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- Persi Diaconis, Xuancheng Shao, Kannan Soundararajan
- The American Mathematical Monthly
- 2014

When numbers are added in the usual way carries occur along the route. These carries cause a mess and it is natural to seek ways to minimize them. This paper proves that balanced arithmetic minimizes the proportion of carries. It also positions carries as cocycles in group theory and shows that if coset representatives for a finite-index normal subgroup H… (More)

- Xuancheng Shao, Steven G. Johnson
- Signal Processing
- 2008

We present algorithms for the discrete cosine transform (DCT) and discrete sine transform (DST), of types II and III, that achieve a lower count of real multiplications and additions than previously published algorithms, without sacrificing numerical accuracy. Asymptotically, the operation count is reduced from ∼ 2N log2 N to ∼ 17 9 N log2 N for a… (More)

- Xuancheng Shao, Steven G. Johnson
- Signal Processing
- 2008

We present algorithms for the type-IV discrete cosine transform (DCT-IV) and discrete sine transform (DSTIV), as well as for the modified discrete cosine transform (MDCT) and its inverse, that achieve a lower count of real multiplications and additions than previously published algorithms, without sacrificing numerical accuracy. Asymptotically, the… (More)

The associahedron is a convex polytope whose face poset is based on nonintersecting diagonals of a convex polygon. In this paper, given an arbitrary simple polygon P , we construct a polytopal complex analogous to the associahedron based on convex diagonalizations of P . We describe topological properties of this complex and provide realizations based on… (More)

- Satyan Devadoss, Rahul Shah, Xuancheng Shao, Ezra Winston
- Contributions to Discrete Mathematics
- 2012

Given an arbitrary polygon P with holes, we construct a polytopal complex analogous to the associahedron based on convex diagonalizations of P . This polytopal complex is shown to be contractible, and a geometric realization is provided based on the theory of secondary polytopes. We then reformulate a combinatorial deformation theory and present an open… (More)

While proving a lower bound of n/ log n membership queries to approximate the volume of a convex body in R, [4] proved the following: Given q1, . . . , qm ∈ R, let Qi = {a ∈ R : |a · qi| ≤ 1} for 1 ≤ i ≤ m. Then the set R 2 − ⋃m i=1 Qi can be partitioned into no more than n m product sets of the form ∏n j=1 Aj , Aj ∈ R . The following question was… (More)

- Xuancheng Shao
- 2008

In this paper, we present some generalizations of Gowers’s result about product-free subsets of groups. For any group G of order n, a subset A of G is said to be product-free if there is no solution of the equation ab = c with a, b, c ∈ A. Previous results showed that the size of any product-free subset of G is at most n/δ1/3, where δ is the smallest… (More)

In this paper, we present some generalizations of Gowers’s result about product-free subsets of groups. For any group G of order n, a subset A of G is said to be product-free if there is no solution of the equation ab = c with a, b, c ∈ A. Previous results showed that the size of any product-free subset of G is at most n/δ1/3, where δ is the smallest… (More)

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