We study the long-time asymptotics of prototypical non-linear diffusion equations. Specifically, we consider the case of a non-degenerate diffusivity function that is a (non-negative) polynomial of the dependent varia...
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In this paper we prove uniform convergence of approximations to p-harmonic functions by using natural p-mean operators on bounded domains of the Heisenberg group H which satisfy an intrinsic exterior corkscrew conditi...
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In this paper we apply a nonconforming rotated bilinear tetrahedral element to the Stokes problem in R3. We show that the element is stable in combination with a piecewise linear, continuous, approximation of the pres...
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This note studies stability of event-triggered control systems with the event-triggered control algorithm proposed in [1]. We construct a novel Halanay-type inequality, which is used to show that sufficient conditions...
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Due to the complex behavior arising from non-uniqueness, symmetry, and bifurcations in the solution space, solving inverse problems of nonlinear differential equations (DEs) with multiple solutions is a challenging ta...
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The area of bio-molecular computing has recently witnessed a major paradigm shift. Rather then being used only as simple calculating units capable of solving hard combinatorial or numerical problems, DNA computers are...
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The area of bio-molecular computing has recently witnessed a major paradigm shift. Rather then being used only as simple calculating units capable of solving hard combinatorial or numerical problems, DNA computers are increasingly becoming more tailored to operate like intelligent ent biological machines with unprecedented potentials. One example of applying DNA computers in such a new setting is in the area of logical control of gene expression levels. For this purpose, DNA computers are designed in such a way as to be able to diagnose some forms of cancer-related irregularities in a cell and release biological strands acting as inhibitors or activators of certain sets of genes. Such a controlaction can also be seen as a form of intra-cell cancer therapy, although it may also have other, more varied, purposes and goals. There are several important problems in the area of coding and network theory that arise in the context of developing DNA computers for controlling gene expressions. The two most important issues are that of minimizing diagnostics failure and of increasing the computational reliability of the system. The first question is intimately related to analyzing the operational principles. of networks of gene interactions, while the second is concerned with relating combinatorial characteristics of single- stranded DNA sequences to their hybridization affinities and secondary structures. In this paper, we will describe the state-of-the-art results and present some new relevant combinatorial and coding theoretic problems in this area.
In this paper, we adopt conformal prediction, a distribution-free uncertainty quantification (UQ) framework, to obtain confidence prediction intervals with coverage guarantees for Deep Operator Network (DeepONet) regr...
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Random graphs are increasingly becoming objects of interest for modeling networks in a wide range of applications. Latent position random graph models posit that each node is associated with a latent position vector, ...
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Quaternion singular value decomposition (QSVD) is a robust technique of digital watermarking which can extract high quality watermarks from watermarked images with low distortion. In this paper, QSVD technique is furt...
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Feature learning is thought to be one of the fundamental reasons for the success of deep neural networks. It is rigorously known that in two-layer fully-connected neural networks under certain conditions, one step of ...
Feature learning is thought to be one of the fundamental reasons for the success of deep neural networks. It is rigorously known that in two-layer fully-connected neural networks under certain conditions, one step of gradient descent on the first layer can lead to feature learning; characterized by the appearance of a separated rank-one component--spike--in the spectrum of the feature matrix. However, with a constant gradient descent step size, this spike only carries information from the linear component of the target function and therefore learning non-linear components is impossible. We show that with a learning rate that grows with the sample size, such training in fact introduces multiple rank-one components, each corresponding to a specific polynomial feature. We further prove that the limiting largedimensional and large sample training and test errors of the updated neural networks are fully characterized by these spikes. By precisely analyzing the improvement in the training and test errors, we demonstrate that these non-linear features can enhance learning.
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