14,187 research outputs found
Stability and convergence analysis of a class of continuous piecewise polynomial approximations for time fractional differential equations
We propose and study a class of numerical schemes to approximate time
fractional differential equations. The methods are based on the approximation
of the Caputo fractional derivative by continuous piecewise polynomials, which
is strongly related to the backward differentiation formulae for the
integer-order case. We investigate their theoretical properties, such as the
local truncation error and global error analyses with respect to a sufficiently
smooth solution, and the numerical stability in terms of the stability region
and -stability by refining the technique proposed in
\cite{LubichC:1986b}. Numerical experiments are given to verify the theoretical
investigations.Comment: 34 pages, 3 figure
A multiset hook length formula and some applications
A multiset hook length formula for integer partitions is established by using
combinatorial manipulation. As special cases, we rederive three hook length
formulas, two of them obtained by Nekrasov-Okounkov, the third one by Iqbal,
Nazir, Raza and Saleem, who have made use of the cyclic symmetry of the
topological vertex. A multiset hook-content formula is also proved.Comment: 19 pages; 3 figure
Image Question Answering using Convolutional Neural Network with Dynamic Parameter Prediction
We tackle image question answering (ImageQA) problem by learning a
convolutional neural network (CNN) with a dynamic parameter layer whose weights
are determined adaptively based on questions. For the adaptive parameter
prediction, we employ a separate parameter prediction network, which consists
of gated recurrent unit (GRU) taking a question as its input and a
fully-connected layer generating a set of candidate weights as its output.
However, it is challenging to construct a parameter prediction network for a
large number of parameters in the fully-connected dynamic parameter layer of
the CNN. We reduce the complexity of this problem by incorporating a hashing
technique, where the candidate weights given by the parameter prediction
network are selected using a predefined hash function to determine individual
weights in the dynamic parameter layer. The proposed network---joint network
with the CNN for ImageQA and the parameter prediction network---is trained
end-to-end through back-propagation, where its weights are initialized using a
pre-trained CNN and GRU. The proposed algorithm illustrates the
state-of-the-art performance on all available public ImageQA benchmarks
Difference operators for partitions under the Littlewood decomposition
The concept of -difference operator for functions of partitions is
introduced to prove a generalization of Stanley's theorem on polynomiality of
Plancherel averages of symmetric functions related to contents and hook
lengths. Our extension uses a generalization of the notion of Plancherel
measure, based on walks in the Young lattice with steps given by the addition
of -hooks. It is well-known that the hook lengths of multiples of can be
characterized by the Littlewood decomposition. Our study gives some further
information on the contents and hook lengths of other congruence classes modulo
.Comment: 24 page
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