Continuity properties and Alexandroff theorem in Vietoris topology

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.

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

Mathematische Zeitschrift ◽

10.1007/s00209-021-02803-w ◽

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.

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Pythagorean fuzzy full implication multiple I method and corresponding applications

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-210527 ◽

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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Coincidence of the upper Vietoris topology and the Scott topology

Topology and its Applications ◽

10.1016/j.topol.2020.107480 ◽

2021 ◽

Vol 288 ◽

pp. 107480 ◽

Cited By ~ 1

Author(s):

Xiaoquan Xu ◽

Zhongqiang Yang

Keyword(s):

Vietoris Topology ◽

Scott Topology

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Singing Transcription from Polyphonic Music Using Melody Contour Filtering

Applied Sciences ◽

10.3390/app11135913 ◽

2021 ◽

Vol 11 (13) ◽

pp. 5913

Author(s):

Zhuang He ◽

Yin Feng

Keyword(s):

Audio Signals ◽

Continuity Properties ◽

Information Retrieval Evaluation ◽

Unique Method ◽

Image Edge ◽

Salience Function ◽

Music Information ◽

Melody Contour ◽

Analysis Platform ◽

Polyphonic Music

Automatic singing transcription and analysis from polyphonic music records are essential in a number of indexing techniques for computational auditory scenes. To obtain a note-level sequence in this work, we divide the singing transcription task into two subtasks: melody extraction and note transcription. We construct a salience function in terms of harmonic and rhythmic similarity and a measurement of spectral balance. Central to our proposed method is the measurement of melody contours, which are calculated using edge searching based on their continuity properties. We calculate the mean contour salience by separating melody analysis from the adjacent breakpoint connective strength matrix, and we select the final melody contour to determine MIDI notes. This unique method, combining audio signals with image edge analysis, provides a more interpretable analysis platform for continuous singing signals. Experimental analysis using Music Information Retrieval Evaluation Exchange (MIREX) datasets shows that our technique achieves promising results both for audio melody extraction and polyphonic singing transcription.

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Randomly perturbed vibrations

Journal of Applied Probability ◽

10.1017/s0021900200102876 ◽

1995 ◽

Vol 32 (02) ◽

pp. 417-428 ◽

Cited By ~ 2

Author(s):

M. Elshamy

Keyword(s):

Dirichlet Boundary ◽

Space Variable ◽

Initial Conditions ◽

Linear Wave ◽

Stochastic Perturbations ◽

Uniform Topology ◽

Linear Wave Equation ◽

Random Forcing ◽

Continuity Properties ◽

Positive Time

Let u ε(t, x) be the position at time t of a point x on a string, where the time variable t varies in an interval I: = [0, T], T is a fixed positive time, and the space variable x varies in an interval J. The string is performing forced vibrations and also under the influence of small stochastic perturbations of intensity ε. We consider two kinds of random perturbations, one in the form of initial white noise, and the other is a nonlinear random forcing which involves the formal derivative of a Brownian sheet. When J has finite endpoints, a Dirichlet boundary condition is imposed for the solutions of the resulting non-linear wave equation. Assuming that the initial conditions are of sufficient regularity, we analyze the deviations u ε(t, x) from u 0(t, x), the unperturbed position function, as the intensity of perturbation ε ↓ 0 in the uniform topology. We also discuss some continuity properties of the realization of the solutions u ε(t, x).

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