There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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Convex Analysisの二,三の進展について

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Economics and Information Theory

At initialization, artificial neural networks (ANNs) are equivalent to Gaussian processes in the infinite-width limit, thus connecting them to kernel methods. We prove that the evolution of an ANN during training can also be described by a kernel: during gradient descent on the parameters of an ANN, the network function (which maps input vectors to output vectors) follows the so-called kernel gradient associated with a new object, which we call the Neural Tangent Kernel (NTK). This kernel is central to describe the generalization features of ANNs. While the NTK is random at initialization and varies during training, in the infinite-width limit it converges to an explicit limiting kernel and stays constant during training. This makes it possible to study the training of ANNs in function space instead of parameter space. Convergence of the training can then be related to the positive-definiteness of the limiting NTK. We then focus on the setting of least-squares regression and show that in the infinite-width limit, the network function follows a linear differential equation during training. The convergence is fastest along the largest kernel principal components of the input data with respect to the NTK, hence suggesting a theoretical motivation for early stopping. Finally we study the NTK numerically, observe its behavior for wide networks, and compare it to the infinite-width limit.

/pdf/neural-tangent-kernel-convergence-and-generalization-in-3ks0qgekbr.pdf

Neural Tangent Kernel: Convergence and Generalization in Neural Networks

In this paper we derive a new inequality of Gruss' type for Riemann- Stieltjes integral and apply it for special means (logarithmic mean, identric mean, etc)

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An Inequality of Grüss Type for Riemann-Stieltjes Integral and Applications for Special Means

Preface Reverse Inequalities Reverses for the Schwarz Inequality Inequalities of the Gruss Type Reverses of Bessel's Inequality Other Inequalities in Inner Product Spaces Generalisations of Bessel's Inequality Some Gruss Type Inequalities for n-Tuples of Vectors Other Inequalities in Inner Product Spaces Bibliography Index.

Advances in Inequalities of the Schwarz, Gruss and Bessel Type in Inner Product Spaces

Some new inequalities of Ostowski-type for the Cebysev functional and applications for Taylor’s expansion and generalised trapezoid formula are pointed out. Mathematics subject classification (2010): Primary 26D15; Secondary 41A55.

http://files.ele-math.com/abstracts/jmi-08-10-abs.pdf

Some new ostrowski-type bounds for the čebyšev functional and applications

A sharp refinement of the Gruss inequality in the general setting of measurable spaces and abstract Lebesgue integrals is proven. Some consequential particular inequalities are mentioned.

A refinement of the Gruss inequality and applications

An inequality of Ostrowski's type for monotonous nondecreasing mappings is given Applications for quadrature formulas are pointed out

Sever S Dragomir

Papers

An Inequality of Grüss Type for Riemann-Stieltjes Integral and Applications for Special Means

Advances in Inequalities of the Schwarz, Gruss and Bessel Type in Inner Product Spaces

Some new ostrowski-type bounds for the čebyšev functional and applications

A refinement of the Gruss inequality and applications

Ostrowski's Inequality for Monotonous Mappings and Applications