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.

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Neural Tangent Kernel: Convergence and Generalization in Neural Networks

Companion results to the Bombieri generalisation of Bessel's inequality in inner product spaces are given

Some Bombieri Type Inequalities in Inner Product

There are many known reverses of the Cauchy-Bunyakovsky-Schwarz
(CBS) inequality in the literature. We obtain here a general integral inequality
comprising some of those results and also provide other related inequalities.
The discrete case, which is of interest in its own turn, is also analysed.

Reverses of the CBS Integral Inequality in Hilbert Spaces and Related Results

In this paper, we establish some refinements of Lah-Ribarić inequality for the general Lebesgue integral on divisions of measurable space. Applications for discrete inequalities and weighted means of positive numbers are also given. Some examples related to Hermite-Hadamard inequality for convex functions are provided as well. MSC 2010. Primary 26D15; Secondary 26D10.

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Refining Lah-Ribarić integral inequality for divisions of measurable space

Let g be a strictly increasing function on (a,b), having a continuous derivative g′ on (a,b). For the Lebesgue integrable function f:(a,b)→C, we define the k-g-left-sided fractional integral of f by S_{k,g,a+}f(x)=∫_{a}^{x}k(g(x)-g(t))g′(t)f(t)dt, x∈(a,b] and the k-g-right-sided fractional integral of f by S_{k,g,b-}f(x)=∫_{x}^{b}k(g(t)-g(x))g′(t)f(t)dt, x∈[a,b), where the kernel k is defined either on (0,∞) or on [0,∞) with complex values and integrable on any finite subinterval. In this paper we establish some new inequalities for the k-g-fractional integrals of functions of bounded variation.Examples for the generalized left- and right-sided Riemann-Liouville fractional integrals of a function f with respect to another function g and a general exponential fractional integral are also provided.

/pdf/further-inequalities-for-the-generalized-k-g-fractional-fkx8suk8jy.pdf

Further inequalities for the generalized k-g-fractional integrals of functions with bounded variation

Some inequalities for functions of bounded variation that provide reverses for the inequality between the integral mean and the p-norm are established. Applications related to the celebrated Landau inequality between the norms of the derivatives of a function are also pointed out.

Sever S Dragomir

Papers

Some Bombieri Type Inequalities in Inner Product

Reverses of the CBS Integral Inequality in Hilbert Spaces and Related Results

Refining Lah-Ribarić integral inequality for divisions of measurable space

Further inequalities for the generalized k-g-fractional integrals of functions with bounded variation

Some Inequalities for Functions of Bounded Variation with Applications to Landau Type Results