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Journal articles on the topic 'Fundamental theorems of LM-logic algebra'

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Published: 3 June 2025

Last updated: 31 July 2025

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1

Solomon, Reed. "Ordered Groups: A Case Study in Reverse Mathematics." Bulletin of Symbolic Logic 5, no. 1 (1999): 45–58. http://dx.doi.org/10.2307/421140.

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The fundamental question in reverse mathematics is to determine which set existence axioms are required to prove particular theorems of mathematics. In addition to being interesting in their own right, answers to this question have consequences in both effective mathematics and the foundations of mathematics. Before discussing these consequences, we need to be more specific about the motivating question.Reverse mathematics is useful for studying theorems of either countable or essentially countable mathematics. Essentially countable mathematics is a vague term that is best explained by an exam

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2

LAMBROPOULOU, SOFIA. "L-MOVES AND MARKOV THEOREMS." Journal of Knot Theory and Its Ramifications 16, no. 10 (2007): 1459–68. http://dx.doi.org/10.1142/s0218216507005919.

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Given a knot theory (virtual, singular, knots in a 3-manifold etc.), there are deep relations between the diagrammatic knot equivalence in this theory, the braid structures and a corresponding braid equivalence. The L-moves between braids, due to their fundamental nature, may be adapted to any diagrammatic situation in order to formulate a corresponding braid equivalence. In this short paper, we discuss and compare various diagrammatic set-ups and results therein, in order to draw the underlying logic relating diagrammatic isotopy, braid structures, Markov theorems and L-move analogues. Finall

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3

Ion, C. Baianu, Georgescu George, F. Glazebrook James, and Brown Ronald. "BRAIN Journal - Lukasiewicz-Moisil Many-Valued Logic Algebra of Highly-Complex Systems." Brain Journal 1, SPECIAL ISSUE ON COMPLEXITY IN SCIENCES AND ARTIFICIAL INTELLIGENCE (2010): 1–11. https://doi.org/10.5281/zenodo.1037321.

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ABSTRACT The fundamentals of ÃLukasiewicz-Moisil logic algebras and their applications to complex genetic network dynamics and highly complex systems are presented in the context of a categorical ontology theory of levels, Medical Bioinformatics and self-organizing, highly complex systems. Quantum Automata were defined in refs.[2] and [3] as generalized, probabilistic automata with quantum state spaces [1]. Their next-state functions operate through transitions between quantum states defined by the quantum equations of motions in the Schr¨odinger representation, with both initial and boundary

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4

Iovino, José. "On the maximality of logics with approximations." Journal of Symbolic Logic 66, no. 4 (2001): 1909–18. http://dx.doi.org/10.2307/2694984.

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In this paper we analyze some aspects of the question of using methods from model theory to study structures of functional analysis.By a well known result of P. Lindström, one cannot extend the expressive power of first order logic and yet preserve its most outstanding model theoretic characteristics (e.g., compactness and the Löwenheim-Skolem theorem). However, one may consider extending the scope of first order in a different sense, specifically, by expanding the class of structures that are regarded as models (e.g., including Banach algebras or other structures of functional analysis), and

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Khan, Wilayat, Muhammad Kamran, Syed Rameez Naqvi, Farrukh Aslam Khan, Ahmed S. Alghamdi, and Eesa Alsolami. "Formal Verification of Hardware Components in Critical Systems." Wireless Communications and Mobile Computing 2020 (February 20, 2020): 1–15. http://dx.doi.org/10.1155/2020/7346763.

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Hardware components, such as memory and arithmetic units, are integral part of every computer-controlled system, for example, Unmanned Aerial Vehicles (UAVs). The fundamental requirement of these hardware components is that they must behave as desired; otherwise, the whole system built upon them may fail. To determine whether or not a component is behaving adequately, the desired behaviour of the component is often specified in the Boolean algebra. Boolean algebra is one of the most widely used mathematical tools to analyse hardware components represented at gate level using Boolean functions.

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Oaknin, David H. "Bypassing the Kochen–Specker Theorem: An Explicit Non-Contextual Statistical Model for the Qutrit." Axioms 12, no. 1 (2023): 90. http://dx.doi.org/10.3390/axioms12010090.

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We describe an explicitly non-contextual statistical model of hidden variables for the qutrit, which fully reproduces the predictions of quantum mechanics, and thus, bypasses the constraints imposed by the Kochen–Specker theorem and its subsequent reformulations. We notice that these renowned theorems crucially rely on the implicitly assumed existence of an absolute frame of reference with respect to which physically indistinguishable tests related by spurious gauge transformations can supposedly be assigned well-defined distinct identities. We observe that the existence of such an absolute fr

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Srivastava, Hari Mohan, Bidu Bhusan Jena, and Susanta Kumar Paikray. "Statistical Riemann and Lebesgue Integrable Sequence of Functions with Korovkin-Type Approximation Theorems." Axioms 10, no. 3 (2021): 229. http://dx.doi.org/10.3390/axioms10030229.

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In this work we introduce and investigate the ideas of statistical Riemann integrability, statistical Riemann summability, statistical Lebesgue integrability and statistical Lebesgue summability via deferred weighted mean. We first establish some fundamental limit theorems connecting these beautiful and potentially useful notions. Furthermore, based upon our proposed techniques, we establish the Korovkin-type approximation theorems with algebraic test functions. Finally, we present two illustrative examples under the consideration of positive linear operators in association with the Bernstein

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8

Rao, N. Seshagiri, and K. Kalyani. "Fixed point results of \((\phi,\psi)\)-weak contractions in ordered \(b\)-metric spaces." Cubo (Temuco) 24, no. 2 (2022): 343–68. http://dx.doi.org/10.56754/0719-0646.2402.0343.

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The purpose of this paper is to prove some results on fixed point, coincidence point, coupled coincidence point and coupled common fixed point for the mappings satisfying generalized \((\phi, \psi)\)-contraction conditions in complete partially ordered \(b\)-metric spaces. Our results generalize, extend and unify most of the fundamental metrical fixed point theorems in the existing literature. A few examples are illustrated to support our findings.

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Taketomi, Nanami, and Takeshi Emura. "Consistency of the Estimator for the Common Mean in Fixed-Effect Meta-Analyses." Axioms 12, no. 5 (2023): 503. http://dx.doi.org/10.3390/axioms12050503.

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Fixed-effect meta-analyses aim to estimate the common mean parameter by the best linear unbiased estimator. Besides unbiasedness, consistency is one of the most fundamental requirements for the common mean estimator to be valid. However, conditions for the consistency of the common mean estimator have not been discussed in the literature. This article fills this gap by clarifying conditions for making the common mean estimator consistent in fixed-effect meta-analyses. In this article, five theorems are devised, which state regularity conditions for the common mean estimator to be consistent. T

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Benkerrouche, Amar, Mohammed Said Souid, Gani Stamov, and Ivanka Stamova. "Multiterm Impulsive Caputo–Hadamard Type Differential Equations of Fractional Variable Order." Axioms 11, no. 11 (2022): 634. http://dx.doi.org/10.3390/axioms11110634.

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In this study, we deal with an impulsive boundary value problem (BVP) for differential equations of variable fractional order involving the Caputo–Hadamard fractional derivative. The fundamental problems of existence and uniqueness of solutions are studied, and new existence and uniqueness results are established in the form of two fixed point theorems. In addition, Ulam–Hyers stability sufficient conditions are proved illustrating the suitability of the derived fundamental results. The obtained results are supported also by an example. Finally, the conclusion notes are highlighted.

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11

Sekatskii, Sergey. "On the Applications of the Generalized Littlewood Theorem Concerning Integrals of the Logarithm of Analytical Functions to Elliptic Functions." Axioms 12, no. 6 (2023): 595. http://dx.doi.org/10.3390/axioms12060595.

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Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain new criteria equivalent to the Riemann hypothesis. Later, the same theorem was applied to calculate certain infinite sums and study the properties of zeroes of a few analytical functions. In this study, we apply this approach to elliptic functions of Jacobi and Weierstrass. Numerous sums over inverse powers of zeroes and poles are calculated, including some results for the Jacobi elliptic functions sn(z, k) and others understood as functions o

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12

Choi, Junesang, Gradimir V. Milovanović, and Arjun K. Rathie. "Generalized Summation Formulas for the KampÉ de FÉriet Function." Axioms 10, no. 4 (2021): 318. http://dx.doi.org/10.3390/axioms10040318.

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By employing two well-known Euler’s transformations for the hypergeometric function 2F1, Liu and Wang established numerous general transformation and reduction formulas for the Kampé de Fériet function and deduced many new summation formulas for the Kampé de Fériet function with the aid of classical summation theorems for the 2F1 due to Kummer, Gauss and Bailey. Here, by making a fundamental use of the above-mentioned reduction formulas, we aim to establish 32 general summation formulas for the Kampé de Fériet function with the help of generalizations of the above-referred summation formulas f

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13

Khan, Shahid, Jong-Suk Ro, Fairouz Tchier, and Nazar Khan. "Applications of Fuzzy Differential Subordination for a New Subclass of Analytic Functions." Axioms 12, no. 8 (2023): 745. http://dx.doi.org/10.3390/axioms12080745.

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This work is concerned with the branch of complex analysis known as geometric function theory, which has been modified for use in the study of fuzzy sets. We develop a novel operator Lα,λm:An→An in the open unit disc Δ using the Noor integral operator and the generalized Sălăgean differential operator. First, we develop fuzzy differential subordination for the operator Lα,λm and then, taking into account this operator, we define a particular fuzzy class of analytic functions in the open unit disc Δ, represented by Rϝλ(m,α,δ). Using the idea of fuzzy differential subordination, several new resu

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HYLAND, J. M. E. "Classical lambda calculus in modern dress." Mathematical Structures in Computer Science 27, no. 5 (2015): 762–81. http://dx.doi.org/10.1017/s0960129515000377.

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Recent developments in the categorical foundations of universal algebra have given an impetus to an understanding of the lambda calculus coming from categorical logic: an interpretation is a semi-closed algebraic theory. Scott's representation theorem is then completely natural and leads to a precise Fundamental Theorem showing the essential equivalence between the categorical and more familiar notions.

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15

Zashev, J. "On the recursion theorem in iterative operative spaces." Journal of Symbolic Logic 66, no. 4 (2001): 1727–48. http://dx.doi.org/10.2307/2694971.

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Abstract.The recursion theorem in abstract partially ordered algebras, such as operative spaces and others, is the most fundamental result of algebraic recursion theory. The primary aim of the present paper is to prove this theorem for iterative operative spaces in full generality. As an intermediate result, a new and rather large class of models of the combinatory logic is obtained.

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Galkanov, Allaberdi G. "ALGEBRAIC EQUATIONS IN UNITARY SPACE AND SHORTEST ALGEBRAIC PROOF OF THE FUNDAMENTAL THEOREM OF ALGEBRA." RSUH/RGGU Bulletin. Series Information Science. Information Security. Mathematics, no. 1 (2022): 83–97. http://dx.doi.org/10.28995/2686-679x-2022-1-83-97.

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The article deals with the presentation of some results obtained in the study of algebraic equations with complex coefficients from a complex variable in a unitary space. In the orthonormal basis, two vectors are introduced, which are called the vector of root and the vector of coefficients of an algebraic polynomial. With the help of these vectors, an algebraic polynomial is represented as a scalar product of them in an orthonormal basis. The criterion of linear independence of a set of root vectors is formulated and proved. A Theorem is formulated and proved that the maximum number of simple

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17

Gill, Richard David. "Does Geometric Algebra Provide a Loophole to Bell’s Theorem?" Entropy 22, no. 1 (2019): 61. http://dx.doi.org/10.3390/e22010061.

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In 2007, and in a series of later papers, Joy Christian claimed to refute Bell’s theorem, presenting an alleged local realistic model of the singlet correlations using techniques from geometric algebra (GA). Several authors published papers refuting his claims, and Christian’s ideas did not gain acceptance. However, he recently succeeded in publishing yet more ambitious and complex versions of his theory in fairly mainstream journals. How could this be? The mathematics and logic of Bell’s theorem is simple and transparent and has been intensely studied and debated for over 50 years. Christian

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18

Toffano, Zeno, and François Dubois. "Adapting Logic to Physics: The Quantum-Like Eigenlogic Program." Entropy 22, no. 2 (2020): 139. http://dx.doi.org/10.3390/e22020139.

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Considering links between logic and physics is important because of the fast development of quantum information technologies in our everyday life. This paper discusses a new method in logic inspired from quantum theory using operators, named Eigenlogic. It expresses logical propositions using linear algebra. Logical functions are represented by operators and logical truth tables correspond to the eigenvalue structure. It extends the possibilities of classical logic by changing the semantics from the Boolean binary alphabet { 0 , 1 } using projection operators to the binary alphabet { + 1 , − 1

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19

М.М., Исакова, Тлупова Р.Г., Канкулова С.Х., Эржибова Ф.А. та Ибрагим А.С. "О СИНТЕТИЧЕСКОМ МЕТОДЕ РЕШЕНИЯ ЗАДАЧ". Журнал "Вестник Челябинского государственного педагогического университета", № 1 (28 лютого 2018): 108–17. https://doi.org/10.25588/cspu.2018.01.11.

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Введение. Степень математической культуры определяется уровнем правильного восприятия и освоения изученного материала. Рассмотрена целесообразность использования синтетического метода при решении задач по математике. Материалы и методы. Определяется уровень сложности и трудности рассматриваемой задачи для правильного выбора метода её решения. Уделено особое внимание развитию логике мышления в применении синтеза. Как основной исследуется синтетический метод решения, неразрывно связанный с аналитическим методом. Их часто объединяют в единое целое и называют аналитико-синтетическим методом. Резул

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20

Junker, Markus. "A note on equational theories." Journal of Symbolic Logic 65, no. 4 (2000): 1705–12. http://dx.doi.org/10.2307/2695070.

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Several attempts have been done to distinguish “positive” information in an arbitrary first order theory, i.e., to find a well behaved class of closed sets among the definable sets. In many cases, a definable set is said to be closed if its conjugates are sufficiently distinct from each other. Each such definition yields a class of theories, namely those where all definable sets are constructible, i.e., boolean combinations of closed sets. Here are some examples, ordered by strength:Weak normality describes a rather small class of theories which are well understood by now (see, for example, [P

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21

Ng, Siu-Ah. "A generalization of forking." Journal of Symbolic Logic 56, no. 3 (1991): 813–22. http://dx.doi.org/10.2178/jsl/1183743730.

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Given a subset A of a fixed saturated model , we let denote the algebra of definable subsets of the domain of M of with parameters from A. Then a complete type p over A can be regarded as a measure on , assigning the value 1 to members of p and 0 to nonmembers. In [5] and [6], Keisler developed a theory of forking concerning probability measures. Therefore it generalizes the ordinary theory. On a different track, we can view the complement of the type p, or the collection of null sets of any measure on , as ideals on . Moreover, ideals and the pseudometric of a measure form examples of the so-

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22

Feldman, Alex. "Recursion theory in a lower semilattice." Journal of Symbolic Logic 57, no. 3 (1992): 892–911. http://dx.doi.org/10.2307/2275438.

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In §3 we construct a universal, ℵ0-categorical recursively presented partial order with greatest lower bound operator. This gives us the unique structure which embeds every countable lower semilattice. In §§5 and 6 we investigate the recursive and recursively enumerable substructures of this structure, in particular finding a suitable definition for the simple-maximal hierarchy and giving an example of an infinite recursively enumerable substructure which does not contain any infinite recursive substructure.The idea of looking at the lattice of recursively enumerable substructures of some recu

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23

Muñoz, Weimar, Olga Lucía León, and Vicenç Font. "A Visualization in GeoGebra of Leibniz’s Argument on the Fundamental Theorem of Calculus." Axioms 12, no. 10 (2023): 1000. http://dx.doi.org/10.3390/axioms12101000.

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In the literature, it is usually assumed that Leibniz described proof for the Fundamental Theorem of Calculus (FTC) in 1693. However, did he really prove it? If the answer is no from today’s perspective, are there works in which Leibniz introduced arguments that can be understood as formulations and justifications of the FTC? In order to answer this question, we used a historiographic methodology with expert triangulation. From the study of Leibniz’s manuscripts describing the inverse problem of tangents and its relationship with the quadrature problem, we found evidence of a geometrical argum

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Imtiaz, Aneeza, та Umer Shuaib. "On conjunctive complex fuzzification of Lagrange's theorem of <i>ξ</i>−CFSG". AIMS Mathematics 8, № 8 (2023): 18881–97. http://dx.doi.org/10.3934/math.2023961.

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<abstract><p>The application of a complex fuzzy logic system based on a linear conjunctive operator represents a significant advancement in the field of data analysis and modeling, particularly for studying physical scenarios with multiple options. This approach is highly effective in situations where the data involved is complex, imprecise and uncertain. The linear conjunctive operator is a key component of the fuzzy logic system used in this method. This operator allows for the combination of multiple input variables in a systematic way, generating a rule base that captures the b

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Srivastava, Hari M., Firdous A. Shah, Huzaifa L. Qadri, Waseem Z. Lone, and Musadiq S. Gojree. "Quadratic-Phase Hilbert Transform and the Associated Bedrosian Theorem." Axioms 12, no. 2 (2023): 218. http://dx.doi.org/10.3390/axioms12020218.

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The Hilbert transform is a commonly used linear operator that separates the real and imaginary parts of an analytic signal and is employed in various fields, such as filter design, signal processing, and communication theory. However, it falls short in representing signals in generalized domains. To address this limitation, we propose a novel integral transform, coined the quadratic-phase Hilbert transform. The preliminary study encompasses the formulation of all the fundamental properties of the generalized Hilbert transform. Additionally, we examine the relationship between the quadratic-pha

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S., Geetha, and S. Jayakumar Dr. "COMPUTER REPRESENTATION OF GRAPHS USING BINARY LOGIC CODES IN DISCRETE MATHEMATICS." International Journal of Multidisciplinary Research and Modern Education 3, no. 1 (2017): 152–57. https://doi.org/10.5281/zenodo.438946.

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Discrete Mathematics is a fundamental component of mathematics and computer science. It is the study of finite systems. The Digital computer is basically a finite structure and many of its properties can be understood and interpreted within the framework of finite mathematical systems. Graph are represented by means of Diagrams. These Graphs may be considered as Graph of certain relation. Graphs, Directed Graphs appear in many areas of Mathematics and Computer Science. Graphs are defined as an abstract mathematical system. Elements of Graph Theory are indispensable in almost all areas of Compu

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Ressayre, J. P. "Jon Barwise and John Schlipf. On recursively saturated models of arithmetic. Model theory and algebra, A memorial tribute to Abraham Robinson, edited by D. H. Saracino and V. B. Weispfenning, Lecture notes in mathematics, vol. 498, Springer-Verlag, Berlin, Heidelberg, and New York, 1975, pp. 42–55. - Patrick Cegielski, Kenneth McAloon, and George Wilmers. Modèles récursivement saturés de l'addition et de la multiplication des entiers naturels. Logic Colloquium '80, Papers intended for the European summer meeting of the Association for Symbolic Logic, edited by D. van Dalen, D. Lascar, and T. J. Smiley, Studies in logic and the foundations of mathematics, vol. 108, North-Holland Publishing Company, Amsterdam, New York, and London, 1982, pp. 57–68. - Julia F. Knight. Theories whose resplendent models are homogeneous. Israel journal of mathematics, vol. 42 (1982), pp. 151–161. - Julia Knight and Mark Nadel. Expansions of models and Turing degrees. The journal of symbolic logic, vol. 47 (1982), pp. 587–604. - Julia Knight and Mark Nadel. Models of arithmetic and closed ideals. The journal of symbolic logic, vol. 47 no. 4 (for 1982, pub. 1983), pp. 833–840. - Henryk Kotlarski. On elementary cuts in models of arithmetic. Fundamenta mathematicae, vol. 115 (1983), pp. 27–31. - H. Kotlarski, S. Krajewski, and A. H. Lachlan. Construction of satisfaction classes for nonstandard models. Canadian mathematical bulletin—Bulletin canadien de mathématiques, vol. 24 (1981), pp. 283–293. - A. H. Lachlan. Full satisfaction classes and recursive saturation. Canadian mathematical bulletin—Bulletin canadien de mathématiques, pp. 295–297. - Leonard Lipshitz and Mark Nadel. The additive structure of models of arithmetic. Proceedings of the American Mathematical Society, vol. 68 (1978), pp. 331–336. - Mark Nadel. On a problem of MacDowell and Specker. The journal of symbolic logic, vol. 45 (1980), pp. 612–622. - C. Smoryński. Back-and-forth inside a recursively saturated model of arithmetic. Logic Colloquium '80, Papers intended for the European summer meeting of the Association for Symbolic Logic, edited by D. van Dalen, D. Lascar, and T. J. Smiley, Studies in logic and the foundations of mathematics, vol. 108, North-Holland Publishing Company, Amsterdam, New York, and London, 1982, pp. 273–278. - C. Smoryński and J. Stavi. Cofinal extension preserves recursive saturation. Model theory of algebra and arithmetic, Proceedings of the Conference on Applications of Logic to Algebra and Arithmetic held at Karpacz, Poland, September 1–7,1979, edited by L. Pacholski, J. Wierzejewski, and A. J. Wilkie, Lecture notes in mathematics, vol. 834, Springer-Verlag, Berlin, Heidelberg, and New York, 1980, pp. 338–345. - George Wilmers. Minimally saturated models. Model theory of algebra and arithmetic, Proceedings of the Conference on Applications of Logic to Algebra and Arithmetic held at Karpacz, Poland, September 1–7, 1979, edited by L. Pacholski, J. Wierzejewski, and A. J. Wilkie, Lecture notes in mathematics, vol. 834, Springer-Verlag, Berlin, Heidelberg, and New York, 1980, pp. 370–380." Journal of Symbolic Logic 52, no. 1 (1987): 279–84. http://dx.doi.org/10.2307/2273884.

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Watkar, V. B., and R. R. Atram. "Comprehensive Study of the General Algebraic Semantics of Logic." Gurukul International Multidisciplinary Research Journal, April 30, 2025. https://doi.org/10.69758/gimrj/2504i5vxiiip0091.

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Abstract This paper delves into the general algebraic semantics of logic, exploring the fundamental connections between logical systems and their corresponding algebraic structures. We investigate how various logical concepts, such as formulas, proofs, and entailment, are mirrored in algebraic constructs like algebras, homomorphisms, and congruences. The paper aims to provide a structured framework for understanding the algebraic semantics of logic, encompassing propositional logic, first-order logic, while highlighting the advantages and limitations of this approach. We will demonstrate how a

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Antić, Christian. "Analogical proportions." Annals of Mathematics and Artificial Intelligence, May 7, 2022. http://dx.doi.org/10.1007/s10472-022-09798-y.

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AbstractAnalogy-making is at the core of human and artificial intelligence and creativity with applications to such diverse tasks as proving mathematical theorems and building mathematical theories, common sense reasoning, learning, language acquisition, and story telling. This paper introduces from first principles an abstract algebraic framework of analogical proportions of the form ‘a is to b what c is to d’ in the general setting of universal algebra. This enables us to compare mathematical objects possibly across different domains in a uniform way which is crucial for AI-systems. It turns

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Emmerson, Parker. "Syntax in Tensor Calculus Applications to Set Theory: A Pure Mathematics of Omega Point Theory." December 14, 2022. https://doi.org/10.5281/zenodo.7710307.

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Abstract :  This provides an AI utility framework for demonstrating semantic ordering theory for subscript syntax structure and how it should be handled when performing calculus operations. After demonstrating how the fundamental theorem of calculus can be written in reverse, we move on to describing the balancing of differentiated meanings of infinity at the, "oneness." Demonstrating the multi - variant applications of non - boolean functions, these infinity meanings extrapolate outward from human origin concept - structure to form tensor relationships which can be collected in

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