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### Fast n-fold Boolean Convolution via Additive Combinatorics [article]

Karl Bringmann, Vasileios Nakos
2021 arXiv   pre-print
At the heart of our approach lies a careful usage of Kneser's theorem from Additive Combinatorics, and a new deterministic almost linear output-sensitive algorithm for non-negative sparse convolution.  ...  We consider the problem of computing the Boolean convolution (with wraparound) of n vectors of dimension m, or, equivalently, the problem of computing the sumset A_1+A_2+...+A_n for A_1,...,A_n ⊆ℤ_m.  ...  necessary. n-Fold Boolean Convolution versus n-Fold Convolution.  ...

### Fast n-Fold Boolean Convolution via Additive Combinatorics

Karl Bringmann, Vasileios Nakos, Nikhil Bansal, Emanuela Merelli, James Worrell
2021
At the heart of our approach lies a careful usage of Kneser's theorem from Additive Combinatorics, and a new deterministic almost linear output-sensitive algorithm for non-negative sparse convolution.  ...  We consider the problem of computing the Boolean convolution (with wraparound) of n vectors of dimension m, or, equivalently, the problem of computing the sumset A₁+A₂+...+A_n for A₁,...,A_n ⊆ ℤ_m.  ...  n-Fold Boolean Convolution via Additive Combinatorics is a divisor of m.  ...

### Symbolic Calculus in Mathematical Statistics: A Review [article]

Elvira Di Nardo
2015 arXiv   pre-print
The symbolic representation of Sheffer polynomial sequences allows us to carry out a unifying theory of classical, Boolean and free cumulants.  ...  One of the main tools is a formal generalization of the convolution of identical probability distributions, which allows us to employ compound Poisson random variables in various topics that are only somewhat  ...  Cumulants A probability measure P is infinitely divisible if for any positive integer n there is a probability measure P n such that P = (P n ) n * , where (P n ) n * denotes the n-fold convolution of  ...

### How Iris Recognition Works

J. Daugman
2004 IEEE transactions on circuits and systems for video technology (Print)
Fig. 11 . 11 The (8 2 8) convolution kernel for fast focus assessment. Fig. 12 . 12 The 2-D Fourier power spectrum of the convolution kernel used for rapid focus assessment.  ...  This "best of n" test skews the distribution to the left and reduces its mean from about 0.5 to 0.458.  ...

### On the Fourier transform of a quantitative trait: Implications for compressive sensing [article]

Stephen Doro, Matthew A. Herman
2021 arXiv   pre-print
It uses the theory of real-valued Boolean functions in a systematic way to translate trait data into the Fourier domain.  ...  This theory provides a theoretical underpinning for systematic use of Boolean function machinery to dissect the dependency of a trait on the genome and environment.  ...  linear measurements via a sensing/measurement matrix A of size M × N, with M < N.  ...

### Counting Problems in Parameterized Complexity

2019 International Symposium on Parameterized and Exact Computation
 , which asserts that counting satisfying assignments to Boolean 3-CNF-formulas requires time 2 Ω(n) .  ...  For computing sub(H, G) and thin patterns H, an additional improvement to f (k) • n 0.46k+2pw(H)+O (1) time is possible, where pw(H) denotes the pathwidth of H  .  ...

### Convex polytopes: Extremal constructions and 𝑓-vector shapes [chapter]

Günter Ziegler
2007 Geometric Combinatorics
of P is boolean.  ...  But let's not get too ambitious too fast.  ...

### Fast Feasible and Unfeasible Matrix Multiplication [article]

Victor Y. Pan
2018 arXiv   pre-print
Fast matrix-by-matrix multiplication (hereafter MM) is a highly recognized research subject.  ...  unbounded sizes, cover the progress in decreasing the exponents of MM, comment on its impact on the theory and practice of computing, and recall various fundamental concepts and techniques supporting fast  ...  Let m = n in order to simplify the estimates. Algorithm 1: Convolution via Evaluation, Interpolation, and FFT. INITIALIZATION.  ...

### Convex Polytopes: Extremal Constructions and f-Vector Shapes [article]

Günter M. Ziegler
2005 arXiv   pre-print
of P is boolean.  ...  (See also Springborn  for an additional discussion of uniqueness.) 1.3. Steinitz' theorem via circle packings Theorem 1.3 (The Koebe-Andreev-Thurston theorem).  ...

### Cumulative index volumes 1–100

1992 Theoretical Computer Science
-295 Blum, N., A Boolean function requiring 3n network size (No&) 28 (3) ( 1984) 337-345 Blum, N., An n( r?)  ...  Robert, Systolic convolution of arithmetic functions 95 (2) ( 1992) 207-229 Quong, R.W., Fast average-case pattern matching by multiplexing sparse tables 92 ( 1) ( 1992) I65- 179 Rabani, Y.  ...

### Euler Calculus with Applications to Signals and Sensing [article]

Justin Curry, Robert Ghrist, Michael Robinson
2012 arXiv   pre-print
We work over R n via induction on n.  ...  Equation (17.1) and compared to true values. (5) Various advanced transforms are available, including addition of noise, smoothing via convolution with Gaussians , and other integral transforms (using  ...

### If the Current Clique Algorithms are Optimal, so is Valiant's Parser [article]

Amir Abboud, Arturs Backurs, Virginia Vassilevska Williams
2015 arXiv   pre-print
Lee showed that any algorithm for a more general parsing problem with running time O(|G|· n^3-ε) can be converted into a surprising subcubic algorithm for Boolean Matrix Multiplication.  ...  complexity of a fundamental problem, our reduction has led us to similar lower bounds for more modern and well-studied cubic time problems for which faster algorithms are highly desirable in practice: RNA Folding  ...  recursive algorithm that is able to utilize many fast boolean matrix multiplications to speed up the computation of the dynamic programming table from the CYK algorithm.  ...

### Abstracts of Presentations

2001 Brain and Cognition
For a boolean ring B, a quatient ring B[X 1 , . . . , X n ]/ X 2 1 −X 1 , . . . , X 2 n −X n is called a boolean polynomial ring.  ...  The arbitrary order of differentiation is found according to the Riemann-Liouville definition, whereas the generalized Cauchy n-fold integral is adopted for arbitrary order of integration.  ...  For evaluating such a series to N digits of accuracy, the numerator and the denominator computed by binary splitting have size O(N log N ) even though the reduced fraction may only have size O(N ).  ...

### A Treatise on Quantum Clifford Algebras [article]

Bertfried Fauser
2002 arXiv   pre-print
We discuss Hopf algebra versus Hopf gebra, the latter emerging naturally from a bi-convolution.  ...  Since we saw that the convolution establishes a group like structure via the convolution and finally the Hopf gebra.  ...  The whole combinatorics of this process is encoded in this singe and harmless looking equation! The crucial point is to investigate what kind of physical reason is behind this additional reordering.  ...

### Quiver Mutations, Seiberg Duality and Machine Learning [article]

Jiakang Bao, Sebastián Franco, Yang-Hui He, Edward Hirst, Gregg Musiker, Yan Xiao
2020 arXiv   pre-print
In addition, we evaluate the relative advantages of Naive Bayes classifiers versus Convolutional Neural Networks.  ...  Finally, we also investigate how the results are affected by the inclusion of additional data, such as ranks of gauge/flavor groups and certain variables motivated by the existence of underlying Diophantine  ...  Figure 2 .1 also illustrates the addition of a meson. 3. Ranks. The rank of the dualized node transforms as N j = N f j − N j , (2.1) where N f j is the number of flavors at the dualized node j.  ...
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