3,320 research outputs found
Thoughts on Eggert's Conjecture
Eggert's Conjecture says that if R is a finite-dimensional nilpotent
commutative algebra over a perfect field F of characteristic p, and R^{(p)} is
the image of the p-th power map on R, then dim_F R \geq p dim_F R^{(p)}.
Whether this very elementary statement is true is not known.
We examine heuristic evidence for this conjecture, versions of the conjecture
that are not limited to positive characteristic and/or to commutative R,
consequences the conjecture would have for semigroups, and examples that give
equality in the conjectured inequality. We pose several related questions, and
briefly survey the literature on the subject.Comment: 12 pages. Copy at http://math.berkeley.edu/~gbergman/papers may be
updated more frequently than arXiv copy. A few misstatements in the first
version have been corrected, and the wording improved in place
Conformal invariance in 2-dimensional discrete field theory
A discretized massless wave equation in two dimensions, on an appropriately
chosen square lattice, exactly reproduces the solutions of the corresponding
continuous equations. We show that the reason for this exact solution property
is the discrete analog of conformal invariance present in the model, and find
more general field theories on a two-dimensional lattice that exactly solve
their continuous limit equations. These theories describe in general
non-linearly coupled bosonic and fermionic fields and are similar to the
Wess-Zumino-Witten model.Comment: 18 pages, RevTeX, 2 figures included; revision of title and
introductio
Square Roots and Continuity in Strictly Linearly Ordered Semigroups on Real Intervals
In this article we show that the semigroup operation of a strictly linearly
ordered semigroup on a real interval is automatically continuous if each
element of the semigroup admits a square root. Hence, by a result of Acz\'el,
such a semigroup is isomorphic to an additive subsemigroup of the real numbers
The Quantum Query Complexity of Algebraic Properties
We present quantum query complexity bounds for testing algebraic properties.
For a set S and a binary operation on S, we consider the decision problem
whether is a semigroup or has an identity element. If S is a monoid, we
want to decide whether S is a group.
We present quantum algorithms for these problems that improve the best known
classical complexity bounds. In particular, we give the first application of
the new quantum random walk technique by Magniez, Nayak, Roland, and Santha
that improves the previous bounds by Ambainis and Szegedy. We also present
several lower bounds for testing algebraic properties.Comment: 13 pages, 0 figure
On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution
Let M denote the space of Borel probability measures on the real line. For
every nonnegative t we consider the transformation
defined for any given element in M by taking succesively the the (1+t) power
with respect to free additive convolution and then the 1/(1+t) power with
respect to Boolean convolution of the given element. We show that the family of
maps {\mathbb B_t|t\geq 0} is a semigroup with respect to the operation of
composition and that, quite surprisingly, every is a homomorphism
for the operation of free multiplicative convolution.
We prove that for t=1 the transformation coincides with the
canonical bijection discovered by Bercovici and
Pata in their study of the relations between infinite divisibility in free and
in Boolean probability. Here M_{inf-div} stands for the set of probability
distributions in M which are infinitely divisible with respect to free additive
convolution. As a consequence, we have that is infinitely
divisible with respect to free additive convolution for any for every in
M and every t greater than or equal to one.
On the other hand we put into evidence a relation between the transformations
and the free Brownian motion; indeed, Theorem 4 of the paper
gives an interpretation of the transformations as a way of
re-casting the free Brownian motion, where the resulting process becomes
multiplicative with respect to free multiplicative convolution, and always
reaches infinite divisibility with respect to free additive convolution by the
time t=1.Comment: 30 pages, minor changes; to appear in Indiana University Mathematics
Journa
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