The present work was primarily motivated by our findings in the literature of some flaws within proof second-order Legendre necessary optimality condition for fractional calculus variations problems. Therefore, we were eager to elaborate a correct and it turns out that this goal is highly nontrivial, especially when considering final constraints. This paper result reflections on subject. Precis...

This paper develops a significant extension of E. Lutwak’s dual Brunn-Minkowski theory, originally applicable only to star-shaped sets, to the class of bounded Borel sets. The focus is on expressions and inequalities involving chord-power integrals, random simplex integrals, and dual affine quermassintegrals. New inequalities obtained include those of isoperimetric and Brunn-Minkowski type. A n...

Inspired by Raychaudhuri’s work, and using the equation named after him as a basic ingredient, I prove a new singularity theorem: open non-rotating everywhere expanding universes with non-vanishing spatial average of the matter variables are totally geodesically incomplete to the past. Another way of stating the same is that, under the same conditions, any singularity-free model must have a van...

Synopsis The concept of bisimulation from concurrency theory Par81, Mil89] is used to reason about recursively deened data types. From two strong-extensionality theorems stating that the equality (resp. inequality) relation is maximal among all bisimulations, a proof principle for the nal coalgebra of an endofunctor on a category of data types (resp. domains) is obtained. As an application of t...

quantum circuits are promising choices for future quantum computing technology, since the multiple-valued quantum system is more compact than the corresponding binary quantum system. Grover's quantum search algorithm requires a sub-circuit called oracle, which takes a set of inputs and gives an output stating whether a given search condition on the inputs is satisfied or not. Equality, less-tha...

there are several suffixes in creating diminutive words with endearing meanings in the russian language. structures and their meaning variations, make it difficult for iranian pupils to comprehend its concepts which leads to a multitude of errors in translating and instructing this important issue. many foreign and russian researchers and linguists have studied and gone through diminutive words...