Continuity properties in constructive mathematics

1992 ◽  
Vol 57 (2) ◽  
pp. 557-565 ◽  
Author(s):  
Hajime Ishihara
Keyword(s):  
Continuous Mapping ◽  
Constructive Mathematics ◽  
Separable Space ◽  
Continuity Properties

AbstractThe purpose of this paper is an axiomatic study of the interrelations between certain continuity properties. We deal with principles which are equivalent to the statements “every mapping is sequentially nondiscontinuous”, “every sequentially nondiscontinuous mapping is sequentially continuous”, and “every sequentially continuous mapping is continuous”. As corollaries, we show that every mapping of a complete separable space is continuous in constructive recursive mathematics (the Kreisel-Lacombe-Schoenfield-Tsejtin theorem) and in intuitionism.

Continuity and nondiscontinuity in constructive mathematics

1991 ◽  
Vol 56 (4) ◽  
pp. 1349-1354 ◽  
Author(s):  
Hajime Ishihara
Keyword(s):  
Constructive Mathematics ◽  
Weak Version ◽  
Church’S Thesis ◽  
Continuity Properties ◽  
Church's Thesis

AbstractThe purpose of this paper is an axiomatic study of the interrelations between certain continuity properties. We show that every mapping is sequentially continuous if and only if it is sequentially nondiscontinuous and strongly extensional, and that “every mapping is strongly extensional”, “every sequentially nondiscontinuous mapping is sequentially continuous”, and a weak version of Markov's principle are equivalent. Also, assuming a consequence of Church's thesis, we prove a version of the Kreisel-Lacombe-Shoenfield-Tseĭtin theorem.

Sequentially continuous linear mappings in constructive analysis

1998 ◽  
Vol 63 (2) ◽  
pp. 579-583 ◽  
Author(s):  
Douglas Bridges ◽  
Ray Mines
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Continuous Mapping ◽  
Constructive Mathematics ◽  
Closed Graph Theorem ◽  
Inverse Mapping ◽  
Inverse Mapping Theorem ◽  
Continuous Mappings ◽  
Sequential Continuity ◽  
Separable Metric Space

A mapping u: X → Y between metric spaces is sequentially continuous if for each sequence (xn) converging to x ∈ X, (u(xn)) converges to u(x). It is well known in classical mathematics that a sequentially continuous mapping between metric spaces is continuous; but, as all proofs of this result involve the law of excluded middle, there appears to be a constructive distinction between sequential continuity and continuity. Although this distinction is worth exploring in its own right, there is another reason why sequential continuity is interesting to the constructive mathematician: Ishihara [8] has a version of Banach's inverse mapping theorem in functional analysis that involves the sequential continuity, rather than continuity, of the linear mappings; if this result could be upgraded by deleting the word “sequential”, then we could prove constructively the standard versions of the inverse mapping theorem and the closed graph theorem.Troelstra [9] showed that in Brouwer's intuitionistic mathematics (INT) a sequentially continuous mapping on a separable metric space is continuous. On the other hand, Ishihara [6, 7] proved constructively that the continuity of sequentially continuous mappings on a separable metric space is equivalent to a certain boundedness principle for subsets of ℕ; in the same paper, he showed that the latter principle holds within the recursive constructive mathematics (RUSS) of the Markov School. Since it is not known whether that principle holds within Bishop's constructive mathematics (BISH), of which INT and RUSS are models and which can be regarded as the constructive core of mathematics, the exploration of sequential continuity within BISH holds some interest.

scholarly journals On the inverse image of Baire spaces

1997 ◽  
Vol 20 (3) ◽  
pp. 423-432
Author(s):  
Mustafa Çiçek
Keyword(s):  
Continuous Mapping ◽  
Inverse Image ◽  
Baire Space ◽  
Open Mapping ◽  
Separable Space

In 1961, Z. Frolik proved that iffis an open and continuous mapping of a metrizable separable spaceXonto Baire spaceYand if the point inverses are Baire spaces, thenXis a Baire space. We give a generalization to semi-continuous and semi-open mapping of this theorem and extended it to the several types of mappings.

Cerebellopontine Angle Surgery Assisted by Continuous Mapping of the Facial Nerve Via the Ultrasonic Aspirator

2020 ◽  
Author(s):  
Marco Cenzato ◽  
Roberto Stefini ◽  
Francesco Zenga ◽  
Maurizio Piparo ◽  
Alberto Debernardi ◽  
...  
Keyword(s):  
Electrical Stimulation ◽  
Facial Nerve ◽  
Cerebellopontine Angle ◽  
Continuous Mapping ◽  
Cost Effective ◽  
Current Intensity ◽  
Vestibular Schwannomas ◽  
Constant Current ◽  
Surgical Workflow ◽  
Ultrasonic Aspirator

Abstract Background Cerebellopontine angle (CPA) surgery carries the risk of lesioning the facial nerve. The goal of preserving the integrity of the facial nerve is usually pursued with intermittent electrical stimulation using a handheld probe that is alternated with the resection. We report our experience with continuous electrical stimulation delivered via the ultrasonic aspirator (UA) used for the resection of a series of vestibular schwannomas. Methods A total of 17 patients with vestibular schwannomas, operated on between 2010 and 2018, were included in this study. A constant-current stimulator was coupled to the UA used for the resection, delivering square-wave pulses throughout the resection. The muscle responses from upper and lower face muscles triggered by the electrical stimulation were displayed continuously on multichannel neurophysiologic equipment. The careful titration of the electrical stimulation delivered through the UA while tapering the current intensity with the progression of the resection was used as the main strategy. Results All operations were performed successfully, and facial nerve conduction was maintained in all patients except one, in whom a permanent lesion of the facial nerve followed a miscommunication to the neurosurgeon. Conclusion The coupling of the electrical stimulation to the UA provided the neurosurgeon with an efficient and cost-effective tool and allowed a safe resection. Positive responses were obtained from the facial muscles with low current intensity (lowest intensity: 0.1 mA). The availability of a resection tool paired with a stimulator allowed the surgeon to improve the surgical workflow because fewer interruptions were necessary to stimulate the facial nerve via a handheld probe.

Topology optimization of continuum structures considering damage based on independent continuous mapping method

2018 ◽  
Vol 35 (2) ◽  
pp. 433-444 ◽  
Author(s):  
Jiazheng Du ◽  
Yunhang Guo ◽  
Zuming Chen ◽  
Yunkang Sui
Keyword(s):  
Topology Optimization ◽  
Continuous Mapping ◽  
Mapping Method ◽  
Continuum Structures

Carathéodory’s thermodynamics and constructive mathematics

1982 ◽  
Vol 34 (2) ◽  
pp. 52-56 ◽  
Author(s):  
A. Drago
Keyword(s):  
Constructive Mathematics

scholarly journals R-continuous functions

1992 ◽  
Vol 15 (1) ◽  
pp. 57-64 ◽  
Author(s):  
Ch. Konstadilaki-Savvapoulou ◽  
D. Janković
Keyword(s):  
Continuous Functions ◽  
Strong Form ◽  
Topological Spaces ◽  
Strong Continuity ◽  
Closed Graph ◽  
Continuity Properties

A strong form of continuity of functions between topological spaces is introduced and studied. It is shown that in many known results, especially closed graph theorems, functions under consideration areR-continuous. Several results in the literature concerning strong continuity properties are generalized and/or improved.

scholarly journals A singular eigenvalue problem for the Dirichlet (p, q)-Laplacian

2021 ◽  
Author(s):  
Yunru Bai ◽  
Nikolaos S. Papageorgiou ◽  
Shengda Zeng
Keyword(s):  
Dirichlet Problem ◽  
Eigenvalue Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Singular Term ◽  
Type Theorem ◽  
Solution Map ◽  
Continuity Properties ◽  
Variational Tools ◽  
Superlinear Reaction

AbstractWe consider a parametric nonlinear, nonhomogeneous Dirichlet problem driven by the (p, q)-Laplacian with a reaction involving a singular term plus a superlinear reaction which does not satisfy the Ambrosetti–Rabinowitz condition. The main goal of the paper is to look for positive solutions and our approach is based on the use of variational tools combined with suitable truncations and comparison techniques. We prove a bifurcation-type theorem describing in a precise way the dependence of the set of positive solutions on the parameter $$\lambda $$ λ . Moreover, we produce minimal positive solutions and determine the monotonicity and continuity properties of the minimal positive solution map.

Pythagorean fuzzy full implication multiple I method and corresponding applications

2021 ◽  
pp. 1-15
Author(s):  
TaiBen Nan ◽  
Haidong Zhang ◽  
Yanping He
Keyword(s):  
Decision Making ◽  
Modus Ponens ◽  
Pythagorean Fuzzy Set ◽  
Implication Operator ◽  
Continuity Properties ◽  
Fuzzy Implication ◽  
Decision Making Problem ◽  
Fuzzy Implication Operator ◽  
Degree Of Similarity ◽  
Fuzzy Modus Ponens

The overwhelming majority of existing decision-making methods combined with the Pythagorean fuzzy set (PFS) are based on aggregation operators, and their logical foundation is imperfect. Therefore, we attempt to establish two decision-making methods based on the Pythagorean fuzzy multiple I method. This paper is devoted to the discussion of the full implication multiple I method based on the PFS. We first propose the concepts of Pythagorean t-norm, Pythagorean t-conorm, residual Pythagorean fuzzy implication operator (RPFIO), Pythagorean fuzzy biresiduum, and the degree of similarity between PFSs based on the Pythagorean fuzzy biresiduum. In addition, the full implication multiple I method for Pythagorean fuzzy modus ponens (PFMP) is established, and the reversibility and continuity properties of the full implication multiple I method of PFMP are analyzed. Finally, a practical problem is discussed to demonstrate the effectiveness of the Pythagorean fuzzy full implication multiple I method in a decision-making problem. The advantages of the new method over existing methods are also explained. Overall, the proposed methods are based on logical reasoning, so they can more accurately and completely express decision information.

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