In the case of dimension 2, sophisticated functional inequalities allow to handle the case (alpha le 8pi ).
Although the physical models are two dimensional, we shall carry out our proofs in the case of dimension.
In the case of dimension two, the density of Φg,K,ν is φ g, K, ν ( x, y ) = C ( g, K ) g h K x y − ν 1 ν 2, ( x, y ) T ∈ ℝ 2. We recall that star-generalized trigonometric functions and random polar coordinates are defined in ([Richter 2011a]) by cos K = cos ϕ h K ( cos ϕ, sin ϕ ), sin K = sin ϕ h K ( cos ϕ, sin ϕ ) and X=R cosK,Y=R sinK, respectively.
where F a, p ( M, r ) = ω E a, p 1 r M ∩ E a, p, r > 0. In the case of dimension n=2, Figure 1 shows the density φg,a,p,ν,O and contours of its superlevel sets where g ( r ) = exp − r p p, a = ( 3, 1 ) T, p takes several values, ν= 0,0) T and O = cos α sin α − sin α cos α, α = 5 π / 3. Figure 1 p -generalized elliptically contoured densities for p =4,1 and 0. 6, from left to right.
The result above was proven in [64, 65] for the case of dimension n ≥ 3. The same basic construction works in the two dimensional case [112].
In 1987, Cheng proved their conjecture in the case of odd dimensions, while for even dimensions he obtained a slightly weaker lower bound for the norm.
Consequently, \ dv^{\,i}_p\) is bi-linear in \ v^{\,i}_p\) and \(dx^{\,i}_p\); that is, where, in the case of four dimensions, the \(4^{3} = 64\) coefficients \ \Gamma^{\,i}_{jk} (x^{i}(p))\) are coordinate functions, that is, functions of \(x^{i}(p)\) \((i = 1, \ldots, 4 \), and the minus sign is introduced to agree with convention.
Even in the case of 16 dimensions, the average recognition time will not exceed 36 ms. For some real-time recognition system, designers can make a tradeoff on the number of sampling dimension according to the sensing resources, data processing time, and average recognition rate.
Our results show that although the problem of searching for similar structures in a database based on the RMSD measure with indels is NP-hard in the case of unbounded dimensions, it can be solved in 3-D by a simple average-case linear time algorithm when the number of indels is bounded by a constant.
The decay rates obtained are optimal in the case of (2+2 -dimensions for any m, while in higher dimensions the result is sharp for m sufficiently large.
