1,074,465 research outputs found
Complete list of Darboux Integrable Chains of the form
We study differential-difference equation of the form with unknown
depending on continuous and discrete variables and . Equation
of such kind is called Darboux integrable, if there exist two functions and
of a finite number of arguments , ,
, such that and , where
is the operator of total differentiation with respect to , and is
the shift operator: . Reformulation of Darboux integrability in
terms of finiteness of two characteristic Lie algebras gives an effective tool
for classification of integrable equations. The complete list of Darboux
integrable equations is given in the case when the function is of the
special form
A Geometric Approach to Noncommutative Principal Torus Bundles
A (smooth) dynamical system with transformation group is a
triple , consisting of a unital locally convex algebra
, the -torus and a group homomorphism
\alpha:\mathbb{T}^n\rightarrow\Aut(A), which induces a (smooth) continuous
action of on . In this paper we present a new, geometrically
oriented approach to the noncommutative geometry of principal torus bundles
based on such dynamical systems. Our approach is inspired by the classical
setting: In fact, after recalling the definition of a trivial noncommutative
principal torus bundle, we introduce a convenient (smooth) localization method
for noncommutative algebras and say that a dynamical system
is called a noncommutative principal
-bundle, if localization leads to a trivial noncommutative
principal -bundle. We prove that this approach extends the
classical theory of principal torus bundles and present a bunch of
(non-trivial) noncommutative examples.Comment: This paper is an extended version of "Smooth Localization in
Noncommutative Geometry", arxiv:1108.4294v1 [math.DG], 22 Aug 2011, with an
application to the noncommutative geometry of principal torus bundles. All
comments are welcome. 43 page
Ideal Tightly Couple (t,m,n) Secret Sharing
As a fundamental cryptographic tool, (t,n)-threshold secret sharing
((t,n)-SS) divides a secret among n shareholders and requires at least t,
(t<=n), of them to reconstruct the secret. Ideal (t,n)-SSs are most desirable
in security and efficiency among basic (t,n)-SSs. However, an adversary, even
without any valid share, may mount Illegal Participant (IP) attack or
t/2-Private Channel Cracking (t/2-PCC) attack to obtain the secret in most
(t,n)-SSs.To secure ideal (t,n)-SSs against the 2 attacks, 1) the paper
introduces the notion of Ideal Tightly cOupled (t,m,n) Secret Sharing (or
(t,m,n)-ITOSS ) to thwart IP attack without Verifiable SS; (t,m,n)-ITOSS binds
all m, (m>=t), participants into a tightly coupled group and requires all
participants to be legal shareholders before recovering the secret. 2) As an
example, the paper presents a polynomial-based (t,m,n)-ITOSS scheme, in which
the proposed k-round Random Number Selection (RNS) guarantees that adversaries
have to crack at least symmetrical private channels among participants before
obtaining the secret. Therefore, k-round RNS enhances the robustness of
(t,m,n)-ITOSS against t/2-PCC attack to the utmost. 3) The paper finally
presents a generalized method of converting an ideal (t,n)-SS into a
(t,m,n)-ITOSS, which helps an ideal (t,n)-SS substantially improve the
robustness against the above 2 attacks
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