5,302 research outputs found
On minors of maximal determinant matrices
By an old result of Cohn (1965), a Hadamard matrix of order n has no proper
Hadamard submatrices of order m > n/2. We generalise this result to maximal
determinant submatrices of Hadamard matrices, and show that an interval of
length asymptotically equal to n/2 is excluded from the allowable orders. We
make a conjecture regarding a lower bound for sums of squares of minors of
maximal determinant matrices, and give evidence in support of the conjecture.
We give tables of the values taken by the minors of all maximal determinant
matrices of orders up to and including 21 and make some observations on the
data. Finally, we describe the algorithms that were used to compute the tables.Comment: 35 pages, 43 tables, added reference to Cohn in v
Probabilistic lower bounds on maximal determinants of binary matrices
Let be the maximal determinant for -matrices, and be the ratio of
to the Hadamard upper bound. Using the probabilistic method,
we prove new lower bounds on and in terms of
, where is the order of a Hadamard matrix and is maximal
subject to . For example, if , and if . By a recent result of Livinskyi, as ,
so the second bound is close to for large . Previous
lower bounds tended to zero as with fixed, except in the
cases . For , our bounds are better for all
sufficiently large . If the Hadamard conjecture is true, then , so
the first bound above shows that is bounded below by a positive
constant .Comment: 17 pages, 2 tables, 24 references. Shorter version of
arXiv:1402.6817v4. Typos corrected in v2 and v3, new Lemma 7 in v4, updated
references in v5, added Remark 2.8 and a reference in v6, updated references
in v
Some binomial sums involving absolute values
We consider several families of binomial sum identities whose definition
involves the absolute value function. In particular, we consider centered
double sums of the form obtaining new results in the cases . We show that there is a close connection between these double sums in the
case and the single centered binomial sums considered by Tuenter.Comment: 15 pages, 19 reference
Bounds on minors of binary matrices
We prove an upper bound on sums of squares of minors of {+1, -1} matrices.
The bound is sharp for Hadamard matrices, a result due to de Launey and Levin
(2009), but our proof is simpler. We give several corollaries relevant to
minors of Hadamard matrices, and generalise a result of Turan on determinants
of random {+1,-1} matrices.Comment: 9 pages, 1 table. Typo corrected in v2. Two references and Theorem 2
added in v
Conformal anomaly of Wilson surface observables - a field theoretical computation
We make an exact field theoretical computation of the conformal anomaly for
two-dimensional submanifold observables. By including a scalar field in the
definition for the Wilson surface, as appropriate for a spontaneously broken
A_1 theory, we get a conformal anomaly which is such that N times it is equal
to the anomaly that was computed in hep-th/9901021 in the large N limit and
which relied on the AdS-CFT correspondence. We also show how the spherical
surface observable can be expressed as a conformal anomaly.Comment: 18 pages, V3: an `i' dropped in the Wilson surface, overall
normalization and misprints corrected, V4: overall normalization factor
corrected, references adde
Stochastic field theory for a Dirac particle propagating in gauge field disorder
Recent theoretical and numerical developments show analogies between quantum
chromodynamics (QCD) and disordered systems in condensed matter physics. We
study the spectral fluctuations of a Dirac particle propagating in a finite
four dimensional box in the presence of gauge fields. We construct a model
which combines Efetov's approach to disordered systems with the principles of
chiral symmetry and QCD. To this end, the gauge fields are replaced with a
stochastic white noise potential, the gauge field disorder. Effective
supersymmetric non-linear sigma-models are obtained. Spontaneous breaking of
supersymmetry is found. We rigorously derive the equivalent of the Thouless
energy in QCD. Connections to other low-energy effective theories, in
particular the Nambu-Jona-Lasinio model and chiral perturbation theory, are
found.Comment: 4 pages, 1 figur
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