关于正、余弦函数的一组恒等式

:利用第二类契贝谢夫多项式的性质得到了关于正余弦函数的一组有趣的恒等式 .     

On the MacWilliams identity

A generalization of the MacWilliams relation between the Hamming weight distributions in a linear subspace of a vector space over a finite field and its dual is deduced from an observation concerning the Fourier transforms of certain members of the group algebra of the additive group of the space.     

Apollonian structure in the Abelian sandpile

The Abelian sandpile process evolves configurations of chips on the integer lattice by toppling any vertex with at least 4 chips, distributing one of its chips to each of its 4 neighbors. When begun from a large stack of chips, the terminal state of the sandpile has a curious fractal structure which has remained unexplained. Using a characterization of the quadratic growths attainable by integer-superharmonic functions, we prove that the sandpile PDE recently shown to characterize the scaling limit of the sandpile admits certain fractal solutions, giving a precise mathematical perspective on the fractal nature of the sandpile.     

On the quintuple product identity

In this note we present a new proof of the quintuple product identity which is based on our study of order theta functions with characteristics and the identities they satisfy. In this context the quintuple product identity is another example of an identity which when phrased in terms of theta functions, rather than infinite products and sums, has a simpler form and is much less mysterious.

     

A generalized collapsing sandpile model

In this paper, we introduce a new model for the collapsing sandpile and we prove existence and uniqueness of a solution for the corresponding initial value problem. Moreover, we prove the convergence of the time-stepping approximation of the solution. We use subgradient flows for variational problems with time dependent gradient constraints. These gradient constraints are interpreted as the critical angles of the sandpile. In particular, our model produces an evolution in time of avalanches in a drying of a sandpile, rather than instantaneous collapse.     

On Weyl's identity

Summary The Weyl Identity (as orignally stated by H. Weyl and used by A. Winter) is not totally correct. Presented here is the correct version of the Identity, a concise proof of it and some applications. Entrata in Redazione il 28 febbraio 1970.     

On the combinatorics of the Pfaff identity

Recently, there has been a revival of interest in the Pfaff identity on hypergeometric series because of the specialization of Simons and a generalization of Munarini. We present combinatorial settings and interpretations of the specialization and the generalization; one is based on free Dyck paths and free Schröder paths, and the other relies on a correspondence of Foata and Labelle between the Meixner endofunctions and bicolored permutations, and an extension of the technique developed by Labelle and Yeh for the Pfaff identity. Applying the involution on weighted Schröder paths, we derive a formula for the Narayana numbers as an alternating sum of the Catalan numbers.     

Conformal covariance of the Abelian sandpile height one field

We study the scaling limit for the height one field of the two-dimensional Abelian sandpile model. The scaling limit for the covariance having height one at two macroscopically distant sites, more generally the centred height one joint moment of a finite number of macroscopically distant sites, is identified and shown to be conformally covariant. The result is based on a representation of the height one joint intensities that is close to a block-determinantal structure.     

On the Schinzel identity ofCyclotomic Polynomial

For integer n > 0, let n.(x) denote the nth cyclotomic polynomialwhere is a primitive nib root of unity and (j, n) denotes the greatest common divisor of j andn.Although 4.(x) is irreducible over the integers, 4.(x) may be reducible over certain quadraticfield. Let n > 1 be an odd square-free number. Aurifeuille and Le Lasseur[1] proved thatLater on Schinzel[2] proved that (1) can be improved aswhere m|n,(3) denotes the Jacobi symbol, andwhere both Pn,m(x) and Qn,m(x) are polynomials with i…     

On the non-persistence of Hamiltonian identity cycles

We study the leading term of the holonomy map of a perturbed plane polynomial Hamiltonian foliation. The non-vanishing of this term implies the non-persistence of the corresponding Hamiltonian identity cycle. We prove that this does happen for generic perturbations and cycles, as well for cycles which are commutators in Hamiltonian foliations of degree two. Our approach relies on the Chen's theory of iterated path integrals which we briefly resume.     

The identity component of the leaf preserving diffeomorphism group is perfect

It is shown that for any smooth foliated manifold the identity component of the group of all leaf preserving diffeomorphisms is perfect. This result generalizes in a sense a well-known theorem of Thurston.     

On the discrete version of Picone's identity

For a real number p with 1<p we consider the first eigenvalues of the p-Laplacian on graphs, and estimates for the solutions of p-Laplace equations on graphs. We provide a discrete version of Picone's identity and its application. More precisely, we prove a Barta-type inequality for graphs with boundary. Finally, we provide a discrete version of the anti-maximum principle.     

Prime rings whose units satisfy a group identity

Let R be a prime algebra over an infinite field and suppose that R is generated by its group of units U(R). We prove that if U(R) satisfies a group identity then R must be an integral domain.     

Group algebras with units satisfying a group identity

Let be the group algebra of a group over a field , and let be its group of units. A conjecture by Brian Hartley asserts that if is a torsion group and satisfies a group identity, then satisfies a polynomial identity. This was verified earlier in case is an infinite field. Here we modify the original proof so that it handles fields of all sizes.

     

On a partition identity

Two proofs are given, one combinatorial, the other by character theory, for the identity, ∏_λa₁! a₂! … a_n! = ∏_λ1^a₁2^a₂ … n^a_n, where λ ranges over all partitions λ = (1^a₁2^a₂ … n^a_n) of n. The two demonstrations yield a simple proof of the known formula, det²T(n) = [∏_λ1^a₁2^a₂ … n^a_n]², where T(n) is the matrix formed by the character table of S_n. Finally a sufficient condition is given so that the permanent of T(n) is zero.     

On a matroid identity

The purpose of this note is to prove an identity for generalized Tutte-Grothendieck invariants, at least two special cases of which have already proved to be of considerable use. In addition, one of these special cases is used to strengthen results of Lindström on the critical exponent of a representable matroid and the chromatic number of a regular matroid.     

Group algebras whose units satisfy a group identity

Let be the group algebra of a torsion group over an infinite field . Let be the group of units of . We prove that if satisfies a group identity, then satisfies a polynomial identity. This confirms a conjecture of Brian Hartley.