102 research outputs found
Factors of sums and alternating sums involving binomial coefficients and powers of integers
We study divisibility properties of certain sums and alternating sums
involving binomial coefficients and powers of integers. For example, we prove
that for all positive integers , , and any
nonnegative integer , there holds {align*} \sum_{k=0}^{n_1}\epsilon^k
(2k+1)^{2r+1}\prod_{i=1}^{m} {n_i+n_{i+1}+1\choose n_i-k} \equiv 0 \mod
(n_1+n_m+1){n_1+n_m\choose n_1}, {align*} and conjecture that for any
nonnegative integer and positive integer such that is odd, where .Comment: 14 pages, to appear in Int. J. Number Theor
Proof of two conjectures of Z.-W. Sun on congruences for Franel numbers
For all nonnegative integers n, the Franel numbers are defined as We confirm two conjectures of Z.-W. Sun on
congruences for Franel numbers: \sum_{k=0}^{n-1}(3k+2)(-1)^k f_k &\equiv 0
\pmod{2n^2}, \sum_{k=0}^{p-1}(3k+2)(-1)^k f_k &\equiv 2p^2 (2^p-1)^2
\pmod{p^5}, where n is a positive integer and p>3 is a prime.Comment: 8 pages, minor changes, to appear in Integral Transforms Spec. Func
Random walk generated by random permutations of {1,2,3, ..., n+1}
We study properties of a non-Markovian random walk , , evolving in discrete time on a one-dimensional lattice of
integers, whose moves to the right or to the left are prescribed by the
\text{rise-and-descent} sequences characterizing random permutations of
. We determine exactly the probability of finding
the end-point of the trajectory of such a
permutation-generated random walk (PGRW) at site , and show that in the
limit it converges to a normal distribution with a smaller,
compared to the conventional P\'olya random walk, diffusion coefficient. We
formulate, as well, an auxiliary stochastic process whose distribution is
identic to the distribution of the intermediate points , ,
which enables us to obtain the probability measure of different excursions and
to define the asymptotic distribution of the number of "turns" of the PGRW
trajectories.Comment: text shortened, new results added, appearing in J. Phys.
Omnibus Sequences, Coupon Collection, and Missing Word Counts
An {\it Omnibus Sequence} of length is one that has each possible
"message" of length embedded in it as a subsequence. We study various
properties of Omnibus Sequences in this paper, making connections, whenever
possible, to the classical coupon collector problem.Comment: 26 page
Avalanche Polynomials of some Families of Graphs
We study the abelian sandpile model on different families of graphs. We introduced the avalanche polynomial which enumerates the size of the avalanches triggered by the addition of a particle on a recurrent configuration. This polynomial is calculated for several families of graphs. In the case of the complete graph, the result involves some known result on Parking function
Diagnostic bactériologique rapide de Bacteridium anthracis par immunofluorescence
Michel Claude, Poussot A., Chabassol Claude, Foata D., Poulet P. Diagnostic bactériologique rapide de Bacteridium anthracis par immunofluorescence. In: Bulletin de l'Académie Vétérinaire de France tome 126 n°8, 1973. pp. 333-342
The critical Ising model via Kac-Ward matrices
The Kac-Ward formula allows to compute the Ising partition function on any
finite graph G from the determinant of 2^{2g} matrices, where g is the genus of
a surface in which G embeds. We show that in the case of isoradially embedded
graphs with critical weights, these determinants have quite remarkable
properties. First of all, they satisfy some generalized Kramers-Wannier
duality: there is an explicit equality relating the determinants associated to
a graph and to its dual graph. Also, they are proportional to the determinants
of the discrete critical Laplacians on the graph G, exactly when the genus g is
zero or one. Finally, they share several formal properties with the Ray-Singer
\bar\partial-torsions of the Riemann surface in which G embeds.Comment: 30 pages, 10 figures; added section 4.4 in version
A novel chromogenic medium for isolation of Pseudomonas aeruginosa from the sputa of cystic fibrosis patients
AbstractBackgroundA novel chromogenic medium for isolation and identification of Pseudomonas aeruginosa from sputa of cystic fibrosis (CF) patients was evaluated and compared with standard laboratory methods.MethodsOne hundred sputum samples from distinct CF patients were cultured onto blood agar (BA), Pseudomonas CN selective agar (CN) and a Pseudomonas chromogenic medium (PS-ID). All Gram-negative morphological variants from each medium were subjected to antimicrobial susceptibility testing, and identification using a combination of biochemical and molecular methods.ResultsP. aeruginosa was isolated from 62 samples after 72Â h incubation. Blood agar recovered P. aeruginosa from 56 samples (90.3%) compared with 59 samples (95.2%) using either CN or PS-ID. The positive predictive value of PS-ID (98.3%) was significantly higher than growth on CN (88.5%) for identification of P. aeruginosa (P<0.05).ConclusionsPS-ID is a promising medium allowing for the isolation and simultaneous identification of P. aeruginosa from sputa of CF patients
Manin matrices and Talalaev's formula
We study special class of matrices with noncommutative entries and
demonstrate their various applications in integrable systems theory. They
appeared in Yu. Manin's works in 87-92 as linear homomorphisms between
polynomial rings; more explicitly they read: 1) elements in the same column
commute; 2) commutators of the cross terms are equal: (e.g. ). We claim
that such matrices behave almost as well as matrices with commutative elements.
Namely theorems of linear algebra (e.g., a natural definition of the
determinant, the Cayley-Hamilton theorem, the Newton identities and so on and
so forth) holds true for them.
On the other hand, we remark that such matrices are somewhat ubiquitous in
the theory of quantum integrability. For instance, Manin matrices (and their
q-analogs) include matrices satisfying the Yang-Baxter relation "RTT=TTR" and
the so--called Cartier-Foata matrices. Also, they enter Talalaev's
hep-th/0404153 remarkable formulas: ,
det(1-e^{-\p}T_{Yangian}(z)) for the "quantum spectral curve", etc. We show
that theorems of linear algebra, after being established for such matrices,
have various applications to quantum integrable systems and Lie algebras, e.g
in the construction of new generators in (and, in general,
in the construction of quantum conservation laws), in the
Knizhnik-Zamolodchikov equation, and in the problem of Wick ordering. We also
discuss applications to the separation of variables problem, new Capelli
identities and the Langlands correspondence.Comment: 40 pages, V2: exposition reorganized, some proofs added, misprints
e.g. in Newton id-s fixed, normal ordering convention turned to standard one,
refs. adde
On -vectors satisfying the Kruskal-Katona inequalities
We present examples of flag homology spheres whose -vectors satisfy
the Kruskal-Katona inequalities. This includes several families of well-studied
simplicial complexes, including Coxeter complexes and the simplicial complexes
dual to the associahedron and to the cyclohedron. In these cases, we construct
explicit simplicial complexes whose -vectors are the -vectors in
question. In another direction, we show that if a flag -sphere has at
most vertices its -vector satisfies the Kruskal-Katona
inequalities. We conjecture that if is a flag homology sphere then
satisfies the Kruskal-Katona inequalities. This conjecture is
a significant refinement of Gal's conjecture, which asserts that such
-vectors are nonnegative.Comment: 18 pages; Our main result and conjectures have been strengthened.
Also we now have explicit constructions of simplicial complexes whose
-vectors are the -vectors in questio
- …