scholarly journals Frames and subspaces

2018 ◽  
Author(s):  
◽  
Desai Cheng
Keyword(s):  
Phase Retrieval ◽  
Wasserstein Distance ◽  
Complex Case ◽  
Exact Constant ◽  
Frame Operator ◽  
Parseval Frame ◽  
Fusion Frames ◽  
Open Questions ◽  
Rank 2 ◽  
Equiangular Lines

This thesis will consist of three parts. In the first part we find the closest probabilistic Parseval frame to a given probabilistic frame in the 2 Wasserstein Distance. It is known that in the traditional [symbol]2 distance the closest Parseval frame to a frame [phi] = {[symbol]i} N[i=1] [symbol] R[d] is [phi[�] = {[symbol] � i }N[i]=1 = {S [-1/2][[symbol]i]} N[i=1] where S is the frame operator of [phi]. We use this fact to prove a similar statement about probabilistic frames in the 2 Wasserstein metric. In the second part, we will associate a complex vector with a rank 2 real projection. Using this association we will answer many open questions in frame theory. In particular we will prove Edidin's theorem in phase retrieval in the complex case, answer a question on mutually unbiased bases, a question on equiangular lines, and a question on fusion frames. In the last part we will give a way to calculate the exact constant for the [symbol]1 � [symbol]2 inequality and use this method to prove a couple of interesting theorems

scholarly journals From real to complex sign pattern matrices

1998 ◽  
Vol 57 (1) ◽  
pp. 159-172 ◽  
Author(s):  
Carolyn A. Eschenbach ◽  
Frank J. Hall ◽  
Zhongshan Li
Keyword(s):  
Matrix Analysis ◽  
Complex Case ◽  
Sign Pattern ◽  
Pattern Class ◽  
Class A ◽  
Open Questions ◽  
Sign Patterns ◽  
Complex Patterns ◽  
Ray Patterns ◽  
Real Matrices

This paper extends some fundamental concepts of qualitative matrix analysis from sign pattern classes of real matrices to sign pattern classes of complex matrices. A complex sign pattern and its corresponding sign pattern class are defined in such a way that they generalize the definitions of a (real) sign pattern and its corresponding sign pattern class. A survey of several qualitative results on complex sign patterns is presented. In particular, sign nonsingular complex patterns are investigated. The type of region in the complex plane representing the distribution of the determinants of the matrices in the sign pattern class of a sign nonsingular complex pattern is identified. Cyclically nonnegative complex patterns and complex patterns that are signature similar to nonnegative patterns are characterized. Extensions of sign stable and sign semistable patterns from the real to the complex case are given. Results on ray patterns are also obtained. Finally, many open questions are mentioned.

scholarly journals On stability of generalized (affine) phase retrieval in the complex case

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Zhitao Zhuang
Keyword(s):  
Lower Bound ◽  
Phase Retrieval ◽  
Complex Case ◽  
Lipschitz Property ◽  
Cramer Rao Lower Bound ◽  
The Stability

AbstractIn this paper, we discuss the stability of generalized phase retrieval and generalized affine phase retrieval in the complex case. By the realification method, we obtain the bi-Lipschitz property in the absence of noise case and Cramer–Rao lower bound under noise conditions.

scholarly journals Controlled K−g−Fusion Frames in Hilbert C∗−Modules

2022 ◽  
Vol 20 ◽  
pp. 1
Author(s):  
Mohamed Rossafi ◽  
Fakhr-dine Nhari
Keyword(s):  
Iterative Algorithms ◽  
Frame Operator ◽  
Numerical Efficiency ◽  
Fusion Frames ◽  
Fusion Frame ◽  
Perturbation Problem ◽  
The Subject

Controlled frames have been the subject of interest because of its ability to improve the numerical efficiency of iterative algorithms for inverting the frame operator. In this paper, we introduce the concepts of controlled g−fusion frame and controlled K−g−fusion frame in Hilbert C∗−modules and we give some properties. Also, we study the perturbation problem of controlled K−g−fusion frame. Moreover, an illustrative example is presented to support the obtained results.

scholarly journals Rank-2 attractors and Deligne’s conjecture

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Wenzhe Yang
Keyword(s):  
Number Theory ◽  
String Theory ◽  
Arithmetic Geometry ◽  
Hodge Structures ◽  
Special Values ◽  
Open Questions ◽  
Deligne's Conjecture ◽  
De Rham ◽  
Rank 2

Abstract In this paper, we will study the arithmetic geometry of rank-2 attractors, which are Calabi-Yau threefolds whose Hodge structures admit interesting splits. We will develop methods to analyze the algebraic de Rham cohomologies of rank-2 attractors, and we will illustrate how our methods work by focusing on an example in a recent paper by Candelas, de la Ossa, Elmi and van Straten. We will look at the interesting connections between rank-2 attractors in string theory and Deligne’s conjecture on the special values of L-functions. We will also formulate several open questions concerning the potential connections between attractors in string theory and number theory.

scholarly journals Lower Semi-frames, Frames, and Metric Operators

2020 ◽  
Vol 18 (1) ◽  
Author(s):  
J.-P. Antoine ◽  
R. Corso ◽  
C. Trapani
Keyword(s):  
Hilbert Space ◽  
Measurable Function ◽  
Adjoint Operator ◽  
Necessary Condition ◽  
Frame Operator ◽  
Parseval Frame ◽  
Lower Semi ◽  
Metric Operator ◽  
Generalized Frame ◽  
Continuous Frame

AbstractThis paper deals with the possibility of transforming a weakly measurable function in a Hilbert space into a continuous frame by a metric operator, i.e., a strictly positive self-adjoint operator. A necessary condition is that the domain of the analysis operator associated with the function be dense. The study is done also with the help of the generalized frame operator associated with a weakly measurable function, which has better properties than the usual frame operator. A special attention is given to lower semi-frames: indeed, if the domain of the analysis operator is dense, then a lower semi-frame can be transformed into a Parseval frame with a (special) metric operator.

Quantitative analysis by reconstruction of HREM focal series

1994 ◽  
Vol 52 ◽  
pp. 740-741
Author(s):  
W. Coene ◽  
A. Thust ◽  
M. Op de Beeck ◽  
D. Van Dyck
Keyword(s):  
Phase Retrieval ◽  
Image Plane ◽  
Initial Guess ◽  
Recursive Least Squares ◽  
Objective Lens ◽  
Electron Wave ◽  
Electron Sources ◽  
Original Algorithm ◽  
Coherent Field ◽  
Direct Interpretation

Compared to conventional electron sources, the use of a highly coherent field-emission gun (FEG) in TEM improves the information resolution considerably. A direct interpretation of this extra information, however, is hampered since amplitude and phase of the electron wave are scrambled in a complicated way upon transfer from the specimen exit plane through the objective lens towards the image plane. In order to make the additional high-resolution information interpretable, a phase retrieval procedure is applied, which yields the aberration-corrected electron wave from a focal series of HRTEM images (Coene et al, 1992).Kirkland (1984) tackled non-linear image reconstruction using a recursive least-squares formalism in which the electron wave is modified stepwise towards the solution which optimally matches the contrast features in the experimental through-focus series. The original algorithm suffers from two major drawbacks : first, the result depends strongly on the quality of the initial guess of the first step, second, the processing time is impractically high.

Limitations to the Spacing Effect

2005 ◽  
Vol 52 (4) ◽  
pp. 257-263 ◽  
Author(s):  
Peter P. J. L. Verkoeijen ◽  
Remy M. J. P. Rikers ◽  
Henk G. Schmidt
Keyword(s):  
Free Recall ◽  
Phase Retrieval ◽  
Incidental Learning ◽  
Factor Model ◽  
Recall Performance ◽  
Spacing Effect ◽  
Word List ◽  
Intentional Learning ◽  
Study Phase ◽  
Learning Condition

Abstract. The spacing effect refers to the finding that memory for repeated items improves when the interrepetition interval increases. To explain the spacing effect in free-recall tasks, a two-factor model has been put forward that combines mechanisms of contextual variability and study-phase retrieval (e.g., Raaijmakers, 2003 ; Verkoeijen, Rikers, & Schmidt, 2004 ). An important, yet untested, implication of this model is that free recall of repetitions should follow an inverted u-shaped relationship with interrepetition spacing. To demonstrate the suggested relationship an experiment was conducted. Participants studied a word list, consisting of items repeated at different interrepetition intervals, either under incidental or under intentional learning instructions. Subsequently, participants received a free-recall test. The results revealed an inverted u-shaped relationship between free recall and interrepetition spacing in both the incidental-learning condition and the intentional-learning condition. Moreover, for intentionally learned repetitions, the maximum free-recall performance was located at a longer interrepetition interval than for incidentally learned repetitions. These findings are interpreted in terms of the two-factor model of spacing effects in free-recall tasks.

What’s Next?

2011 ◽  
Vol 23 (1) ◽  
pp. 60-63 ◽  
Author(s):  
Peter Vorderer
Keyword(s):  
Cultural Differences ◽  
Factor Model ◽  
New Developments ◽  
Theoretical Concepts ◽  
Open Questions ◽  
Entertainment Products ◽  
Entertainment Theory

This paper points to new developments in the context of entertainment theory. Starting from a background of well-established theories that have been proposed and elaborated mainly by Zillmann and his collaborators since the 1980s, a new two-factor model of entertainment is introduced. This model encompasses “enjoyment” and “appreciation” as two independent factors. In addition, several open questions regarding cultural differences in humans’ responses to entertainment products or the usefulness of various theoretical concepts like “presence,” “identification,” or “transportation” are also discussed. Finally, the question of why media users are seeking entertainment is brought to the forefront, and a possibly relevant need such as the “search for meaningfulness” is mentioned as a possible major candidate for such an explanation.

Study-Phase Retrieval: Suppressing False Recognition in a Repetition Paradigm

2007 ◽  
Author(s):  
Peter M. Wessels ◽  
Jonathan Schnader ◽  
Allison Smith ◽  
Christopher Thomas ◽  
Haley Titus
Keyword(s):  
False Recognition ◽  
Phase Retrieval ◽  
Study Phase
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