7,642 research outputs found

### UHF flows and the flip automorphism

A UHF flow is an infinite tensor product type action of the reals on a UHF
algebra $A$ and the flip automorphism is an automorphism of $A\otimes A$
sending $x\otimes y$ into $y\otimes x$. If $\alpha$ is an inner perturbation of
a UHF flow on $A$, there is a sequence $(u_n)$ of unitaries in $A\otimes A$
such that $\alpha_t\otimes \alpha_t(u_n)-u_n$ converges to zero and the flip is
the limit of \Ad u_n. We consider here whether the converse holds or not and
solve it with an additional assumption: If $A\otimes A\cong A$ and $\alpha$
absorbs any UHF flow $\beta$ (i.e., $\alpha\otimes\beta$ is cocycle conjugate
to $\alpha$), then the converse holds; in this case $\alpha$ is what we call a
universal UHF flow.Comment: 18 page

### Trace-scaling automorphisms of certain stable AF algebras

Trace scaling automorphisms of stable AF algebras with dimension group
totally ordered are outer conjugate if the scaling factors are the same (not
equal to one). This is an adaptation of a similar result for the AFD type
II_infty factor by Connes and extends the previous result for stable UHF
algebras.Comment: 12 pages, late

### Exploring Vacuum Structure around Identity-Based Solutions

We explore the vacuum structure in bosonic open string field theory expanded
around an identity-based solution parameterized by $a(>=-1/2)$. Analyzing the
expanded theory using level truncation approximation up to level 20, we find
that the theory has the tachyon vacuum solution for $a>-1/2$. We also find
that, at $a=-1/2$, there exists an unstable vacuum solution in the expanded
theory and the solution is expected to be the perturbative open string vacuum.
These results reasonably support the expectation that the identity-based
solution is a trivial pure gauge configuration for $a>-1/2$, but it can be
regarded as the tachyon vacuum solution at $a=-1/2$.Comment: 12 pages, 5 figures; new numerical data up to level (20,60) included;
Contribution to the proceedings of "Second International Conference on String
Field Theory and Related Aspects" (Steklov Mathematical Institute, Moscow,
Russia, April 12-19, 2009

### Continuous and discrete flows on operator algebras

Let $(N,\R,\theta)$ be a centrally ergodic W* dynamical system. When $N$ is
not a factor, we show that, for each $t\not=0$, the crossed product induced by
the time $t$ automorphism $\theta_t$ is not a factor if and only if there exist
a rational number $r$ and an eigenvalue $s$ of the restriction of $\theta$ to
the center of $N$, such that $rst=2\pi$. In the C* setting, minimality seems to
be the notion corresponding to central ergodicity. We show that if
$(A,\R,\alpha)$ is a minimal unital C* dynamical system and $A$ is either prime
or commutative but not simple, then, for each $t\not=0$, the crossed product
induced by the time $t$ automorphism $\alpha_t$ is not simple if and only if
there exist a rational number $r$ and an eigenvalue $s$ of the restriction of
$\alpha$ to the center of $A$, such that $rst=2\pi$.Comment: 7 page

### Computationally efficient algorithms for the two-dimensional Kolmogorov-Smirnov test

Goodness-of-fit statistics measure the compatibility of random samples against some theoretical or reference probability distribution function. The classical one-dimensional Kolmogorov-Smirnov test is a non-parametric statistic for comparing two empirical distributions which defines the largest absolute difference between the two cumulative distribution functions as a measure of disagreement. Adapting this test to more than one dimension is a challenge because there are 2^d-1 independent ways of ordering a cumulative distribution function in d dimensions. We discuss Peacock's version of the Kolmogorov-Smirnov test for two-dimensional data sets which computes the differences between cumulative distribution functions in 4n^2 quadrants. We also examine Fasano and Franceschini's variation of Peacock's test, Cooke's algorithm for Peacock's test, and ROOT's version of the two-dimensional Kolmogorov-Smirnov test. We establish a lower-bound limit on the work for computing Peacock's test of
Omega(n^2.lg(n)), introducing optimal algorithms for both this and Fasano and Franceschini's test, and show that Cooke's algorithm is not a faithful implementation of Peacock's test. We also discuss and evaluate parallel algorithms for Peacock's test

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