf02784506.

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8

Greene, R. E. "Homotopy finiteness theorems and Helly’s theorem." Journal of Geometric Analysis 4, no. 3 (September 1994): 317–25. http://dx.doi.org/10.1007/bf02921584.

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9

Chen, Han. "On the Proof of Sylow’s Theorems in Group Theory." Highlights in Science, Engineering and Technology 47 (May 11, 2023): 63–66. http://dx.doi.org/10.54097/hset.v47i.8165.

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The Sylow’s theorems are significant principles for analysis of special subgroups of a finite group, and they are significant theories in finite group research area. It is firmly established that every group owns more than one Sylow group pertaining to every prime factor of . The purpose of this article is to summarize and generalize the existing research progress of Sylow’s theorems. A summary of them will help more people comprehend and apply group theory to analyze problems. In this paper, the basic information of three distinct Sylow’s theorems are introduced, including their definitions and detailed proof procedures. After that, some examples and application that are related to the Sylow’s theorems are shown one by one. After that, the relationship between the Sylow’s theorems to that of the orbit-stabilizer theorem is discussed. This work will potentially stimulate more research efforts on the basis theorems in group theory.

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10

KO, HWEI-MEI, and KOK-KEONG TAN. "COINCIDENCE THEOREMS AND MATCHING THEOREMS." Tamkang Journal of Mathematics 23, no. 4 (December 1, 1992): 297–309. http://dx.doi.org/10.5556/j.tkjm.23.1992.4553.

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Two coincidence theorems of Ky Fan are first slightly generalized. As applications, new matching theorems are obtained, one of which has several equivalent forms, including the classical Knaster Kuratowski-Mazurkiewicz theorem.

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11

Tantau, Till. "Weak cardinality theorems." Journal of Symbolic Logic 70, no. 3 (September 2005): 861–78. http://dx.doi.org/10.2178/jsl/1122038917.

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AbstractKummer's Cardinality Theorem states that a language A must be recursive if a Turing machine can exclude for any n words , …, one of the n + 1 possibilities for the cardinality of {, …, }⋂ A. There was good reason to believe that this theorem is a peculiarity of recursion theory: neither the Cardinality Theorem nor weak forms of it hold for resource-bounded computational models like polynomial time. This belief may be flawed. In this paper it is shown that weak cardinality theorems hold for finite automata and also for other models. An explanation is proposed as to why recursion-theoretic and automata-theoretic weak cardinality theorems hold, but not corresponding 'middle-ground theorems': The recursion- and automata-theoretic weak cardinality theorems are instantiations of purely logical weak cardinality theorems. The logical theorems can be instantiated for logical structures characterizing recursive computations and finite automata computations. A corresponding structure characterizing polynomial time computations does not exist.

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12

Sobel, Jordan Howard. "Partition-Theorems for Causal Decision Theories." Philosophy of Science 56, no. 1 (March 1989): 70–93. http://dx.doi.org/10.1086/289473.

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13

Taubes, Clifford Henry. "Unique continuation theorems in gauge theories." Communications in Analysis and Geometry 2, no. 1 (1994): 35–52. http://dx.doi.org/10.4310/cag.1994.v2.n1.a2.

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14

Matsumura, Shin-ichi. "A Nadel vanishing theorem via injectivity theorems." Mathematische Annalen 359, no. 3-4 (February 12, 2014): 785–802. http://dx.doi.org/10.1007/s00208-014-1018-6.

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15

Chang, S. S., X. Wu, and S. W. Xiang. "A Topological KKM Theorem and Minimax Theorems." Journal of Mathematical Analysis and Applications 182, no. 3 (March 1994): 756–67. http://dx.doi.org/10.1006/jmaa.1994.1119.

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16

Park, Sehie. "Equivalents of maximum principles for several spaces." Topological Algebra and its Applications 10, no. 1 (January 1, 2022): 68–76. http://dx.doi.org/10.1515/taa-2022-0113.

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Abstract According to our long-standing Metatheorem, certain maximum theorems can be equivalently reformulated to various types of fixed point theorems, and conversely. As examples of such theorems, in this paper, we list Zermelo’s fixed point theorem, Brøndsted’s principle, Fang’s F-type theorem, related theorems for locally convex spaces and quasi-uniform spaces. Further we review some few works concerned with our Metatheorem.

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17

Park, Sehie. "Fixed points, intersection theorems, variational inequalities, and equilibrium theorems." International Journal of Mathematics and Mathematical Sciences 24, no. 2 (2000): 73–93. http://dx.doi.org/10.1155/s0161171200002593.

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From a fixed point theorem for compact acyclic maps defined on admissible convex sets in the sense of Klee, we first deduce collectively fixed point theorems, intersection theorems for sets with convex sections, and quasi-equilibrium theorems. These quasi-equilibrium theorems are applied to give simple and unified proofs of the known variational inequalities of the Hartman-Stampacchia-Browder type. Moreover, from our new fixed point theorem, we deduce new variational inequalities which can be used to obtain fixed point results for convex-valued maps. Finally, various general economic equilibrium theorems are deduced in the forms of the Nash type, the Tarafdar type, and the Yannelis-Prabhakar type. Our results are stated for not-necessarily locally convex topological vector spaces and for abstract economies with arbitrary number of commodities and agents. Our new results extend a lot of known works with much simpler proofs.

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18

MacCaull, W. A. "On the validity of hilbert's nullstellensatz, artin's theorem, and related results in grothendieck toposes." Journal of Symbolic Logic 53, no. 4 (December 1988): 1177–87. http://dx.doi.org/10.1017/s0022481200028000.

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Using formally intuitionistic logic coupled with infinitary logic and the completeness theorem for coherent logic, we establish the validity, in Grothendieck toposes, of a number of well-known, classically valid theorems about fields and ordered fields. Classically, these theorems have proofs by contradiction and most involve higher order notions. Here, the theorems are each given a first-order formulation, and this form of the theorem is then deduced using coherent or formally intuitionistic logic. This immediately implies their validity in arbitrary Grothendieck toposes. The main idea throughout is to use coherent theories and, whenever possible, find coherent formulations of formulas which then allow us to call upon the completeness theorem of coherent logic. In one place, the positive model-completeness of the relevant theory is used to find the necessary coherent formulas.The theorems here deal with polynomials or rational functions (in s indeterminates) over fields. A polynomial over a field can, of course, be represented by a finite string of field elements, and a rational function can be represented by a pair of strings of field elements. We chose the approach whereby results on polynomial rings are reduced to results about the base field, because the theory of polynomial rings in s indeterminates over fields, although coherent, is less desirable from a model-theoretic point of view. Ultimately we are interested in the models.This research was originally motivated by the works of Saracino and Weispfenning [SW], van den Dries [Dr], and Bunge [Bu], each of whom generalized some theorems from algebraic geometry or ordered fields to (commutative, von Neumann) regular rings (with unity).

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19

AHMED, M. A. "SOME FIXED POINT THEOREMS." International Journal of Geometric Methods in Modern Physics 08, no. 01 (February 2011): 1–8. http://dx.doi.org/10.1142/s0219887811004926.

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This paper has three objectives. First, we establish a fixed point theorem for a generalized contraction in dislocated quasi-metric spaces. Second, we present a characterization of a unique fixed point for any mapping. Third, we prove another fixed point theorem in complete dislocated quasi-metric spaces. These theorems generalize known results, especially some theorems in [1–3, 5, 7, 8, 11–14, 16, 19, 20, 22]. Also, we give some comments on [17, Theorem 3] and [21, Theorem 1].

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20

Walters, Peter. "Topological Wiener–Wintner ergodic theorems and a randomL2ergodic theorem." Ergodic Theory and Dynamical Systems 16, no. 1 (February 1996): 179–206. http://dx.doi.org/10.1017/s0143385700008762.

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AbstractWe give some topological ergodic theorems inspired by theWiener-Wintnerergodic theorem. These theorems are used to give results for uniquely ergodic transformations and to study unique equilibrium states for shift maps. These latter results give randomL2ergodic theorems for a finite set of commuting measure-preserving transformations.

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21

Gan, Wenbin, Xinguo Yu, Ting Zhang, and Mingshu Wang. "Automatically Proving Plane Geometry Theorems Stated by Text and Diagram." International Journal of Pattern Recognition and Artificial Intelligence 33, no. 07 (June 7, 2019): 1940003. http://dx.doi.org/10.1142/s0218001419400032.

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This paper presents an algorithm for proving plane geometry theorems stated by text and diagram in a complementary way. The problem of proving plane geometry theorems involves two challenging subtasks, being theorem understanding and theorem proving. This paper proposes to consider theorem understanding as a problem of extracting relations from text and diagram. A syntax–semantics (S2) model method is proposed to extract the geometric relations from theorem text, and a diagram mining method is proposed to extract geometry relations from diagram. Then, a procedure is developed to obtain a set of relations that is consistent with the given theorem with high confidence. Finally, theorem proving is conducted by using the existing proving methods which take the extracted geometric relations as input. The experimental results show that the proposed theorem proving algorithm can prove 86% of plane geometry theorems in the test dataset of 200 theorems, which is all the theorems in the popular textbook. The proposed algorithm outperforms the existing algorithms mainly because it can extract relations not only from text but also from diagram.

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22

Feng, Yaoshen, Ziyue Song, and Baojie Xu. "Cauchy Theorem, Sylow Theorems, and Orbit-Stabilizer Theorem in Group Theory." Highlights in Science, Engineering and Technology 47 (May 11, 2023): 58–62. http://dx.doi.org/10.54097/hset.v47i.8164.

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With the rapid development of modern mathematics, math researchers have an increasing demand to take the advantage of group theory in the latest field. The group action is one of the essential parts of the group theory. In order to have a better understand of group action, this paper will describe the insight development and logic in it step by step. For this purpose, three essential theorems, which are Cauchy theorem, Sylow theorems, and orbit-stabilizer theorem, are chosen as representative examples. The proof and applications in some fields of modern science of these three theorems are discussed. In the proof, this paper emphasizes that group action can be used conveniently to solve the problem in group theory. In the application, this paper includes as many fields as possible to attach importance to group action. This paper is expected to give the opinion of how the field of group action is established and its far-reaching influence to the modern science.

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23

Öztürk, Özkan, Raegan Higgins, and Georgia Kittou. "Oscillation of three-dimensional time scale systems with fixed point theorems." Filomat 35, no. 6 (2021): 1915–25. http://dx.doi.org/10.2298/fil2106915o.

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Oscillation and nonoscillation theories play very important roles in gaining information about the long-time behavior of solutions of a system. Therefore, we investigate the asymptotic behavior of nonoscillatory solutions as well as the existence of such solutions so that one can determine the limit behavior. For the existence, we use some fixed point theorems such as Schauder?s fixed point theorem and the Knaster fixed point theorem.

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24

Zolotarev, V. M. "Limit Theorems as Stability Theorems." Theory of Probability & Its Applications 34, no. 1 (January 1990): 153–63. http://dx.doi.org/10.1137/1134006.

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25

Jiahe, Jiang. "Coincidence theorems and minimax theorems." Acta Mathematica Sinica 5, no. 4 (December 1989): 307–20. http://dx.doi.org/10.1007/bf02107708.

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26

Laskowski, Michael Chris. "Uncountable theories that are categorical in a higher power." Journal of Symbolic Logic 53, no. 2 (June 1988): 512–30. http://dx.doi.org/10.1017/s0022481200028437.

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AbstractIn this paper we prove three theorems about first-order theories that are categorical in a higher power. The first theorem asserts that such a theory either is totally categorical or there exist prime and minimal models over arbitrary base sets. The second theorem shows that such theories have a natural notion of dimension that determines the models of the theory up to isomorphism. From this we conclude that I(T,ℵα,) = ℵ0 + ∣α∣ where ℵα = the number of formulas modulo T-equivalence provided that T is not totally categorical. The third theorem gives a new characterization of these theories.

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27

Li, Chengze, Yunhan Lin, and Shichen Xiang. "Complex Analysis and Residue Theorem." Highlights in Science, Engineering and Technology 38 (March 16, 2023): 736–45. http://dx.doi.org/10.54097/hset.v38i.5938.

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Mathematics is knitted into our lives by laying the fundamental base of many things. But people still think it is unnecessary to study math deeply due to its complexity. A lot of the great work from mathematicians is not noticed by other fields. Among all the essential mathematical methods and theorems, the Residue Theorem is one of the most significant ones. Therefore, this article will interpret the Residue Theorem step by step with its related theorems. Before jumping into the Residue Theorem, the article will introduce a series of mathematical methods, including the Triangle Inequality, the Euler's Formula, the Analytic function, the Taylor Series, and the Laurent Series. The applications and graphs of these mathematical methods will help to understand them further. Consequently, this article will demonstrate the most crucial part of this article — the Residue Theorem by its definition, formula, and example. Last but not least, this article will finish with how the Residue Theorem is used in our wind power generation system to express further how math correlates to our daily life. The result of this article shows that the Residue Theorem can be used to help explain other theorems and to extend new mathematical theories. It can be used not only in mathematics but also in a lot of fields, such as computer science, physics, and engineering. This article hopes a more extensive population can acknowledge the contribution of math to our world through the Residue Theorem.

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28

Alani, Ivo, and Osvaldo P. Santillán. "Cosmological singularity theorems forf(R) gravity theories." Journal of Cosmology and Astroparticle Physics 2016, no. 05 (May 10, 2016): 023. http://dx.doi.org/10.1088/1475-7516/2016/05/023.

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29

MAJID, SHAHN. "RECONSTRUCTION THEOREMS AND RATIONAL CONFORMAL FIELD THEORIES." International Journal of Modern Physics A 06, no. 24 (October 10, 1991): 4359–74. http://dx.doi.org/10.1142/s0217751x91002100.

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We obtain an explicit reconstruction theorem for rational conformal field theories and other situations where we are presented with a braided or quasitensor category [Formula: see text]. It takes the form of a generalized Fourier transform. The reconstructed object turns out to be a quantum group in a generalized sense. Our results include both the Tannaka-Krein case where there is a functor [Formula: see text], and the case where there is no functor at all.

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30

Lerche, W., and N. P. Warner. "Index theorems in N=2 superconformal theories." Physics Letters B 205, no. 4 (May 1988): 471–78. http://dx.doi.org/10.1016/0370-2693(88)90980-x.

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31

Sethi, Savdeep, and Mark Stern. "Invariance theorems for supersymmetric Yang–Mills theories." Advances in Theoretical and Mathematical Physics 4, no. 2 (2000): 487–501. http://dx.doi.org/10.4310/atmp.2000.v4.n2.a8.

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32

Ume, Jeong Sheok, and Jong Kyu Kim. "Common fixed point theorems for two mappings satisfying some conditions." Bulletin of the Australian Mathematical Society 62, no. 1 (August 2000): 75–85. http://dx.doi.org/10.1017/s0004972700018499.

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In this paper, using the concept of w-distance, we first prove common fixed point theorems in a complete metric space. Then these theorems are used to improve Kannan's fixed point theorem, Ćirić's fixed point theorem, Kada, Suzuki and Takahashi's fixed point theorem and Ume's fixed point theorem.

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33

Darzi, Rahmat, Rostamian Delavar, and Mehdi Roohi. "Fixed point theorems in minimal generalized convex spaces." Filomat 25, no. 4 (2011): 165–76. http://dx.doi.org/10.2298/fil1104165d.

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This paper deals with coincidence and fixed point theorems in minimal generalized convex spaces. By establishing a kind of KKM Principle in minimal generalized convex space, we obtain some results on coincidence point and fixed point theorems. Generalized versions of Ky Fan?s lemma, Fan-Browder fixed point theorem, Nash equilibrium theorem and some Urai?s type fixed point theorems in minimal generalized convex spaces are given.

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34

Burkett, Shawn T. "A Jordan–Hölder type theorem for supercharacter theories." Journal of Group Theory 23, no. 3 (May 1, 2020): 399–414. http://dx.doi.org/10.1515/jgth-2019-0028.

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AbstractThe Jordan–Hölder theorem is a general term given to a collection of theorems about maximal chains in suitably nice lattices. For example, the well-known Jordan–Hölder type theorem for chief series of finite groups has been rather useful in studying the structure of finite groups. In this paper, we present a Jordan–Hölder type theorem for supercharacter theories of finite groups, which generalizes the one for chief series of finite groups.

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35

Lin, Tzu-Chu. "Approximation Theorems and Fixed Point Theorem in Cones." Proceedings of the American Mathematical Society 102, no. 3 (March 1988): 502. http://dx.doi.org/10.2307/2047211.

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36

Kennedy, Paul, and Kenneth Evans. "The Pythagorean Theorem and Area: Postulates into Theorems." Humanistic Mathematics Network Journal 1, no. 25 (August 2001): 33–37. http://dx.doi.org/10.5642/hmnj.200101.25.13.

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37

Eisenbaum, Nathalie. "Dynkin's isomorphism theorem and the Ray-Knight theorems." Probability Theory and Related Fields 99, no. 2 (June 1994): 321–35. http://dx.doi.org/10.1007/bf01199028.

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38

Chen, ZhiHua, Min Ru, and QiMing Yan. "The truncated Second Main Theorem and uniqueness theorems." Science China Mathematics 53, no. 3 (March 2010): 605–16. http://dx.doi.org/10.1007/s11425-010-0039-1.

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39

Khanh, Phan Quoc. "An induction theorem and general open mapping theorems." Journal of Mathematical Analysis and Applications 118, no. 2 (September 1986): 519–34. http://dx.doi.org/10.1016/0022-247x(86)90279-9.

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40

Fritsch, Rudolf. "Remarks on orthocenters, Pappus’ theorem and Butterfly theorems." Journal of Geometry 107, no. 2 (December 11, 2015): 305–16. http://dx.doi.org/10.1007/s00022-015-0304-0.

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41

Guo, Jiaming, and Biran Song. "Relationship Between Cauchy Integral Theorem and Residue Theorems." Highlights in Science, Engineering and Technology 38 (March 16, 2023): 775–80. http://dx.doi.org/10.54097/hset.v38i.5944.

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Cauchy integral theorem belongs to an extremely important part of complex functions, which is a fundamental bridge, and people can derive Cauchy integral theorem from the residue theorem. Cauchy's integral theorem is generally applied in many higher mathematics, is an important theorem concerning path integrals of fully pure functions. It claims that if there are two different paths from a point to another point and the function is fully pure between these two different paths, then it can be derived that the two path integrals of the function are equal. It is widely believed that the a generalization of Cauchy integral theorem and Cauchy integral formula is just like the residue theorem, and the flexible use of the residue theorem can easily solve some difficult problems in complex functions. Therefore, this paper will use the definition and derivation process of Cauchy residue theorem and integral theorem to demonstrate the specific connection between them and their practical application.

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42

Berkani, M., and H. Zariouh. "Extended Weyl type theorems." Mathematica Bohemica 134, no. 4 (2009): 369–78. http://dx.doi.org/10.21136/mb.2009.140669.

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43

Brattka, Vasco, and Guido Gherardi. "Effective Choice and Boundedness Principles in Computable Analysis." Bulletin of Symbolic Logic 17, no. 1 (March 2011): 73–117. http://dx.doi.org/10.2178/bsl/1294186663.

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AbstractIn this paper we study a new approach to classify mathematical theorems according to their computational content. Basically, we are asking the question which theorems can be continuously or computably transferred into each other? For this purpose theorems are considered via their realizers which are operations with certain input and output data. The technical tool to express continuous or computable relations between such operations is Weihrauch reducibility and the partially ordered degree structure induced by it. We have identified certain choice principles such as co-finite choice, discrete choice, interval choice, compact choice and closed choice, which are cornerstones among Weihrauch degrees and it turns out that certain core theorems in analysis can be classified naturally in this structure. In particular, we study theorems such as the Intermediate Value Theorem, the Baire Category Theorem, the Banach Inverse Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem. We also explore how existing classifications of the Hahn–Banach Theorem and Weak Kőnig's Lemma fit into this picture. Well-known omniscience principles from constructive mathematics such as LPO and LLPO can also naturally be considered as Weihrauch degrees and they play an important role in our classification. Based on this we compare the results of our classification with existing classifications in constructive and reverse mathematics and we claim that in a certain sense our classification is finer and sheds some new light on the computational content of the respective theorems. Our classification scheme does not require any particular logical framework or axiomatic setting, but it can be carried out in the framework of classical mathematics using tools of topology, computability theory and computable analysis. We develop a number of separation techniques based on a new parallelization principle, on certain invariance properties of Weihrauch reducibility, on the Low Basis Theorem of Jockusch and Soare and based on the Baire Category Theorem. Finally, we present a number of metatheorems that allow to derive upper bounds for the classification of the Weihrauch degree of many theorems and we discuss the Brouwer Fixed Point Theorem as an example.

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44

C. anak, ˙Ibrah˙im, and Üm˙it Totur. "A note on Tauberian Theorems for regularly generated sequences." Tamkang Journal of Mathematics 39, no. 2 (June 30, 2008): 187–91. http://dx.doi.org/10.5556/j.tkjm.39.2008.29.

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45

He, Rong-Hua. "Some Results for the Family KKM(X,Y) and the Φ-Mapping in FC-Spaces." ISRN Applied Mathematics 2011 (October 23, 2011): 1–13. http://dx.doi.org/10.5402/2011/696917.

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We first establish a fixed point theorem for a k-set contraction map on the family KKM(X,X), which does not need to be a compact map. Next, we present the KKM type theorems, matching theorems, coincidence theorems, and minimax theorems on the family KKM(X,Y) and the Φ-mapping in FC-spaces. Our results improve and generalize some recent results.

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46

Lin, Lai-Jiu, and Hsin I. Chen. "Coincidence theorems for families of multimaps and their applications to equilibrium problems." Abstract and Applied Analysis 2003, no. 5 (2003): 295–309. http://dx.doi.org/10.1155/s1085337503210034.

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We apply some continuous selection theorems to establish coincidence theorems for a family of multimaps under various conditions. Then we apply these coincidence theorems to study the equilibrium problem withmfamilies of players and2mfamilies of constraints on strategy sets. We establish the existence theorems of equilibria of this problem and existence theorem of equilibria of abstract economics with two families of players.

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47

Lupaş, Alina Alb. "Applications of the Fractional Calculus in Fuzzy Differential Subordinations and Superordinations." Mathematics 9, no. 20 (October 15, 2021): 2601. http://dx.doi.org/10.3390/math9202601.

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The fractional integral of confluent hypergeometric function is used in this paper for obtaining new applications using concepts from the theory of fuzzy differential subordination and superordination. The aim of the paper is to present new fuzzy differential subordinations and superordinations for which the fuzzy best dominant and fuzzy best subordinant are given, respectively. The original theorems proved in the paper generate interesting corollaries for particular choices of functions acting as fuzzy best dominant and fuzzy best subordinant. Another contribution contained in this paper is the nice sandwich-type theorem combining the results given in two theorems proved here using the two theories of fuzzy differential subordination and fuzzy differential superordination.

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48

Hartono, Hartono. "Resolution of Dynamic Optimization Problems Constrained by the Fraction Penalty Method." Indonesian Journal of Information Systems 2, no. 2 (February 28, 2020): 51. http://dx.doi.org/10.24002/ijis.v2i2.3223.

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This article discusses the application of fractional penalty method to solve dynamic optimization problem with state constraints. The main theories supporting the use of the method are described in some theorem and corollary. The theorems give sufficient conditons for the application of the method. Therefore, if all conditions mentioned in the theorems are met then the resulted solution will converge to the analytic solution. In addition, there are some examples to support the theory. The numerical simulation shows that the accuracy of the method is quite good. Hence, this method can play a role as an alternative method for solving dynamic optimization problem with state constrints.

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Grewal, Brijesh, and Meenu Goyal. "Approximation by a family of summation-integral type operators preserving linear functions." Filomat 36, no. 16 (2022): 5563–72. http://dx.doi.org/10.2298/fil2216563g.

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This article investigates the approximation properties of a general family of positive linear operators defined on the unbounded interval [0,?). We prove uniform convergence theorem and Voronovskayatype theorem for functions with polynomial growth. More precisely, we study weighted approximation i.e basic convergence theorems, quantitative Voronovskaya-asymptotic theorems and Gr?ss Voronovskayatype theorems in weighted spaces. Finally, we obtain the rate of convergence of these operators via a suitable weighted modulus of continuity.

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50

Boukhrisse, H., and M. Moussaoui. "Critical point theorems." International Journal of Mathematics and Mathematical Sciences 29, no. 7 (2002): 429–38. http://dx.doi.org/10.1155/s0161171202011249.

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LetHbe a Hilbert space such thatH=V⊕W, whereVandWare two closed subspaces ofH. We generalize an abstract theorem due to Lazer et al. (1975) and a theorem given by Moussaoui (1990-1991) to the case whereVandWare not necessarily finite dimensional. We give two mini-max theorems where the functionalΦ:H→ℝis of class𝒞2and𝒞1, respectively.

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