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

/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

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

The Hermite-Hadamard double inequality is the first fundamental result for convex functions defined on a interval of real numbers with a natural geometrical interpretation and a loose number of applications for particular inequalities. In this monograph we present the basic facts related to Hermite- Hadamard inequalities for convex functions and a large number of results for special means which can naturally be deduced. Hermite-Hadamard type inequalities for other concepts of convexities are also given. The properties of a number of functions and functionals or sequences of functions which can be associated in order to refine the result are pointed out. Recent references that are available online are mentioned as well.

Selected Topics on Hermite-Hadamard Inequalities and Applications

Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula

An inequality of Hadamard’s type for convex functions and convex functions on the co-ordinates defined in a rectangle from the plane and some applications are given.

/pdf/on-the-hadamard-s-inequlality-for-convex-functions-on-the-co-5cl3mxddet.pdf

On the hadamard’s inequlality for convex functions on the co-ordinates in a rectangle from the plane

In this paper we derive a new inequality of Ostrowski-Gruss' type and apply it to the estimation of error bounds for some special means and for some numerical quadrature rules.

An inequality of Ostrowski-Grüss' type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules

Some Gronwall type inequalities for kernels of L-type and application in qualitative theory of Volterra integral equations and for systems of differential equations are presented.

/pdf/some-gronwall-type-inequalities-and-applications-32cl4aw807.pdf

Sever S Dragomir

Papers

Selected Topics on Hermite-Hadamard Inequalities and Applications

Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula

On the hadamard’s inequlality for convex functions on the co-ordinates in a rectangle from the plane

An inequality of Ostrowski-Grüss' type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules

Some Gronwall Type Inequalities and Applications