4,845 research outputs found
A Family of Partially Ordered Sets with Small Balance Constant
Given a finite poset and two distinct elements and , we
let denote the fraction of linear
extensions of in which precedes . The balance constant
of is then defined by The
- conjecture asserts that whenever
is not a chain, but except from certain trivial examples it is not
known when equality occurs, or even if balance constants can approach .
In this paper we make some progress on the conjecture by exhibiting a
sequence of posets with balance constants approaching
, answering a question of
Brightwell. These provide smaller balance constants than any other known
nontrivial family.Comment: 11 pages, 4 figure
Multiplicative and Exponential Variations of Orthomorphisms of Cyclic Groups
An orthomorphism is a permutation of for which
is also a permutation on . Eberhard,
Manners, Mrazovi\'c, showed that the number of such orthomorphisms is
for odd and zero otherwise.
In this paper we prove two analogs of these results where is
replaced by (a "multiplicative orthomorphism") or with
(an "exponential orthomorphism"). Namely, we show that no
multiplicative orthomorphisms exist for but that exponential
orthomorphisms exist whenever is twice a prime such that is
squarefree. In the latter case we then estimate the number of exponential
orthomorphisms.Comment: 11 pages, 1 figur
Elliptic Curve Variants of the Least Quadratic Nonresidue Problem and Linnik's Theorem
Let and be -nonisogenous, semistable
elliptic curves over , having respective conductors and
and both without complex multiplication. For each prime , denote
by the trace of Frobenius. Under the
assumption of the Generalized Riemann Hypothesis (GRH) for the convolved
symmetric power -functions where , we prove an explicit result that can be stated
succinctly as follows: there exists a prime such that
and
This improves and
makes explicit a result of Bucur and Kedlaya.
Now, if is a subinterval with Sato-Tate measure and if
the symmetric power -functions are functorial and
satisfy GRH for all , we employ similar techniques to prove an
explicit result that can be stated succinctly as follows: there exists a prime
such that and
Comment: 30 page
On Logarithmically Benford Sequences
Let be an infinite subset, and let
be a sequence of nonzero real numbers indexed by
such that there exist positive constants for which
for all . Furthermore, let be defined by for each , and suppose the 's are equidistributed in with
respect to a continuous, symmetric probability measure . In this paper, we
show that if is not too sparse, then the
sequence fails to obey Benford's Law with respect
to arithmetic density in any sufficiently large base, and in fact in any base
when is a strictly convex function of . Nonetheless,
we also provide conditions on the density of
under which the sequence satisfies Benford's Law
with respect to logarithmic density in every base.
As an application, we apply our general result to study Benford's Law-type
behavior in the leading digits of Frobenius traces of newforms of positive,
even weight. Our methods of proof build on the work of Jameson, Thorner, and
Ye, who studied the particular case of newforms without complex multiplication.Comment: 10 page
Linnik's Theorem for Sato-Tate Laws on Elliptic Curves with Complex Multiplication
Let be an elliptic curve with complex multiplication (CM), and
for each prime of good reduction, let
denote the trace of Frobenius. By the Hasse bound, for a unique . In this paper, we prove that
the least prime such that
satisfies where is
the conductor of and the implied constant and exponent are absolute
and effectively computable. Our result is an analogue for CM elliptic curves of
Linnik's Theorem for arithmetic progressions, which states that the least prime
for satisfies for an absolute
constant .Comment: 11 pages; made minor modification
Video TFRC
TCP-friendly rate control (TFRC) is a congestion control technique that trade-offs responsiveness to the network
conditions for a smoother throughput variation. We take
advantage of this trade-off by calculating the rate gap between the theoretical TCP throughput and the smoothed TFRC throughput. Any rate gain from this rate gap is then
opportunistically used for video coding. We define a frame
complexity measure to determine the additional rate to be used from the rate gap and then perform a rate negotiation to determine the target rate for the encoder and the final sending rate. Results show that although this method has a more aggressive sending rate compared to TFRC, it is still TCP friendly, does not contribute too much to network congestion and achieves a reasonable video quality gain over the conventional method
Interpretation of Angular Distributions of -boson Production at Colliders
High precision data of dilepton angular distributions in
production were reported recently by the CMS Collaboration covering a broad
range of the dilepton transverse momentum, , up to GeV.
Pronounced dependencies of the and parameters,
characterizing the and angular distributions, were
found. Violation of the Lam-Tung relation was also clearly observed. We show
that the dependence of allows a determination of the relative
contributions of the annihilation versus the Compton process.
The violation of the Lam-Tung relation is attributed to the presence of a
non-zero component of the axis in the direction normal to the
"hadron plane" formed by the colliding hadrons. The magnitude of the violation
of the Lam-Tung relation is shown to reflect the amount of this
`non-coplanarity". The observed dependencies of and from
the CMS and the earlier CDF data can be well described using this approach.Comment: 5 pages, 3 figure
On the Rotational Invariance and Non-Invariance of Lepton Angular Distributions in Drell-Yan and Quarkonium Production
Several rotational invariant quantities for the lepton angular distributions
in Drell-Yan and quarkonium production were derived several years ago, allowing
the comparison between different experiments adopting different reference
frames. Using an intuitive picture for describing the lepton angular
distribution in these processes, we show how the rotational invariance of these
quantities can be readily obtained. This approach can also be used to determine
the rotational invariance or non-invariance of various quantities specifying
the amount of violation for the Lam-Tung relation. While the violation of the
Lam-Tung relation is often expressed by frame-dependent quantities, we note
that alternative frame-independent quantities are preferred.Comment: 4 pages, 1 figure, revised version, to appear in Phys. Lett
Optimal use of Charge Information for the HL-LHC Pixel Detector Readout
The pixel detectors for the High Luminosity upgrades of the ATLAS and CMS
detectors will preserve digitized charge information in spite of extremely high
hit rates. Both circuit physical size and output bandwidth will limit the
number of bits to which charge can be digitized and stored. We therefore study
the effect of the number of bits used for digitization and storage on single
and multi-particle cluster resolution, efficiency, classification, and particle
identification. We show how performance degrades as fewer bits are used to
digitize and to store charge. We find that with limited charge information (4
bits), one can achieve near optimal performance on a variety of tasks.Comment: 27 pages, 20 figure
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