In this note, we consider the number of k’s in all the partitions of n in order to provide a new proof of a classical identity involving Euler’s partition function p(n) and the sum of the positive ...divisors function a(n). New relations connecting classical functions of multiplicative number theory with the partition function p(n) from additive number theory are introduced in this context. The fascinating feature of these relations is their common nature. A new identity for the number of 1’s in all the partitions of n is derived in this context.
With the expression ''structure of an odd integer'' we mean the set of properties of the integer n which specifies the odd integer 2n+1 and brings about its behaviour. These properties of n, for both ...composite and prime numbers, are expounded in detail, together with their geometrical implications. In this context, a set, in a two dimensional space, where all the composite odd integers in 2a+1, 2n+1 are localized, is illustrated.
In this paper we present an a-posteriori KAM theorem for the existence of an (n−d)-parameter family of d-dimensional isotropic invariant tori with Diophantine frequency vector ω∈Rd, of type (γ,τ), ...for n degrees of freedom Hamiltonian systems with (n−d) independent first integrals in involution. If the first integrals induce a Hamiltonian action of the (n−d)-dimensional torus, then we can produce n-dimensional Lagrangian tori with frequency vector of the form (ω,ωp), with ωp∈Rn−d. In the light of the parameterization method, we design a (modified) quasi-Newton method for the invariance equation of the parameterization of the torus, whose proof of convergence from an initial approximation, and under appropriate non-degeneracy conditions, is the object of this paper. We present the results in the analytic category, so the initial torus is real-analytic in a certain complex strip of width ρ, and the corresponding error in the functional equation is ɛ. We heavily use geometric properties and the so called automatic reducibility to deal directly with the functional equation and get convergence if γ−2ρ−2τ−1ɛ is small enough, in contrast with most of KAM results based on the parameterization method, that get convergence if γ−4ρ−4τɛ is small enough. The approach is suitable to perform computer assisted proofs.
•KAM theorem for Hamiltonian Systems.•First integrals reduction.•Tight constants.
Let S be a (multiplicative) commutative semigroup with 0, Z(S) the set of zero-divisors of S, and n a positive integer. The zero-divisor graph of S is the (simple) graph
with vertices
and distinct ...vertices x and y are adjacent if and only if xy = 0. In this paper, we introduce and study the n-zero-divisor graph of S as the (simple) graph
with vertices
and distinct vertices x and y are adjacent if and only if xy = 0. Thus each
is an induced subgraph of
We pay particular attention to
and the case when S is a commutative ring with
We also consider several other types of "n-zero-divisor" graphs and commutative rings such that some power of every element (or zero-divisor) is idempotent.
We show that the smooth theta divisors of general principally polarised abelian varieties can be chosen as irreducible algebraic representatives of the coefficients of the Chern-Dold character in ...complex cobordisms and describe the action of the Landweber-Novikov operations on them. We introduce a quantisation of the complex cobordism theory with the dual Landweber-Novikov algebra as the deformation parameter space and show that the Chern-Dold character can be interpreted as the composition of quantisation and dequantisation maps. Some smooth real-analytic representatives of the cobordism classes of theta divisors are described in terms of the classical Weierstrass elliptic functions. The link with the Milnor-Hirzebruch problem about possible characteristic numbers of irreducible algebraic varieties is discussed.
We develop Milne’s theory of Lefschetz motives for general adequate equivalence relations and over a not necessarily algebraically closed base field. The corresponding categories turn out to enjoy ...all properties predicted by standard and less standard conjectures, in a stronger way: algebraic and numerical equivalences agree in this context. We also compute the Tannakian group associated to a Weil cohomology in a different and more conceptual way than Milne’s case-by-case approach.
Let
R be a commutative ring with
Nil
(
R
)
its ideal of nilpotent elements,
Z
(
R
)
its set of zero-divisors, and
Reg
(
R
)
its set of regular elements. In this paper, we introduce and investigate ...the
total graph of
R, denoted by
T
(
Γ
(
R
)
)
. It is the (undirected) graph with all elements of
R as vertices, and for distinct
x
,
y
∈
R
, the vertices
x and
y are adjacent if and only if
x
+
y
∈
Z
(
R
)
. We also study the three (induced) subgraphs
Nil
(
Γ
(
R
)
)
,
Z
(
Γ
(
R
)
)
, and
Reg
(
Γ
(
R
)
)
of
T
(
Γ
(
R
)
)
, with vertices
Nil
(
R
)
,
Z
(
R
)
, and
Reg
(
R
)
, respectively.
Tropical fans and normal complexes Nathanson, Anastasia; Ross, Dustin
Advances in mathematics (New York. 1965),
05/2023, Volume:
420
Journal Article
Peer reviewed
Open access
Associated to any divisor in the Chow ring of a simplicial tropical fan, we construct a family of polytopal complexes, called normal complexes, which we propose as an analogue of the well-studied ...notion of normal polytopes from the setting of complete fans. We describe certain closed convex polyhedral cones of divisors for which the “volume” of each divisor in the cone—that is, the degree of its top power—is equal to the volume of the associated normal complexes. For the Bergman fan of any matroid with building set, we prove that there exists an open family of such cones of divisors with nonempty interiors. We view the theory of normal complexes developed in this paper as a polytopal model underlying the combinatorial Hodge theory pioneered by Adiprasito, Huh, and Katz.
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