4,411 research outputs found
Counting Derangements, Non Bijective Functions and the Birthday Problem
The article provides counting derangements of finite sets and counting non bijective functions. We provide a recursive formula for the number of derangements of a finite set, together with an explicit formula involving the number e. We count the number of non-one-to-one functions between to finite sets and perform a computation to give explicitely a formalization of the birthday problem. The article is an extension of [10].Institut fĂŒr Informatik I4, Technische UniversitĂ€t MĂŒnchen, BoltzmannstraĂe 3 85748 Garching, GermanyGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.CzesĆaw ByliĆski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.CzesĆaw ByliĆski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.CzesĆaw ByliĆski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Agata DarmochwaĆ. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.RafaĆ Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.Yatsuka Nakamura and Hisashi Ito. Basic properties and concept of selected subsequence of zero based finite sequences. Formalized Mathematics, 16(3):283-288, 2008, doi:10.2478/v10037-008-0034-y.Karol PÄ
k. Cardinal numbers and finite sets. Formalized Mathematics, 13(3):399-406, 2005.Konrad Raczkowski and Andrzej NÄdzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213-216, 1991.Konrad Raczkowski and PaweĆ Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Piotr Rudnicki and Andrzej Trybulec. Abian's fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997.MichaĆ J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825-829, 2001.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255-263, 1998
More on Divisibility Criteria for Selected Primes
This paper is a continuation of [19], where the divisibility criteria for initial prime numbers based on their representation in the decimal system were formalized. In the current paper we consider all primes up to 101 to demonstrate the method presented in [7].Naumowicz Adam - Institute of Informatics University of BiaĆystok Sosnowa 64, 15-887 BiaĆystok PolandPiliszek RadosĆaw - Institute of Informatics University of BiaĆystok Sosnowa 64, 15-887 BiaĆystok PolandGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281-290, 1990.Grzegorz Bancerek. Increasing and continuous ordinal sequences. Formalized Mathematics, 1(4):711-714, 1990.Grzegorz Bancerek. Veblen hierarchy. Formalized Mathematics, 19(2):83-92, 2011. doi:10.2478/v10037-011-0014-5.C.C. Briggs. Simple divisibility rules for the 1st 1000 prime numbers. arXiv preprint arXiv:math/0001012, 2000.CzesĆaw Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.CzesĆaw Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.CzesĆaw Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.CzesĆaw Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata DarmochwaĆ. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Krzysztof Hryniewiecki. Recursive definitions. Formalized Mathematics, 1(2):321-328, 1990.Magdalena Jastrzžebska and Adam Grabowski. Some properties of Fibonacci numbers. Formalized Mathematics, 12(3):307-313, 2004.Artur KorniĆowicz. On the real valued functions. Formalized Mathematics, 13(1):181-187, 2005.RafaĆ Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.RafaĆ Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990.Yatsuka Nakamura and Hisashi Ito. Basic properties and concept of selected subsequence of zero based finite sequences. Formalized Mathematics, 16(3):283-288, 2008. doi:10.2478/v10037-008-0034-y.Adam Naumowicz. On the representation of natural numbers in positional numeral systems. Formalized Mathematics, 14(4):221-223, 2006. doi:10.2478/v10037-006-0025-9.Karol Pak. Stirling numbers of the second kind. Formalized Mathematics, 13(2):337-345, 2005.Piotr Rudnicki and Andrzej Trybulec. Abianâs fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.MichaĆ J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825-829, 2001.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990
Overcomplete sets in non-separable Banach spaces
We introduce and study the notion of overcomplete set in a Banach space, that
subsumes and extends the classical concept of overcomplete sequence in a
(separable) Banach space. We give existence and non-existence results of
overcomplete sets for a wide class of (non-separable) Banach spaces and we
study to which extent properties of overcomplete sequences are retained by
every overcomplete set
Flexary Operations
AbstractIn this article we introduce necessary notation and definitions to prove the Eulerâs Partition Theorem according to H.S. Wilfâs lecture notes [31]. Our aim is to create an environment which allows to formalize the theorem in a way that is as similar as possible to the original informal proof. Eulerâs Partition Theorem is listed as item #45 from the âFormalizing 100 Theoremsâ list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/ [30].Institute of Informatics, University of BiaĆystok, CioĆkowskiego 1M, 15-245 BiaĆystok, PolandGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. Tarskiâs classes and ranks. Formalized Mathematics, 1(3):563-567, 1990.Grzegorz Bancerek. Monoids. Formalized Mathematics, 3(2):213-225, 1992.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.CzesĆaw Bylinski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.CzesĆaw Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.CzesĆaw Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.CzesĆaw Bylinski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.CzesĆaw Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Marco B. Caminati. Preliminaries to classical first order model theory. Formalized Mathematics, 19(3):155-167, 2011. doi:10.2478/v10037-011-0025-2. [Crossref]Marco B. Caminati. First order languages: Further syntax and semantics. Formalized Mathematics, 19(3):179-192, 2011. doi:10.2478/v10037-011-0027-0. [Crossref]Agata DarmochwaĆ. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Scalar multiple of Riemann definite integral. Formalized Mathematics, 9(1):191-196, 2001.Yoshinori Fujisawa, Yasushi Fuwa, and Hidetaka Shimizu. Public-key cryptography and Pepinâs test for the primality of Fermat numbers. Formalized Mathematics, 7(2):317-321, 1998.Artur KorniĆowicz. Arithmetic operations on functions from sets into functional sets. Formalized Mathematics, 17(1):43-60, 2009. doi:10.2478/v10037-009-0005-y. [Crossref]RafaĆ Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.Yatsuka Nakamura and Hisashi Ito. Basic properties and concept of selected subsequence of zero based finite sequences. Formalized Mathematics, 16(3):283-288, 2008. doi:10.2478/v10037-008-0034-y. [Crossref]Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Konrad Raczkowski and Andrzej Nedzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213-216, 1991.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.MichaĆ J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990.Wojciech A. Trybulec. Binary operations on finite sequences. Formalized Mathematics, 1 (5):979-981, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825-829, 2001.Freek Wiedijk. Formalizing 100 theorems.Herbert S. Wilf. Lectures on integer partitions.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990
Criteria for the determination of time dependent scalings in the Fock quantization of scalar fields with a time dependent mass in ultrastatic spacetimes
For Klein-Gordon fields, it is well known that there exist an infinite number
of nonequivalent Fock representations of the canonical commutation relations
and, therefore, of inequivalent quantum theories. A context in which this kind
of ambiguities arises and prevents the derivation of robust results is, e.g.,
in the quantum analysis of cosmological perturbations. In these situations,
typically, a suitable scaling of the field by a time dependent function leads
to a description in an auxiliary static background, though the nonstationarity
still shows up in a time dependent mass. For such a field description, and
assuming the compactness of the spatial sections, we recently proved in three
or less spatial dimensions that the criteria of a natural implementation of the
spatial symmetries and of a unitary time evolution are able to select a unique
class of unitarily equivalent vacua, and hence of Fock representations. In this
work, we succeed to extend our uniqueness result to the consideration of all
possible field descriptions that can be reached by a time dependent canonical
transformation which, in particular, involves a scaling of the field by a
function of time. This kind of canonical transformations modify the dynamics of
the system and introduce a further ambiguity in its quantum description,
exceeding the choice of a Fock representation. Remarkably, for any compact
spatial manifold in less than four dimensions, we show that our criteria
eliminate any possible nontrivial scaling of the field other than that leading
to the description in an auxiliary static background. Besides, we show that
either no time dependent redefinition of the field momentum is allowed or, if
this may happen, the redefinition does not introduce any Fock representation
that cannot be obtained by a unitary transformation.Comment: 37 pages. Modified title. Improved discussion concerning the spatial
symmetry group. New section (section VI
Polynomially Bounded Sequences and Polynomial Sequences
AbstractIn this article, we formalize polynomially bounded sequences that plays an important role in computational complexity theory. Class P is a fundamental computational complexity class that contains all polynomial-time decision problems [11], [12]. It takes polynomially bounded amount of computation time to solve polynomial-time decision problems by the deterministic Turing machine. Moreover we formalize polynomial sequences [5].Hiroyuki Okazaki - Shinshu University, Nagano, JapanYuichi Futa - Japan Advanced Institute of Science and Technology,
Ishikawa, JapanGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek. Increasing and continuous ordinal sequences. Formalized Mathematics, 1(4):711-714, 1990.Grzegorz Bancerek and Piotr Rudnicki. Two programs for SCM. Part I - preliminaries. Formalized Mathematics, 4(1):69-72, 1993.E.J. Barbeau. Polynomials. Springer, 2003.CzesĆaw Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.CzesĆaw Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.CzesĆaw Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.CzesĆaw Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata DarmochwaĆ. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Jon Kleinberg and Eva Tardos. Algorithm Design. Addison-Wesley, 2005.Donald E. Knuth. The Art of Computer Programming, Volume 1: Fundamental Algorithms, Third Edition. Addison-Wesley, 1997.Artur KorniĆowicz. On the real valued functions. Formalized Mathematics, 13(1):181-187, 2005.JarosĆaw Kotowicz. The limit of a real function at infinity. Formalized Mathematics, 2 (1):17-28, 1991.JarosĆaw Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Richard Krueger, Piotr Rudnicki, and Paul Shelley. Asymptotic notation. Part I: Theory. Formalized Mathematics, 9(1):135-142, 2001.Richard Krueger, Piotr Rudnicki, and Paul Shelley. Asymptotic notation. Part II: Examples and problems. Formalized Mathematics, 9(1):143-154, 2001.Yatsuka Nakamura and Hisashi Ito. Basic properties and concept of selected subsequence of zero based finite sequences. Formalized Mathematics, 16(3):283-288, 2008. doi:10.2478/v10037-008-0034-y. [Crossref]Jan PopioĆek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263-264, 1990.Konrad Raczkowski and Andrzej Nedzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213-216, 1991.Konrad Raczkowski and Andrzej Nedzusiak. Series. Formalized Mathematics, 2(4):449-452, 1991.Konrad Raczkowski and PaweĆ Sadowski. Real function differentiability. Formalized Mathematics, 1(4):797-801, 1990.Yasunari Shidama. The Taylor expansions. Formalized Mathematics, 12(2):195-200, 2004.MichaĆ J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825-829, 2001.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990
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