The function u(t) need not be a member of C, but can be any Borel measurable function defined on OSj/^1, and may even be permitted to take on infinite values. The Wiener integral of a functional is simply the Lebesgue integral of the functional with respect to Wiener's measure [4]1 in C. This measure is not invariant under translations, but is in other respects a Lebesgue measure based on inter...

Games often provide a good introduction to interesting phenomena in mathematics. In this note, we examine three variations of an iterative sharing game played around a circular (or not so circular) table. More precisely, for each variation we study the tendency toward equal distribution among the players. In the first variation, the players have discrete amounts at each step. The second variati...

thorough knowledge of the morphology and internal anatomy of the root canal system is essential, because it determines the successful outcome of endodontic treatment. the main goal of endodontic treatment is to prevent apical periodontitis and/or to promote the healing of periapical lesion. presence of two canals or roots on the palatal side of the first maxillary molar has rarely been reported...

We derive a formula for the first variation of horizontal perimeter measure for C2 hypersurfaces of completely general sub-Riemannian manifolds, allowing for the existence of characteristic points. When the manifold admits dilations, we establish a sub-Riemannian Minkowski formula. For C2 hypersurfaces in vertically rigid sub-Riemannian manifolds we also produce a second variation formula for v...

Let q be a prime power and fix a ∈ Fq2 \ Fq. In this note it is proved that, provided q > 5, the set Sa = {a− α : α ∈ Fq} contains both a square and a non-square of Fq2 . In particular, every a ∈ Fq2 \ Fq can be written as a sum of a square of Fq2 and a square of Fq. § 1. The problem and the main result Throughout we use Fq to denote the finite field of q elements and Fq to denote the non-zero ...

A. In this paper, by applying the first variation formula of f -bi-energy given in [OND], we study f -biharmonic maps between doubly warped product manifolds M ×(μ,λ) N. Under imposing existence condition concerning proper f -biharmonic maps, we derive f -biharmonicity’s characteristic equations for the inclusion maps: iy0 : (M, g) → (M ×(μ,λ) N, ḡ), ix0 : (N, h) → (M ×(μ,λ) N, ḡ) and th...

DNA’s shape mostly lies on its total energy F . Its corresponding equilibrium shape equations can be got by classical variation method: letting the first energy variation δF = 0. Here we not only provide the first variation δF but also give the second variation δF in planar case. Moreover, the general shape equations of DNA are obtained and a mistake in [Zhang, et al. Phys. Rev. E 70 051902 (20...

The weak mean curvature is lower semicontinuous under weak convergence of varifolds that is if μk → μ weakly as varifolds then ‖ ~ Hμ ‖Lp(μ)≤ lim infk→∞ ‖ ~ Hμk ‖Lp(μk) . In contrast, if Tk → T weakly as integral currents then μT may not have locally bounded first variation even if ‖ ~ HμTk ‖L∞(μk) is bounded. In 1999, Luigi Ambrosio asked the question whether lower semicontinuity of the weak m...

In this paper, we show that the conditional transform with respect to the Gaussian process involving the first variation can be expressed in terms of the conditional transform without the first variation. We then use this result to obtain various integration formulas involving the conditional $diamond$-product and the first variation.