Moments and absolute continuity properties of superdiffusions

2004 ◽  
pp. 49-60
Author(s):  
E. Dynkin
Keyword(s):  
Absolute Continuity ◽  
Continuity Properties

scholarly journals Operators with absolute continuity properties: an application to quasinormality

2013 ◽  
Vol 215 (1) ◽  
pp. 11-30 ◽  
Author(s):  
Zenon Jan Jabłoński ◽  
Il Bong Jung ◽  
Jan Stochel
Keyword(s):  
Absolute Continuity ◽  
Continuity Properties

scholarly journals Absolute continuity and hyponormal operators

1981 ◽  
Vol 4 (2) ◽  
pp. 321-335
Author(s):  
C. R. Putnam
Keyword(s):  
Absolute Continuity ◽  
Hyponormal Operator ◽  
Unitary Operators ◽  
Absolutely Continuous ◽  
Continuity Properties ◽  
Hyponormal Operators ◽  
Dilation Theory ◽  
Polar Factor ◽  
Polar Factorization ◽  
Continuous Operators

LetTbe a completely hyponormal operator, with the rectangular representationT=A+iB, on a separable Hilbert space. If0is not an eigenvalue ofT*thenTalso has a polar factorizationT=UP, withUunitary. It is known thatA,BandUare all absolutely continuous operators. Conversely, given an arbitrary absolutely continuous selfadjointAor unitaryU, it is shown that there exists a corresponding completely hyponormal operator as above. It is then shown that these ideas can be used to establish certain known absolute continuity properties of various unitary operators by an appeal to a lemma in which, in one interpretation, a given unitary operator is regarded as a polar factor of some completely hyponormal operator. The unitary operators in question are chosen from a number of sources: the F. and M. Riesz theorem, dissipative and certain mixing transformations in ergodic theory, unitary dilation theory, and minimal normal extensions of subnormal contractions.

Lower dimensional absolute continuity properties of quasiconformal mappings

1975 ◽  
Vol 78 (1) ◽  
pp. 81-93 ◽  
Author(s):  
F. W. Gehring
Keyword(s):  
Jordan Curve ◽  
Absolute Continuity ◽  
Quasiconformal Mappings ◽  
Outer Measure ◽  
Counting Measure ◽  
Continuity Properties ◽  
Image Position ◽  
Lower Dimensional ◽  
Natural Way

Let Rn denote Euclidean n-space. For p ∈ [0, ∞), the normalized Hausdorff p-dimensional outer measure of a set E ⊂ Rn is defined aswhere the infimum is taken over all countable coverings of E by sets Ej withand(See, for example (4).) Then ℋP (E) measures the p-dimensional size of E. For example, ℋ0 (E) ≤ ∞ if and only if E is finite, in which case ℋ0(E) = card (E). Next if E is an arc or Jordan curve, then ℋ1(E) ≤ ∞ if and only if E is rectifiable, in which case ℋ1(E) = length (E). Finally if p is an integer in [1, n] and if E lies in a p-dimensional hyperplane T in Rn, thenwhere mp denotes the Lebesgue p-dimensional outer measure in T. Thus as p varies from 0 to n, the measures ℋp interpolate in a natural way between the counting measure and Lebesgue outer measure in Rn.

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.

scholarly journals Singing Transcription from Polyphonic Music Using Melody Contour Filtering

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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