Discontinuous Bifurcation of a Soft-Impact System

In this paper, we investigate discontinuous bifurcations of a soft-impact system, which is nonsmooth at the switching boundary consisting of two parts. We find that there are no periodic orbits located only in the impact zone, and when grazing bifurcation on one part of the switching boundary occurs, the tangency point changes may occur for different bifurcation parameters, which is also verified by numerical simulation. In particular, we discover degenerate inner and external corner bifurcations, which can produce chaotic behavior, for example, period-doubling cascades and a degenerate inner corner bifurcation that induce chaotic responses. In this way, our investigation reveals the presence of narrow bands of chaotic motion induced by the afore mentioned dynamical phenomena.

Bifurcation Analysis in Planar Quadratic Differential Systems with Boundary

Given a planar quadratic differential system delimited by a straight line, we are interested in studying the bifurcation phenomena that can arise when the position on the boundary of two tangency points are considered as parameters of bifurcation. First, under generic conditions, we find a two-parametric family of quadratic differential systems with at least one tangency point. After that, we find a normal form for this parameterized family. Next, we study two subfamilies, one of them characterized by the existence of two fold points of different nature, and the other one, characterized by the existence of one fold point and one boundary equilibrium point. For the first family, we find sufficient conditions for the existence of stationary bifurcations: saddle-node, transcritical and pitchfork, while for the second family, the existence of the called transcritical Bogdanov–Takens bifurcation is proved. Finally, the results are illustrated with two examples.

Loci of Points Equally Spaced From Two Given Geometrical Figures. Part 3

The loci (L) equally spaced from a sphere and a straight line, and from a conic surface and a plane, are considered. The following options have been considered. The straight line passes through the center of the sphere (a = 0), at the same time completely at spheres’ positive radiuses a surface of rotation is obtained, forming which the parabola is, and a rotation axis – this straight line. The parabola’s top forms the biggest parallel on the site points of intersection of the parabola’s forming with the rotation axis. Let's call such paraboloid a perpendicular paraboloid of rotation. The straight line crosses the sphere, but does not pass through the center (0 < a < R/2) – a perpendicular paraboloid, at that the surface is also completely obtained at radiuses’ positive values. The straight line is tangent to the sphere (a = R/2) – a surface which projections are parabolas, lemniscates and circles, and a piece from a tangency point to the sphere center – at radiuses positive values; a beam from the sphere center, perpendicular to this straight line – at radiuses negative values, at that the beam and the piece belong to one straight line. The straight line lies out of the sphere (α > R/2) – two different surfaces, having the general properties with a hyperbolic paraboloid, are obtained, one of which is obtained at radius positive values, and another one – at radius negative values. It has been noticed that loci, equally spaced from a sphere and a straight line, and from a cylinder and a point, coincide at equal radiuses and distances from axes to points and straight lines if to take into account the surfaces obtained both at positive, and negative values of radiuses. Locus, equally spaced from the conic surface of rotation and the plane, are two elliptic conic surfaces which in case 7.4.1 degenerate in the conic surfaces of rotation. In cases 7.4.3 and 7.4.4 one elliptic conic surface degenerates in a plane and a parabolic cylinder respectively.

Competence Approach in Teaching the Topic "Tangent Plane and Normal"

Qualified presentation of the topic "Tangent Plane and Surface Normal" in terms of competence approach is possible with the proper level for students' attention focusing on both intra-subject and inter-subject relations of descriptive geometry. Intra-subject connections follow from the position that the contingence is a particular (limit) case of intersection. Therefore, the line of intersection of the tangent plane and the surface, or two touching surfaces, has a special point at the tangency point. It is known from differential geometry [1] that this point can be nodal, return, or isolated one. In turn, this point’s appearance depends on differential properties of the surface(s) in this point’s vicinity. That's why, for the competent solution of the considered positional problem account must be also taken of the inter-subject connections for descriptive and differential geometry. In the training courses of descriptive geometry tangent planes are built only to the simplest surfaces, containing, as a rule, the frames of straight lines and circles. Therefore, the tangent plane is defined by two tangents drawn at the tangency point to two such lines. In engineering practice, as such lines are used cross-sections a surface by planes parallel to any two coordinate planes. That is, from the standpoints for the course of higher mathematics, the problem is reduced to calculation for partial derivatives. Although this topic is studied after the course of descriptive geometry, it seems possible to give geometric explanation for computation of partial derivatives in a nutshell. It also seems that the study of this topic will be stimulated by a story about engineering problems, which solution is based on construction of the tangent plane and the normal to the technical surface. In this paper has been presented an example for the use of surface curvature lines for programming of milling processing for 3D-harness surfaces.

KONFLIK KEWENANGAN ABSOLUT PENGADILAN AKIBAT PENENTUAN POKOK SENGKETA YANG BERBEDA

ABSTRAKPutusan Nomor 454/PDT.G/2005/PA.LMG menarik untuk dianalisis karena dua hal. Pertama, terkait dengan titik singgung kewenangan mengadili sengketa hibah/waris pada dua lembaga peradilan dengan adanya Putusan Nomor 163/PDT.G/2008/PT.SBY, dan kedua, tidak ada amar bersifat condemnatoir pada putusan tersebut. Adanya dua putusan pada dua lembaga peradilan yang saling berlawanan terhadap objek dan subjek yang sama menyebabkan penyelesaian perkara ini belum berakhir hingga kini dan tidak adanya kepastian hukum bagi masyarakat pancari keadilan walaupun perkara ini telah melalui proses panjang (sejak tahun 2005 sampai saat ini di tahun 2017). Untuk menganalisis masalah tersebut, ada dua masalah pokok yang diangkat dalam penelitian ini. Apakah pertimbangan hukum pengadilan negeri dalam menerima dan mengadili perkara ini dapat dibenarkan menurut kompetensi absolut yang dimilikinya? Apakah asas ne bis in idem dapat diterapkan dalam hal pengadilan negeri mengadili perkara yang sudah diputus oleh pengadilan agama? Adapun metode yang digunakan dalam penelitian ini adalah metode penelitian hukum normatif. Hasil penelitian menunjukkan bahwa pokok sengketa yang harus diangkat oleh pengadilan negeri maupun pengadilan agama adalah keabsahan hibah dengan jalan pewarisan. Asas ne bis in idem tidak dapat diterapkan oleh pengadilan negeri dalam mengadili perkara tersebut karena putusan pengadilan agama belum berkekuatan hukum tetap.Kata kunci: hibah, waris, kewenangan, peradilan, pokok sengketa. ABSTRACT Court Decision Number 454/PDT.G/2005/PA.LMG is thought-provoking to examine for two things. Firstly, it is related to the authority tangency point in adjudicating grant/heir disputes at two judicial institutions with the Court Decision Number 163/PDT.G/2008/PT.SBY, and secondly, the ruling of the decision is not condemnatory. Two decisions on two opposing jurisdictions against the same object and subject cause the case to remain unresolved until now. There is no legal certainty for the justice seekers, although the case has gone through a long process (since 2005 until now in 2017). To analyze the problem, there are two main issues elaborated in this study. Could the legal considerations of a district court in accepting and adjudicating cases be justified according to their absolute competence? Could the principle of nebis in idem be applied in the case of a district court adjudicating a case which has been decided by a religious court? This research uses normative legal research methods. The results of the study indicate that the subject of the dispute that should be examined by the district court as well as the religious court is the validity of the grant through inheritance. The district court cannot apply the nebis in idem principle in the proceedings as the decision of the religious court has not been permanently enforced.Keywords: grant, inheritance, authority, judiciary, subject matter dispute.

Higher Order, Planar Tangent-Line Envelope Curvature Theory

Following the analogy of higher order point-path generation, new concepts for tangent-line higher order envelope curvature theory are introduced. Mathematical relationships are derived to define, using the instantaneous invariants and the stretch rotation concepts, the characteristic numbers λ1 and λ2 for third and fourth-order contacts at the tangency point of a tangent-line generating an envelope. The newly developed tangent-line higher order envelope curvature theory is applied to demonstrate its application in synthesis of mechanisms. Examples involving circular, elliptic and involute are generation using a mechanism are presented.