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We study the Rouquier dimension of wrapped Fukaya categories of Liouville manifolds and pairs, and apply this invariant to various problems in algebraic and symplectic geometry. On the algebro-geometric side, we introduce a new method based on symplectic flexibility and mirror symmetry to bound the Rouquier dimension of derived categories of coherent sheaves on certain complex algebraic varieties and stacks. These bounds are sharp in dimension at most $$3$$ . As an application, we resolve a well-known conjecture of Orlov for new classes of examples (e.g. toric $$3$$ -folds, certain log Calabi–Yau surfaces). We also discuss applications to non-commutative motives on partially wrapped Fukaya categories. On the symplectic side, we study various quantitative questions including the following. (1) Given a Weinstein manifold, what is the minimal number of intersection points between the skeleton and its image under a generic compactly supported Hamiltonian diffeomorphism? (2) What is the minimal number of critical points of a Lefschetz fibration on a Liouville manifold with Weinstein fibers? We give lower bounds for these quantities which are to the best of the authors’ knowledge the first to go beyond the basic flexible/rigid dichotomy.
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This content will become publicly available on May 1, 2026
On Computing Linear, Positive-Wrapped (Circular), and Negative-Wrapped Convolutions in the Frequency Domain [Tips & Tricks]
Convolution is a fundamental operation with diverse applications in signal processing, computer vision, and machine learning. This article reviews three distinct convolutions: linear convolution (also referred to as aperiodic convolution), positive-wrapped convolution (PWC) (also known as circular convolution), and negative-wrapped convolution (NWC). Additionally, we propose an alternative approach to computing linear convolution without zero padding by leveraging the PWC and NWC. We compare two fast Fourier transform (FFT)-based methods to compute linear convolution: the traditional zero-padded PWC method and a new method based on the PWC and NWC. Through a detailed analysis of the flowgraphs (FGs), we demonstrate the equivalence of these methods while highlighting their unique characteristics. We show that computing the NWC using the weighted PWC method is equivalent to a part of the linear convolution computation with zero padding. Furthermore, it is possible to extract the PWC and NWC from structures to compute linear convolution with zero padding, where the last butterfly stage can be eliminated. This article aims to establish a clear connection among PWC, NWC, and linear convolution, illustrating new perspectives on computing different convolutions.
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- Award ID(s):
- 2243053
- PAR ID:
- 10636746
- Publisher / Repository:
- IEEE
- Date Published:
- Journal Name:
- IEEE Signal Processing Magazine
- Volume:
- 42
- Issue:
- 3
- ISSN:
- 1053-5888
- Page Range / eLocation ID:
- 77 to 82
- Subject(s) / Keyword(s):
- Linear Convolution Signal Processing Fast Fourier Transform Convolution Operation Post-processing Step Point-wise Multiplication Zero Padding Positive Wrapped Convolution Negative Wrapped Convolution
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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