Narrow Results

We present a toolbox to compute and extract information from inhomogeneous (i.e. unequally spaced) time series. The toolbox contains a large set of operators, mapping from the space of inhomogeneous time series to itself. These operators are computationally efficient (time and memory-wise) and suitable for stochastic processes. This makes them attractive for processing high-frequency data in finance and other fields. Using a basic set of operators, we easily construct more powerful combined operators which cover a wide set of typical applications.

The operators are classified as either macroscopic operators (that have a limit value when the sampling frequency goes to infinity) or microscopic operators (that strongly depend on the actual sampling). For inhomogeneous data, macroscopic operators are more robust and more important. Examples of macroscopic operators are (exponential) moving averages, differentials, derivatives, moving volatilities, etc.…

The purpose of this paper is to introduce a new category of fuzzy automata based on complete residuated lattice. We introduce and study the categorical concepts such as product, equalizer and their duals in this category. Finally, we establish a construction of a minimal fuzzy automaton for a given fuzzy language in a categorical framework. The construction of such fuzzy automaton is based on derivative of a given fuzzy language.

Yıldırım obtained an asymptotic formula of the discrete moment of $|\zeta (\frac{1}{2}+it)|$ over the zero of the higher derivatives of Hardy’s $Z$-function. We give a generalization of his result on Hardy’s $Z$-function.

Let $H_d[x_1,\dots ,x_m]$ be the complex vector space of homogeneous polynomials of degree $d$ with the independent variables $x_1,\dots ,x_m$. Let $V$ be the complex vector space of homogeneous linear polynomials in the variables $x_1,\dots ,x_m$. For any linear operator $T$ acting on $V$, there is a (unique) induced operator $P(T)$ acting on $H_d[x_1,\dots ,x_m]$ satisfying

This paper illustrates a derivative of a derivative (i.e. "delta") of an exchange option in the U.S. real estate investment trust (REIT) industry. So far, it seems that there is no study that derives a "delta" of an exchange option using Laplace transform. First, the "delta" illustrates that change in exchange option is "driven" by at least one variable which in turn influences the curvature of a "delta" distribution. The empirical results show that each "delta" is 0.03 on average. Lastly, it seems that REIT mergers and acquisitions (M&As) occurred both for strategic and financial reasons.

This work considers Chaos aspects in a setting of arithmetic provided by Observer's Mathematics (see http://www.mathrelativity.com). We prove that the physical speed is a random variable, cannot exceed some constant, and this constant does not depend on an inertial coordinate system. Certain results and communications pertaining to these theorems are provided.

The research presented here developed from rather mysterious observations, originally made by the authors independently and in different circumstances, that Lebesgue null sets may have uniquely defined tangent directions that are still seen even if the set is much enlarged (but still kept Lebesgue null). This phenomenon appeared, for example, in the rank-one property of derivatives of BV functions and, perhaps in its most striking form, in attempts to decide whether Rademacher's theorem on differentiability of Lipschitz functions may be strengthened or not.

We describe the non-differentiability sets of Lipschitz functions on ℝⁿ and use this description to explain the development of the ideas and various approaches to the definition of the tangent fields to null sets. We also indicate connections to other current results, including results related to the study of structure of sets of small measure, and present some of the main remaining open problems.