100 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
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
Parasitofauna study of the brown trout, Salmo trutta (Pisces, Teleostei) from Corsica (Mediterranean island) rivers
Corsica is a mediterranean island characterised by a great number of rivers. Salmonides are the main fishes which populate these rivers. Very appreciated by fishermen, Salmonides are represented by three species in the insular hydrographical network, among which an autochthonous species, the brown trout (Salmo trutta). In the present work, we have analysed the parasitofauna of this species. According to our knowledge, this research has never been carried out in Corsica. In a first step, we drew up an inventory of the parasites found in this freshwater fish. In a second step, we studied the differences which appeared in the composition of parasite communities of this species
Parasitofauna study of the brown trout,
Corsica is a mediterranean island characterised by a great number of rivers. Salmonides are the main fishes which populate these rivers. Very appreciated by fishermen, Salmonides are represented by three species in the insular hydrographical network, among which an autochthonous species, the brown trout (Salmo trutta). In the present work, we have analysed the parasitofauna of this species. According to our knowledge, this research has never been carried out in Corsica. In a first step, we drew up an inventory of the parasites found in this freshwater fish. In a second step, we studied the differences which appeared in the composition of parasite communities of this species
Ultrastructure of the spermatozoon of Calliobothrium verticillatum (Cestoda, Tetraphyllidea, Oncobothriidae)
International audienceThe ultrastructure of the spermatozoon of Calliobothrium verticillatum (Cestoda, Tetraphyllidea, Oncobothriidae), parasite of the smoothhound shark, Mustelus mustelus L. (Pisces, Carcharhiniformes), was studied by transmission electron microscopy. This spermatozoon presents five regions characterized by several ultrastructural elements: an apical cone, a crested body, two axonemes of 9 + 1 pattern, electron-dense granules, a nucleus and cortical microtubules. In the present study, three of these features were the subject of a detailed attention. The first is the presence of two axonemes, which confirms that the Tetraphyllidea, Oncobothriidae possess two axonemes whereas the Tetraphyllidea, Phyllobothriidae possess only one axoneme. The second is the presence of one crested body, a criterion homogeneous in the Tetraphyllidea but heterogeneous among the different orders of Cestoda. The third is the number and the disposition of cortical microtubules. These three criteria seem to be interesting for phylogeny
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