12,172 research outputs found
Partial Differentiation of Real Ternary Functions
In this article, we shall extend the result of [19] to discuss partial differentiation of real ternary functions (refer to [8] and [16] for partial differentiation).Inoué Takao - Inaba 2205, Wing-Minamikan Nagano, Nagano, JapanXie Bing - Qingdao University of Science and Technology, ChinaLiang Xiquan - Qingdao University of Science and Technology, ChinaGrzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces Rn. Formalized Mathematics, 15(2):65-72, 2007, doi:10.2478/v10037-007-0008-5.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Xiquan Liang, Piqing Zhao, and Ou Bai. Vector functions and their differentiation formulas in 3-dimensional Euclidean spaces. Formalized Mathematics, 18(1):1-10, 2010, doi: 10.2478/v10037-010-0001-2.Konrad Raczkowski and Paweł Sadowski. Real function continuity. Formalized Mathematics, 1(4):787-791, 1990.Konrad Raczkowski and Paweł Sadowski. Real function differentiability. Formalized Mathematics, 1(4):797-801, 1990.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Walter Rudin. Principles of Mathematical Analysis. MacGraw-Hill, 1976.Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Bing Xie, Xiquan Liang, and Hongwei Li. Partial differentiation of real binary functions. Formalized Mathematics, 16(4):333-338, 2008, doi:10.2478/v10037-008-0041-z
Second-Order Partial Differentiation of Real Ternary Functions
In this article, we shall extend the result of [17] to discuss second-order partial differentiation of real ternary functions (refer to [7] and [14] for partial differentiation).Inaba 2205, Wing-Minamikan Nagano, Nagano, JapanGrzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces Rn. Formalized Mathematics, 15(2):65-72, 2007, doi:10.2478/v10037-007-0008-5.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Konrad Raczkowski and Paweł Sadowski. Real function continuity. Formalized Mathematics, 1(4):787-791, 1990.Konrad Raczkowski and Paweł Sadowski. Real function differentiability. Formalized Mathematics, 1(4):797-801, 1990.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Walter Rudin. Principles of Mathematical Analysis. MacGraw-Hill, 1976.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Bing Xie, Xiquan Liang, and Hongwei Li. Partial differentiation of real binary functions. Formalized Mathematics, 16(4):333-338, 2008, doi:10.2478/v10037-008-0041-z.Bing Xie, Xiquan Liang, and Xiuzhuan Shen. Second-order partial differentiation of real binary functions. Formalized Mathematics, 17(2):79-87, 2009, doi: 10.2478/v10037-009-0009-7
The cubic chessboard
We present a survey of recent results, scattered in a series of papers that
appeared during past five years, whose common denominator is the use of cubic
relations in various algebraic structures. Cubic (or ternary) relations can
represent different symmetries with respect to the permutation group S_3, or
its cyclic subgroup Z_3. Also ordinary or ternary algebras can be divided in
different classes with respect to their symmetry properties. We pay special
attention to the non-associative ternary algebra of 3-forms (or ``cubic
matrices''), and Z_3-graded matrix algebras. We also discuss the Z_3-graded
generalization of Grassmann algebras and their realization in generalized
exterior differential forms. A new type of gauge theory based on this
differential calculus is presented. Finally, a ternary generalization of
Clifford algebras is introduced, and an analog of Dirac's equation is
discussed, which can be diagonalized only after taking the cube of the
Z_3-graded generalization of Dirac's operator. A possibility of using these
ideas for the description of quark fields is suggested and discussed in the
last Section.Comment: 23 pages, dedicated to A. Trautman on the occasion of his 64th
birthda
Hypersymmetry: a Z_3-graded generalization of supersymmetry
We propose a generalization of non-commutative geometry and gauge theories
based on ternary Z_3-graded structures. In the new algebraic structures we
define, we leave all products of two entities free, imposing relations on
ternary products only. These relations reflect the action of the Z_3-group,
which may be either trivial, i.e. abc=bca=cab, generalizing the usual
commutativity, or non-trivial, i.e. abc=jbca, with j=e^{(2\pi i)/3}. The usual
Z_2-graded structures such as Grassmann, Lie and Clifford algebras are
generalized to the Z_3-graded case. Certain suggestions concerning the eventual
use of these new structures in physics of elementary particles are exposed
Ternary Syndrome Decoding with Large Weight
The Syndrome Decoding problem is at the core of many code-based
cryptosystems. In this paper, we study ternary Syndrome Decoding in large
weight. This problem has been introduced in the Wave signature scheme but has
never been thoroughly studied. We perform an algorithmic study of this problem
which results in an update of the Wave parameters. On a more fundamental level,
we show that ternary Syndrome Decoding with large weight is a really harder
problem than the binary Syndrome Decoding problem, which could have several
applications for the design of code-based cryptosystems
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