273,646 research outputs found
Sums of Two Generalized Tetrahedral Numbers
Expressing whole numbers as sums of figurate numbers, including tetrahedral numbers, is a longstanding problem in number theory. Pollock's tetrahedral number conjecture states that every positive integer can be expressed as the sum of at most five tetrahedral numbers. Here we explore a generalization of this conjecture to negative indices. We provide a method for computing sums of two generalized tetrahedral numbers up to a given bound, and explore which families of perfect powers can be expressed as sums of two generalized tetrahedral numbers
Log-concavity and LC-positivity
A triangle of nonnegative numbers is LC-positive
if for each , the sequence of polynomials is
-log-concave. It is double LC-positive if both triangles and
are LC-positive. We show that if is LC-positive
then the log-concavity of the sequence implies that of the sequence
defined by , and if is
double LC-positive then the log-concavity of sequences and
implies that of the sequence defined by
. Examples of double LC-positive triangles
include the constant triangle and the Pascal triangle. We also give a
generalization of a result of Liggett that is used to prove a conjecture of
Pemantle on characteristics of negative dependence.Comment: 16 page
A Minimum problem for finite sets of real numbers with non-negative sum
Let and be two integers such that ; we denote by
[] the minimum [maximum] number of the non-negative
partial sums of a sum , where are
real numbers arbitrarily chosen in such a way that of them are
non-negative and the remaining are negative. Inspired by some interesting
extremal combinatorial sum problems raised by Manickam, Mikl\"os and Singhi in
1987 \cite{ManMik87} and 1988 \cite{ManSin88} we study the following two
problems:
\noindent {\it which are the values of and
for each and , ?}
\noindent {\it if is an integer such that , can we find real numbers , such that of
them are non-negative and the remaining are negative with , such that the number of the non-negative sums formed from these
numbers is exactly ?}
\noindent We prove that the solution of the problem is given by
and . We provide a partial
result of the latter problem showing that the answer is affirmative for the
weighted boolean maps. With respect to the problem such maps (that we
will introduce in the present paper) can be considered a generalization of the
multisets with . More precisely we
prove that for each such that there
exists a weighted boolean map having exactly positive boolean values.Comment: 15 page
Exact Dynamics of the SU(K) Haldane-Shastry Model
The dynamical structure factor of the SU(K) (K=2,3,4)
Haldane-Shastry model is derived exactly at zero temperature for arbitrary size
of the system. The result is interpreted in terms of free quasi-particles which
are generalization of spinons in the SU(2) case; the excited states relevant to
consist of K quasi-particles each of which is characterized by a
set of K-1 quantum numbers. Near the boundaries of the region where
is nonzero, shows the power-law singularity. It is
found that the divergent singularity occurs only in the lowest edges starting
from toward positive and negative q. The analytic result
is checked numerically for finite systems via exact diagonalization and
recursion methods.Comment: 35 pages, 3 figures, youngtab.sty (version 1.1
Untwisting information from Heegaard Floer homology
The unknotting number of a knot is the minimum number of crossings one must
change to turn that knot into the unknot. We work with a generalization of
unknotting number due to Mathieu-Domergue, which we call the untwisting number.
The p-untwisting number is the minimum number (over all diagrams of a knot) of
full twists on at most 2p strands of a knot, with half of the strands oriented
in each direction, necessary to transform that knot into the unknot. In
previous work, we showed that the unknotting and untwisting numbers can be
arbitrarily different. In this paper, we show that a common route for
obstructing low unknotting number, the Montesinos trick, does not generalize to
the untwisting number. However, we use a different approach to get conditions
on the Heegaard Floer correction terms of the branched double cover of a knot
with untwisting number one. This allows us to obstruct several 10 and
11-crossing knots from being unknotted by a single positive or negative twist.
We also use the Ozsv\'ath-Szab\'o tau invariant and the Rasmussen s invariant
to differentiate between the p- and q-untwisting numbers for certain p,q > 1.Comment: 21 pages, 11 figures; final version, accepted for publication in
Algebraic & Geometric Topolog
Some Generalizations and Properties of Balancing Numbers
The sequence of balancing numbers admits generalization in two different ways. The first way is through altering coefficients occurring in its binary recurrence sequence and the second way involves modification of its defining equation, thereby allowing more than one gap. The former generalization results in balancing-like numbers that enjoy all important properties of balancing numbers. The second generalization gives rise to gap balancing
numbers and for each particular gap, these numbers are realized in multiple sequences. The definition of gap balancing numbers allow further generalization resulting in higher order gap balancing numbers but unlike gap balancing numbers, these numbers are scarce, the existence of these numbers are often doubtful. The balancing zeta function—a variant of Riemann zeta function—permits analytic continuation to the entire complex plane, while the series converges to irrational numbers at odd negative integers. The periods of balancing numbers modulo positive integers exhibits many wonderful properties. It coincides with the modulus of congruence if calculated modulo any power of two. There are three known primes such that the period modulo any one of these primes is equal to the period modulo its square. The sequence of balancing numbers remains stable modulo half of the primes, while modulo other half, the sequence is unstable
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