THESIS
2020

x, 88 pages : illustrations (some color) ; 30 cm

**Abstract**
Convolution plays an important role in theory and application of mathematics.
Typical convolutions include discrete convolution and continuous convolution and
have close relations with Fourier transform. The concept of convolution could be
extended to series, measure theory and functionals when replacing the measures.
In this thesis, we introduce the convolution ideas in the studies of mixed volumes,
Neural Networks and convolution algebra.

Mixed Volumes, defined on convex bodies, is related to Alexandrov-Fenchel inequality,
one of the most fundamental results in convex geometry. The convolution
with respect to Euler-Schanuel integral leads to a new perspective of mixed
volumes. Following the convolution idea, we extend mixed volumes to the vector
space spanned by the indicato...[

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Convolution plays an important role in theory and application of mathematics.
Typical convolutions include discrete convolution and continuous convolution and
have close relations with Fourier transform. The concept of convolution could be
extended to series, measure theory and functionals when replacing the measures.
In this thesis, we introduce the convolution ideas in the studies of mixed volumes,
Neural Networks and convolution algebra.

Mixed Volumes, defined on convex bodies, is related to Alexandrov-Fenchel inequality,
one of the most fundamental results in convex geometry. The convolution
with respect to Euler-Schanuel integral leads to a new perspective of mixed
volumes. Following the convolution idea, we extend mixed volumes to the vector
space spanned by the indicator functions of bounded semi-algebraic sets.

Deep Learning is a popular topic in recent years. Convolutional Neural Network
has achieved much success in computer vision tasks. The interpretability of CNN
remains an open problem. For CNN with ReLU as activation function, considering
the role of convolution, we suggest that the activation status of the neurons are
important and propose randomized methods to study the phenomenon. Numerical
results are shown and in some extent support the points.

Polynomial multiplication is a discrete convolution. Following the same way complex
algebra could be generalized. For a complete lattice L and a relational structure
π = (X, (R

_{i})

_{I}), a new algebra L

^{π} called convolution algebra is constructed.
By setting f

_{i}(Ξ±

_{1},...,Ξ±

_{ni})(π) = Vο½Ξ±

_{1}(π

_{1})
β§β―β§Ξ±

_{ni}(π

_{ni}) : (π

_{1},β¦,π

_{ni},π) β R

_{i})ο½for Ξ±

_{1},...,Ξ±

_{ni} β L

^{X} and π β X, the algebra L

^{π} consists of the lattice L

^{X} equipped with an n

_{i}-ary operation f

_{i} for each (n

_{i}+1)-ary relation R

_{i} of π. This construction is bifunctorial and extensions are made through meets instead of joins.

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