دالة ارتدادية
في الرياضيات، الدالة الارتدادية[1] أو الدالة ذاتية الانعكاس إنگليزية: involutory function هي دالة تساوي دالتها العكسية. رياضياً: الدالة تُعدُّ ارتداديَّةً إذا وفقط إذا:
for all x in the domain of f.[2] Equivalently, applying f twice produces the original value.
السمات العامة
Any involution is a bijection.
The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (), reciprocation (), and complex conjugation () in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the Beaufort polyalphabetic cipher.
The composition g ∘ f of two involutions f and g is an involution if and only if they commute: g ∘ f = f ∘ g.[3]
الارتداد في الفئات المحدودة
The number of involutions, including the identity involution, on a set with n = 0, 1, 2, ... elements is given by a recurrence relation found by Heinrich August Rothe in 1800:
- and for
The first few terms of this sequence are 1, 1, 2, 4, 10, 26, 76, 232 (المتتالية A000085 في OEIS); these numbers are called the telephone numbers, and they also count the number of Young tableaux with a given number of cells.[4] The number can also be expressed by non-recursive formulas, such as the sum
The number of fixed points of an involution on a finite set and its number of elements have the same parity. Thus the number of fixed points of all the involutions on a given finite set have the same parity. In particular, every involution on an odd number of elements has at least one fixed point. This can be used to prove Fermat's two squares theorem.[5]
الارتداد في مجالات الرياضيات
ما قبل التفاضل
Some basic examples of involutions include the functions
These are not the only pre-calculus involutions. Another one within the positive reals is
The graph of an involution (on the real numbers) is symmetric across the line . This is due to the fact that the inverse of any general function will be its reflection over the line . This can be seen by "swapping" with . If, in particular, the function is an involution, then its graph is its own reflection.
Other elementary involutions are useful in solving functional equations.
الهندسة الاقليدية
A simple example of an involution of the three-dimensional Euclidean space is reflection through a plane. Performing a reflection twice brings a point back to its original coordinates.
Another involution is reflection through the origin; not a reflection in the above sense, and so, a distinct example.
These transformations are examples of affine involutions.
الهندسة الوصفية
لمعرفة ما إذا كانت علاقة إسقاطية معينة التفافية، من الضروري، التحقق مما إذا كان كل زوج من العناصر المتقابلة يتكون من عناصر متقابلة بطريقة مزدوجة.[6] وفي حالة القطوع المخروطية يمكن أن يقتصر التحقق على زوج واحد، بحكم النظرية التالية:
إذا نقطتين لشكلين متطابقين، يتقابلان بطريقة مزدوجة، فإن أي نقطتين آخرتين يتقابلان أيضا بطريقة مزدوجة.
مثلا معلوم رباعي A, B و C, D، وحيث تم تحديد بطريقة تحكمية نقطة X. مطلوب إنشاء نجمة ثمانية بحيث يكون بين رؤوسها علافة التفافية.
في هذه الحالة يكمن الحل في اعتماد اثنتين من العلاقات التقابلية الالتفافية: واحدة مركزها النقطة P ومحورها الخط p؛ والعلاقة الأخرى مركزها U ومحورها u. علما بأن النقاط المتقابلة تكون مصطفة مع مركز التقابل والخطوط المتقابلة تلتقي على طول محور التقابل.[7]
معرض صور
انظر أيضاً
مراجع
- ^ Robert Alexander Adams, Calculus: Single Variable, 2006, ISBN 0321307143, p. 165
- ^ Russell, Bertrand (1903), Principles of mathematics (2nd ed.), W. W. Norton & Company, Inc, p. 426, ISBN 9781440054167, https://books.google.com/books?id=63ooitcP2osC&q=involution%20subject%3A%&pg=PR3
- ^ Kubrusly, Carlos S. (2011), The Elements of Operator Theory, Springer Science & Business Media, Problem 1.11(a), p. 27, ISBN 9780817649982, https://books.google.com/books?id=g-UFYTO8SbMC&pg=PA27.
- ^ Knuth, Donald E. (1973), The Art of Computer Programming, Volume 3: Sorting and Searching, Reading, Mass.: Addison-Wesley, pp. 48, 65.
- ^ Zagier, D. (1990), "A one-sentence proof that every prime p≡ 1 (mod 4) is a sum of two squares", American Mathematical Monthly 97 (2): 144, doi:.
- ^ Lezioni di geometria By Guido Castelnuovo Archived 2023-02-18 at the Wayback Machine
- ^ The problem of tangency to three non-homothetic conics Archived 2022-06-28 at the Wayback Machine
للاستزادة
- Ell, Todd A.; Sangwine, Stephen J. (2007). "Quaternion involutions and anti-involutions". Computers & Mathematics with Applications. 53 (1): 137–143. arXiv:math/0506034. doi:10.1016/j.camwa.2006.10.029. S2CID 45639619.
- Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre (1998), The book of involutions, Colloquium Publications, 44, With a preface by J. Tits, Providence, RI: American Mathematical Society, ISBN 0-8218-0904-0
- Hazewinkel, Michiel, ed. (2001), "Involution", Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104