On Schurity of Finite Abelian Groups

A finite group G is called a Schur group, if any Schur ring over G is associated in a natural way with a subgroup of Sym(G) that contains all right translations. Recently, the authors have completely identified the cyclic Schur groups. In this article, it is shown that any abelian Schur group belongs to one of several explicitly given families only. In particular, any noncyclic abelian Schur group of odd order is isomorphic to ?₃ × ?_{3 ^k} or ?₃ × ?₃ × ? _p where k ≥ 1 and p is a prime. In addition, we prove that ?₂ × ?₂ × ? _p is a Schur group for every prime p.     

Nodal Curves with General Moduli on K3 Surfaces

We investigate the modular properties of nodal curves on a low genus K3 surface. We prove that a general genus g curve C is the normalization of a δ-nodal curve X sitting on a primitively polarized K3 surface S of degree 2p ? 2, for 2 ≤ g = p ? δ < p ≤ 11. The proof is based on a local deformation-theoretic analysis of the map from the stack of pairs (S, X) to the moduli stack of curves ? _g that associates to X the isomorphism class [C] of its normalization.     

Some Examples of Simple Small Singularities

We provide some examples of simple small singularities of higher dimensional algebraic varieties. One of them is an E ₆ type singularity w ² ? z ³ + xy ³ ? 3x ² yz ? x ⁵ + xzw ? x ⁴ y = 0 in ?⁴. We also treat small contractions of curves with higher genera whose normal bundles are not negative.     

When the v-Operation Distributes over Intersections

Let D be an integral domain. We investigate when (∩ A_α) ?¹ = ∑ A_α ?¹ or (∩ A_α) ?¹ =(∑ A_α ?¹)_v (equivalently, (∩ A _α) _v  = ∩(A _α) _v ) for certain families {A _α} of nonzero fractional ideals of D.     

An Explicit Computation of a Family of Trivializing Étale Covers

We explicitly compute étale covers of the smooth Fermat curves Y _p+1 = Proj k[u, v, w]/(u ^p+1 + v ^p+1 ? w ^p+1) which trivialize the vector bundles Syz(u ², v ², w ²)(3), where k is a field of characteristic p ≥ 3.     

Level Structures on the Weierstrass Family of Cubics

Let W → 𝔸 ² be the universal Weierstrass family of cubic curves over ?. For each N ≥ 2, we construct surfaces parameterizing the three standard kinds of level N structures on the smooth fibers of W. We then complete these surfaces to finite covers of 𝔸 ². Since W → 𝔸 ² is the versal deformation space of a cusp singularity, these surfaces convey information about the level structure on any family of curves of genus g degenerating to a cuspidal curve. Our goal in this note is to determine for which values of N these surfaces are smooth over (0, 0). From a topological perspective, the results determine the homeomorphism type of certain branched covers of S ³ with monodromy in SL₂ (?/N).     

A Construction Method for Albert Algebras over Algebraic Varieties

We construct cubic Jordan algebras over an integral proper scheme X such that 2, 3 ∈ H ⁰(X, 𝒪 _X ), generalizing a construction by B. N. Allison and J. R. Faulkner. In the process, we obtain admissible cubic algebras and pseudocomposition algebras over X. Results on the structure of these algebras are obtained, as well as examples over elliptic curves.     

The Real Solutions to a System of Quaternion Matrix Equations with Applications

In this article we establish necessary and sufficient conditions for the existence and the expressions of the general real solutions to the classical system of quaternion matrix equations A ₁ XB ₁ = C ₁, A ₂ XB ₂ = C ₂. Moreover, formulas of the maximal and minimal ranks of four real matrices X ₁, X ₂, X ₃, and X ₄ in solution X = X ₁ + X ₂ i + X ₃ j + X ₄ k to the system mentioned above are derived. As applications, we give necessary and sufficient conditions for the quaternion matrix equations A ₁ XB ₁ = C ₁, A ₂ XB ₂ = C ₂, A ₃ XB ₃ = C ₃ to have common real solutions. In addition, the maximal and minimal ranks of four real matrices E, F, G, and H in the common generalized inverse of A ₁ + B ₁ i + C ₁ j + D ₁ k and A ₂ + B ₂ i + C ₂ j + D ₂ k, which can be expressed as E + Fi + Gj + Hk are also presented.     

Small Data Scattering for the One-Dimensional Nonlinear Dirac Equation with Power Nonlinearity

We study scattering problems for the one-dimensional nonlinear Dirac equation (?_t + α?_x + iβ)Φ = λ|Φ|^p?1Φ. We prove that if p > 3 (resp. p > 3 + 1/6), then the wave operator (resp. the scattering operator) is well-defined on some 0-neighborhood of a weighted Sobolev space. In order to prove these results, we use linear operators D(t)xD(?t) and t?_x + x?_t ? α/2, where {D(t)}_t∈? is the free Dirac evolution group. For the reader's convenience, in an appendix we list and prove fundamental properties of D(t)xD(?t) and t?_x + x?_t ? α/2.     

Rings with Zassenhaus Families of Ideals

Let R be a ring with identity. We call a family ? of left ideals of R a Zassenhaus family if the only additive endomorphisms of R that leave all members of ? invariant are the left multiplications by elements of R. Moreover, if R is torsion-free and there is some left R-module M such that R ? M ? R?_?? and End _?(M) = R we call R a “Zassenhaus ring”. It is well known that all Zassenhaus rings have Zassenhaus families. We will give examples to show that the converse does not hold even for torsion-free rings of finite rank.     

Asymptotic zero distribution of a class of 3F2 hypergeometric functions

We study the asymptotic behavior of the zeros of certain families of ₃F₂ functions. Classical tools are used to analyse the asymptotic behavior of the zeros of the polynomial In addition, families of ₃F₂ functions that are connected in a formulaic sense with Gauss hypergeometric polynomials of the form and are investigated. Numerical evidence of the clustering o zeros on certain curves is generated by Mathematica.     

Finite free resolutions of length two and koszul generalized complex

Abstract

In 1956, Ehrenfeucht proved that a polynomial f ₁(x ₁) + · + f _n (x _n ) with complex coefficients in the variables x ₁, …, x _n is irreducible over the field of complex numbers provided the degrees of the polynomials f ₁(x ₁), …, f _n (x _n ) have greatest common divisor one. In 1964, Tverberg extended this result by showing that when n ≥ 3, then f ₁(x ₁) + · + f _n (x _n ) belonging to K[x ₁, …, x _n ] is irreducible over any field K of characteristic zero provided the degree of each f _i is positive. Clearly a polynomial F = f ₁(x ₁) + · + f _n (x _n ) is reducible over a field K of characteristic p ≠ 0 if F can be written as F = (g ₁(x ₁)) ^p  + (g ₂(x ₂)) ^p  + · + (g _n (x _n )) ^p  + c[g ₁(x ₁) + g ₂(x ₂) + · + g _n (x _n )] where c is in K and each g _i (x _i ) is in K[x _i ]. In 1966, Tverberg proved that the converse of the above simple fact holds in the particular case when n = 3 and K is an algebraically closed field of characteristic p > 0. In this article, we prove an extension of Tverberg's result by showing that this converse holds for any n ≥ 3.     

Rings of Formal Power Series in an Infinite Set of Indeterminates

Let α be an infinite cardinal number, Λ be an index set of cardinality > α, and {X _λ}_λ∈Λ be a set of indeterminates over an integral domain D. It is well known that there are three ways of defining the ring of formal power series in {X _λ}_λ∈Λ over D, say, D[[{X _λ}]] _i for i = 1, 2, 3. In this paper, we let D[[{X _λ}]]_α = ∪ {D[[{X _λ}_λ∈Γ]]₃ | Γ ? Λ and |Γ| ≤ α}, and we then show that D[[{X _λ}]]_α is an integral domain such that D[[{X _λ}]]₂ ? D[[{X _λ}]]_α ? D[[{X _λ}]]₃. We also prove that (1) D is a Krull domain if and only if D[[{X _λ}]]_α is a Krull domain and (2) D[[{X _λ}]]_α is a unique factorization domain (UFD) (resp., π-domain) if and only if D[[X ₁,…, X _n ]] is a UFD (resp., π-domain) for every integer n ≥ 1.     

Min-Wise Independent Families with Respect to any Linear Order

A set of permutations 𝒮 on a finite linearly ordered set Ω is said to be k-min-wise independent, k-MWI for short, if Pr (min (π(X)) = π(x)) = 1/|X| for every X ? Ω such that |X| ≤ k and for every x ∈ X. (Here π(x) and π(X) denote the image of the element x or subset X of Ω under the permutation π, and Pr refers to a probability distribution on 𝒮, which we take to be the uniform distribution.) We are concerned with sets of permutations which are k-MWI families for any linear order. Indeed, we characterize such families in a way that does not involve the underlying order. As an application of this result, and using the Classification of Finite Simple Groups, we deduce a complete classification of the k-MWI families that are groups, for k ≥ 3.     

An Example of Constructing Versal Deformation for Leibniz Algebras

In this work we compute a versal deformation of the 3-dimensional nilpotent Leibniz algebra over ?, defined by the nontrivial brackets [e ₁, e ₃] = e ₂ and [e ₃, e ₃] = e ₁.     

On Generators of the Tame Automorphism Group of Free Metabelian Lie Algebras

We prove that the tame automorphism group TAut(M _n ) of a free metabelian Lie algebra M _n in n variables over a field k is generated by a single nonlinear automorphism modulo all linear automorphisms if n ≥ 4 except the case when n = 4 and char(k) ≠ 3. If char(k) = 3, then TAut(M ₄) is generated by two automorphisms modulo all linear automorphisms. We also prove that the tame automorphism group TAut(M ₃) cannot be generated by any finite number of automorphisms modulo all linear automorphisms.     

Vector Bundles Near Negative Curves: Moduli and Local Euler Characteristic

We study moduli of vector bundles on a two-dimensional neighbourhood Z _k of an irreducible curve ? ? ?¹ with ?² = ?k and give an explicit construction of their moduli stacks. For the case of instanton bundles, we stratify the stacks and construct moduli spaces. We give sharp bounds for the local holomorphic Euler characteristic of bundles on Z _k and prove existence of families of bundles with prescribed numerical invariants. Our numerical calculations are performed using a Macaulay 2 algorithm, which is available for download at http://www.maths.ed.ac.uk/~s0571100/Instanton/.     

A Symmetric Presentation for J 1

Abstract

We give a computer-free proof that the sporadic simple group J ₁ is a isomorphic to the progenitor 2*⁵ : A ₅ factorized over a single relation. Precisely, we prove that J ₁ is defined by the presentation ?x, y, t ∣ x ⁵ = y ³ = (xy)² = 1 = t ² = [y, t] = [y, t ^{x 3} ] = (xt)⁷?.     

Counting Function of Characteristic Values and Magnetic Resonances

We consider the meromorphic operator-valued function I ? K(z) = I ? A(z)/z where A is holomorphic on the domain 𝒟 ? ?, and has values in the class of compact operators acting in a given Hilbert space. Under the assumption that A(0) is a selfadjoint operator which can be of infinite rank, we study the distribution near the origin of the characteristic values of I ? K, i.e. the complex numbers w ≠ 0 for which the operator I ? K(w) is not invertible, and we show that generically the characteristic values of I ? K converge to 0 with the same rate as the eigenvalues of A(0).

We apply our abstract results to the investigation of the resonances of the operator H = H ₀ + V where H ₀ is the shifted 3D Schrödinger operator with constant magnetic field of scalar intensity b > 0, and V: ?³ → ? is the electric potential which admits a suitable decay at infinity. It is well known that the spectrum σ(H ₀) of H ₀ is purely absolutely continuous, coincides with [0, + ∞[, and the so-called Landau levels 2bq with integer q ≥ 0, play the role of thresholds in σ(H ₀). We study the asymptotic distribution of the resonances near any given Landau level, and under generic assumptions obtain the main asymptotic term of the corresponding resonance counting function, written explicitly in the terms of appropriate Toeplitz operators.