On integers which are the sum of a power of 2 and a polynomial value

Here, we show that if f (x) ∈ ?[x] has degree at least 2 then the set of integers which are of the form 2 ^k + f (m) for some integers k ≥ 0 and m is of asymptotic density 0. We also make some conjectures and prove some results about integers not of the form |2 ^k ± m ^a (m ? 1)|.     

A note on the kernels of higher derivations

Let k ? k′ be a field extension. We give relations between the kernels of higher derivations on k[X] and k′[X], where k[X]:= k[x ₁,…, x _n ] denotes the polynomial ring in n variables over the field k. More precisely, let D = {D _n } _n=0 ^∞ a higher k-derivation on k[X] and D′ = {D′ _n } _n=0 ^∞ a higher k′-derivation on k′[X] such that D′ _m (x _i ) = D _m (x _i ) for all m ? 0 and i = 1, 2,…, n. Then (1) k[X] ^D = k if and only if k′[X] ^D′ = k′; (2) k[X] ^D is a finitely generated k-algebra if and only if k′[X] ^D′ is a finitely generated k′-algebra. Furthermore, we also show that the kernel k[X] ^D of a higher derivation D of k[X] can be generated by a set of closed polynomials.     

Uniqueness of analytic continuation: Necessary and sufficient conditions

Necessary and sufficient conditions for uniqueness of analytic continuation are investigated for a system of m ? 1 first-order linear homogeneous partial differential equations in one unknown, with complex-valued $\text{b}$^∞ coefficients, in some connected open subset of $\text{R}$^k, k ? 2. The type of system considered is one for which there exists a real k-dimensional, $\text{b}$^∞, connected C-R submanifold M^k of $\text{C}$ⁿ, for k, n ? 2, such that the system may be identified with the induced Cauchy-Riemann operators on M^k. The question of uniqueness of analytic continuation for a system of partial differential equations is thus transformed to the question of uniqueness of analytic continuation for C-R functions on the manifold M^k ? Cⁿ. Under the assumption that the Levi algebra of M^k has constant dimension, it is shown that if the excess dimension of this algebra is maximal at every point, then M^k has the property of uniqueness of analytic continuation for its C-R functions. Conversely, under certain mild conditions, it is shown that if M^k has the property of uniqueness of analytic continuation for all $\text{b}$^∞ C-R functions, and if the Levi algebra has constant dimension on all of M^k, then the excess dimension must be maximal at every point of M^k.     

Identification of the unknown diffusion coefficient in a linear parabolic equation by the semigroup approach

In this article, we study the semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(x) in the linear parabolic equation ut(x,t)=(k(x)uxx(x,t)), with Dirichlet boundary conditions u(0,t)=ψ₀, u(1,t)=ψ₁. Main goal of this study is to investigate the distinguishability of the input-output mappings Φ[⋅]:K→C¹[0,T], Ψ[⋅]:K→C¹[0,T] via semigroup theory. In this paper, we show that if the null space of the semigroup T(t) consists of only zero function, then the input-output mappings Φ[⋅] and Ψ[⋅] have the distinguishability property. Moreover, the values k(0) and k(1) of the unknown diffusion coefficient k(x) at x=0 and x=1, respectively, can be determined explicitly by making use of measured output data (boundary observations) f(t):=k(0)ux(0,t) or/and h(t):=k(1)ux(1,t). In addition to these, the values k^′(0) and k^′(1) of the unknown coefficient k(x) at x=0 and x=1, respectively, are also determined via the input data. Furthermore, it is shown that measured output dataf(t) and h(t) can be determined analytically, by an integral representation. Hence the input-output mappings Φ[⋅]:K→C¹[0,T], Ψ[⋅]:K→C¹[0,T] are given explicitly in terms of the semigroup. Finally by using all these results, we construct the local representations of the unknown coefficient k(x) at the end points x=0 and x=1.     

Super Jack-Laurent Polynomials

Let \(\mathcal {D}_{n,m}\) be the algebra of quantum integrals of the deformed Calogero-Moser-Sutherland problem corresponding to the root system of the Lie superalgebra \(\frak {gl}(n,m)\). The algebra \(\mathcal {D}_{n,m}\) acts naturally on the quasi-invariant Laurent polynomials and we investigate the corresponding spectral decomposition. Even for general value of the parameter k the spectral decomposition is not multiplicity free and we prove that the image of the algebra \(\mathcal {D}_{n,m}\) in the algebra of endomorphisms of the generalised eigenspace is k[ε]^?r where k[ε] is the algebra of dual numbers. The corresponding representation is the regular representation of the algebra k[ε]^?r.     

The inversion formulae for automorphisms of polynomial algebras and rings of differential operators in prime characteristic

Let K be an arbitrary field of characteristic p>0. We find an explicit formula for the inverse of any algebra automorphism of any of the following algebras: the polynomial algebra P_n?K[x₁,…,x_n], the ring of differential operators D(P_n) on P_n, D(P_n)⊗P_m, the n’th Weyl algebra A_n or A_n⊗P_m, the power series algebra K[[x₁,…,x_n]], the subalgebra T_k1,…,kn of D(P_n) generated by P_n and the higher derivations , 0≤j<p^k_i, i=1,…,n (where k₁,…,k_n∈N), T_k1,…,kn⊗P_m or an arbitrary central simple (countably generated) algebra over an arbitrary field.     

On the completion of excellent rings in characteristicp>0

Si considera il seguente problema posto da Grothendieck (E.G.A.): SeA è un anello eccellente edm un ideale diA, (A, m) ^{^}=m-adico completamento diA è eccellente? Si mostra che la risposta è positiva nei seguenti casi:

1. A=algebra di tipo finito su un DVR completo di caratteristicap>0;
2. A=algebra di tipo finito su un DVRC contenente un corpok di caratteristicap>0 e finito suk [C ^p ] oppure tale che:

1. per ogni sottocampok′ dik contenentek ^p tale che [k:k′]<∞, il modulo universale finito dei differenzialiD _{k′ (C)} esiste;
2. il corpo residuoK diC soddisfa rank _KK ? _K/k <∞
3. C ha una Der-base.

     

Spin invariants of multivectors

LetCl(p, q) be a real universal Clifford algebra which is isomorphic to a full matrix algebra ?(2m). In this paper we show that on the linear subspaceCl ^k(p, q) ofk-vectors the determinant can be written as a product of two polynomialsd _i of degreem and that on the subset ofdecomposable k-vectors we have det=±Q ^m for some quadratic formQ. The polynomialsd _i andQ are examples of a spin invariant, the latter being defined as a functionJ:Cl ^k (p,q) → ? for whichJ(sus^?1)=J(u) for allu ∈Cl ^k(p, q) ands ∈Spin(p, q). In the last section we identify the ‘fundamental’ spin invariants on the bivector spacesCl ²(p, p) forp=2 andp=3.     

Blocker sets, orthogonal arrays and their application to combination locks

Let A denote a set of order m and let X be a subset of A^k+1. Then X will be called a blocker (of A^k+1) if for any element say (a₁,a₂,…,a_k,a_k+1) of A^k+1, there is some element (x₁,x₂,…,x_k,x_k+1) of X such that x_i equals a_i for at least two i. The smallest size of a blocker set X will be denoted by α(m,k)and the corresponding blocker set will be called a minimal blocker. Honsberger (who credits Schellenberg for the result) essentially proved that α(2n,2) equals 2n² for all n. Using orthogonal arrays, we obtain precise numbers α(m,k) (and lower bounds in other cases) for a large number of values of both k and m. The case k=2 that is three coordinate places (and small m) corresponds to the usual combination lock. Supposing that we have a defective combination lock with k+1 coordinate places that would open if any two coordinates are correct, the numbers α(m,k) obtained here give the smallest number of attempts that will have to be made to ensure that the lock can be opened. It is quite obvious that a trivial upper bound for α(m,k) is m² since allowing the first two coordinates to take all the possible values in A will certainly obtain a blocker set. The results in this paper essentially prove that α(m,k) is no more than about m²/k in many cases and that the upper bound cannot be improved. The paper also obtains precise values of α(m,k) whenever suitable orthogonal arrays of strength two (that is, mutually orthogonal Latin squares) exist.     

Numerical computation and analysis of the Titchmarsh–Weyl mα(λ) function for some simple potentials

This article is concerned with the Titchmarsh–Weyl m_α(λ) function for the differential equation d²y/dx²+[λ−q(x)]y=0. The test potential q(x)=x², for which the relevant m_α(λ) functions are meromorphic, having simple poles at the points λ=4k+1 and λ=4k+3, is studied in detail. We are able to calculate the m_α(λ) function both far from and near to these poles. The calculation is then extended to several other potentials, some of which do not have analytical solutions. Numerical data are given for the Titchmarsh–Weyl m_α(λ) function for these potentials to illustrate the computational effectiveness of the method used.     

On tense modules for extended algebras over rational fields

Let k(x) be the field of fractions of the polynomial algebra k[x] over the field k. We prove that, for an arbitrary finite dimensional k-algebra Λ, any finitely generated Λ ⊗_k k(x)-module M such that its minimal projective presentation admits no non-trivial selfextension is of the form M ≅ N^k(x), for some finitely generated Λ-module N. Some consequences are derived for tilting modules over the rational algebra Λ ⊗_k k(x) and for some generic modules for Λ. Received: 24 November 2003; revised: 11 February 2005     

Dimensions of some fractals defined via the semigroup generated by 2 and 3

We compute the Hausdorff and Minkowski dimension of subsets of the symbolic space Σ _m ={0, ...,m?1}^? that are invariant under multiplication by integers. The results apply to the sets {x∈Σ _m :? k, x _k x _2k ... x _nk =0}, where n ≥ 3. We prove that for such sets, the Hausdorff and Minkowski dimensions typically differ.     

Algebraic Theory of Causal Double Products

Corresponding to each “rectangular” double product in the form of a formal power series R[h] with coefficients in the tensor product ?(?)⊙ ? (?) with itself of the Itô Hopf algebra, we construct “triangular” elements T[h] of ?(?) satisfying ΔT[h] = T[h]⁽¹⁾ R[h]T{h]⁽²⁾. In Fock space representations of ?(?) by iterated quantum stochastic integrals when ? is the algebra of Itô differentials of the calculus, these correspond to “causal” double product integrals in a single Fock space.     

A note on pi-rings

The classical result that an algebra which satisfies a polynomial identity satisfies a powerS _2n [x] ^m =0 of the standard identity, is generalized to arbitrary rings.     

Images of locally finite derivations of polynomial algebras in two variables

In this paper we show that the image of any locally finite k-derivation of the polynomial algebra k[x,y] in two variables over a field k of characteristic zero is a Mathieu subspace. We also show that the two-dimensional Jacobian conjecture is equivalent to the statement that the image of every k-derivation D of k[x,y] such that and is a Mathieu subspace of k[x,y].     

Theorems on equalization and monomiality in a relatively free Grassmann algebra

In this paper, we prove theorems on equalization and monomiality, which are essential for developing the structural theory of T-spaces in a relatively free algebra k〈1, x ₁,…, x _i ,…〉/([[x ₁ , x ₂], x ₃]) ^T over an infinite field k of characteristic p > 2. Additionally, some specifics of the case p = 2 are considered.     

Class Partition Algebras as Centralizer Algebras of Wreath Products

In this article, we study an important subalgebra of the tensor product partition algebra P _k (x)? P _k (y), denoted by P _k (x, y) and called “Class Partition Algebra.” We show that the algebra P _k (n, m) is the centralizer algebra of the wreath product S _m ? S _n . Furthermore, the algebra P _k (x, y) and the tensor product partition algebra P _k (x)? P _k (y) are subalgebras of the G-colored partition algebra P _k (x;G) and G-vertex colored partition algebra P _k (x, G) respectively, for every group G with |G|=y ≥ 2k.     

The Jacobian problem as a system of ordinary differential equations

LetP=x ⁿ +P _n?1(y)x ^n?1+…+P ₀(y),Q=x ^m +Q _m?2(y)x ^m?2+…+Q ₀(y) belong toK[x, y], whereK is a field of characteristic zero. The main result of this paper is the following: Assume thatP _x Q _y ?P _y Q _x =1. Then:*

1. K[Q _m?2(y), …,Q ₀(y)]=K[y],
2. K[P, Q]=K[x, y] ifQ=x ^m +Q _k (y)x ^k +Q _r (y)x ^r

     

Hodge Star as Braided Fourier Transform

We study super-braided Hopf algebras Λ primitively generated by finite-dimensional right crossed (or Drinfeld-Radford-Yetter) modules Λ¹ over a Hopf algebra A which are quotients of the augmentation ideal A ⁺ as a crossed module by right multiplication and the adjoint coaction. Here super-bosonisation Open image in new window provides a bicovariant differential graded algebra on A. We introduce Λ _{m a x} providing the maximal prolongation, while the canonical braided-exterior algebra Λ _min = B _?(Λ¹) provides the Woronowicz exterior calculus. In this context we introduce a Hodge star operator ? by super-braided Fourier transform on B _?(Λ¹) and left and right interior products by braided partial derivatives. Our new approach to the Hodge star (a) differs from previous approaches in that it is canonically determined by the differential calculus and (b) differs on key examples, having order 3 in middle degree on k[S ₃] with its 3D calculus and obeying the q-Hecke relation ?² = 1 + (q ? q ^?1)? in middle degree on k _q [S L ₂] with its 4D calculus. Our work also provides a Hodge map on quantum plane calculi and a new starting point for calculi on coquasitriangular Hopf algebras A whereby any subcoalgebra \(\mathcal {L}\subseteq A\) defines a sub-braided Lie algebra and \({\Lambda }^{1}\subseteq \mathcal {L}^{*}\) provides the required data A ⁺ → Λ¹.