In the previous unit, we gave a rigorous proof, that all the elements of the set of single valued functions which are defined by eqn. (1) for any nonzero natural number n coincide with the **Heaviside** **function** over the set of real numbers. Here, one may pinpoint that to define H(0) as 1/2 something that is taken for granted in many explicit approximations of this **function**, cannot be in consensus with the proposed formula in the present work. On the contrary, according to the performed algebraic representation it was definitely shown that f ( 0 ) = 1

In remote sensing image analysis, segmentation of an image is an important aspect. It classifies similar pixels within the image. Image Segmentation is helpful in analyzing the patterns, objects, and edges within an image. There are many ways for performing image segmentation. In this paper, we are segmenting a satellite image using Multiphase Chan-Vese model. Chan-Vese models are based on ‘Active Contours without edges’. Active contour model is also known as Snake and Energy-Based Model, which is finding local minima in the equivalent energy **function**. Chan- Vese model gives effective results of segmented image. The multiphase level set construction is mechanized to avoid the drawback of overlap and vacuum; it can also signify edges with convoluted topologies. Researchers conclude in this paper with the findings that the multiphase CV method can give a sensible segmented image of satellite imagery with 2D-DWT, when they manipulate **Heaviside** **function**.

As we have shown in §4 the replacement of a sigmoidal firing rate by a **Heaviside** func- tion can lead to highly tractable models for which substantial analytical results can be obtained (with the use of matrix exponentials and saltation matrices). However, at the network level the mathematical differences between the treatment of smooth and nons- mooth firing rates are considerably amplified relative to those at the single node level. At the node level it is well known that regarding the **Heaviside** **function** as the steep limit of a sigmoidal **function** can lead to arbitrarily many different non-equivalent dynamical systems. This is simply due to the non-uniqueness of the singular limits by which smooth functions may tend towards discontinuities. For a recent perspective on this issue see the work of Jeffrey [41]. Thus there is no reason to assume that taking the limit → 0 for the PWL network considered in §3 will be relevant to a Wilson-Cowan network with a **Heaviside** nonlinearity. Namely the approximation of a **Heaviside** **function** by a contin- uous **function** such that H(x) = lim →0 F(x), where F (x) is given by (2.2), may have

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and vice versa. Hence, since the first and the second terms in (21) change places, (21) remains in force. Note also that the **Heaviside** **function** in (21) is responsible for shielding and reflecting the plane wave. One should take into account the plane waves also from the first edge. Obviously, the angle ϑ 0 in (22) (Fig. 1) corresponds to the geometrical boundary of a shadow from the strip edge.

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and the purpose of this paper is to analyze the limit process β → ∞. This inves- tigation is motivated by the fact that the stable numerical solution of an ill-posed problem is very difficult, if not to say impossible, see, e.g., [10, 11]. Consequently, such models must be regularized to obtain a sequence of well-posed equations which, at least in principle, can be approximately solved by a computer. Also, steep firing rate functions, or even the **Heaviside** **function**, are often used in simulations. It is thus important to explore the limit process β → ∞ rigorously.

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towards this formula will be presented. The approach is based on a continuous 3D extension of the **Heaviside** **function**, with this extension drawing on the phase-field concept of diffuse interfaces. Entropy enters into the local, statistical description of contrast respectively gradient distributions in the transition region of the extended **Heaviside** **function** definition. The structure of the Bekenstein- Hawking formula eventually is derived for a geometric sphere based on mere geometric-statistic considerations.

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taken either as a **Heaviside** or a more general sigmoidal form. Introducing a spec- tral wave-vector k then this product is simply w( | k | )f (k), where functions with arguments k denote two-dimensional spatial Fourier transforms. We may evaluate w( | k | )f (k) directly, at every time step, using fast Fourier transforms (FFTs). Note that w( | k | ) can be pre-computed, by FFT or here even analytically, so that the proce- dure iterated over time amounts to computing f (k) by FFT, followed by a (complex) multiplication with w( | k | ), and finally an inverse FFT to obtain the result of the inte- gral. We wish to employ a parallel compute cluster for rapid computation over large grids, and hence use the free software package FFTW 3.3 [42], which includes a parallel MPI-C version. Note that the use of Fourier methods implies that the dis- cretization grid has periodic boundaries, or in other words, the solution is effectively computed on a torus. We use a grid spacing of about 0.03 or better in our computa- tions here.

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In this thesis we show that when ¯ f is of KPP, Nagumo or Unstable type, and solutions to (1.4) start from the **Heaviside** initial condition, there exists a unique (up to translation) stationary travelling wave. This is a new result and conﬁrms our intuition that the addition of noise in the KPP and Nagumuo type equations does not destroy the existence of a stationary travelling wave. It also proves that the addition of noise in the Unstable type stabilises the solution. This is intuitive given that the condition g(a) ̸ = 0 means if any large ﬂat patch at the wave marker level a were to form, then this would be destroyed by the noise. We do not conduct any computer simulations of these results but explore equation (1.4) in a purely theoretical way and whilst a large number of our arguments are soft, the results are extremely informative.

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Of course, there is a well known indetermination of this expression that can be absorbed in the definition of R or W . It is necessary to mention that up to know non experiment has been made to try to find the W gravitational field (something similar when one has a current in a wire and see what happen to the motion of a charge nearby), and this is the gravitational field so called h by **Heaviside** [21], defined on purpose to obtain Maxwell-like equations within his approach. So far, one knows ∇ ⋅ R and ∇ ×W of the vector fields R and W . To know ∇ × R and ∇ ⋅W , let us take the curl of (4) and use the known expression ∇ × ∇ × ( ) = ∇ ∇ ⋅ − ∇ ( ) 2

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The spatial-dynamic formulation developed in [5, 45] for the Swift–Hohenberg equation allows to predict snaking branches of localized patterns from the bifurcation structure of fronts connecting the trivial (background) state to the core state. We can not directly apply this theory to our case, in that system (3.5) is non-autonomous and (0, 0) is not an equilibrium. However, we shall see that the ideas presented in those papers remain valid: in the limit of **Heaviside** firing rate, which gives rise to a non- smooth spatial-dynamic formulation, we are able to compute explicit expressions for connecting orbits and, hence, for the snaking bifurcation diagram, which we partially present in Figure 3.3; for sigmoidal firing rates we will adopt numerical continuation and compute snaking bifurcation branches solving the boundary-value problem (3.2) and the associated stability problem (3.3).

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To overcome the first problem, several fast techniques have been proposed. Some of them are designed especially for computation acceleration of a specific objective **function** [11-13], while others are designed to be used with a general purpose. Sequential dichotomization [14-16] use an iterative scheme and meta-heuristic optimization methods[17, 18] to realize the quickly compute of threshold. Furthermore, in another approach, in order to accelerate the calculation of thresholds research, the resolution of the histogram is first reduced using the wavelets transform. After that, the optimal thresholds are determined faster by optimizing the objective **function** based on an exhaustive search [19] or through meta-heuristics. For the same purpose, Chang and Wang [20] use a low-pass and high-pass filter repeatedly for adjusting the peaks or valleys to a desired number of classes. The valleys in the filtered histogram are then considered as threshold values. In [21], the authors propose the histogram smoothing by convoluting it with a Gaussian filter for extracting the peaks and the valleys. In [22], Arifin and Asano use hierarchical clustering method to solve the multilevel thresholding problem. Each non-empty gray level of the histogram is, initially, considered as a separate mode representing a cluster. Then, the similarities between adjacent clusters are computed and the most similar pair is merged. The estimated thresholds which are defined as the highest gray levels of the clusters are obtained by iterating this operation until the desired number of clusters is found. In [23], Chen and Wang use the support vector regression as a fitting tool rather than other histogram smoothing techniques and the support vectors on boundary (BSV) are determined to provide candidate thresholds. The optimal threshold values are then obtained by seeking among the BSV set where negative to positive transition of the first- order derivatives of the fitted histogram is occurring. It can be pointed out that the Otsu’s **function** is modified by incorporating a weight in order to ensure that the threshold values will always stand in the valleys ofthe histogram[24, 25].

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d t f t ≠ d t g t . This appears understandably nonsensical within the current mathematics. However, we have figured out how such outcomes could be possible by adopting the smooth approximate representation of the **Heaviside** step **function** H k ( t − t 0 ) given by (13). For the case in which the indicies included in T n are all the same τ , we have shown that while all the recursive **Heaviside** step functions U t i ( ) , τ have the same shape given by H ( − + t τ ) , their derivatives can be represented by the powers of the Dirac delta **function** δ ( ) i ( − + t τ ) and that they are also all different. We have also shown that by using the examples of U 2 ( ) t , τ , U 2 A ( ) t T , 2 A , and U 2 B ( ) t T , 2 B that shapes of all the recursive **Heaviside** step functions U n ( t T , n ) are the same if the values of the minimum indicies are the same while their derivatives are different when the order n is different, or the order of the indicies placed within T n is different. Thus, in this article, we have presented an unavoidable evidence that it is possible to construct func- tions with the same shape but different derivatives.

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respiratory system including patient history, physical examinations, and tests of pulmonary **function**. with insulin-dependent diabetes compared with age-matched control subjects, all lifelong nonsmokers. Lung CO transfer capacity is significantly affected by the integrity of lung capillary endothelium and, therefore, the findings of Sandler et al. focused attention on pulmonary vascular changes. The concept of the lung as a target organ for diabetic microangiopathy received continuing attention. Reports of lung **function** tests in patients with diabetes over the next 15 years have focused largely on pulmonary microangiopathy with relatively few studies of pulmonary mechanical **function** Diabetes and Lung **Function** Test : Some studies showed that all the pulmonary parameters, that is, FVC, FEV 1 , FEF 25 , FEF 50 , FEF 75 , FEF 25–75 , FEF 0.2–1.2 , and PEFR were significantly reduced except

Synaptic plasticity facilitates the modulation of connection strength between different neu- rons in an NMS. It is the primary mechanism of learning and adaptation employed in these systems. Within an NMS, it is critical to have efficient circuits that facilitate synaptic plas- ticity. Synapses are modified via a particular training algorithm which can be unsupervised, supervised, or semi-supervised. However, there are some common characteristics among each of these. One commonality is that each algorithm typically involves the product of two quantities, such as a pre-synaptic neuron’s output and an error value. Multiplication of dig- ital values is easily achieved using an AND gate, but analog multiplication is more costly, requiring at least a Gilbert Multiplier. This chapter discusses novel training algorithms to reduce the cost of training circuits. First, a stochastic least-mean-squares algorithm is de- veloped for both online and batch mode training. Then, another approach to is developed, where multiplication on the unit square is approximated by a minimum **function**

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can be taken either as a **Heaviside** or a more general sigmoidal form. Introducing a spectral wave-vector k then this product is simply w(|k|)f (k), where functions with arguments k denote two-dimensional spatial Fourier transforms. We may evaluate w(|k|)f (k) directly, at every time step, using fast Fourier transforms (FFTs). Note that w(|k|) can be pre-computed, by FFT or here even analytically, so that the procedure iterated over time amounts to computing f (k) by FFT, followed by a (complex) multiplication with w(|k|), and finally an inverse FFT to obtain the result of the integral. We wish to employ a parallel compute cluster for rapid computation over large grids, and hence use the free software package FFTW 3.3 [42], which includes a parallel MPI-C version. Note that the use of Fourier methods implies that the discretization grid has periodic boundaries, or in other words, the solution is effectively computed on a torus. We use a grid spacing of about 0.03 or better in our computations here.

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Into the frame of classical electrodynamics, the explanation of the created force in the coil-ring system may be detailed considering the retarded regime. Thus, the current pulse generates a spectrum of electromagnetic waves that leave the coil, covers the coil-to-ring distance at light speed and reaches the conductive ring surface. Here, it induces eddy currents which become the source of repulsive electromotive force. The newly created electromagnetic spectrum covers back the ring-to-coil distance at light speed to produce an induced current in the coil and so on. The electromagnetic waves in retarded regime for time dependent current distributions are entirely described by the generalized Jefimenko equations [4] (also called **Heaviside**-Feynman formula [5]). These equations take into account retardation effects becoming important at extremely high frequencies, by essence. At lower frequencies the retardation can be neglected. It is shown in the paper that only high frequencies (> 3 GHz) are involved, by looking at the spatial scales. The retarded fields were mainly studied in particle physics [6–9] and for moving atomic dipoles [10, 11] More recently, Jim´enez et al. [12] gave the exact electromagnetic field in retarded and static regime produced by a finite wire. So far, the electromagnetic field produced by a single circular ring in retarded coordinates with a time dependent current has not been reported in the literature. In this study, the dimension of wire sections is very small compared to the diameter of rings.

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The preceding considerations imply that the electromagnetic force acting on the comoving charge q in the field of the **Heaviside** ellipsoid (9) is simply the Lorentz force by which the electromagnetic field due to the two comoving charges Q + qa/b and −qa/b acts on q. Using the fact that the electric field E of a uniformly moving point charge ˜ Q is given by expression (17), and its magnetic field is

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This paper is organized as follows. First, we construct the Green **function** using Fourier series expansion in Sections and and derive a Sobolev type inequality from a solution formula to the time-periodic boundary value problem in Section . Using the solution formula, we compute the best constant and best **function** of the Sobolev type inequality as in []. The best constant is expressed as a **function** of a , a , . . . , a n– . Several concrete

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This is how the second virial coefficient is related to the excluded volume of two particles. Therefore, based on Onsager theory, the excess free energy of hard-body systems can be written in terms of the excluded volume between two particles. The isotropic to nematic first order phase transition is attributed to either Maier-Saupe theory of long range forces or Onsager theory of short range repulsive forces (Tjipto‐Margo and Evans 1990). The Onsager model shows that based on virial expansion, the two particles excluded volume effect can predict the nematic ordering (Tjipto‐Margo and Evans 1990). It is proved by Frenkel that only the second virial coefficient is needed to predict the phase transition of LC (Tjipto‐Margo and Evans 1990). According to Onsager theory, the orientational distribution **function** is obtained by minimization of the free energy of a fluid of hard rods (Tjipto‐Margo and Evans 1990) which will be discussed later in this chapter.

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In this paper we have developed the numerical tools to explore and continue travelling wave solutions for non-local neural field models with space-dependent axonal delays. Moreover, we have validated our approach against the analytically tractable case of a **Heaviside** firing rate and shown how bifurcation diagrams of this special case are modified as one moves toward more physiologically realistic shallower sigmoidal firing rate shapes. Interestingly we have shown that as well as pulses and fronts expected for excitatory networks with inhibitory feedback, also anti-pulses are a robust travelling wave solu- tion. Moreover, the bifurcation diagram for travelling localised states, i.e. fronts, pulses, and anti-pulses, is organised around a co-dimension 2 heteroclinic cycle bifurcation. Our main result, however, has been the numerical construction of dispersion curves. We have shown that they offer similar insight into the behaviour and instability of periodic travel- ling waves as originally found in their application to excitable reaction diffusion systems. Namely, that the eigen-spectrum of fixed points for a homoclinic orbit corresponding to a

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