scholarly journals Pedal Curves of the Mixed-Type Curves in the Lorentz-Minkowski Plane

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2852
Author(s):  
Xin Zhao ◽  
Donghe Pei
Keyword(s):  
Fixed Point ◽  
Orthogonal Projection ◽  
Mixed Type ◽  
Minkowski Plane ◽  
Type Curve ◽  
Type Curves ◽  
Base Curve ◽  
Relationship Of ◽  
The Relationship ◽  
Pedal Curve

In this paper, we consider the pedal curves of the mixed-type curves in the Lorentz–Minkowski plane R12. The pedal curve is always given by the pseudo-orthogonal projection of a fixed point on the tangent lines of the base curve. For a mixed-type curve, the pedal curve at lightlike points cannot always be defined. Herein, we investigate when the pedal curves of a mixed-type curve can be defined and define the pedal curves of the mixed-type curve using the lightcone frame. Then, we consider when the pedal curves of the mixed-type curve have singular points. We also investigate the relationship of the type of the points on the pedal curves and the type of the points on the base curve.

scholarly journals Evolutoids of the Mixed-Type Curves

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Xin Zhao ◽  
Donghe Pei
Keyword(s):  
Mixed Type ◽  
Minkowski Plane ◽  
Singular Points ◽  
Regular Curve ◽  
Type Curve ◽  
Type Curves ◽  
Definition Of ◽  
Relationship Of ◽  
Lightlike Vector ◽  
The Relationship

The evolutoid of a regular curve in the Lorentz-Minkowski plane ℝ 1 2 is the envelope of the lines between tangents and normals of the curve. It is regarded as the generalized caustic (evolute) of the curve. The evolutoid of a mixed-type curve has not been considered since the definition of the evolutoid at lightlike point can not be given naturally. In this paper, we devote ourselves to consider the evolutoids of the regular mixed-type curves in ℝ 1 2 . As the angle of lightlike vector and nonlightlike vector can not be defined, we introduce the evolutoids of the nonlightlike regular curves in ℝ 1 2 and give the conception of the σ -transform first. On this basis, we define the evolutoids of the regular mixed-type curves by using a lightcone frame. Then, we study when does the evolutoid of a mixed-type curve have singular points and discuss the relationship of the type of the points of the mixed-type curve and the type of the points of its evolutoid.

Evolutes of the (n,m)-cusp mixed-type curves in the Lorentz–Minkowski plane

2020 ◽  
Vol 18 (01) ◽  
pp. 2150001
Author(s):  
Xin Zhao ◽  
Tongchang Liu ◽  
Donghe Pei ◽  
Cuilian Zhang
Keyword(s):  
Mixed Type ◽  
Minkowski Plane ◽  
Regular Curve ◽  
Type Curves ◽  
Base Curve

The evolute of a regular curve in the Lorentz–Minkowski plane is given by the locus of centers of osculating pseudo-circle of the base curve. But the case when a curve has singularities is not very clear. In this paper, we use lightcone frame to define the [Formula: see text]-cusp mixed-type curves and their evolutes in Lorentz–Minkowski plane. In order to attain this goal, we define the [Formula: see text]-cusp non-lightlike curves and their evolutes in Lorentz–Minkowski plane first. Then we study the behaviors of the evolutes of the [Formula: see text]-cusp mixed-type curves at the [Formula: see text]-cusp.

Mixed-type curves and the lightcone frame in Minkowski 3-space

2020 ◽  
Vol 17 (06) ◽  
pp. 2050088 ◽  
Author(s):  
Tongchang Liu ◽  
Donghe Pei
Keyword(s):  
Mixed Type ◽  
Frenet Frame ◽  
Type Curve ◽  
Type Curves ◽  
Physical Research ◽  
Null Curves

In this paper, we study mixed-type curves in Minkowski 3-space. Mixed-type curves are regular curves, and there are both non-lightlike points and lightlike points in a mixed-type curve. For non-lightlike curves and null curves in Minkowski 3-space, we can study them by a Frenet frame or a Cartan frame, respectively. But for mixed-type curves, the two frames will not work. As far as we know, no one has yet given a frame to study them in Minkowski 3-space. So, we give the lightcone frame in order to provide a tool for studying this type curves in mathematical and physical research. As an application of the lightcone frame, we define an evolute of a mixed-type curve. We also give some examples to show the evolutes.

A Note on the Ancient World-Systems

2018 ◽  
Vol 22 (2) ◽  
pp. 160-167
Author(s):  
Ervand Margaryan
Keyword(s):  
Mixed Type ◽  
World System ◽  
Ancient World ◽  
World Systems ◽  
Time And Space ◽  
The World ◽  
Relationship Of ◽  
The Relationship

The article deals with the relationship of such concepts as the world-system and civilisation, both living independently and co-existing in time and space. World-systems and civilisations may be forced to unite into hyper-systems, or world-empires of different kind—self-sufficient, militarist-parasitic, and mixed type. Militarist empires-parasites can be settled and nomadic. Nomadic or bivouac empires are empires-armies, which exist only in movement. Stopping leads either to the death of the empire-army, or to the transformation into one but usually several stationary empires, mostly also militarist-parasitic.

The relationship of the scleractinian corals to the rugose corals

Paleobiology ◽  
1980 ◽  
Vol 6 (02) ◽  
pp. 146-160 ◽  
Author(s):  
William A. Oliver
Keyword(s):  
Critical Points ◽  
Scleractinian Corals ◽  
Convincing Evidence ◽  
Early Triassic ◽  
Rugose Corals ◽  
History Of ◽  
The Subject ◽  
Relationship Of ◽  
The Relationship ◽  
Biologic Factors

The Mesozoic-Cenozoic coral Order Scleractinia has been suggested to have originated or evolved (1) by direct descent from the Paleozoic Order Rugosa or (2) by the development of a skeleton in members of one of the anemone groups that probably have existed throughout Phanerozoic time. In spite of much work on the subject, advocates of the direct descent hypothesis have failed to find convincing evidence of this relationship. Critical points are:(1) Rugosan septal insertion is serial; Scleractinian insertion is cyclic; no intermediate stages have been demonstrated. Apparent intermediates are Scleractinia having bilateral cyclic insertion or teratological Rugosa.(2) There is convincing evidence that the skeletons of many Rugosa were calcitic and none are known to be or to have been aragonitic. In contrast, the skeletons of all living Scleractinia are aragonitic and there is evidence that fossil Scleractinia were aragonitic also. The mineralogic difference is almost certainly due to intrinsic biologic factors.(3) No early Triassic corals of either group are known. This fact is not compelling (by itself) but is important in connection with points 1 and 2, because, given direct descent, both changes took place during this only stage in the history of the two groups in which there are no known corals.

Skeletal elements of crinoid echinoderms: High-resolution structural and microanalytical results

1983 ◽  
Vol 41 ◽  
pp. 194-195
Author(s):  
D. F. Blake ◽  
L. F. Allard ◽  
D. R. Peacor
Keyword(s):  
High Resolution ◽  
Marine Invertebrates ◽  
Sea Urchins ◽  
Structural Features ◽  
Sea Stars ◽  
High Resolution Tem ◽  
Relationship Of ◽  
The Relationship ◽  
Insight Into

Echinodermata is a phylum of marine invertebrates which has been extant since Cambrian time (c.a. 500 m.y. before the present). Modern examples of echinoderms include sea urchins, sea stars, and sea lilies (crinoids). The endoskeletons of echinoderms are composed of plates or ossicles (Fig. 1) which are with few exceptions, porous, single crystals of high-magnesian calcite. Despite their single crystal nature, fracture surfaces do not exhibit the near-perfect {10.4} cleavage characteristic of inorganic calcite. This paradoxical mix of biogenic and inorganic features has prompted much recent work on echinoderm skeletal crystallography. Furthermore, fossil echinoderm hard parts comprise a volumetrically significant portion of some marine limestones sequences. The ultrastructural and microchemical characterization of modern skeletal material should lend insight into: 1). The nature of the biogenic processes involved, for example, the relationship of Mg heterogeneity to morphological and structural features in modern echinoderm material, and 2). The nature of the diagenetic changes undergone by their ancient, fossilized counterparts. In this study, high resolution TEM (HRTEM), high voltage TEM (HVTEM), and STEM microanalysis are used to characterize tha ultrastructural and microchemical composition of skeletal elements of the modern crinoid Neocrinus blakei.

Studies on Leukemogenic and Sarcomagenic Viruses

1970 ◽  
Vol 28 ◽  
pp. 156-157
Author(s):  
Leon Dmochowski
Keyword(s):  
Electron Microscopy ◽  
Electron Microscope ◽  
Morphological Properties ◽  
Mode Of Development ◽  
Relationship Of ◽  
The Relationship

Electron microscopy has proved to be an invaluable discipline in studies on the relationship of viruses to the origin of leukemia, sarcoma, and other types of tumors in animals and man. The successful cell-free transmission of leukemia and sarcoma in mice, rats, hamsters, and cats, interpreted as due to a virus or viruses, was proved to be due to a virus on the basis of electron microscope studies. These studies demonstrated that all the types of neoplasia in animals of the species examined are produced by a virus of certain characteristic morphological properties similar, if not identical, in the mode of development in all types of neoplasia in animals, as shown in Fig. 1.

The relationship of IGE receptor topography to signaling activity in RBL-2H3 mast cells

1991 ◽  
Vol 49 ◽  
pp. 236-237
Author(s):  
J.R. Pfeiffer ◽  
J.C. Seagrave ◽  
C. Wofsy ◽  
J.M. Oliver
Keyword(s):  
Mast Cells ◽  
Protein A ◽  
Scattered Electron ◽  
Ige Receptor ◽  
Receptor Complexes ◽  
Ige Binding ◽  
Electron Imaging ◽  
Small Clusters ◽  
Relationship Of ◽  
The Relationship

In RBL-2H3 rat leukemic mast cells, crosslinking IgE-receptor complexes with anti-IgE antibody leads to degranulation. Receptor crosslinking also stimulates the redistribution of receptors on the cell surface, a process that can be observed by labeling the anti-IgE with 15 nm protein A-gold particles as described in Stump et al. (1989), followed by back-scattered electron imaging (BEI) in the scanning electron microscope. We report that anti-IgE binding stimulates the redistribution of IgE-receptor complexes at 37“C from a dispersed topography (singlets and doublets; S/D) to distributions dominated sequentially by short chains, small clusters and large aggregates of crosslinked receptors. These patterns can be observed (Figure 1), quantified (Figure 2) and analyzed statistically. Cells incubated with 1 μg/ml anti-IgE, a concentration that stimulates maximum net secretion, redistribute receptors as far as chains and small clusters during a 15 min incubation period. At 3 and 10 μg/ml anti-IgE, net secretion is reduced and the majority of receptors redistribute rapidly into clusters and large aggregates.

Sign in / Sign up

Export Citation Format

Share Document