We present a new fractional Taylor formula for singular functions whose Caputo derivatives are of bounded variation. It bridges and ``interpolates" the usual formulas with two consecutive integer orders. This enables us to obtain an analogous Legendre expansion coefficient this type functions, further derive optimal (weighted) $L^\infty$-estimates $L^2$-estimates polynomial approximations. set ...

In the paper we discuss conformable derivative behavior in arbitrary Banach spaces and clear connection between two derivatives of different order. As a consequence obtain important result that an abstract function has at point (which does not coincide with lower terminal derivative) if only it first order same point. application obtained results prove existence weak solution mixed (initial/bou...

One of the ongoing issues with fractional-order diffusion models is the design of efficient numerical schemes for the space and time discretizations. Until now, most models have relied on a low-order finite difference method to discretize both the fractional-order space and time derivatives. While the finite difference method is simple and straightforward to solve integer-order differential equ...

We review the application of differential operators of noninteger order to the modeling of dynamic systems. We compare all the definitions of Variable Order VO operators recently proposed in literature and select the VO operator that has the desirable property of continuous transition between integer and non-integer order derivatives. We use the selected VO operator to connect the meaning of fu...

We introduce MISO, the Mixed-Integer Surrogate Optimization framework. MISO aims at solving computationally expensive black-box optimization problems with mixed-integer variables. Although encountered in many applications, such as optimal reliability design or structural optimization, for example, where time consuming simulation codes have to be run in order to obtain an objective function valu...

We consider the problem of minimizing a continuously differentiable function of several variables subject to simple bound constraints where some of the variables are restricted to take integer values. We assume that the first order derivatives of the objective function can be neither calculated nor approximated explicitly. This class of mixed integer nonlinear optimization problems arises frequ...

Rectangular hybrid automatamodel digital control programs of analog plant environments. We study rectangular hybrid automata where the plant state evolves continuously in real-numbered time, and the controller samples the plant state and changes the control state discretely, only at the integer points in time. We prove that rectangular hybrid automata have nite bisimilarity quotients when all c...

Substituted phenylazo and phenylazoxy compounds were systematically prepared and their anti-androgenic activity was measured in terms of (1) the growth-inhibiting effect on an androgen-dependent cell line, SC-3, and (2) the binding affinity to nuclear androgen receptor. Generally, azo/azoxy compounds showed cell toxicity, and the growth-inhibiting effects on SC-3 cells correlated with the toxic...

We propose a closed form solution for segmenting mixtures of 2-D translational and 2-D affine motion models directly from the image intensities. Our approach exploits the fact that the spatial-temporal image derivatives generated by a mixture of these motion models must satisfy a bi-homogeneous polynomial called the multibody brightness constancy constraint (MBCC). We show that the degrees of t...

The purpose of this presentation is to describe a recent family of basis functions—the fractional B-splines—which appear to be intimately connected to fractional calculus. Among other properties, we show that they are the convolution kernels that link the discrete (finite differences) and continuous (derivatives) fractional differentiation operators. We also provide simple closed forms for the ...