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Concept# Special case

Summary

In logic, especially as applied in mathematics, concept A is a special case or specialization of concept B precisely if every instance of A is also an instance of B but not vice versa, or equivalently, if B is a generalization of A. A limiting case is a type of special case which is arrived at by taking some aspect of the concept to the extreme of what is permitted in the general case. A degenerate case is a special case which is in some way qualitatively different from almost all of the cases allowed.
Examples
Special case examples include the following:

- All squares are rectangles (but not all rectangles are squares); therefore the square is a special case of the rectangle.
- Fermat's Last Theorem, that an + bn {{=}} cn has no solutions in positive integers with n > 2, is a special case of Beal's conjecture, that ax +

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The effect of an elastic deformation on optical properties of an aeolotropic medium (piezo-optic effect) is explicitly established from an energy function. Then is deduced the number of distinct coefficients ruling this effect in the special case of quartz. The author gives a new method to measure the absolute dephasage ; method based on photometric exploration of the three beams diffracted image, around the focus. By this method a phase variation of 2π/2000 has been disclosed. This method has been used for the direct measure of plaza-optic effect in quartz. With a single type of cubic sample, 5 out of the 8 coefficients of quartz have been accurately settled, and an upper limit has been given for the one of others. Other applications of this measurement method are considered.

Seyed Hossein Nassajianmojarrad

In the present thesis, we delve into different extremal and algebraic problems arising from combinatorial geometry. Specifically, we consider the following problems. For any integer $n\ge 3$, we define $e(n)$ to be the minimum positive integer such that any set of $e(n)$ points in general position in the plane contains $n$ points in convex position. In 1935, Erd\H{o}s and Szekeres proved that $e(n) \le {2n-4 \choose n-2}+1$ and later in 1961, they obtained the lower bound $2^{n-2}+1 \le e(n)$, which they conjectured to be optimal. We prove that $e(n) \le {2n-5 \choose n-2}-{2n-8 \choose n-3}+2$. In a recent breakthrough, Suk proved that $e(n) \le 2^{n+O\left(n^{2/3}\log n\right)}$. We strengthen this result by extending it to pseudo-configurations and also improving the error term. Combining our results with a theorem of Dobbins et al., we significantly improve the best known upper bounds on the following two functions, introduced by Bisztriczky and Fejes T'{o}th and by Pach and T'{o}th, respectively. Let $c(n)$ (and $c'(n)$) denote the smallest positive integer $N$ such that any family of $N$ pairwise disjoint convex bodies in general position (resp., $N$ convex bodies in general position, any pair of which share at most two boundary points) has an $n$ members in convex position. We show that $c(n)\le c'(n)\le 2^{n+O\left(\sqrt{n\log n}\right)}$. Given a point set $P$ in the plane, an ordinary circle for $P$ is defined as a circle containing exactly three points of $P$. We prove that any set of $n$ points in the plane, not all on a line or a circle, determines at least $\frac{1}{4}n^2-O(n)$ ordinary circles. We determine the exact minimum number of ordinary circles for all sufficiently large $n$, and characterize all point sets that come close to this minimum. We also consider the orchard problem for circles, where we determine the maximum number of circles containing four points of a given set and describe the extremal configurations. A special case of the Schwartz-Zippel lemma states that given an algebraic curve $C\subset \mathbb{C}^2$ of degree $d$ and two finite sets $A,B\subset \mathbb{C}$, we have $|C\cap (A\times B)|=O_d(|A|+|B|)$. We establish a two-dimensional version of this result, and prove upper bounds on the size of the intersection $|X\cap (P\times Q)|$ for a variety $X\subset \mathbb{C}^4$ and finite sets $P,Q\subset \mathbb{C}^2$. A key ingredient in our proofs is a two-dimensional version of a special case of Alon's combinatorial Nullstellensatz. As corollaries, we generalize the Szemer'edi-Trotter point-line incidence theorem and several known bounds on repeated and distinct Euclidean distances. We use incidence geometry to prove some sum-product bounds over arbitrary fields. First, we give an explicit exponent and improve a recent result of Bukh and Tsimerman by proving that $\max \{ |A+A|, |f(A, A)|\}\gg |A|^{6/5}$ for any small set $A\subset \mathbb{F}_p$ and quadratic non-degenerate polynomial $f(x, y)\in \mathbb{F}_p[x, y]$. This generalizes the result of Roche-Newton et al. giving the best known lower bound for the term $\max \{ |A+A|, |A \cdot A |\}$. Secondly, we improve and generalize the sum-product results of Hegyv'{a}ri and Hennecart on $\max\{ |A+B|, |f(B,C)|\}$, for a specific type of function $f$. Finally, we prove that the number of distinct cubic distances generated by any small set $A\times A\subset \mathbb{F}_p^2$ is $\Omega(|A|^{8/7})$, which improves a result of Yazici et al.

The minimal faithful permutation degree (G) of a finite group G is the least nonnegative integer n such that G embeds in the symmetric group Sym(n). Clearly (G x H) (G) + (H) for all finite groups G and H. In 1975, Wright ([10]) proved that equality occurs when G and H are nilpotent and exhibited an example of strict inequality where G x H embeds in Sym(15). In 2010 Saunders ([7]) produced an infinite family of examples of permutation groups G and H where (G x H) < (G) + (H), including the example of Wright's as a special case. The smallest groups in Saunders' class embed in Sym(10). In this article, we prove that 10 is minimal in the sense that (G x H)=(G) + (H) for all groups G and H such that (G x H)

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