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Furthermore, there are examples of infinite cyclic models solving the problem of unbounded entropy growth which are geodesically complete. [14] In both of these studies, the authors argue that the previous investigations often did not use mathematically precise formulations of the BGV theorem and thus reached incomplete conclusions.
All other isotopes have half-lives less than 4 days, most less than an hour. There are also many isomers , the longest-lived of which is 120m1 Sb with half-life 5.76 days. Except for beryllium , antimony is the lightest element observed to have isotopes capable of alpha decay ; 104 Sb is seen to undergo this mode of decay.
In mathematics, Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who experimented with soap films. The problem is considered part of the calculus of variations.
The material forms when Sb 2 O 3 is heated in air: [5] Sb 2 O 3 + 0.5 O 2 → Sb 2 O 4 ΔH = −187 kJ/mol. At 800 °C, antimony(V) oxide loses oxygen to give the same material: Sb 2 O 5 → Sb 2 O 4 + 0.5 O 2 ΔH = −64 kJ/mol. The material is mixed valence, containing both Sb(V) and Sb(III) centers.
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
Boundary value problems are similar to initial value problems.A boundary value problem has conditions specified at the extremes ("boundaries") of the independent variable in the equation whereas an initial value problem has all of the conditions specified at the same value of the independent variable (and that value is at the lower boundary of the domain, thus the term "initial" value).
The calculus of variations may be said to begin with Newton's minimal resistance problem in 1687, followed by the brachistochrone curve problem raised by Johann Bernoulli (1696). [2] It immediately occupied the attention of Jacob Bernoulli and the Marquis de l'Hôpital , but Leonhard Euler first elaborated the subject, beginning in 1733.
The Short Integer Solution (SIS) problem is an average case problem that is used in lattice-based cryptography constructions. Lattice-based cryptography began in 1996 from a seminal work by Ajtai [ 1 ] who presented a family of one-way functions based on the SIS problem.