Equipartition of sphere measures by hyperplanes

Blagojević, Pavle V. M.; Dimitrijević-Blagojević, Aleksandra S.; Milosevic, Marko

doi:10.2298/fil0601001b

Cited by 2 publications

(9 citation statements)

References 8 publications

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“…We use methods introduced in [1], and briefly review them. The CS / TM scheme is the standard method for solving measure partition problems.…”

Section: Configuration Space / Test Map Schemementioning

confidence: 99%

“…As in the [1], the problem of the existence of the D 2n -equivariant map V 2 (R 3 ) → W n \ A n can be substituted with the following equivalent problem.…”

Section: Configuration Space / Test Map Schemementioning

confidence: 99%

“…To identify the obstruction element, as in the papers [4], [1], [3] we use general position map scheme. Briefly, the procedure can be divided in four steps.…”

Section: Theoremmentioning

confidence: 99%

“…Let µ be a proper Borel probability measure on the sphere S 2 . Then there are three planes in the fan position such that the ratio of measure µ in angular sectors cut by planes is (A) (1, 1, 2, 1, 1, 2) (C) (1,1,3,1,1,3) This result is a generalization of the Makeev result [7]. …”

mentioning

confidence: 99%

“…In this paper we use the CS / TM scheme stated in [1] for the V.V. Makeev equipartition problem [7] to prove the existence of new weighted partitions.…”

mentioning

confidence: 99%

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Weighted partitions of sphere measures by hyperplanes

Blagojević¹,

Dimitrijević-Blagojević²,

Milosevic³

2006

Filomat

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Abstract. In this paper we use the CS / TM scheme stated in [1] for the V.V. Makeev equipartition problem [7] to prove the existence of new weighted partitions. For the first time computations of the associated equivariant problem are done by codes written in the package Mathematica 5.0 [2]. Statement of the main resultLet H 1 , H 2 and H 3 be planes in R 3 through the origin. They are in the fan position if H 1 ∩ H 2 = H 1 ∩ H 3 = H 2 ∩ H 3 . Planes in the fan position cut the sphere S 2 in six parts σ 1 , .., σ 6 which can be naturally oriented up to a cyclic permutation. A measure µ is the proper measure µ if µ([a, b]) = 0 for any circular arc [a, b] ⊂ S 2 , and µ(U ) > 0 for each nonempty open set U ⊂ S 2 . We prove the existence of the following measure partitions. Theorem 1. Let µ be a proper Borel probability measure on the sphere S 2 . Then there are three planes in the fan position such that the ratio of measure µ in angular sectors cut by planes is (A) (1, 1, 2, 1, 1, 2) (C) (1,1,3,1,1,3) This result is a generalization of the Makeev result [7].

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“…We use methods introduced in [1], and briefly review them. The CS / TM scheme is the standard method for solving measure partition problems.…”

Section: Configuration Space / Test Map Schemementioning

confidence: 99%

“…As in the [1], the problem of the existence of the D 2n -equivariant map V 2 (R 3 ) → W n \ A n can be substituted with the following equivalent problem.…”

Section: Configuration Space / Test Map Schemementioning

confidence: 99%

“…To identify the obstruction element, as in the papers [4], [1], [3] we use general position map scheme. Briefly, the procedure can be divided in four steps.…”

Section: Theoremmentioning

confidence: 99%

mentioning

confidence: 99%

“…In this paper we use the CS / TM scheme stated in [1] for the V.V. Makeev equipartition problem [7] to prove the existence of new weighted partitions.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Weighted partitions of sphere measures by hyperplanes

Blagojević¹,

Dimitrijević-Blagojević²,

Milosevic³

2006

Filomat

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The “Odious Scourge”: Evolving Interpretations of the Crime of Genocide

Schabas

2006

Genocide Studies and Prevention

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The crime of genocide was defined in the 1948 Convention on the Prevention and Punishment of the Crime of Genocide in reaction to the concept of crimes against humanity developed at Nuremberg, which insisted upon a connection with aggressive war in prosecutions for atrocity crimes. The convention stated genocide could be committed in time of peace, but it also narrowed the scope of the crime itself to the intentional destruction of a national, ethnic, racial, or religious group. Cultural genocide was intentionally excluded. Although the text of the definition remains unchanged, judicial interpretation has broadened it significantly. Recent decisions have held that there is no requirement of a state plan or policy. They have also set out a subjective approach to identification of the protected group. Although cultural genocide in an extensive sense is still not recognized within the definition, there is a definite tendency to extend the concept to what is colloquially called ''ethnic cleansing.'' These broadening definitions influence determinations about genocides, even those committed many years ago. When hundreds of thousands of Armenians living within the Ottoman Empire perished in 1915, 1 the governments of France, Great Britain, and Russia responded with an unprecedented declaration. Dated 24 May 1915, it asserts that ''in the presence of these new crimes of Turkey against humanity and civilization, the allied Governments publicly inform the Sublime Porte that they will hold personally responsible for the said crimes all members of the Ottoman Government as well as those of its agents who are found to be involved in such massacres.'' 2 It has been suggested that this constitutes the first use, at least within an international law context, of the term ''crimes against humanity.'' 3 According to the Treaty of Sèvres, signed on 10 August 1920, Turkey recognized the right of trial ''notwithstanding any proceedings or prosecution before a tribunal in Turkey'' and was obliged to surrender ''all persons accused of having committed an act in violation of the laws and customs of war, who are specified either by name or by rank, office or employment which they held under Turkish authorities.'' 4 This formulation is similar to the war crimes clauses in the Treaty of Versailles. 5 But the Treaty of Sèvres contains a major innovation, contemplating prosecution for the massacres committed within Turkey by the Turkish regime, as well as of war crimes committed against Allied soldiers or civilians within occupied territories. Pursuant to article 230, The Turkish Government undertakes to hand over to the Allied Powers the persons whose surrender may be required by the latter as being responsible for the massacres committed during the continuance of the state of war on territory which formed part of the Turkish Empire on the 1st August, 1914. The Allied Powers reserve to themselves

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Equipartition of sphere measures by hyperplanes

Abstract: 2 by three planes. The advantage of our approach is that it can be applied on other more complicated questions of the similar nature.

Cited by 2 publications

References 8 publications

Weighted partitions of sphere measures by hyperplanes

Weighted partitions of sphere measures by hyperplanes

The “Odious Scourge”: Evolving Interpretations of the Crime of Genocide

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