
An analysis of geological-geophysical, metallogenic, geochronological, and seismic tomographic studies in territories joining Southeast Russia, East Mongolia, and Northeast China led to the conclusion that deep geodynamics significantly influenced the formation of highly productive ore-magmatic systems in the Late Jurassic-Early Cretaceous. This influence was likely manifested through the initiation of decompression processes around stagnant slab boundaries in the Late Mesozoic. Decompression and advection, which are particularly active near the natural boundaries of the slab, act as triggers for the intense interaction of under and over subduction asthenospheric fluids with adjacent sections of the mantle and for the directed upwelling of powerful flows of matter and energy into the lithosphere. These flows determine the locations of intermediate and peripheral magma chambers: Primary chambers in the lower lithosphere among the metasomatized mantle and lower crust and associated chambers in the middle and upper cratonized parts of the lithosphere. Large ore clusters containing noble metals (Au, PGE), uranium, fluorite, and Cu-Mo-porphyry deposits are associated with late- and postmagmatic derivatives of the emerging magma chambers over the frontal and peripheral (paleotransform) boundaries of a stagnant Pacific slab. These large Late Mesozoic ore clusters and districts form a distinctive "necklace" of strategic materials in East Asia.
Citation: Natalia Boriskina. A «necklace» of large clusters of strategic raw materials over a stagnant oceanic slab in East Asia[J]. AIMS Geosciences, 2024, 10(4): 864-881. doi: 10.3934/geosci.2024040
[1] | Yasin Ünlütürk, Talat Körpınar, Muradiye Çimdiker . On k-type pseudo null slant helices due to the Bishop frame in Minkowski 3-space E13. AIMS Mathematics, 2020, 5(1): 286-299. doi: 10.3934/math.2020019 |
[2] | Ufuk Öztürk, Zeynep Büşra Alkan . Darboux helices in three dimensional Lie groups. AIMS Mathematics, 2020, 5(4): 3169-3181. doi: 10.3934/math.2020204 |
[3] | Zuhal Küçükarslan Yüzbașı, Dae Won Yoon . On geometry of isophote curves in Galilean space. AIMS Mathematics, 2021, 6(1): 66-76. doi: 10.3934/math.2021005 |
[4] | Esra Damar . Adjoint curves of special Smarandache curves with respect to Bishop frame. AIMS Mathematics, 2024, 9(12): 35355-35376. doi: 10.3934/math.20241680 |
[5] | Gülnur Şaffak Atalay . A new approach to special curved surface families according to modified orthogonal frame. AIMS Mathematics, 2024, 9(8): 20662-20676. doi: 10.3934/math.20241004 |
[6] | Kemal Eren, Hidayet Huda Kosal . Evolution of space curves and the special ruled surfaces with modified orthogonal frame. AIMS Mathematics, 2020, 5(3): 2027-2039. doi: 10.3934/math.2020134 |
[7] | Yanlin Li, Kemal Eren, Kebire Hilal Ayvacı, Soley Ersoy . Simultaneous characterizations of partner ruled surfaces using Flc frame. AIMS Mathematics, 2022, 7(11): 20213-20229. doi: 10.3934/math.20221106 |
[8] | Semra Kaya Nurkan, İlkay Arslan Güven . Construction of vectorial moments via direction curves. AIMS Mathematics, 2023, 8(6): 12857-12871. doi: 10.3934/math.2023648 |
[9] | Beyhan YILMAZ . Some curve pairs according to types of Bishop frame. AIMS Mathematics, 2021, 6(5): 4463-4473. doi: 10.3934/math.2021264 |
[10] | Bahar UYAR DÜLDÜL . On some new frames along a space curve and integral curves with Darboux q-vector fields in E3. AIMS Mathematics, 2024, 9(7): 17871-17885. doi: 10.3934/math.2024869 |
An analysis of geological-geophysical, metallogenic, geochronological, and seismic tomographic studies in territories joining Southeast Russia, East Mongolia, and Northeast China led to the conclusion that deep geodynamics significantly influenced the formation of highly productive ore-magmatic systems in the Late Jurassic-Early Cretaceous. This influence was likely manifested through the initiation of decompression processes around stagnant slab boundaries in the Late Mesozoic. Decompression and advection, which are particularly active near the natural boundaries of the slab, act as triggers for the intense interaction of under and over subduction asthenospheric fluids with adjacent sections of the mantle and for the directed upwelling of powerful flows of matter and energy into the lithosphere. These flows determine the locations of intermediate and peripheral magma chambers: Primary chambers in the lower lithosphere among the metasomatized mantle and lower crust and associated chambers in the middle and upper cratonized parts of the lithosphere. Large ore clusters containing noble metals (Au, PGE), uranium, fluorite, and Cu-Mo-porphyry deposits are associated with late- and postmagmatic derivatives of the emerging magma chambers over the frontal and peripheral (paleotransform) boundaries of a stagnant Pacific slab. These large Late Mesozoic ore clusters and districts form a distinctive "necklace" of strategic materials in East Asia.
One of the most researched areas of differential geometry is curve theory, with studies presented in different dimensions and spaces. Generally, the geometric location of the different positions taken by a moving point in space during motion is a curve. In this motion, the parameter range of the curve represents the time elapsed through the motion. In addition, the parameter of the curve has an important place to characterize the curve. For example, the parameter of any given curve has many differences in terms of whether the curve is parameterized with respect to arc length. On the other hand, the characterization of a regular curve is one of the most important problems in curve studies in differential geometry, where curvature and torsion play a major role in solving the problem. The definitions of some special curves can be made by using the curvature and torsion [3,4,5,6,7,8,9,10,11]. One of them is the helices definition given by Lancret in 1806 [18]. Lancret defined the classical definition of helices as "curves whose tangent vector makes a constant angle with a constant direction" [18]. Starting from this definition, the helices drawn on a right circular cylinder were considered, and they were called a circular helix. It is among the first characterizations indicating that curvature and torsion are constants separately in these helices. Later, although these curvatures are not constant, curves with constant ratios were found, which were called the general helices, and the proof of this theory was proved by Venant in 1845 [11]. Helices have an impressive place both in nature and science. Fractal geometry, the arrangement of molecules in the structure of DNA, and its applications in physics and civil and mechanical engineering are some examples. In addition to that, Izumiya and Takeuchi named the curves slant helices, where the angle between the principal normal vector field and a constant direction is constant at every point of the curve, and they obtained that the necessary and sufficient condition for curves to be slant helices is that the geodesic curvature of the principal normals indicator of the given curves is a constant function [14]. Then, some studies about slant helices were conducted [2,24]. Ali and Lopez [1] studied slant helices in Lorenz space and investigated the tangent indicatrix on the binormal indicatrix of a slant helix in Minkowski space. On the other hand, Kula and Yaylı investigated the tangential and binormal indicatrices of slant helices and showed that these spherical indicatrices are spherical helices [17]. Later, Önder et al. introduced B2-slant helices in E4 and determined the properties of these curves [21]. In addition, Yaylı and Zıplar examined the relationship between slant helices and spherical helices in En space [26]. Then, Gök et al. defined Vn-slant helices in the space En [12], K\i z\i ltu\u{g} et al. [16] obtained the characterization of slant helices using a type-2 bishop frame and Lucas and Ortega-Yagües [20] investigated the relationship between helix surfaces and slant helices in a sphere S3. Recently, Uzuno\u{g}lu et al. [25] explained a recent alternative orthogonal frame as {N,C=N′‖N′‖,W} and gave a characterization of C-slant helices. Moreover, they investigated curves of C-constant precession according to an alternative frame [25]. After that, Kaya and Önder et al. [15] defined C-partner curve with respect to the frame {N,C,W}. On the other hand, the modified orthogonal frame has been defined by Sasai [23], and the helices and Bertrand curves for modified orthogonal frames in E3 have been given in [19].
In this section, we give some basic definitions of curves in Euclidean 3-space. Let α:I⊂R⟶E3 be a regular-unit speed curve with the arc-length parameter s (‖α′(s)‖=1). Therefore, the curve (α) is called a Frenet curve if it has non-zero curvature (α″(s)≠0). Moreover, the derivative formulas of the Frenet frame of a regular-unit speed curve can be given as
T′=κN,N′=−κT+τB,B′=−τN, |
where {T,N,B} constitutes the Frenet frame and κ=‖α″‖=⟨T′,N⟩; and, τ=⟨N′,B⟩ are the curvature and torsion of the unit speed curve (α), respectively. In addition to the Frenet frame, Uzunoǧlu et al. [25] explained a new alternative orthogonal frame such that {N,C,W}, where N is the principal normal vector field, C=N′‖N′‖, W=τT+κB√κ2+τ2 and W is the Darboux vector. Therefore, the derivative formulae of this alternative frame {N,C,W} are provided by
[N′C′W′]=[0f0−f0g0−g0][NCW], | (2.1) |
where
f=√κ2+τ2=κ√1+H2,g=κ2κ2+τ2(τκ)′, |
κ and τ are the curvature and torsion of the curve (α), respectively, and H is the harmonic curvature. Moreover, the relationship between the new alternative frame {N,C,W} and Frenet frame {T,N,B} are given as follows:
T=−κC+τW√κ2+τ2,B=τC+κW√κ2+τ2, |
where the vector field N is common [15].
Now, for a regular-unit speed curve α=α(s), we can say that, if the vector field T of the curve (α) makes a constant angle with a fixed-unit vector field, α=α(s) is called a general helix [11,18]. Moreover, a helix curve is characterized by κτ=constant, where κ and τ are the curvature and torsion of the curve (α), respectively [11]. Then, Izumiya and Takeuchi described a slant helix as "a curve α whose principal normal vector make a constant angle with such a constant direction, that is
⟨N,U⟩=cosθ |
for every s∈I and θ≠π2" [7]. At the same time, they also stated that (α) is a slant helix where the necessary and sufficient condition is that its curvatures satisfy the equation
κ2(κ2+τ2)32(τκ)′=constant. |
After that, Uzunoǧlu et al. [25] gave the following definition of a slant helix according to the alternative orthogonal frame {N,C,W}.
Definition 1. Let (α) be a unit speed curve in E3 and {N,C,W} be the alternative frame of the curve (α). In that case, (α) is called a C-slant helix according to the alternative frame if the vector field C makes a constant angle with a constant direction V, that is, if
⟨C,V⟩=cosδ,(δ≠π2) |
for all s∈I. Therefore, the ratio
fg=constant | (2.2) |
for the adapted curvatures fandg [25].
In this section, we first define the modified adapted frame; later, we give the definitions of the N∗-slant helix and C∗-slant helix and some of their characterizations using the modified adapted frame in E3.
Let α:I⊂R⟶E3 be a unit speed curve that can be parameterized in terms of the arc length s. Now, we define a new and different frame i.e., a modified adapted frame. The motivation for us to define this new frame in this study is that the frame defined by Uzunoǧlu et al. [25] cannot be created at every position of the curve. If the equation f(s)=‖N′(s)‖=0 holds in the frame {N,C,W} defined by them, then the vector field C=N′f=N′‖N′‖ cannot be defined. Therefore, the frame {N,C,W} cannot be constructed for ∀s∈I. In this case, the alternative frame given by them [25] needs to be modified. In this study, we modified this frame and named this new frame the "modified adapted frame". As a result, we defined this new frame {N∗,C∗,W∗} such that
N∗=κN,C∗=fC,W∗=fW, | (3.1) |
where C∗(s0)=W∗(s0)=0 if f(s0)=0. Now, we give the derivative formulas of the modified adapted frame {N∗,C∗,W∗}. For this, if we use Eqs (2.1) and (3.1), then we get
[(N∗)′(C∗)′(W∗)′]=[κ′κκ0−f2κf′fg0−gf′f][N∗C∗W∗], | (3.2) |
where ⟨N∗,N∗⟩=κ2,⟨C∗,C∗⟩=f2,⟨W∗,W∗⟩=f2, f=√κ2+τ2, κ and τ are the Frenet curvature and torsion of the curve (α), respectively, and f and g are the modified adapted curvatures. Moreover, the relationship between the modified adapted frame {N∗,C∗,W∗} and Frenet frame {T,N,B} can be written as
T=−κC∗+τW∗κ2+τ2,N=N∗κ,B=τC∗+κW∗κ2+τ2. |
Now, we give some characterizations of the N∗-slant helices and C∗-slant helices with the modified adapted frame {N∗,C∗,W∗}.
Theorem 1. Let (α) be a unit speed curve with the modified adapted apparatus {N∗,C∗,W∗,f,g} in E3. In that case, if the unit speed curve (α) is an N∗-slant helix according to the modified adapted frame, then the axis of the N∗-slant helix (α) is the constant unit vector X, which can be written as
X=κcosζN∗+f2gcosζW∗, |
where κ is the Frenet curvature of (α), f=√κ2+τ2 and ζ is the angle between the vector fields N∗ and X.
Proof. Suppose that (α) is an N∗-slant helix with respect to the modified adapted frame {N∗,C∗,W∗} in E3. Then, for the vector fields N∗ and X, we get
⟨N∗,X⟩=κcosζ, | (3.3) |
where ζ is the constant angle between the vector fields N∗ and X. In that case, differentiating (3.3) according to s with the aid of Eq (3.2), we obtain
⟨C∗,X⟩=0. | (3.4) |
Thus, we can write the constant unit vector X as
X=κcosζN∗+λW∗, | (3.5) |
where λ is a differentiable function. If we apply the differentiation process again to Eq (3.4), we have
−f2κ⟨N∗,X⟩+f′f⟨C∗,X⟩+g⟨W∗,X⟩=0 |
and
⟨W∗,X⟩=f2gcosζ. | (3.6) |
Finally, considering (3.5) and (3.6), we obtain
X=κcosζN∗+f2gcosζW∗. |
Theorem 2. Consider that (α) is a unit speed curve with the modified adapted apparatus {N∗,C∗,W∗,f,g} in E3; then, (α) is a C∗-slant helix such that the constant unit vector Y characterizing this C∗-slant helix is given by
Y=μcosηN∗+fcosηC∗+δcosηW∗, |
where f=√κ2+τ2, η is the angle between the vector fields C∗ and Y, μ=gκ(f2+g2)g′f−f′g and δ=f2(f2+g2)g′f−f′g.
Proof. Suppose that (α) is a C∗-slant helix with respect to the modified adapted frame {N∗,C∗,W∗}. In that case, with aid of Definition 1, we have
⟨C∗,Y⟩=fcosη | (3.7) |
such that η is the constant angle between the vector fields C∗ and Y. Now, we take successive derivatives of Eq (3.7) with respect to the arc-length parameter s of (α); thus, we obtain
−f2κ⟨N∗,Y⟩+f′f⟨C∗,Y⟩+g⟨W∗,Y⟩=f′cosη, |
−2ff′κ2⟨N∗,Y⟩−(f2+g2)⟨C∗,Y⟩+(g′+gf′f)⟨W∗,Y⟩=0, |
and, as a result,
⟨N∗,Y⟩=gκ(f2+g2)g′f−f′gcosη, |
⟨W∗,Y⟩=f2(f2+g2)g′f−f′gcosη, |
respectively. Consequently, considering μ=gκ(f2+g2)g′f−f′g and δ=f2(f2+g2)g′f−f′g, we can easily get
Y=μcosηN∗+fcosηC∗+δcosηW∗. |
In this section, we define the C∗-partner curves considering the modified adapted frame {N∗,C∗,W∗} and we give some characterizations of these curves.
Let α:I⊂R⟶E3 and β:J⊂R⟶E3 be considered as two different unit speed curves with arc-length parameters s∈I and s1∈J, respectively. Moreover, we consider that the Frenet frames and curvatures of the curves (α) and (β) are {T,N,B,κ,τ} and {T1,N1,B1,κ1,τ1}, respectively. In that case, the following definitions and theorems can be given.
Definition 2. Consider that α:I⟶E3 and β:J⟶E3 are two regular curves and the modified adapted frames and curvature of the curves (α) and (β) are {N∗,C∗,W∗,f,g} and {N∗1,C∗1,W∗1,f1,g1}, respectively. In that case, if the vector fields C∗ and C∗1 are linear-dependent at the corresponding noncoincident points of these curves, then (α) and (β) are called C∗-partner curves.
Theorem 3. Suppose that (α) and (β) are C∗-partner curves in E3. In that case, the distance between the corresponding noncoincident points of (α) and (β) is not constant.
Proof. Suppose that (α) and (β) are C∗-partner curves in E3. {N∗,C∗,W∗} and {N∗1,C∗1,W∗1} are the modified adapted frames of the curves (α) and (β), respectively. In that case, the relationship between the C∗-partner curves at the corresponding noncoincident points can be written as
β(s)=α(s)+D(s)C∗(s), | (3.8) |
where D=D(s) is the distance function and the arc-length parameter of the curve (α) is s∈I. If we take the derivative of Eq (3.8) with respect to the parameter s∈I, we get
T1ds1ds=T+D′C∗+D(C∗)′, |
where the tangent vectors of the C∗-partner curves (α) and (β) are, respectively, T and T1, and the arc-length parameter of the curve (β) is s1. Now, we use the relationships between the Frenet and modified adapted frames of the C∗-partner curves. Therefore, we have
T1ds1ds=T+(D′+Df′f)C∗−Df2κN∗+DgW∗ |
and
−κ1f21ds1dsC∗1+τ1f21ds1dsW∗1=(−κf2+D′+Df′f)C∗−Df2κN∗+(τf2+Dg)W∗, |
where {κ,τ} and {κ1,τ1} are the Frenet curvatures of C∗-partner curves (α) and (β), respectively. By considering the linear-dependent pair {C∗,C∗1}, the equation
κ−κ1ds1dsf=(Df)′ |
holds. Consequently, we see that the distance function D=D(s) is not constant.
Now, we assume that the vector fields C∗ and C∗1 satisfy the equation C∗=C∗1. In this case, for the adapted curvatures f and f1 of the C∗-partner curves, the equation f2=f21 holds. As a result, the following theorem can be given:
Theorem 4. Suppose that (α) and (β) are C∗-partner curves. In that case, if the angle between N∗ and N∗1 is ϱ, then the relationships between the modified adapted frames {N∗,C∗,W∗} and {N∗1,C∗1,W∗1} of the curves (α) and (β), respectively, are
[N∗C∗W∗]=[κκ1cosϱ0κf1sinϱ010−fκ1sinϱ0cosϱ][N∗1C∗1W∗1], | (3.9) |
where κ,κ1 and f,f1 are the Frenet curvatures and adapted curvatures of the C∗-partner curves (α,β), respectively.
Proof. Consider that (α,β) are C∗-partner curves and C∗=C∗1. In that case, if we consider that the angle between the vector fields N∗ and N∗1 is ϱ, we can write
⟨N∗,N∗1⟩=κκ1cosϱ,⟨W∗,W∗1⟩=ff1cosϱ,⟨N∗,W∗1⟩=κf1sinϱ,⟨W∗,N∗1⟩=−κ1fsinϱ. |
Consequently, considering the basic arrangements in differential geometry, we obtain
N∗=κκ1cosϱN∗1+κf1sinϱW∗1,W∗=−fκ1sinϱN∗1+cosϱW∗1. |
Alternatively, we can write
N∗1=κ1κcosϱN∗−κ1fsinϱW∗,W∗1=fκsinϱN∗+cosϱW∗, | (3.10) |
where f=f1.
Theorem 5. Consider that (α) and (β) are C∗-partner curves in E3. Therefore, the angle between N∗ and N∗1 of these curves is constant.
Proof. We know that (α) and (β) are C∗-partner curves, and that there is the relationship
N∗1=κ1κcosϱN∗−κ1fsinϱW∗ | (3.11) |
from (3.10), where {N∗,C∗,W∗,f,g} and {N∗1,C∗1,W∗1,f1,g1} are the modified adapted frames and κ and κ1 are the Frenet curvatures of the C∗-partner curves (α,β), respectively. In that case, by differentiating (3.11) with respect to s, we obtain
dN∗1ds1ds1ds=(κ1cosϱκ)′N∗+κ1cosϱκ(N∗)′−(κ1sinϱf)′W∗−κ1sinϱf(W∗)′, |
and by considering Eqs (3.2) and (3.11), we can write
κ′1κ1cosϱds1dsN∗+κ1ds1dsC∗1−κ′1fsinϱds1dsW∗=(κ′1cosϱ−κ1ϱ′sinϱκ)N∗+(κ1cosϱ+κ1gfsinϱ)C∗−(κ′1sinϱ+κ1ϱ′cosϱf)W∗. |
As a result, we get
(κ1cosϱ)′=κ′1cosϱds1ds | (3.12) |
and
(κ1sinϱ)′=κ′1sinϱds1ds. | (3.13) |
Moreover, we obtain from Eqs (3.12) and (3.13) the equations
(κ1cosϱ)′cosϱ=(κ1sinϱ)′sinϱ |
and
κ1ϱ′=0. |
Consequently, for κ1≠0, we have ϱ′=0, and it is easy to see that the angle ϱ is constant.
Theorem 6. Let the curves (α) and (β) be considered C∗-partner curves and D=D(s) be the distance function between corresponding noncoincident points of these curves. Therefore, the distance D is a constant necessary and sufficient condition, i.e.,
(τf3)′f2(τf2g)′g2=−tanϱ=constant, |
where τ and f and g are the Frenet torsion and adapted curvatures of the curve (α), respectively, and ϱ is the angle between the vector fields N∗ and N∗1.
Proof. If (α) and (β) are C∗-partner curves, then we know that the relationship between these curves is
β(s)=α(s)+D(s)C∗(s). |
In that case, considering the derivative of the last equation with respect to s, we get
−κ1f21ds1dsC∗1+τ1f21ds1dsW∗1=−Df2κN∗+(−κf2+D′+Df′f)C∗+(τf2+Dg)W∗. |
If we make necessary arrangements using (3.9), then we have
−κ1f21ds1dsC∗1+τ1f21ds1dsW∗1=(−Df2κ1cosϱ−τfκ1sinϱ−Dgfκ1sinϱ)N∗1+(−κf2+D′+Df′f)C∗+(−Df2f1sinϱ+τf2cosϱ+Dgcosϱ)W∗1. |
The vector field C∗ and C∗1 are linear-dependent; therefore, the equation
Df3cosϱ+Dgf2sinϱ+τsinϱ=0 |
is satisfied. In that case, we obtain
D=−τsinϱf2(fcosϱ+gsinϱ) |
and
D′=−τ′sinϱ(f2gsinϱ+f3cosϱ)+τsinϱ((2ff′g+f2g′)sinϱ+3f2f′cosϱ)f4(fcosϱ+gsinϱ)2. |
Now, we assume that the distance function D(s) is a constant necessary and sufficient condition, where D′(s)=0. As a result, we get
τ′sinϱfg+τ′cosϱf2−2τsinϱf′g−τsinϱfg′−3τff′cosϱ=0 |
from the last equation. Consequently, we obtain
(τf3)′f2(τf2g)′g2=−tanϱ=constant, |
where ϱ is constant.
Let (α) and (β) be considered C∗-partner curves. Now, we give some applications to show the character of the C∗-partner curve, where any given curve (α) or (β) is helix or slant helix. For this, the following theorems can be given.
Theorem 7. Let (α,β) be C∗-partner curves in E3. In that case, the modified adapted curvatures {f1,g1} of the curve (β) in terms of {f,g} of the curve (α) is given by
g1f1=gfcosϱ−sinϱgfsinϱ+cosϱ, | (3.14) |
where ϱ is the constant angle between N∗ and N∗1.
Proof. Suppose that (α) and (β) are C∗-partner curves. Therefore, if we differentiate W∗1 in (3.10) with respect to s∈I of (α), we obtain
−g1ds1dsC∗1+f′1f1ds1dsW∗1=f′sinϱκN∗+(fsinϱ−gcosϱ)C∗+f′cosϱfW∗, | (3.15) |
where, by using the derivative formulas given by Eqs (3.2) and (3.10) in (3.15), the equation
−g1ds1dsC∗+f′1κsinϱds1dsN∗+f′1f1cosϱds1dsW∗=f′κsinϱN∗+(fsinϱ−gcosϱ)C∗+f′fcosϱW∗ |
holds. We know that C∗=C∗1; therefore, we obtain
−g1ds1ds=fsinϱ−gcosϱ; |
consequently, we get
g1f1=gcosϱ−fsinϱgsinϱ+fcosϱ. |
The following corollaries can be given as consequences of Theorem 7.
Corollary 1. Consider that (α,β) are C∗-partner curves in E3. In addition, let (τ,τ1) be the Frenet torsions of the C∗-partner curves (α,β) and {f,g} and {f1,g1} be the modified adapted curvatures of the curves (α,β), respectively. Therefore, the torsions (τ,τ1) can be written by modified adapted curvatures such that
τ1τ=cosϱ−g1f1sinϱcosϱ+gfsinϱ. | (3.16) |
Corollary 2. Let the curves (α,β) be C∗-partner curves. If the curve (α) is a slant helix, then the necessary and sufficient condition is that (β) is a slant helix.
Proof. Consider that (α,β) are C∗-partner curves and the curve (α) is a slant helix. In that case, we know that gf=constant from (2.2). If we consider Eq (3.14), we have g1f1=constant. As a result, (β) is a slant helix. Similarly, if the curve (β) is a slant helix, then g1f1=constant. Therefore, from Eq (3.14), we obtain
gf=−g1f1cosϱ−sinϱg1f1sinϱ−cosϱ. | (3.17) |
Consequently, we have gf=constant and the curve (α) is a slant helix.
Corollary 3. Suppose that the curves (α,β) are C∗-partner curves. If (α) is a helix, then (β) is a slant helix. If (β) is a helix, then (α) is a slant helix.
Proof. We assume that the curve (α) is a helix. The Frenet curvature ratio τκ=constant and g=0; therefore, considering Eq (3.14) we obtain
g1f1=−tanϱ(ϱ=constant), |
and the curve (β) is a slant helix. Similarly, if we take that the curve (β) is a helix, then g1=0. Consequently, from Eq (3.17), we see that gf=tanϱ=constant and the curve (α) is a slant helix.
Corollary 4. Consider that the curves (α,β) are the C∗-partner curves. In that case, if the curve (α) (or(β)) is a slant helix, then τ1τ=constant where (τ,τ1) are the Frenet torsions of the C∗-partner curves (α) and (β), respectively.
Proof. Let the curve (α) (or(β)) be a slant helix from the C∗-partner curves (α,β). Therefore, we know that (α) is a slant helix necessary and sufficient condition whereby the curve (β) is a slant helix from Corollary 2. As a result, the ratio gf=constant with the necessary and sufficient condition of g1f1=constant. Consequently, consider Eq (3.16); we obtain τ1τ=constant.
Example 1. Consider a circular helix ω given by the parameterization
ω(ρ)=1√2(−cosρ,sinρ,ρ). | (3.18) |
Then, let the C∗-partner curves of the curve ω be ψ∗, ψ∗∗ for the distance function D(s);
i) for D(ρ)=√3ρ, the C∗-partner curve ψ∗ of ω is obtained as
ψ∗(s)=(−cosρ√2−√3ρsinρ2,sinρ√2+√3ρcosρ2,ρ2). | (3.19) |
Figure 1 shows the graph of the curve ω and the C∗-partner curve ψ∗.
ii) for D(ρ)=√3ρ, the C∗-partner curve ψ∗∗ of ω is obtained as
ψ∗∗(s)=(−cosρ√2−(√2+ρ2)sinρ6,sinρ√2+(√2+ρ2)cosρ6,ρ2). | (3.20) |
Figure 2 shows the graph of the curve ω and the C∗-partner curve ψ∗∗.
Example 2. Consider a circular helix ω given by the parameterization
ω(ρ)=(112cos4ρ+112cos2ρ,112sin4ρ+112sin2ρ,2√23). | (3.21) |
Then, let the C∗-partner curves of the curve ω be ψ∗, ψ∗∗ for the distance function D(s);
i) for D(ρ)=√32ρ, the C∗-partner curve ψ∗ of ω is obtained as ψ∗(s)=ω(s)+D(s)C∗(s)
ψ∗(s)=(112cos4ρ+112cos2ρ+2ρsin3ρ,112sin4ρ+112sin2ρ−2ρcos3ρ,2√23). | (3.22) |
Figure 3 shows the graph of the curve ω and the C∗-partner curve ψ∗.
ii) for D(ρ)=8√2+ρ26√2ρ, the C∗-partner curve ψ∗∗ of ω is obtained as
ψ∗∗(s)=(112cos4ρ+112cos2ρ+8√2+ρ23sin3ρ,112sin4ρ+112sin2ρ−8√2+ρ23cos3ρ,2√23). | (3.23) |
Figure 4 shows the graph of the curve ω and the C∗-partner curve ψ∗∗.
The fundamental theorem of regular curves is that there is a unit speed curve with curvature and torsion κ and τ, respectively, where κ>0 and τ are differentiable functions. In addition, the principal normal and binormal vector fields of the curve are not continuous at the zero points of the curvature. In this case, at a point where the curvature of a generally considered space curve disappears, the Frenet frame for this curve causes uncertainty. To solve this problem, Hord [13] and Sasai [22,23] found a simple but useful way and introduced a new orthogonal frame i.e., a the modified frame. On the other hand, another frame, as an alternative to the Frenet frame i.e., an adapted frame, was defined by Uzunoǧlu et. al [25] with the help of the Darbuox vector and Frenet normal vector of a regular curve. However, if the equation f=0 holds in the adapted frame {N,C,W} defined by them, then the vector field C cannot be defined. Therefore, the frame {N,C,W} cannot be constructed for all points on the curve. In that case, the alternative frame given by them [25] needs to be modified. So, the purpose of our study was to define a new modified adapted frame that removes all singular points in a curve. Moreover, with this new frame we have defined in this study, we believe that the curve theory, which will be studied with this frame in future studies, will be open to many innovations by bringing a different perspective to the curve theory in differential geometry. In addition, with this new frame, curve theory will gain a different perspective in many fields such as engineering and physics. Our theoretical and practical studies on this new frame continue.
The authors declare that they have no conflict of interests.
[1] |
Khomich VG, Boriskina NG (2013) The deep geodynamics of Southeast Russia and the setting of platinum-bearing basite-hyperbasite massifs. J Volcanolog Seismol 7: 328–337. http://dx.doi.org/10.1134/S0742046313040040 doi: 10.1134/S0742046313040040
![]() |
[2] |
Khomich VG, Boriskina NG (2014) Deep geodynamics and uranium giants of Southeastern Russia. Dokl Earth Sci 458: 1226–1229. http://dx.doi.org/10.1134/S1028334X14100249 doi: 10.1134/S1028334X14100249
![]() |
[3] |
Khomich VG, Boriskina NG, Santosh M (2018) Super large mineral deposits and deep mantle dynamics: The scenario from Southeast Trans-Baikal region, Russia. Geol J 53: 412–423. https://doi.org/10.1002/gj.2908 doi: 10.1002/gj.2908
![]() |
[4] |
Khomich VG, Boriskina NG, Santosh M (2014) A geodynamic perspective of world-class gold deposits in East Asia. Gondwana Res 26: 816–833. https://doi.org/10.1016/j.gr.2014.05.007 doi: 10.1016/j.gr.2014.05.007
![]() |
[5] |
Khomich VG, Boriskina NG, Santosh M (2015) Geodynamics of Late Mesozoic PGE, Au, and U mineralization in the Aldan shield, North Asian Craton. Ore Geol Rev 68: 30–42. http://dx.doi.org/10.1016/j.oregeorev.2015.01.007 doi: 10.1016/j.oregeorev.2015.01.007
![]() |
[6] |
Khomich VG, Boriskina NG (2019) Paleovolcanic necks and extrusions: Indicators of large uranium orebelts in the territories joining Russia, Mongolia, and China. J Volcanol Geotherm Res 383: 88–102. https://doi.org/10.1016/j.jvolgeores.2018.05.004 doi: 10.1016/j.jvolgeores.2018.05.004
![]() |
[7] | Seminskiy ZV (2021) Clusters of mineral deposits of the Southern East Siberia and prospects for their development: an overview of the problem. Geodyn Tectonophysics 12: 754–768. |
[8] | Zonenshain LP, Kuzmin MI, Natapov LM (1990) Tectonics of lithosphere plates of the USSR territory: in 2 books, Moscow: Nedra. Available from: https://www.geokniga.org/books/6515 |
[9] | Parfenov LM, Berzin NA, Khanchuk AI, et al. (2003) Model of formation of orogenic belts of Central and North-East Asia. Tikhookeanskaya Geologiya 22: 7–41. |
[10] | Shatkov GA, Volsky AS (2004) Tectonics, deep structure, and minerageny of the Amur river region and neighboring areas. St. Petersburg: VSEGEI. |
[11] | Gordienko IV (2014) Metallogeny of various geodynamic settings of the Mongolia-Transbaikalia region. Geol Miner Resour Sib S3–1: 7–13. |
[12] |
Yarmolyuk VV, Kudryashov EA, Kozlovsky AM, et al. (2011) Late Cenozoic volcanic province in Central and East Asia. Petrology 19: 327–347. http://dx.doi.org/10.1134/S0869591111040072 doi: 10.1134/S0869591111040072
![]() |
[13] | Didenko AI, Malyshev YuF, Saksin BG (2010) Deep structure and metallogeny of East Asia, Vladivostok: Dalnauka. Available from: https://search.rsl.ru/ru/record/01004906622?ysclid = m2mxy27n73631732383. |
[14] |
Didenko AN, Nosyrev MY, Gil'manova GZ (2022) A gravity-derived Moho model for the Sikhote Alin orogenic belt. Pure Appl Geophys 179: 3967–3988. https://doi.org/10.1007/s00024-021-02842-8 doi: 10.1007/s00024-021-02842-8
![]() |
[15] | Grachev AF (1996) The main problems of the latest tectonics and geodynamics of Northern Eurasia. Izv Phys Solid Earth 12: 5–36. |
[16] |
Larin AM (2014) Ulkan-Dzhugdzhur ore-bearing anorthosite-rapakivi granite-peralkaline granite association, Siberian Craton: Age, tectonic setting, sources, and metallogeny. Geol Ore Deposits 56: 257–280. https://doi.org/10.1134/S1075701514040047 doi: 10.1134/S1075701514040047
![]() |
[17] |
Larin AM, Kotov AB, Salnikova EB, et al. (2023) Age and tectonic setting of the kopri-type granitoids at the Junction Zone of the Dzhugdzhur-Stanovoi and Western Stanovoi superterranes of the Central Asian fold belt. Dokl Earth Sc 509: 111–117. https://doi.org/10.1134/S1028334X22601821 doi: 10.1134/S1028334X22601821
![]() |
[18] |
Sorokin AA, Sorokin AP, Ponomorchuk VA, et al. (2009) Late Mesozoic volcanism of the eastern part of the Argun superterrane (Far East): Geochemistry and 40Ar/39Ar geochronology. Stratigr Geol Correl 17: 645–658. https://doi.org/10.1134/S0869593809060069 doi: 10.1134/S0869593809060069
![]() |
[19] |
Li XC, Fan HR, Santosh M, et al. (2012) An evolving magma chamber within extending lithosphere: An integrated geochemical, isotopic and zircon U–Pb geochronological study of the Gushan granite, eastern North China Craton. J Asian Earth Sci 50: 27–43. https://doi.org/10.1016/j.jseaes.2012.01.016 doi: 10.1016/j.jseaes.2012.01.016
![]() |
[20] |
Ouyang HG, Mao JW, Zhou ZH, et al. (2015) Late Mesozoic metallogeny and intracontinental magmatism, southern Great Xing'an Range, northeastern China. Gondwana Res 27: 1153–1172. http://dx.doi.org/10.1016/j.gr.2014.08.010 doi: 10.1016/j.gr.2014.08.010
![]() |
[21] |
Zhou JB, Wilde SA, (2013) The crustal accretion history and tectonic evolution of the NE China segment of the Central Asian Orogenic Belt. Gondwana Res 23: 1365–1377. http://dx.doi.org/10.1016/j.gr.2012.05.012 doi: 10.1016/j.gr.2012.05.012
![]() |
[22] |
Zhao R, Wang QF, Liu XF, et al. (2016) Architecture of the Sulu crustal suture between the North China Craton and Yangtze Craton: Constraints from Mesozoic granitoids. Lithos 266–267: 348–361. http://dx.doi.org/10.1016/j.lithos.2016.10.018 doi: 10.1016/j.lithos.2016.10.018
![]() |
[23] |
Larin AM, Kotov AB, Salnikova EB, et al. (2021) Age and tectonic setting of granitoids of the Uda Complex of the Dzhugdzhur block of the Stanovoy suture: new data on the formation of giant magmatic belts in Eastern Asia. Dokl Earth Sci 498: 362–366 https://doi.org/10.1134/S1028334X21050081 doi: 10.1134/S1028334X21050081
![]() |
[24] |
Fan WM, Guo F, Wang YJ, et al. (2003) Late Mesozoic calc-alkaline volcanism of post-orogenic extension in the northern Da Hinggan Mountains, northeastern China. J Volcanol Geotherm Res 121: 115–135. https://doi.org/10.1016/S0377-0273(02)00415-8 doi: 10.1016/S0377-0273(02)00415-8
![]() |
[25] |
Fan HR, Hu FF, Yang JH, et al. (2007) Fluid evolution and large-scale gold metallogeny during Mesozoic tectonic transition in the Jiaodong Peninsula, eastern China. Geol Soc London Spec Publ 280: 303–316. https://doi.org/10.1144/SP280.16 doi: 10.1144/SP280.16
![]() |
[26] |
Gordienko IV (2001) Geodynamic evolution of the Central-Asian and Mongol-Okhotsk fold belts and formation of the endogenic deposits. Geosci J 5: 233–241. https://doi.org/10.1007/BF02910306 doi: 10.1007/BF02910306
![]() |
[27] |
Sun JG, Han SJ, Zhang Y, et al. (2013) Diagenesis and metallogenetic mechanisms of the Tuanjiegou gold deposit from the Lesser Xing'an Range, NE China: Zircon U-Pb geochronology and Lu-Hf isotopic constraints. J Asian Earth Sci 62: 373–388. https://doi.org/10.1016/j.jseaes.2012.10.021 doi: 10.1016/j.jseaes.2012.10.021
![]() |
[28] |
Ren YS, Chen C, Zou XT, et al. (2016) The age, geological setting, and types of gold deposits in the Yanbian and adjacent areas, NE China. Ore Geol Rev 73: 284–297. https://doi.org/10.1016/j.oregeorev.2015.03.013 doi: 10.1016/j.oregeorev.2015.03.013
![]() |
[29] |
Chen YJ, Zhang C, Wang P, et al. (2017) The Mo deposits of Northeast China: A powerful indicator of tectonic settings and associated evolutionary trends. Ore Geol Rev 81: 602–640. http://dx.doi.org/10.1016/j.oregeorev.2016.04.017 doi: 10.1016/j.oregeorev.2016.04.017
![]() |
[30] |
Li Q, Santosh M, Li SR, et al. (2015) Petrology, geochemistry and zircon U–Pb and Lu–Hf isotopes of the Cretaceous dykes in the central North China Craton: Implications for magma genesis and gold metallogeny. Ore Geol Rev 67: 57–77. https://doi.org/10.1016/j.oregeorev.2014.11.015 doi: 10.1016/j.oregeorev.2014.11.015
![]() |
[31] |
Andreeva OV, Petrov VA, Poluektov VV (2020) Mesozoic acid magmatites of Southeastern Transbaikalia: petrogeochemistry and relationship with metasomatism and ore formation. Geol Ore Deposits 62: 69–96. https://doi.org/10.1134/S1075701520010018 doi: 10.1134/S1075701520010018
![]() |
[32] |
Zhao D, Pirajno F, Dobretsov NL, et al. (2010) Mantle structure and dynamics under East Russia and adjacent regions. Russ Geol Geophys 51: 925–938. https://doi.org/10.1016/j.rgg.2010.08.003 doi: 10.1016/j.rgg.2010.08.003
![]() |
[33] |
Deng J, Wang Q (2016) Gold mineralization in China: Metallogenic provinces, deposit types and tectonic framework. Gondwana Res 36: 219–274. https://doi.org/10.1016/j.gr.2015.10.003 doi: 10.1016/j.gr.2015.10.003
![]() |
[34] |
Zonenshain LP, Savostin LA (1981) Geodynamics of the Baikal Rift System and Plate Tectonics of Asia. Tectonophysics 76: 1–45. https://doi.org/10.1016/0040-1951(81)90251-1 doi: 10.1016/0040-1951(81)90251-1
![]() |
[35] |
Kepezhinskas PK, Kepezhinskas NP, Berdnikov NV, et al. (2020) Native metals and intermetallic compounds in subduction-related ultramafic rocks from the Stanovoy mobile belt (Russian Far East): implications for redox heterogeneity in subduction zones. Ore Geol Rev 127: 103800. https://doi.org/10.1016/j.oregeorev.2020.103800 doi: 10.1016/j.oregeorev.2020.103800
![]() |
[36] |
Kepezhinskas P, Berdnikov N, Kepezhinskas N, et al. (2022) Adakites, high-Nb basalts and copper-gold depositions in magmatic arcs and collictional orogens: an overview. Geosciences 12: 29. https://doi.org/10.3390/geosciences12010029 doi: 10.3390/geosciences12010029
![]() |
[37] |
Malyshev YF, Podgornyi VY, Shevchenko BF, et al. (2007) Deep structure of the Amur lithospheric plate border zone. Russ J Pac Geol 1: 107–119. https://doi.org/10.1134/S1819714007020017 doi: 10.1134/S1819714007020017
![]() |
[38] |
Maruyama S, Santosh M, Zhao D (2007) Superplume, supercontinent, and post-perovskite: Mantle dynamics and antiplate tectonics on the Core-Mantle Boundary. Gondwana Res 11: 7–37. http://dx.doi.org/10.1016/j.gr.2006.06.003 doi: 10.1016/j.gr.2006.06.003
![]() |
[39] | Li C, van der Hilst RD (2010) Structure of the upper mantle and transition zone beneath Southeast Asia from traveltime tomography. J Geophys Res Solid Earth 115: B07308. https://doi.org/10.1029/2009JB006882 |
[40] |
Li J, Wang X, Wang X, et al. (2013) P and SH velocity structure in the upper mantle beneath Northeast China: Evidence for a stagnant slab in hydrous mantle transition zone. Earth Planet Sci Lett 367: 71–81. https://doi.org/10.1016/j.epsl.2013.02.026 doi: 10.1016/j.epsl.2013.02.026
![]() |
[41] | Zhao D, Tian Y (2013) Changbai intraplate volcanism and deep earthquakes in East Asia: a possible link? Geophys J Int 195: 706–724. |
[42] |
Jiang G, Zhang G, Zhao D, et al. (2015) Mantle dynamics and cretaceous magmatism in east-central China: Insight from teleseismic tomograms. Tectonophysics 664: 256–268. http://dx.doi.org/10.1016/j.tecto.2015.09.019 doi: 10.1016/j.tecto.2015.09.019
![]() |
[43] |
Khomich VG, Boriskina NG, Kasatkin SA (2019) Geology, magmatism, metallogeny, and geodynamics of the South Kuril Islands. Ore Geol Rev 105: 151–162. https://doi.org/10.1016/j.oregeorev.2018.12.015 doi: 10.1016/j.oregeorev.2018.12.015
![]() |
[44] |
Khomich VG, Boriskina NG, Santosh M (2016) Geodynamic framework of large unique uranium orebelts in Southeast Russia and East Mongolia. J Asian Earth Sci 119: 145–166. http://dx.doi.org/10.1016/j.jseaes.2016.01.018 doi: 10.1016/j.jseaes.2016.01.018
![]() |
[45] |
Khutorskoi MD, Polyak BG (2017) Special features of heat flow in transform faults of the North Atlantic and Southeast Pacific. Geotectonics 51: 152–162. https://doi.org/10.1134/S0016852117010022 doi: 10.1134/S0016852117010022
![]() |
[46] |
Goldfarb RJ, Santosh M (2014) The dilemma of the Jiaodong gold deposits: are they unique? Geosci Front 5: 139–153. http://dx.doi.org/10.1016/j.gsf.2013.11.001 doi: 10.1016/j.gsf.2013.11.001
![]() |
[47] |
Mao XC, Ren J, Liu ZK, et al. (2019) Three-dimensional prospectivity modeling of the Jiaojia-type gold deposit, Jiaodong Peninsula, Eastern China: A case study of the Dayingezhuang deposit. J Geochem Explor 203: 27–44. https://doi.org/10.1016/j.gexplo.2019.04.002 doi: 10.1016/j.gexplo.2019.04.002
![]() |
[48] |
Chen Y, Guo G, Li X (1998) Metallogenic geodynamic background of Mesozoic gold deposits in granite-greenstone terrains of North China Craton. Sci China Ser D-Earth Sci 41: 113–120. https://doi.org/10.1007/BF02932429 doi: 10.1007/BF02932429
![]() |
[49] | Zorin YA, Turutanov EK, Kozhevnikov VM, et al. (2006) On the nature of Cenozoic upper mantle plumes in Eastern Siberia (Russia) and Central Mongolia. Russ Geol Geophys 47: 1056–1070. |
[50] |
Khomich VG, Boriskina NG (2009) Relationship between the gold bearing areas and gradient sones of the gravity field of southeastern regions of Russia. Dokl Earth Sc 428: 1100–1104. http://dx.doi.org/10.1134/S1028334X09070137 doi: 10.1134/S1028334X09070137
![]() |
[51] | Laverov NP, Velichkin VI, Vlasov BP, et al. (2012) Uranium and molybdenum-uranium deposits in the areas of continental intracrustal magmatism: geology, geodynamic and physico-chemical conditions of formation. Moscow: IFSN RAN IGEM RAN. |
[52] |
Berzina AP, Berzina AN, Gimon VO, et al. (2015) The Zhireken porphyry Mo ore-magmatic system (Eastern Transbaikalia): U-Pb age, sources, and geodynamic setting. Russ Geol Geophys 56: 446–465. https://doi.org/10.1016/j.rgg.2015.02.006 doi: 10.1016/j.rgg.2015.02.006
![]() |
[53] | Mashkovtsev GA, Korotkov VV, Rudnev VV (2020) Ore-bearing of the Mesozoic tectonic-magmatic activation area of the Eastern Siberia and the Far East. Prospect Prot Miner Resour 11: 5–7. |
[54] | Vetluzhskikh VG, Kazansky VI, Kochetkov AY, et al. (2002) Central Aldan gold deposits. Geol Ore Deposits 44: 405–434. Available from: https://www.pleiades.online/cgi-perl/search.pl?type = abstract & name = geolore & number = 6 & year = 2 & page = 405. |
[55] | Shatov VV, Molchanov AV, Shatova NV, et al. (2012) Petrography, geochemistry and isotopic (U-Pb and Rb-Sr) dating of alkaline magmatic rocks of the Ryabinovy massif (South Yakutia). Reg Geol Metallog 51: 62–78. |
[56] |
Chernyshev IV, Prokof'ev VY, Bortnikov NS, et al. (2014) Age of granodiorite porphyry and beresite from the Darasun gold field, Eastern Transbaikal region, Russia. Geol Ore Deposits 56: 1–14. https://doi.org/10.1134/S1075701514010036 doi: 10.1134/S1075701514010036
![]() |
[57] | Andreeva OV, Golovin VA, Kozlova PS, et al. (1996) Evolution of mesozoic magmatism and ore-forming metasomatic processes in the Southeastern Transbaikal region (Russia). Geol Ore Deposits 38: 101–113. |
[58] |
Stupak FM, Kudryashova EA, Lebedev VA, et al. (2018) The structure, composition, and conditions of generation for the early cretaceous Mongolia—East-Transbaikalia volcanic belt: the Durulgui–Torei area (Southern Transbaikalia, Russia). J Volcanolog Seismol 12: 34–46. https://doi.org/10.1134/S0742046318010074 doi: 10.1134/S0742046318010074
![]() |
[59] |
Ponomarchuk AV, Prokopyev IR, Svetlitskaya TV, et al. (2019) 40Ar/39Ar geochronology of alkaline rocks of the Inagli massif (Aldan Shield, Southern Yakutia). Russ Geol Geophys 60: 33–44. https://doi.org/10.15372/RGG2019003 doi: 10.15372/RGG2019003
![]() |
[60] |
Gaskov IV, Borisenko AS, Borisenko ID, et al. (2023) Chronology of alkaline magmatism and gold mineralization in the Central Aldan ore district (Southern Yakutia). Russ Geol Geophys 64: 175–191. https://doi.org/10.2113/RGG20214427 doi: 10.2113/RGG20214427
![]() |
[61] |
Shukolyukov YA, Yakubovich OV, Mochalov AG, et al. (2012) New geochronometer for the direct isotopic dating of native platinum minerals (190Pt-4He method). Petrology 20: 491–505. http://dx.doi.org/10.1134/S0869591112060033 doi: 10.1134/S0869591112060033
![]() |
[62] |
Ronkin YL, Efimov AA, Lepikhina GA, et al. (2013) U-Pb dating of the baddeleytte-zircon system from Pt-bearing dunite of the Konder massif, Aldan Shield: New data. Dokl Earth Sc 450: 607–612. https://doi.org/10.1134/S1028334X13060135 doi: 10.1134/S1028334X13060135
![]() |
[63] |
Mochalov AG, Yakubovich OV, Bortnikov NS (2016) 190Pt–4He age of PGE ores in the alkaline–ultramafic Konder massif (Khabarovsk district, Russia). Dokl Earth Sc 469: 846–850. https://doi.org/10.1134/S1028334X16080134 doi: 10.1134/S1028334X16080134
![]() |
[64] |
Mochalov AG, Yakubovich OV, Bortnikov NS (2022) 190Pt–4He dating of placer-forming minerals of platinum from the Chad alkaline-ultramafic massif: new evidence of the polycyclic nature of ore formation. Dokl Earth Sc 504: 240–247. https://doi.org/10.1134/S1028334X22050105 doi: 10.1134/S1028334X22050105
![]() |
[65] | Kazansky VI (2004) The unique Central Aldan gold-uranium ore district (Russia). Geol Ore Deposits 46: 167–181. |
[66] | Mashkovtsev GA, Konstantinov AK, Miguta AK, et al. (2010) Uranium of Russian entrails, Moscow: VIMS. Available from: https://elib.biblioatom.ru/text/uran-rossiyskih-nedr_2010/p0/. |
[67] | Afanasyev GV, Mironov YB, Pinsky EM (2018) Role of combined paleohydrogeological systems in forming uranium mineralization in volcano-tectonic depressions of the Central Asian mobile belt. Reg Geol Metallog 3: 77–87. |
[68] | Rui GZ (2010) Discussion on the geological characteristics and origin of the 460 large uranium-molybdenum deposit. World Nucl Geol 27: 149–154. |
[69] | Wu JH, Ding H, Niu ZL, et al. (2015) SHRIMP zircon U-Pb dating of country rock in Zhangmajing U-Mo deposit in Guyuan, Hebei Province, and its geological significance. Mineral Deposit 4: 757–768. |
[70] |
Wang JQ, Liu GJ, Dong S, et al. (2023) The "trinity" prospecting prediction geological model of the Zhangmajing uranium molybdenum deposit in Guyuan County, Hebei Province. Geol Bull China 42: 931–940. http://dx.doi.org/10.12097/j.issn.1671-2552.2023.06.006 doi: 10.12097/j.issn.1671-2552.2023.06.006
![]() |
[71] | Rybalov BL (2000) Evolutionary rows of Late Mesozoic ore deposits of the Eastern Transbaikal region (Russia). Geol Ore Deposits 42: 340–350. |
[72] |
Berzina AP, Berzina AN, Gimon VO (2014) Geochemical and Sr–Pb–Nd isotopic characteristics of the Shakhtama porphyry Mo-Cu system (Eastern Transbaikalia, Russia). J Asian Earth Sci 79: 655–665. http://dx.doi.org/10.1016/j.jseaes.2013.07.028 doi: 10.1016/j.jseaes.2013.07.028
![]() |
[73] |
Berzina AP, Berzina AN, Gimon VO (2016) Paleozoic-Mesozoic porphyry Cu (Mo) and Mo (Cu) deposits within the southern margin of the Siberian Craton: geochemistry, geochronology, and petrogenesis (a Review). Minerals 6: 125. http://dx.doi.org/10.3390/min6040125 doi: 10.3390/min6040125
![]() |
[74] | Pushkarev YD, Kostoyanov AI, Orlova MP, et al. (2002) Features of Rb-Sr, Sm-Nd, Pb-Pb, Re-Os and K-Ar isotope systems in the Konder massif: mantle substrate enriched in platinum group metals. Reg Geol Metallog 16: 80–91. |
[75] |
Simonov VA, Prikhodko VS, Kovyazin SV (2011) Genesis of platiniferous massifs in the Southeastern Siberian platform. Petrology 19: 549–567. http://dx.doi.org/10.1134/S0869591111050043 doi: 10.1134/S0869591111050043
![]() |
[76] |
Shatkov GA, Antonov AV, Butakov PM, et al. (2014) Uranyl molybdates in fluorites and uranium-carbonate ores from the Streltsovskoe ore field of the Argun deposit. Dokl Earth Sc 456: 764–768. https://doi.org/10.1134/S1028334X14060373 doi: 10.1134/S1028334X14060373
![]() |
[77] | Konstantinov MM, Politov VK, Novikov VP, et al. (2002) Geological structure of gold districts of volcano-plutonic belts in Eastern Russia. Geol Ore Deposits 44: 252–266. |
[78] |
Dobretsov NL, Koulakov I, Kukarina EV, et al. (2015) An integrate model of subduction: contributions from geology, experimental petrology, and seismic tomography. Russ Geol Geophys 56: 13–38. https://doi.org/10.1016/j.rgg.2015.01.002 doi: 10.1016/j.rgg.2015.01.002
![]() |
[79] | Pechenkin IG (2016) Relationship between uranium metallogeny and geodynamic processes in the marginal parts of Eurasia. Ores Metals 2: 5–17. |
1. | Ana Savić, Kemal Eren, Soley Ersoy, Vladimir Baltić, Alternative View of Inextensible Flows of Curves and Ruled Surfaces via Alternative Frame, 2024, 12, 2227-7390, 2015, 10.3390/math12132015 | |
2. | ESRA PARLAK, TEVFİK ŞAHİN, GEOMETRIC PERSPECTIVE OF BERRY'S PHASE ACCORDING TO ALTERNATIVE ORTHOGONAL MODIFIED FRAME, 2025, 25, 2068-3049, 11, 10.46939/J.Sci.Arts-25.1-a02 |