Although the types of **singular** **points** of solutions are classiﬁed in detail [1, ch. 2] and processes of the shock waves formation are studied [2], constructing asymptotic series in the small viscosity parameter ε for an equation of the general form (0.1) is a separate problem in every speciﬁc case. First substantial results for several types of singularities, including the Whitney fold singularity A 3 ,

The core-delta relation deduced in Part II is used as a global restriction for selecting the best set of final **singular** **points**. In real applications, many fingerprint images captured by optical or capacitive sensors are not complete. Often they will lose one or two deltas. In this case, the number of cores is not necessarily equal to the number of deltas. Nevertheless, (3) still presents us a global topological restriction for **singular** **points**. Suppose the effective region of the fingerprints is Ω. By computing we can know that only a few combinations of the **singular** **points** are valid. In Table 1, most of the possible combinations of **singular** **points** are listed for fingerprints with the Poincare´ Index and the possible types (PA—plain arch, TA— tented arch, LL—left loop, RL—right loop, TL—twin loop).

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Recently, it has turned out that the singularity has a nega- tive effect on learning dynamics in the real-valued neural networks [12,13,15]. That is, the hierarchical structure or a symmetric property on exchange of weights of the the real-valued neural networks have **singular** **points**. For example, if a weight v between a hidden neuron and an output neuron is eaual to zero, then no value of the weight vector w between the hidden neuron and the input neurons affects the output value of the real-valued neural network. Then, the weight v is called an unidentifiable parameter, which is a kind of **singular** point. It has been proved that **singular** **points** affect the learning dynamics of learning models, and that they can cause a standstill in learning.

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Of course, in a living cell the membrane nanotube is sel- dom open and is usually closed at the tube's end point. Practical examples for such situation can be found in Refs.[1,4]. Geometrically this can be realized by letting the curve C converge gradually (i.e. r → 0) and tangentially to a point at the tube axis. Hence the cell or vesicle with a closed membrane nanotube may be abstracted as a closed surface with a **singular** point (Fig. 3a). In practice, more than one **singular** point may exist on a cell or vesicle. A typical example for two **singular** **points** on a vesicle has been reported in Ref.[4] and may be schematically expressed in Fig. 3b. A cell with a group of **singular** **points** is displayed in Ref.[5]. If the total number of **singular** **points** on the cell or vesicle is n point , Eq.(7) and Eq.(8) may be rewritten as:

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Abstract—An applicable and convenient method is critical for calculating the RCS (Radar Cross Sections) of chaﬀ clouds. An improved method based on direct method is proposed in this paper to promote eﬃciency, which is called SPMDM (**Singular** **Points** Meshing Direct Method). The tanh-sinh method is applied in SPMDM to compute the complex **singular** function in which the integral domain is meshed by **singular** **points**. The practicability and accuracy of the SPMDM are conﬁrmed through comparison with direct method. Results indicate that the SPMDM can signiﬁcantly decrease calculation time and increase computing eﬃciency, especially in large-scale case or small relative error region.

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Another feature that is often used for distinguishing fingerprint classes is the existence and location of **singular** **points**. The **singular** **points** are classified into core and delta as depicted by Figure 1.7. The difficulties faced by singularities-based are: the **singular** **points** may not appear in the image, especially if the image is small; the noise in the fingerprint images makes the **singular** **points** extraction unreliable, including missing or wrong detection. There are several methods have been proposed to locate the **singular** **points**. However, the most common and widely used is the Poincaré index (Li et al., 2007), but this method is very sensitive to noise, low contrast and quality of fingerprint images.

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3. The motion of **singular** **points**. Let X(µ) be a Lie rotated vector ﬁeld, we require the **singular** **points** of X(µ) to be strictly moved as parameter µ is changed, and permit the **singular** **points** that have been moved disappear or decompose, but require the **singular** **points** that have been decomposed to be at most limited in number, which do not coincide with the **singular** **points** of the original vector ﬁeld.

In this section, we consider Darboux surface F (Z,X) (s, u) = Z(s) + uX(s) of the unit speed curve α. Then we show that the Darboux surface is a developable surface which has zero Gaussian curvature. There is no any geometric interpretation of **singular** **points** in previous papers. Hence we investigate the **singular** locus of the surface and we show that it coincide with striction line of the surface.

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V The symmetric part of the dynamical Kelvin doublet gives us another fundamental solution which will be called the dynamical stresslet... The corresponding dis-.[r]

This paper is organized as follows: First we show the construction of a class of cubic parametric curves with a variable shape factor. Ball curve, cubic Bézier curve and cubic Timmer curve are special cases of the curve. In Section 3, the inflection **points** and singularities of the space cubic parametric curves are discussed. In Section 4, shape features of the planar cubic parametric curves are proposed by using the method based on the theory of envelope and topological mapping. Necessary and sufficient conditions are derived for this curve to have one or two inflection **points**, a loop or a cusp, to be locally or globally convex. The results are summarized in a shape diagram. At last, the influences of shape factor on the shape diagram and their ability for adjusting the shape of the curve are analyzed.

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a limit process expansion based at the turning point together with a twovariable expansion leads to a uniformly valid expansion... This is equivalent.[r]

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Abstract. The paradigm of ideal MHD is investigated in the vicinity of null **points** of flows and magnetic fields. These null **points** determine the location and geometrical shape of the heliopause (or other astropauses). We investigate the question whether regular and stable solutions of the ideal MHD equations in the vicinity of null **points** of flow and magnetic field exist. This is done to test the validity of ideal MHD in the vicinity of flow and magnetic field of the plasma boundaries of stellar winds and their local inter- stellar medium. We calculate the general solutions of ideal MHD in the vicinity of magnetic null **points** and use the stan- dard form of stagnation point flows to analyse all possible time evolutions of these plasma environments. We show that the solution space in 2-D consists almost exclusively of either exponentially (in time) growing velocity or mag- netic fields, or collapse solutions. Regular solutions must be three-dimensional and seem to be unstable with respect to small perturbations. This is an argument that reconnection has to take place in such regions and that therefore nonideal terms in Ohm’s law are necessary, allowing for reconnec- tion. We conclude that the use of ideal MHD in the vicinity of **singular** **points** of flow and magnetic field has to be anal- ysed very carefully with respect to simulation results as those simulations show numerical dissipation (resistivity). These simulations can therefore produce unphysical reconnection regimes. Thus one has to search for a realistic Ohm’s law, allowing for reconnection at the heliospheric boundaries.

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always assume that the singularities are isolated? in the sense that each such point is contained in a neighbourhood which contains no other* We see? therefore? by applying the Weierstrass theorem to the compact space S, that the **singular** **points** form a finite set* Xt is a woll-known fact that if there are no **singular** **points** associated with f? then S is of genus one (the homeomorph of a torus),

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interval containing . A point is an ordinary point of the ODE (2), if the functions P(x)and Q(x) are analytic at . Otherwise is a **singular** point of the ODE. On the other hand if P(x) or Q(x) are not analytic at then is said to be a **singular** point [12- 13].There is at present, numerical method for solving problems with regular **singular** **points** using Hermite interpolation method with interval [0 1].

Input fingerprint image Fingerprint enhancement Quincunx orientation field estimation Location of Singular Points SPs using Ridge Flow Codes technique Types of SPs using Pointcare index [r]

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The heuristic rule-based fingerprint classification technique, and sometimes is also called model-based approach, uses the number and the locations of **singular** **points** to classify a fingerprint. This approach was first introduced by Henry [18] in his manual classification in the early 1990s. Later, the idea is adopted by Karu and Jain [1] to automatically classify the fingerprints. Their approach consists of three major steps: (i) computation of the ridge directions using 9 9 mask, (ii) finding the singularities in the directional image using Poincare index, and (iii) classification of the fingerprint based on the detected number and location of **singular** **points**. The classifier was tested on 4,000 and 5,400 images in the NIST-Db-4 and DB9, respectively. For both databases, classification accuracies of 85.4% for the five - class and 91.1% for the four-class problems, respectively, are reported. Since the method solely relies on the **singular** **points**, failure to locate them will result in classification errors. In other words, the method is only suitable for good quality fingerprints. Due to the limitations mentioned above, most of the recent studies combine singularities with another feature, such as ridge orientation field [11].

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Continuous Legendre Transform, Inverse Legendre Transform, Inversion ormulae, Shannon Sampling Theorem, Truncation Error Estimate, Behaviour of Legendre Functions Near Singular Points.. [r]

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The geometric theory of symmetric Hamiltonian systems is based on Poisson and symplectic geometries. The symmetry leads to the conservation of certain quantities and to the reduction of these systems. We take special attention to the reduction at **singular** **points** of the momentum map. We survey the **singular** reduction procedures and we give a method of reducing a symmetric Hamiltonian system in a neighbourhood of a group orbit which is valid even when the momentum map is **singular**. This reduc tion process, which we called slice reduction, enables us to partially reduce the (local) dynamics to the dynamics of a system defined on a symplectic manifold which is the product of a symplectic vector space (symplectic slice) with a coadjoint orbit for the original symmetry group. The reduction represents the local dynamics as a coupling between vibrational motion on the vector space and generalized rigid body dynamics on the coadjoint group orbits. Some applications of the slice reduction are described, namely the application to the bifurcation of relative equilibria.

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In diﬀerent areas of applied mathematics and physics many problems arise in the form of boundary value problems involving transmission conditions at the interior **singular** **points**. Such problems are called boundary value-transmission problems (BVTPs). For example, in electrostatics and magnetostatics the model problem which describes the heat trans- fer through an inﬁnitely conductive layer is a transmission problem (see [] and references therein). Another completely diﬀerent ﬁeld is that of ‘hydraulic fracturing’ (see []) used in order to increase the ﬂow of oil from a reservoir into a producing oil well. Some problems with transmission conditions arise in thermal conduction problems for a thin laminated plate (i.e. a plate composed by materials with diﬀerent characteristics piled in the thick- ness; see []). Some aspects of spectral problems for diﬀerential equations having singu- larities with classical boundary conditions at the endpoints were studied among others in [–] and references therein.

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However, when constructing a re-envelopment that does this, it is necessary to make use of the detailed structure of the singularity, and hence there is no general recipe for unravelling essential singularities. For this reason, the question of whether or not every essential singularity covers a pure singularity has never been settled, although a counterexample to this rule of thumb would seem at first glance to be very counterintuitive. It would consist of an essential singularity, from which, regardless of how much it has been “spread out”, non-**singular** **points** could always be extracted from it, like rabbits out of a hat. However, after having given the problem a considerable amount of thought, it would seem that the conjecture might not be true in general. The following example highlights what might go wrong, and is presented as a suspected counterexample.

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