Difference of Function on Vector Space over F

In [11], the definitions of forward difference, backward difference, and central difference as difference operations for functions on R were formalized. However, the definitions of forward difference, backward difference, and central difference for functions on vector spaces over F have not been formalized. In cryptology, these definitions are very important in evaluating the security of cryptographic systems [3], [10]. Differential cryptanalysis [4] that undertakes a general purpose attack against block ciphers [13] can be formalized using these definitions. In this article, we formalize the definitions of forward difference, backward difference, and central difference for functions on vector spaces over F. Moreover, we formalize some facts about these definitions.Arai Kenichi - Tokyo University of Science Chiba, JapanWakabayashi Ken - Shinshu University Nagano, JapanOkazaki Hiroyuki - Shinshu University Nagano, 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.E. Biham and A. Shamir. Differential cryptanalysis of DES-like cryptosystems. Lecture Notes in Computer Science, 537:2-21, 1991.E. Biham and A. Shamir. Differential cryptanalysis of the full 16-round DES. Lecture Notes in Computer Science, 740:487-496, 1993.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.X. Lai. Higher order derivatives and differential cryptoanalysis. Communications and Cryptography, pages 227-233, 1994.Bo Li, Yan Zhang, and Xiquan Liang. Difference and difference quotient. Formalized Mathematics, 14(3):115-119, 2006. doi:10.2478/v10037-006-0014-z.Michał Muzalewski and Wojciech Skaba. From loops to Abelian multiplicative groups with zero. Formalized Mathematics, 1(5):833-840, 1990.Hiroyuki Okazaki and Yasunari Shidama. Formalization of the data encryption standard. Formalized Mathematics, 20(2):125-146, 2012. doi:10.2478/v10037-012-0016-y.Beata Perkowska. Functional sequence from a domain to a domain. Formalized Mathematics, 3(1):17-21, 1992.Christoph Schwarzweller. The binomial theorem for algebraic structures. Formalized Mathematics, 9(3):559-564, 2001.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.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.Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171-175, 1992

An Inner Product on Adelic Measures: With Applications to the Arakelov-Zhang Pairing

We define an inner product on a vector space of adelic measures over a number field. We find that the norm induced by this inner product governs weak convergence at each place of $K$. The canonical adelic measure associated to a rational map is in this vector space, and the square of the norm of the difference of two such adelic measures is the Arakelov-Zhang pairing from arithmetic dynamics. We prove a sharp lower bound on the norm of adelic measures with points of small adelic height. We find that the norm of a canonical adelic measure associated to a rational map is commensurate with the Arakelov height on the space of rational functions with fixed degree. As a consequence, the Arakelov-Zhang pairing of two rational maps $f$ and $g$ can be bounded from below as a function of $g$

Difference Operator Approach to the Moyal Quantization and Its Application to Integrable Systems

Inspired by the fact that the Moyal quantization is related with nonlocal operation, I define a difference analogue of vector fields and rephrase quantum description on the phase space. Applying this prescription to the theory of the KP-hierarchy, I show that their integrability follows to the nature of their Wigner distribution. Furthermore the definition of the ``expectation value'' clarifies the relation between our approach and the Hamiltonian structure of the KP-hierarchy. A trial of the explicit construction of the Moyal bracket structure in the integrable system is also made.Comment: 19 pages, to appear in J. Phys. Soc. Jp

Evolutionary dynamics in heterogeneous populations: a general framework for an arbitrary type distribution

A general framework of evolutionary dynamics under heterogeneous populations is presented. The framework allows continuously many types of heterogeneous agents, heterogeneity both in payoff functions and in revision protocols and the entire joint distribution of strategies and types to influence the payoffs of agents. We clarify regularity conditions for the unique existence of a solution trajectory and for the existence of equilibrium. We confirm that equilibrium stationarity in general and equilibrium stability in potential games are extended from the homogeneous setting to the heterogeneous setting. In particular, a wide class of admissible dynamics share the same set of locally stable equilibria in a potential game through local maximization of the potential

Box splines and the equivariant index theorem

In this article, we start to recall the inversion formula for the convolution with the Box spline. The equivariant cohomology and the equivariant K-theory with respect to a compact torus G of various spaces associated to a linear action of G in a vector space M can be both described using some vector spaces of distributions, on the dual of the group G or on the dual of its Lie algebra. The morphism from K-theory to cohomology is analyzed and the multiplication by the Todd class is shown to correspond to the operator (deconvolution) inverting the semidiscrete convolution with a box spline. Finally, the multiplicities of the index of a G-transversally elliptic operator on M are determined using the infinitesimal index of the symbol.Comment: 44 page