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Journal articles on the topic 'Minkowski'

Author: Grafiati

Published: 4 June 2021

Last updated: 11 March 2023

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1

Alrimawi, Fadi, and Fuad A. Abushaheen. "Minkowski–Clarkson’s type inequalities." Analysis 41, no. 3 (May 19, 2021): 163–68. http://dx.doi.org/10.1515/anly-2019-0020.

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Abstract In this paper, we give some Minkowski–Clarkson’s type inequalities related to two finite sequences of real nonnegative numbers. In particular, we prove two inequalities which in some sense can be regarded as inverse Minkowski’s inequalities concerning the cases p ≥ 2 {p\geq 2} and 0 < p ≤ 1 {0<p\leq 1} . Moreover, for 1 < p < 2 {1<p<2} we prove another Minkowski–Clarkson’s type inequality.

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2

Ali, Farhad, Muhammad Asif Jan, Wali Khan Mashwani, Rashida Adeeb Khanum, and Hidayat Ullah Khan. "The Physical Significance of Time Conformal Minkowski Spacetime." April 2020 39, no. 2 (April 1, 2020): 365–70. http://dx.doi.org/10.22581/muet1982.2002.12.

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The Minkowsiki spacetime is flat and there is no source of gravitation. The time conformal factor is adding some cuvature to this spacetime which introduces some source of gravitation to the spacetime. For the Minkowski spacetime the Einstein Field equation tells nothing, because all the components of the Ricci curvature tensor are zero, but for the time conformal Minkowski spacetime some of them are non zero. Calculating the components of the Ricci tensor and using the Einstein field equations, expressions for the cosmological constant are cacultaed. These expressions give some information for the cosmological constant. Generally, the Noether symmetry generator corresponding to the energy content in the spacetime disapeares by introducing the time conformal factor, but our investigations in this paper reveals that it appears somewhere with some re-scale factor. The appearance of the time like isometry along with some re-scaling factor will rescale the energy content in the corresponding particular time conformal Minkowski spacetime. A time conformal factor of the form () is introduced in the Minkowski spacetime for the invistigation of the cosmological constant. The Noether symmetry equation is used for the Lagrangian of general time conformal Minkowski spacetime to find all those particular Minkowski spacetimes that admit the time conformal factor. Besides the Noether symmetries the cosmology constant is calculated in the corresponding spacetimes.

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3

Zhao, Chang-Jian, and Wing Sum Cheung. "On reverse Hölder and Minkowski inequalities." Mathematica Slovaca 70, no. 4 (August 26, 2020): 821–28. http://dx.doi.org/10.1515/ms-2017-0395.

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AbstractIn the paper, we give new improvements of the reverse Hölder and Minkowski integral inequalities. These new results in special case yield the Pólya-Szegö’s inequality and reverse Minkowski’s inequality, respectively.

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4

Martini, Horst, and Margarita Spirova. "Covering Discs in Minkowski Planes." Canadian Mathematical Bulletin 52, no. 3 (September 1, 2009): 424–34. http://dx.doi.org/10.4153/cmb-2009-046-2.

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AbstractWe investigate the following version of the circle covering problem in strictly convex (normed or) Minkowski planes: to cover a circle of largest possible diameter by k unit circles. In particular, we study the cases k = 3, k = 4, and k = 7. For k = 3 and k = 4, the diameters under consideration are described in terms of side-lengths and circumradii of certain inscribed regular triangles or quadrangles. This yields also simple explanations of geometric meanings that the corresponding homothety ratios have. It turns out that basic notions from Minkowski geometry play an essential role in our proofs, namely Minkowskian bisectors, d-segments, and the monotonicity lemma.

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5

Kjeldsen, Tinne Hoff. "Egg-Forms and Measure-Bodies: Different Mathematical Practices in the Early History of the Modern Theory of Convexity." Science in Context 22, no. 1 (March 2009): 85–113. http://dx.doi.org/10.1017/s0269889708002081.

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ArgumentTwo simultaneous episodes in late nineteenth-century mathematical research, one by Karl Hermann Brunn (1862–1939) and another by Hermann Minkowski (1864–1909), have been described as the origin of the theory of convex bodies. This article aims to understand and explain (1) how and why the concept of such bodies emerged in these two trajectories of mathematical research; and (2) why Minkowski's – and not Brunn's – strand of thought led to the development of a theory of convexity. Concrete pieces of Brunn's and Minkowski's mathematical work in the two episodes will, from the perspective of the above questions, be presented and analyzed with the use of the methodological framework of epistemic objects, techniques, and configurations as adapted from Hans-Jörg Rheinberger's work on empirical sciences to the historiography of mathematics by Moritz Epple. Based on detailed descriptions and a comparison of the objects and techniques that Brunn and Minkowski studied and used in these pieces it will be concluded that Brunn and Minkowski worked in different epistemic configurations, and it will be argued that this had a significant influence on the mathematics they developed for those bodies, which can provide answers to the two research questions listed above.

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6

Fillastre, François. "Christoffel and Minkowski problems in Minkowski space." Séminaire de théorie spectrale et géométrie 32 (2015): 97–114. http://dx.doi.org/10.5802/tsg.305.

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7

Xue, F., and C. Zong. "Minkowski bisectors, Minkowski cells and lattice coverings." Geometriae Dedicata 188, no. 1 (November 16, 2016): 123–39. http://dx.doi.org/10.1007/s10711-016-0208-7.

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8

Bonsante, Francesco, and François Fillastre. "The equivariant Minkowski problem in Minkowski space." Annales de l’institut Fourier 67, no. 3 (2017): 1035–113. http://dx.doi.org/10.5802/aif.3105.

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9

Speer, C. P. "Alexandre Minkowski." Neonatology 86, no. 3 (2004): 183. http://dx.doi.org/10.1159/000079519.

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10

Ludwig, Monika. "Minkowski valuations." Transactions of the American Mathematical Society 357, no. 10 (October 28, 2004): 4191–213. http://dx.doi.org/10.1090/s0002-9947-04-03666-9.

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11

Roux, Stéphane Le, Arno Pauly, and Jean-François Raskin. "Minkowski Games." ACM Transactions on Computational Logic 19, no. 3 (September 18, 2018): 1–29. http://dx.doi.org/10.1145/3230741.

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12

Dorrek, Felix. "Minkowski Endomorphisms." Geometric and Functional Analysis 27, no. 3 (March 30, 2017): 466–88. http://dx.doi.org/10.1007/s00039-017-0405-z.

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13

Carbó-Dorca, Ramon, and Tanmoy Chakraborty. "Extended Minkowski spaces, zero norms, and Minkowski hypersurfaces." Journal of Mathematical Chemistry 59, no. 8 (July 10, 2021): 1875–79. http://dx.doi.org/10.1007/s10910-021-01266-y.

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14

Chen, Feixiang, and Gangsong Leng. "Orlicz–Brunn–Minkowski inequalities for Blaschke–Minkowski homomorphisms." Geometriae Dedicata 187, no. 1 (September 19, 2016): 137–49. http://dx.doi.org/10.1007/s10711-016-0193-x.

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15

Freidel, Laurent, Jerzy Kowalski-Glikman, and Sebastian Nowak. "From noncommutative κ-Minkowski to Minkowski space–time." Physics Letters B 648, no. 1 (April 2007): 70–75. http://dx.doi.org/10.1016/j.physletb.2007.02.056.

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16

Bahn, Hyoungsick, and Paul Ehrlich. "A Brunn-Minkowski Type Theorem on the Minkowski Spacetime." Canadian Journal of Mathematics 51, no. 3 (June 1, 1999): 449–69. http://dx.doi.org/10.4153/cjm-1999-020-0.

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AbstractIn this article, we derive a Brunn-Minkowski type theorem for sets bearing some relation to the causal structure on the Minkowski spacetime . We also present an isoperimetric inequality in the Minkowski spacetime as a consequence of this Brunn-Minkowski type theorem.

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17

Feng, Yibin, Weidong Wang, and Jun Yuan. "Inequalities of quermassintegrals about mixed Blaschke Minkowski homomorphisms." Tamkang Journal of Mathematics 46, no. 3 (September 30, 2015): 217–27. http://dx.doi.org/10.5556/j.tkjm.46.2015.1689.

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In this article, we establish some inequalities of quermassintegrals associated with mixed Blaschke Minkowski homomorphisms. In particular, Minkowski and Brunn-Minkowski type inequalities for quermassintegrals differences of mixed Blaschke Minkowski homomorphisms are established. In addition, we also give an isolated form of Brunn-Minkowski type inequality of quermassintegrals established by Schuster.

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18

MENDIVIL, F., and J. C. SAUNDERS. "ON MINKOWSKI MEASURABILITY." Fractals 19, no. 04 (December 2011): 455–67. http://dx.doi.org/10.1142/s0218348x11005506.

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Two "pathological" properties of Minkowski content are that countable sets can have positive content (unlike Hausdorff measures) and the property of a set being Minkowski measurable is quite rare. In this paper, we explore both of these issues. In particular, for each d ∈ (0,2) we give an explicit construction of a countable Minkowski measurable subset of ℝ2 of Minkowski dimension d and arbitrary positive Minkowski content. We also indicate how this construction can be extended to ℝn, to construct a countable subset with arbitrary positive Minkowski content of any dimension in (0, n). Furthermore, we give an example of a strictly increasing C1 function which takes a Minkowski measurable subset of [0,1] onto a set which is not Minkowski measurable but of the same dimension.

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19

Zhao, Chang-Jian, and Mihály Bencze. "Lp -dual three mixed quermassintegrals." Analele Universitatii "Ovidius" Constanta - Seria Matematica 29, no. 2 (June 1, 2021): 265–74. http://dx.doi.org/10.2478/auom-2021-0030.

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Abstract In the paper, the concept of Lp -dual three-mixed quermassintegrals is introduced. The formula for the Lp -dual three-mixed quermassintegrals with respect to the p-radial addition is proved. Inequalities of Lp -Minkowski, and Brunn-Minkowski type for the Lp -dual three-mixed quermassintegrals are established. The new Lp -Minkowski inequality is obtained that generalize a family of Minkowski type inequalities. The Lp -Brunn-Minkowski inequality is used to obtain a series of Brunn-Minkowski type inequalities.

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20

Zhao, Chang-Jian. "Orlicz dual affine quermassintegrals." Forum Mathematicum 30, no. 4 (July 1, 2018): 929–45. http://dx.doi.org/10.1515/forum-2017-0174.

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Abstract In the paper, our main aim is to generalize the dual affine quermassintegrals to the Orlicz space. Under the framework of Orlicz dual Brunn–Minkowski theory, we introduce a new affine geometric quantity by calculating the first-order variation of the dual affine quermassintegrals, and call it the Orlicz dual affine quermassintegral. The fundamental notions and conclusions of the dual affine quermassintegrals and the Minkoswki and Brunn–Minkowski inequalities for them are extended to an Orlicz setting, and the related concepts and inequalities of Orlicz dual mixed volumes are also included in our conclusions. The new Orlicz–Minkowski and Orlicz–Brunn–Minkowski inequalities in a special case yield the Orlicz dual Minkowski inequality and Orlicz dual Brunn–Minkowski inequality, which also imply the {L_{p}} -dual Minkowski inequality and Brunn–Minkowski inequality for the dual affine quermassintegrals.

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21

Zekraoui, Hanifa, Zeyad Al-Zhour, and Cenap Özel. "Some New Algebraic and Topological Properties of the Minkowski Inverse in the Minkowski Space." Scientific World Journal 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/765732.

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We introduce some new algebraic and topological properties of the Minkowski inverseA⊕of an arbitrary matrixA∈Mm,n(including singular and rectangular) in a Minkowski spaceμ. Furthermore, we show that the Minkowski inverseA⊕in a Minkowski space and the Moore-Penrose inverseA+in a Hilbert space are different in many properties such as the existence, continuity, norm, and SVD. New conditions of the Minkowski inverse are also given. These conditions are related to the existence, continuity, and reverse order law. Finally, a new representation of the Minkowski inverseA⊕is also derived.

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22

Benaissa, Bouharket. "More on reverses of Minkowski’s inequalities and Hardy’s integral inequalities." Asian-European Journal of Mathematics 13, no. 03 (December 13, 2018): 2050064. http://dx.doi.org/10.1142/s1793557120500643.

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In 2012, Sulaiman [Reverses of Minkowski’s, Hölder’s, and Hardy’s integral inequalities, Int. J. Mod. Math. Sci. 1(1) (2012) 14–24] proved integral inequalities concerning reverses of Minkowski’s and Hardy’s inequalities. In 2013, Banyat Sroysang obtained a generalization of the reverse Minkowski’s inequality [More on reverses of Minkowski’s integral inequality, Math. Aeterna 3(7) (2013) 597–600] and the reverse Hardy’s integral inequality [A generalization of some integral inequalities similar to Hardy’s inequality, Math. Aeterna 3(7) (2013) 593–596]. In this article, two results are given. First one is further improvement of the reverse Minkowski inequality and second is further generalization of the integral Hardy inequality.

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23

Altunkaya, Bülent. "Mappings that preserve helices in the n-dimensional Minkowski spaces." International Journal of Geometric Methods in Modern Physics 17, no. 10 (June 23, 2020): 2050107. http://dx.doi.org/10.1142/s0219887820501078.

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We introduce two types of mappings that preserve nonnull helices in Minkowski spaces. The first type constructs helices in the [Formula: see text]-dimensional Minkowski space from helices in the same Minkowski space. The second type constructs helices in the [Formula: see text]-dimensional Minkowski space from helices in the [Formula: see text]-dimensional Minkowski space. Furthermore, we study invariants of these mappings and present examples.

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24

Gardner, R. J., and S. Vassallo. "The Brunn–Minkowski Inequality, Minkowski's First Inequality, and Their Duals." Journal of Mathematical Analysis and Applications 245, no. 2 (May 2000): 502–12. http://dx.doi.org/10.1006/jmaa.2000.6774.

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25

Zhao, Chang-Jian. "Orlicz-Brunn-Minkowski inequality for radial Blaschke-Minkowski homomorphisms." Quaestiones Mathematicae 41, no. 7 (February 9, 2018): 937–50. http://dx.doi.org/10.2989/16073606.2017.1417336.

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26

Chen, Feixiang, and Gangsong Leng. "Brunn-Minkowski type inequalities for Lp Blaschke- Minkowski homomorphisms." Journal of Nonlinear Sciences and Applications 09, no. 12 (December 10, 2016): 6034–40. http://dx.doi.org/10.22436/jnsa.009.12.10.

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27

Zhao, Chang-Jian. "On Blaschke–Minkowski Homomorphisms and Radial Blaschke–Minkowski Homomorphisms." Journal of Geometric Analysis 26, no. 2 (March 13, 2015): 1523–38. http://dx.doi.org/10.1007/s12220-015-9598-2.

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28

Lone, Mohd Saleem, and D. Krishnaswamy. "Representation of projectors involving Minkowski inverse in Minkowski space." Indian Journal of Pure and Applied Mathematics 48, no. 3 (September 2017): 369–89. http://dx.doi.org/10.1007/s13226-017-0238-3.

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29

Legland, David, Kiên Kiêu, and Marie-Françoise Devaux. "COMPUTATION OF MINKOWSKI MEASURES ON 2D AND 3D BINARY IMAGES." Image Analysis & Stereology 26, no. 2 (May 3, 2011): 83. http://dx.doi.org/10.5566/ias.v26.p83-92.

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Minkowski functionals encompass standard geometric parameters such as volume, area, length and the Euler-Poincaré characteristic. Software tools for computing approximations of Minkowski functionals on binary 2D or 3D images are now available based on mathematical methods due to Serra, Lang and Ohser. Minkowski functionals can not be used to describe spatial heterogeneity of structures. This description can be performed by using Minkowski measures, which are local versions of Minkowski functionals. In this paper, we discuss how to extend the computation of Minkowski functionals to the computation of Minkowski measures. Approximations of Minkowski measures are computed using fltering and look-up table transformations. The final result is represented as a grey-level image. Approximation errors are investigated based on numerical examples. Convergence and non convergence of the measure approximations are discussed. The measure of surface area is used to describe spatial heterogeneity of a synthetic structure, and of an image of tomato pericarp.

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30

Zhao, Chang-Jian, and Wing-Sum Cheung. "Orlicz Mean Dual Affine Quermassintegrals." Journal of Function Spaces 2018 (2018): 1–13. http://dx.doi.org/10.1155/2018/8123924.

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Our main aim is to generalize the mean dual affine quermassintegrals to the Orlicz space. Under the framework of dual Orlicz-Brunn-Minkowski theory, we introduce a new affine geometric quantity by calculating the first Orlicz variation of the mean dual affine quermassintegrals and call it the Orlicz mean dual affine quermassintegral. The fundamental notions and conclusions of the mean dual affine quermassintegrals and the Minkowski and Brunn-Minkowski inequalities for them are extended to an Orlicz setting. The related concepts and inequalities of dual Orlicz mixed volumes are also included in our conclusions. The new Orlicz isoperimetric inequalities in special case yield theLp-dual Minkowski inequality and Brunn-Minkowski inequality for the mean dual affine quermassintegrals, which also imply the dual Orlicz-Minkowski inequality and dual Orlicz-Brunn-Minkowski inequality.

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31

Chang-Jian, Zhao. "On polars of Blaschke-Minkowski homomorphisms." MATHEMATICA SCANDINAVICA 111, no. 1 (September 1, 2012): 147. http://dx.doi.org/10.7146/math.scand.a-15220.

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32

FENG, ZHIGANG, and GANG CHEN. "ON THE MINKOWSKI DIMENSION OF FUNCTIONAL DIGRAPH." Fractals 11, no. 01 (March 2003): 87–92. http://dx.doi.org/10.1142/s0218348x03001616.

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Functional digraphs are sometimes fractal sets. As a special kind of fractal sets, the dimension properties of the functional digraph are studied in this paper. Firstly, the proof of a Minkowski dimension theorem is discussed and a new proof is given. Secondly, according to this dimension theorem, the Minkowski dimensions of the digraphs of the sum, deviation, product and quotient of two functions are discussed. And the relations between these Minkowski dimensions and the Minkowski dimensions of the digraphs of the two functions are established. In the conclusion, the maximum Minkowski dimension of the two functional digraphs plays a decisive part in the Minkowski dimensions of the digraphs of the sum, deviation, product and quotient of two functions.

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33

Fioresi, R. "Quantizations of Flag Manifolds and Conformal Space Time." Reviews in Mathematical Physics 09, no. 04 (May 1997): 453–65. http://dx.doi.org/10.1142/s0129055x9700018x.

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In this paper we work out the deformations of some flag manifolds and of complex Minkowski space viewed as an affine big cell inside G(2,4). All the deformations come in tandem with a coaction of the appropriate quantum group. In the case of the Minkowski space this allows us to define the quantum conformal group. We also give two involutions on the quantum complex Minkowski space, that respectively define the real Minkowski space and the real euclidean space. We also compute the quantum De Rham complex for both real (complex) Minkowski and euclidean space.

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34

Stunzhas, L. P. "Projective Minkowski set." Moscow University Mathematics Bulletin 63, no. 2 (April 2008): 39–43. http://dx.doi.org/10.3103/s0027132208020010.

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35

Schuster, Franz E., and Thomas Wannerer. "Even Minkowski valuations." American Journal of Mathematics 137, no. 6 (2015): 1651–83. http://dx.doi.org/10.1353/ajm.2015.0041.

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36

Förster-Beuthan, Yvonne. "Kant meets Minkowski." Deutsche Zeitschrift für Philosophie 61, no. 1 (April 2013): 162–66. http://dx.doi.org/10.1524/dzph.2013.0013.

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37

Margulis, G. A. "Random minkowski theorem." Problems of Information Transmission 47, no. 4 (December 2011): 398–402. http://dx.doi.org/10.1134/s0032946011040077.

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38

Nakazato, Hiromichi, and Yoshiya Yamanaka. "Minkowski stochastic quantization." Physical Review D 34, no. 2 (July 15, 1986): 492–96. http://dx.doi.org/10.1103/physrevd.34.492.

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39

Dajani, Karma, Mathijs R. de Lepper, and E. Arthur Robinson. "Introducing Minkowski normality." Journal of Number Theory 211 (June 2020): 455–76. http://dx.doi.org/10.1016/j.jnt.2019.10.017.

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40

Moon, Hwan Pyo. "Minkowski Pythagorean hodographs." Computer Aided Geometric Design 16, no. 8 (September 1999): 739–53. http://dx.doi.org/10.1016/s0167-8396(99)00016-3.

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41

Rinaldi, Gloria. "Minkowski near— planes." Results in Mathematics 16, no. 1-2 (August 1989): 162–67. http://dx.doi.org/10.1007/bf03322652.

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42

Gerlach, Ulrich H. "Minkowski Bessel modes." Physical Review D 38, no. 2 (July 15, 1988): 514–21. http://dx.doi.org/10.1103/physrevd.38.514.

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43

Rinaldi, Globria. "Minkowski semi-planes." Journal of Geometry 35, no. 1-2 (July 1989): 158–66. http://dx.doi.org/10.1007/bf01222271.

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44

Wang, Wei. "L p Brunn–Minkowski type inequalities for Blaschke–Minkowski homomorphisms." Geometriae Dedicata 164, no. 1 (August 21, 2012): 273–85. http://dx.doi.org/10.1007/s10711-012-9772-7.

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45

Shen, Zhonghuan, and Weidong Wang. "Lp Radial Blaschke-Minkowski Homomorphisms and Lp Dual Affine Surface Areas." Mathematics 7, no. 4 (April 10, 2019): 343. http://dx.doi.org/10.3390/math7040343.

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Schuster introduced the notion of radial Blaschke-Minkowski homomorphism and considered the Busemann-Petty problem for volume forms. Whereafter, Wang, Liu and He presented the L p radial Blaschke-Minkowski homomorphisms and extended Schuster’s results. In this paper, associated with L p dual affine surface areas, we give an affirmative and a negative form of the Busemann-Petty problem and establish two Brunn-Minkowski inequalities for the L p radial Blaschke-Minkowski homomorphisms.

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46

Zhang, Juan, and Weidong Wang. "Some inequalities for the (p,q)-mixed geominimal surface areas and Lp radial Blaschke-Minkowski homomorphisms." Filomat 35, no. 4 (2021): 1393–403. http://dx.doi.org/10.2298/fil2104393z.

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Wang et al. introduced Lp radial Blaschke-Minkowski homomorphisms based on Schuster?s radial Blaschke-Minkowski homomorphisms. In 2018, Feng and He gave the concept of (p,q)-mixed geominimal surface area according to the Lutwak, Yang and Zhang?s (p,q)-mixed volume. In this article, associated with the (p,q)-mixed geominimal surface areas and the Lp radial Blaschke-Minkowski homomorphisms, we establish some inequalities including two Brunn-Minkowski type inequalities, a cyclic inequality and two monotonic inequalities.

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47

Richterova, M., J. Olivova, M. Popela, and V. Blazek. "Minkowski fractal antenna based on 3D printing." IOP Conference Series: Materials Science and Engineering 1254, no. 1 (September 1, 2022): 012019. http://dx.doi.org/10.1088/1757-899x/1254/1/012019.

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Abstract This paper describes the manufacturing of a Minkowski fractal antenna using a 3D commercial printer. The Minkowski fractal antenna was chosen for its simplicity and practical versatility. The paper also describes a manufacturing procedure for Minkowski fractal antenna by using metalic sprays. The design of the Minkowski fractal antenna in the MATLAB application using the Antenna Toolbox extension is also described, including 3D printing procedures, post processing procedures (plating) and practical testing of its functionality. The measured results are compared to simulations and then analysed.

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48

Wang, Weidong, Heping Chen, and Yuanyuan Zhang. "Busemann-Petty problem for the i-th radial Blaschke-Minkowski homomorphisms." Filomat 32, no. 19 (2018): 6819–27. http://dx.doi.org/10.2298/fil1819819w.

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Schuster introduced the notion of radial Blaschke-Minkowski homomorphism and considered its Busemann-Petty problem. In this paper, we further study the Busemann-Petty problem for the radial Blaschke-Minkowski homomorphisms and give the affirmative and negative forms of Busemann-Petty problem for the i-th radial Blaschke-Minkowski homomorphisms.

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49

Saad, Anis, and Attia Mostafa. "TYPES OF 3D SURFACE OF ROTATIONS EMBEDDED IN 4D MINKOWSKI SPACE." EPH - International Journal of Applied Science 2, no. 1 (March 27, 2016): 29–35. http://dx.doi.org/10.53555/eijas.v2i1.34.

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The geometry of surfaces of rotation in three dimensional Euclidean spaces has been studied widely. The rotational surfaces in three dimensional Euclidean spaces are generated byrotating an arbitrary curve about an arbitrary axis. Which should be using a type of matrices called matrices of rotation. But they are should be created by one parameter group of isometry. On the other hand, the Minkowski spaces have shorter history. In 1908 Minkowski [1864-1909] gave his talk on four dimensional real vector space, with a symmetric form of signature (+,+,+,-). In this space there are different types of vectors/ axes (space-liketime- like and null) as well as different types of curves (space-like- time-like and null). The relationship between Euclidean and Minkowskian geometry has many intriguing aspects, one of which is the manner in which formal similarity can co-exist with significantgeometric disparity. There has been considerable interest in the comparison of these twogeometries, as can be seen in the lecture notes of L’opez. In this manuscript we produce different types of surfaces of rotation in four dimensionalMinkowski spaces. And then we will provide a brief description of surfaces of rotation of 4D Minkowski spaces. Firstly consider the beginning by creating different type of matrices of rotation corresponding to the appropriate subgroup of the Lorentz group, and then generate all types of surfaces of rotation. The new work here is the spherical symmetric case which is nonabeliansubalgebra isomorphic to lie algebra. This case is known by expectation.

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Boskoff, Wladimir-Georges, and Salvatore Capozziello. "Recovering the cosmological constant from affine geometry." International Journal of Geometric Methods in Modern Physics 16, no. 10 (October 2019): 1950161. http://dx.doi.org/10.1142/s0219887819501615.

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A gravity theory without masses can be constructed in Minkowski spaces using a geometric Minkowski potential. The related affine spacelike spheres can be seen as the regions of the Minkowski spacelike vectors characterized by a constant Minkowski gravitational potential. These spheres point out, for each dimension [Formula: see text], spacetime models, the de Sitter ones, which satisfy Einstein’s field equations in absence of matter. In other words, it is possible to generate geometrically the cosmological constant. Even if a lot of possible parameterizations have been proposed, each one highlighting some geometric and physical properties of the de Sitter space, we present here a new natural parameterization which reveals the intrinsic geometric nature of cosmological constant relating it with the invariant affine radius coming from the so-called Minkowski–Tzitzeica surfaces theory.

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