Sequence Of Nonnegative Integers Research Articles

Finite-dimensional algebras with smallest resolutions of simple modules

Let $X$ be a finitely generated left module over a left artinian ring $R$, and let $p(X)=\{l_i\}$ be the infinite sequence of nonnegative integers where $l_i$ is the length of the $i$-th term of the minimal projective resolution of $X$. We introduce a preorder relation $\le$ on the set $\{p(X)\}$ and characterize the elementary finite-dimensional algebras $\Lambda$ with the following property. Let $S$ be a simple $\Lambda$-module, and let $T$ be a finitely generated module over an arbitrary left artinian ring $R$. If the projective dimension of $S$ does not exceed the projective dimension of $T$, then $p(S)\le p(T)$. We characterize the indicated algebras by quivers with relations.

Derived endo-discrete artin algebras

Let ${\mit\Lambda} $ be an artin algebra. We prove that for each sequence $(h_{i})_{i\in \mathbb{Z}}$ of non-negative integers there are only a finite number of isomorphism classes of indecomposables $X\in \mathcal{D}^{\rm b}({\mit\Lambda} ),$ the boun

Certain Classes of Automorphisms of Infinite Rank Affine Lie Algebras#

ABSTRACT An infinite rank affine Lie algebra 𝔤 is a Kac-Moody algebra associated with an infinite affine matrix. For each nonnegative integer l, 𝔤 contains a subalgebra 𝔤l which is a classical finite dimensional simple Lie algebra, 𝔤 0 ⊂ 𝔤 1 ⊂ ± b ··· and 𝔤 is the inductive limit of the set {𝔤i, i = 0, 1,…} of these subalgebras. In the present article, we will determine all automorphisms of 𝔤 leaving 𝔤ni invariant for each {ni} in a set {ni}, where the set {ni, i = 1, 2,…} is any given nonnegative integer sequence with n1 < n2 < ···. These automorphisms are generalizations of automorphisms of classical finite dimensional Lie algebras.

On integer sequences whose first iterates are linear

In this paper we discuss the functional equation a(a(n)) = dn, where (a(n))n≥0 is an increasing sequence of non-negative integers. Mallows observed this equation has a unique solution for d = 2, and Propp observed the same thing for d = 3. We show that the equation has uncountably many solutions for d ≥ 4. Further, we give a complete description for the lexicographically least such sequence, showing that the first difference sequence can be generated by a finite automaton.

Generating Function for $K$-Restricted Jagged Partitions

We present a natural extension of Andrews' multiple sums counting partitions of the form $(\lambda_1,\cdots,\lambda_m)$ with $\lambda_i\geq \lambda_{i+k-1}+2$. The multiple sum that we construct is the generating function for the so-called $K$-restricted jagged partitions. A jagged partition is a sequence of non-negative integers $(n_1,n_2,\cdots , n_m)$ with $n_m\geq 1$ subject to the weakly decreasing conditions $n_i\geq n_{i+1}-1$ and $n_i\geq n_{i+2}$. The $K$-restriction refers to the following additional conditions: $n_i \geq n_{i+K-1} +1$ or $n_i = n_{i+1}-1 = n_{i+K-2}+1= n_{i+K-1}$. The corresponding generalization of the Rogers-Ramunjan identities is displayed, together with a novel combinatorial interpretation.

Artinian Gorenstein algebras of embedding dimension four: components of [formula omitted] for [formula omitted

A Gorenstein sequence H is a sequence of nonnegative integers H = ( 1 , h 1 , … , h j = 1 ) symmetric about j / 2 that occurs as the Hilbert function in degrees less or equal j of a standard graded Artinian Gorenstein algebra A = R / I , where R is a polynomial ring in r variables and I is a graded ideal. The scheme P Gor ( H ) parametrizes all such Gorenstein algebra quotients of R having Hilbert function H and it is known to be smooth when the embedding dimension satisfies h 1 ⩽ 3 . The authors give a structure theorem for such Gorenstein algebras of Hilbert function H = ( 1 , 4 , 7 , … ) when R = K [ w , x , y , z ] and I 2 ≅ 〈 wx , wy , wz 〉 (Theorems 3.7 and 3.9). They also show that any Gorenstein sequence H = ( 1 , 4 , a , … ) , a ⩽ 7 satisfies the condition Δ H ⩽ j / 2 is an O -sequence (Theorems 4.2 and 4.4). Using these results, they show that if H = ( 1 , 4 , 7 , h , b , … , 1 ) is a Gorenstein sequence satisfying 3 h - b - 17 ⩾ 0 , then the Zariski closure C ( H ) ¯ of the subscheme C ( H ) ⊂ P Gor ( H ) parametrizing Artinian Gorenstein quotients A = R / I with I 2 ≅ 〈 wx , wy , wz 〉 is a generically smooth component of P Gor ( H ) (Theorem 4.6). They show that if in addition 8 ⩽ h ⩽ 10 , then such P Gor ( H ) have several irreducible components (Theorem 4.9). M. Boij and others had given previous examples of certain P Gor ( H ) having several components in embedding dimension four or more (Pacific J. Math. 187(1) (1999) 1–11). The proofs use properties of minimal resolutions, the smoothness of P Gor ( H ′ ) for embedding dimension three (J.O. Kleppe, J. Algebra 200 (1998) 606–628), and the Gotzmann Hilbert scheme theorems (Math. Z. 158(1) (1978) 61–70).

The Berman conjecture is true for nilpotent extensions of regular semigroups

Let S be any semigroup. A term operation of S is an n-ary operation of S which is induced by some nonempty word w by substitution of letters. Such an operation is essentially n-ary if it depends on all of its variables. For n ≥ 1, we denote the set of all essentially n-ary term operations of S by ES n , while ES 0 is the set of all constant unary term operations of S. Let pn(S) = |ES n | for all n ≥ 0. It is easy to see that the sequence pn(S), n ≥ 0, consists entirely of finite cardinals if and only if S generates a locally finite semigroup variety. Of course, all these concepts can be generalized for arbitrary algebras, cf. the survey paper by Gratzer and Kisielewicz [7]. The theory of pn-sequence of general algebras was founded by E. Marczewski and his ‘Wroc law School’ back in the sixties (see [9]). They identified as the main goal of this theory to characterize all sequences of non-negative integers which are representable as the pn-sequence of some algebra (or some particular kind of algebra). Later on, most of the research was limited to finite algebras. One of the most intriguing hypotheses in this direction was given by J. Berman, who conjectured in [1] that the pn-sequence of any finite algebra is either bounded above by a constant, or eventually strictly increasing. While this assertion, referred to as the Berman conjecture, is easily shown to be true for finite monoids, groups, rings, modules, lattices, Boolean algebras, etc., R. Willard refuted this conjecture in [10] by constructing a very nice counterexample, a 4-element algebra with finitely many basic operations whose pn-sequence is (0, 3, 2, 5, 4, 7, 6, 9, 8, . . .). On the other hand, there are several papers (co-authored by the first named author of this note) investigating the (still open) question whether the Berman conjecture holds in the class of all finite semigroups. Namely, in [5] all finite semigroups whose pn-sequences have constant bounds are determined. In [6],

Particle Content of the $(\boldsymbol k, 3)$-configurations

For all k , we construct a bijection between the set of sequences of non-negative integers \mathbf a = (a_i)_{i\in \mathbf Z_{≥0}} satisfying a_i + a_{i+1} + a_{i+2} ≤ k and the set of rigged partitions (λ,ρ) . Here λ = (λ_1, . . . , λ_n) is a partition satisfying k ≥ λ_1 ≥ · · · ≥ λ_n ≥ 1 and ρ = (ρ_1, . . . , ρ_n ) \in \mathbf Z^n_{≥0} is such that ρ_j ≥ ρ_{j+1} if λ_j = λ_{j+1} . One can think of λ as the particle content of the configuration \mathbf a and ρ_j as the energy level of the j -th particle, which has the weight λ_j . The total energy \sum_i ia_i is written as the sum of the two-body interaction term \sum_{j&lt;j'} A_{λ_j,λ_j'} and the free part \sum_j ρ_j . The bijection implies a fermionic formula for the one-dimensional configuration sums \sum_{\mathbf a}q^{\sum_i ia_i} . We also derive the polynomial identities which describe the configuration sums corresponding to the configurations with prescribed values for a_0 and a_1 , and such that a_i = 0 for all i &gt; N .

Statistical Convergent Topological Sequence Entropy Maps of the Circle

A continuous map f of the interval is chaotic iff there is an increasing of nonnegative integers T such that the topological sequence entropy of f relative to T, hT(f), is positive [4]. On the other hand, for any increasing sequence of nonnegative integers T there is a chaotic map f of the interval such that hT(f)=0 [7]. We prove that the same results hold for maps of the circle. We also prove some preliminary results concerning statistical convergent topological sequence entropy for maps of general compact metric spaces.

The Family G T of Graded Artinian Quotients of k[x, y] of Given Hilbert Function#

Let R = k[x, y] be the polynomial ring over an algebraically closed field k. Let Tbe a sequence of nonnegative integers that occurs as the Hilbert function of a length-nArtinian quotient of R. The nonsingular projective variety G T parametrizes all graded ideals Iof R = k[x, y] for which the Hilbert function H(R/I) = T(see Iarrobino, A. (1977). Punctual Hilbert Schemes. Mem. Amer. Math. Soc. Vol. 10, #188, Providence: American Mathematical Society). We show that G T is birational to a certain product SGrass(T) of small Grassmann varieties (Proposition 3.15), and that over k = ℂ the birational map induces an additive ℤ-isomorphism τ : H*(G T ) → H*(SGrass(T)) of homology groups (Theorem 3.29). The map τ is not usually an isomorphism of rings. We determine the ring H*(G T ) when where G T ⊂ ℙυ×ℙ j (Theorem 4.5). In this case G T is a desingularisation of the υ-secant bundle Sec(υ, j) of the degree jrational normal curve. We use this ring H*(G T ) to determine the number of ideals satisfying an intersection of ramification conditions at different points (Example 4.6). We also determine the classes in H*(G T ) of the pullback of the singular locus of Sec(υ, j) and of the pullbacks of the higher singular loci (Theorem 4.12). Let Ebe a monomial ideal of R, satisfying H(R/E) = T, where ∣ T ∣ = n: it corresponds to a partition P(E) of nhaving diagonal lengths T. A main tool is that the family of graded ideals having initial monomials Eis a cell 𝕍(E). We connect these cells to ramification conditions, using the Wronskian determinant, and to a “hook code” for P(E). Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.

Some problems on the global attractivity of linear nonautonomous difference equations

This paper is to solve several problems on the global attractivity of the zero solution of the nonautonomous difference equation \(x_{n + 1} - x_n + P_n x_{n - k_n } = 0,n \in \mathbb{Z} (0)\), where P n is a sequence of nonnegative real numbers, and k n is a sequence of nonnegative integers with n − k n →∞ as n → ∞.

On regular polytope numbers

Lagrange proved a theorem which states that every nonnegative integer can be written as a sum of four squares. This result can be generalized as the polygonal number theorem and the Hilbert-Waring problem. In this paper, we shall generalize Lagrange’s sum of four squares theorem further. To each regular polytope $V$ in a Euclidean space, we will associate a sequence of nonnegative integers which we shall call regular polytope numbers, and consider the problem of finding the order $g(V)$ of the set of regular polytope numbers associated to $V$.

Quadrature formulae connected to σ-orthogonal polynomials

Let d λ( t) be a given nonnegative measure on the real line R , with compact or infinite support, for which all moments μ k= ∫ R t k dλ(t), k=0,1,…, exist and are finite, and μ 0>0. Quadrature formulas of Chakalov–Popoviciu type with multiple nodes ∫ R f(t) dλ(t)= ∑ ν=1 n ∑ i=0 2s ν A i,νf (i)(τ ν)+R(f), where σ= σ n =( s 1, s 2,…, s n ) is a given sequence of nonnegative integers, are considered. A such quadrature formula has maximum degree of exactness d max=2∑ ν=1 n s ν +2 n−1 if and only if ∫ R ∏ ν=1 n (t−τ ν) 2s ν+1 t k dλ(t)=0, k=0,1,…,n−1. The proof of the uniqueness of the extremal nodes τ 1, τ 2,…, τ n was given first by Ghizzetti and Ossicini (Rend. Mat. 6(8) (1975) 1–15). Here, an alternative simple proof of the existence and the uniqueness of such quadrature formulas is presented. In a study of the error term R( f), an influence function is introduced, its relevant properties are investigated, and in certain classes of functions the error estimate is given. A numerically stable iterative procedure, with quadratic convergence, for determining the nodes τ ν, ν=1,2,…,n , which are the zeros of the corresponding σ-orthogonal polynomial, is presented. Finally, in order to show a numerical efficiency of the proposed procedure, a few numerical examples are included.

A General Comparison Result for Higher Order Nonlinear Difference Equations With Deviating Arguments

The authors consider m -th order nonlinear difference equations of the form D m p x n + i h j ( n , x s j ( n ) )=0, j =1,2,( E j ) where m S 1, n ] N 0 ={0,1,2,…}, D 0 p x n = x n , D i p x n = p n i j ( D i m 1 p x n ), i =1,2,…, m , j x n = x n +1 m x n , { p n 1 },…,{ p n m } are real sequences, p n i >0, and p n m L 1. In Eq. ( E 1 ) , p = a and p n i = a n i , and in Eq. ( E 2 ) , p = A and p n i = A n i , i =1,2,…, m . Here, { s j ( n )} are sequences of nonnegative integers with s j ( n ) M X as n M X , and h j : N 0 2 R M R is continuous with uh j ( n , u )>0 for u p 0. They prove a comparison result on the oscillation of solutions and the asymptotic behavior of nonoscillatory solutions of Eq. ( E j ) for j =1,2. Examples illustrating the results are also included.

On an additive property of sequences of nonnegative integers

On an additive property of sequences of nonnegative integers

Projective representations of symmetric groups via Sergeev duality

In this article, we determine the irreducible projective representations of the symmetric group Sd and the alternating group Ad over an algebraically closed field of characteristic p 6= 2. These matters are well understood in the case p = 0, thanks to the fundamental work of Schur [24] in 1911, as well as the much more recent work of Nazarov [19, 20], Sergeev [25, 26] and others. So the focus here is primarily on the case of positive characteristic, where surprisingly little is known as yet. In particular, we obtain a natural combinatorial labelling of the irreducibles in terms of a certain set RPp(d) of restricted p-strict partitions of d. Such partitions arose recently in work of Kashiwara, Miwa, Peterson and Yung [11] and Leclerc and Thibon [14] on the q-deformed Fock space of the affine Kac-Moody algebra of type A p−1. Leclerc and Thibon proposed that RPp(d) should label the irreducible projective representations in some natural way, and we show here how this can be done. Note that for p = 3, 5, the labelling problem was solved in [1, 3], while if p = 2 all projective representations of Sd and Ad are linear so do not need to be considered further here. To be more precise, recall that λ is a partition of d if λ = (λ1, λ2, . . . ) is a non-increasing sequence of non-negative integers summing to d. Call λ p-strict if in addition

Sets of random variables with a given uncorrelation structure

Let ξ 1,…, ξ n be random variables having finite expectations. Denote i k ≔ # (j 1,…,j k) : 1⩽j 1<⋯<j k⩽n and E ∏ l=1 k ξ j l = ∏ l=1 k E ξ j l , k=2,…,n. The finite sequence ( i 2,…, i n ) is called the uncorrelation structure of ξ 1,…, ξ n . It is proved that for any given sequence of nonnegative integers ( i 2,…, i n ) satisfying 0⩽i k⩽( n k ) and any given nondegenerate probability distributions P 1,…, P n there exist random variables η 1,…, η n with respective distributions P 1,…, P n such that ( i 2,…, i n ) is their uncorrelation structure.

Quasi-Parabolic Subgroups of G(m,1,r)

We study the properties of cosets and double cosets of quasi-parabolic subgroups via tableaux. We characterize these double cosets by n × n arrays of non-negative integer sequences. For each double coset, we construct and count all shortest and longest representatives and find their lengths. We also find the number of double cosets in G ( m , 1, r ) of each pair of quasi-parabolic subgroups.

On the refined lecture hall theorem

A lecture hall partition of length n is a sequence (λ 1,λ 2, …, λ n) of nonnegative integers satisfying 0⩽ λ 1/1⩽⋯⩽ λ n / n. M. Bousquet-Mélou and K. Eriksson showed that there is an one to one correspondence between the set of all lecture hall partitions of length n and the set of all partitions with distinct parts between 1 and n, and possibly multiple parts between n+1 and 2 n. In this paper, we construct a bijection which is an identity mapping in the limiting case.

On Periodic Sequences for Algebraic Numbers

For each positive integer n⩾2, a new approach to expressing real numbers as sequences of nonnegative integers is given. The n=2 case is equivalent to the standard continued fraction algorithm. For n=3, it reduces to a new iteration of the triangle. Cubic irrationals that are roots x3+kx2+x−1 are shown to be precisely those numbers with purely periodic expansions of period length one. For general positive integers n, it reduces to a new iteration of an n dimensional simplex.