On the structure of the augmentation quotient group for some nonabelian 2-groups

Let G be a finite nonabelian group, ℤG its associated integral group ring, and Δ(G) its augmentation ideal. For the semidihedral group and another nonabelian 2-group the problem of their augmentation ideals and quotient groups Q _n (G) = Δ ⁿ (G)/Δ ⁿ⁺¹(G) is deal with. An explicit basis for the augmentation ideal is obtained, so that the structure of its quotient groups can be determined.     

Unit groups of group algebras of some small groups

Let FG be a group algebra of a group G over a field F and U (FG) the unit group of FG. It is a classical question to determine the structure of the unit group of the group algebra of a finite group over a finite field. In this article, the structure of the unit group of the group algebra of the non-abelian group G with order 21 over any finite field of characteristic 3 is established. We also characterize the structure of the unit group of FA ₄ over any finite field of characteristic 3 and the structure of the unit group of FQ ₁₂ over any finite field of characteristic 2, where Q ₁₂ = 〈x, y; x ⁶ = 1, y ² = x ³, x ^y = x ^?1〉.     

Solvable absolute Galois groups are metabelian

We show that solvable absolute Galois groups have an abelian normal subgroup such that the quotient is the direct product of two finite cyclic and a torsion-free procyclic group. In particular, solvable absolute Galois groups are metabelian. Moreover, any field with solvable absolute Galois group G admits a non-trivial henselian valuation, unless each Sylow-subgroup of G is either procyclic or isomorphic to Z ₂⋊Z/2Z. A complete classification of solvable absolute Galois groups (up to isomorphism) is given. Oblatum 22-IV-1998 & 1-IX-2000?Published online: 30 October 2000     

Ruling out (160, 54, 18) difference sets in some nonabelian groups

We prove the following theorems. Theorem A. Let G be a group of order 160 satisfying one of the following conditions. (1) G has an image isomorphic to D₂₀ × Z₂ (for example, if G ≃ D₂₀ × K). (2) G has a normal 5‐Sylow subgroup and an elementary abelian 2‐Sylow subgroup. (3) G has an abelian image of exponent 2, 4, 5, or 10 and order greater than 20. Then G cannot contain a (160, 54, 18) difference set. Theorem B. Suppose G is a nonabelian group with 2‐Sylow subgroup S and 5‐Sylow subgroup T and contains a (160, 54, 18) difference set. Then we have one of three possibilities. (1) T is normal, |ϕ(S)| = 8, and one of the following is true: (a) G = S × T and S is nonabelian; (b) G has a D₁₀ image; or (c) G has a Frobenius image of order 20. (2) G has a Frobenius image of order 80. (3) G is of index 6 in A Γ L(1, 16). To prove the first case of Theorem A, we find the possible distribution of a putative difference set with the stipulated parameters among the cosets of a normal subgroup using irreducible representations of the quotient; we show that no such distribution is possible. The other two cases are due to others. In the second (due to Pott) irreducible representations of the elementary abelian quotient of order 32 give a contradiction. In the third (due to an anonymous referee), the contradiction derives from a theorem of Lander together with Dillon's “dihedral trick.” Theorem B summarizes the open nonabelian cases based on this work. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 221–231, 2000     

Low frequency asymptotics for the reduced wave equation in two-dimensional exterior spaces

We consider the Dirichlet problem for the reduced wave equation ΔU_x + x²U_x = 0 in a two-dimensional exterior domain with boundary C, where C consists of a finite number of smooth closed curves C₁,…,C_m. The question of interest is the behavior of U_x as ? → 0. We show that U converges to the solution of the corresponding exterior Dirichlet problem of potential theory if the boundary data converge to a limit uniformly on C. This generalizes a well-known result of R. C. MacCamy for the case m = 1.     

A Green's Function Approach to Wave Propagation and Potential Flow around Obstacles in Infinite Cylindrical Channels

We study (a) acoustic waves generated by a time-harmonic force distribution and (b) the potential flow with prescribed velocity at infinity in an infinite cylinder Ω₀ = Ω′×ℝ with bounded cross-section Ω′⊂ℝ² in the presence of m embedded obstacles B₁,…,B_m. By using Green's function G_κ(x,y) of the Neumann problem for the reduced wave equation ΔU+κ²U = 0 in the unperturbed domain Ω₀, both problems can be reduced to integral equations over the boundaries of the obstacles. The main properties of G_κ(x,y), which are required for this approach, are derived in the first part of this paper.     

Graded Nilpotent Algebras and Eggert's Conjecture

A group G is (l,m,n)-generated if it is a quotient group of the triangle group T(l,m,n) = (x,y,z|x ^l= y ^m= z ⁿ= xyz= 1). In [8] the problem is posed to find all possible (l,m,n)-generations for the non-abelian finite simple groups. In this paper we partially answer this question for the Janko group J ₃. We find all (2, 3, t)-generations as well as (2, 2,2,p)-generations, p a prime, for J ₃     

Inverse problem for N × N hyperbolic systems on the plane and the N-wave interactions

We study regiorously the solvability of the direct and inverse problems associated with Ψ_x–JΨ_y = QΨ,(x,y) ∈ ?², where (i) Ψ is an N × N-matrix-valued function on ?² (N ≦ 2), (ii) J is a constant, real, diagonal N × N matrix with entries, J₁ > J₂ > …? > J_N and (iii) Q is off-diagonal with rapidly decreasing (Schwartz) component functions. In particular we show that the direct problem is always solvable and give a small norm condition for the solvability of the inverse problem. In the particular case that Q is skew Hermitian the inverse problem is solvable without the small norm assumption. Furthermore we show how these results can be used to solve certain Cauchy problems for the associated nonlinear evolution equations. For concreteness we consider the N-wave interactions and show that if a certain norm of Q(x, y, 0) is smallor if Q(x, y, 0) is skew Hermitian the N-wave interations equation has a unique global solution.     

On the global existence and uniqueness of solutions to Prandtl’s system

In this paper, we consider the Prandtl system for the non-stationary boundary layer in the vicinity of a point where the outer flow has zero velocity. It is assumed that U（t, x, y） = x^mU1（t, x）, where 0 〈 x 〈 L and m 〉 1. We establish the global existence of the weak solution to this problem. Moreover the uniqueness of the weak solution is proved.     

Geodesics in Graphs,an Extremal Set Problem,and Perfect Hash Families

 A set U of vertices of a graph G is called a geodetic set if the union of all the geodesics joining pairs of points of U is the whole graph G. One result in this paper is a tight lower bound on the minimum number of vertices in a geodetic set. In order to obtain that result, the following extremal set problem is solved. Find the minimum cardinality of a collection 𝒮 of subsets of [n]={1,2,…,n} such that, for any two distinct elements x,y∈[n], there exists disjoint subsets A _x ,A _y ∈𝒮 such that x∈A _x and y∈A _y . This separating set problem can be generalized, and some bounds can be obtained from known results on families of hash functions. Received: May 19, 2000 Final version received: July 5, 2001     

ON SYMMETRIC UNITS IN GROUP ALGEBRAS

Let U(KG) be the group of units of the group ring KG of the group G over a commutative ring K. The anti-automorphism g → g ^?1 of G can be extended linearly to an anti-automorphism a → a ^* of KG. Let S _* (KG) = {x ∈ U(KG) | x ^* = x} be the set of all symmetric units of U(KG). We consider the following question: for which groups G and commutative rings K it is true that S _* (KG) is a subgroup in U(KG). We answer this question when either a) G is torsion and K is a commutative G-favourable integral domain of characteristic p≥ 0 or b) G is non-torsion nilpotent group and KG is semiprime.     

On Certain Weak Engel-Type Conditions in Groups

Let w(x, y) be a word in two variables and 𝔚 the variety determined by w. In this paper we raise the following question: if for every pair of elements a, b in a group G there exists g ∈ G such that w(a ^g , b) = 1, under what conditions does the group G belong to 𝔚? In particular, we consider the n-Engel word w(x, y) = [x, _n y]. We show that in this case the property is satisfied when the group G is metabelian. If n = 2, then we extend this result to the class of all solvable groups.     

Monogenic functions in the biharmonic boundary value problem

We consider a commutative algebra over the field of complex numbers with a basis {e₁,e₂} satisfying the conditions , . Let D be a bounded domain in the Cartesian plane xOy and D_ζ={xe₁+ye₂:(x,y)∈D}. Components of every monogenic function Φ(xe₁+ye₂) = U₁(x,y)e₁+U₂(x,y)ie₁+U₃(x,y)e₂+U₄(x,y)ie₂ having the classic derivative in D_ζ are biharmonic functions in D, that is, Δ²U_j(x,y) = 0 for j = 1,2,3,4. We consider a Schwarz‐type boundary value problem for monogenic functions in a simply connected domain D_ζ. This problem is associated with the following biharmonic problem: to find a biharmonic function V(x,y) in the domain D when boundary values of its partial derivatives ?V/?x, ?V/?y are given on the boundary ?D. Using a hypercomplex analog of the Cauchy‐type integral, we reduce the mentioned Schwarz‐type boundary value problem to a system of integral equations on the real axes and establish sufficient conditions under which this system has the Fredholm property. Copyright © 2015 John Wiley & Sons, Ltd.     

A Polytope Related to Empirical Distributions, Plane Trees, Parking Functions, and the Associahedron

Abstract. The volume of the n -dimensional polytope Π _n (x):= {y ∈ R ⁿ : y _i ≥ 0 and y ₁ + · · · + y _i ≤ x ₁ + · · ·+ x _i for all 1 ≤ i ≤ n } for arbitrary x:=(x ₁ , . . ., x _n ) with x _i >0 for all i defines a polynomial in variables x _i which admits a number of interpretations, in terms of empirical distributions, plane partitions, and parking functions. We interpret the terms of this polynomial as the volumes of chambers in two different polytopal subdivisions of Π _n (x) . The first of these subdivisions generalizes to a class of polytopes called sections of order cones. In the second subdivision the chambers are indexed in a natural way by rooted binary trees with n+1 vertices, and the configuration of these chambers provides a representation of another polytope with many applications, the associahedron .     

Nonlinear equations with mixed derivatives

The nonlinear hyperbolic equation ∂²u(x, y)/∂x ∂y + g(x, y)f(u(x, y)) = 0 with u(x, 0) = φ(x) and u(0, y) = Ψ(y), considered by [1.], 31–45) under appropriate smoothness conditions, is solvable by the author's decomposition method (“Stochastic Systems,” Academic Press, 1983 and “Nonlinear Stochastic Operator Equations,” Academic Press, 1986).     

Existence and Uniqueness of Strong Solutions for a Class of Quasi‐Linear Hyperbolic Equations with Order Degeneration

Let k(y) > 0, 𝓁(y) > 0 for y > 0, k(0) = 𝓁(0) = 0 and lim_{y → 0}k(y)/𝓁(y) exists; then the equation L(u) ≔ k(y)u_xx – ∂_y(𝓁(y)u_y) + a(x, y)u_x = f(x, y, u) is strictly hyperbolic for y > 0 and its order degenerates on the line y = 0. Consider the boundary value problem Lu = f(x, y, u) in G, u|_AC = 0, where G is a simply connected domain in ℝ² with piecewise smooth boundary ∂G = AB∪AC∪BC; AB = {(x, 0) : 0 ≤ x ≤ 1}, AC : x = F(y) = ∫^y₀(k(t)/𝓁(t))^1/2dt and BC : x = 1 – F(y) are characteristic curves. Existence of generalized solution is obtained by a finite element method, provided f(x, y, u) satisfies Carathéodory condition and |f(x, y, u)| ≤ Q(x, y) + b|u| with Q ∈ L²(G), b = const > 0. It is shown also that each generalized solution is a strong solution, and that fact is used to prove uniqueness under the additional assumption |f(x, y, u₁) – f(x, y, u₂| ≤ C|u₁ – u₂|, where C = const > 0.     

Recognition of some simple groups by their noncommuting graphs

Let G be a nonabelian group and associate a noncommuting graph ∇(G) with G as follows: The vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. Abdollahi et al. (J Algebra 298(2):468–492, 2006) put forward a conjecture called AAM’s Conjecture in as follows: If M is a finite nonabelian simple group and G is a group such that ∇(G) ≅ ∇(M), then G ≅ M. Even though this conjecture is well known to hold for all simple groups with nonconnected prime graphs and the alternating group A ₁₀ [see Darafsheh (Groups with the same non-commuting graph. Discrete Appl Math (2008) doi:), Wang and Shi (Commun Algebra 36(2):523–528, 2008)], it is still unknown for all simple groups with connected prime graphs except A ₁₀. In the present paper, we prove that this conjecture is also true for the projective special linear simple group L ₄(9). The new method used in this paper also works well in the cases L ₄(4), L ₄(7), U ₄(7), etc.     

Product recurrent properties, disjointness and weak disjointness

Let F be a collection of subsets of ℝ₊ and (X, T) be a dynamical system; x ∈ X is F-recurrent if for each neighborhood U of x, {n ∈ ℝ₊: T ⁿ x ∈ U} ∈ F; x is F-product recurrent if (x, y) is recurrent for any F-recurrent point y in any dynamical system (Y, S). It is well known that x is {infinite}-product recurrent if and only if it is minimal and distal. In this paper it is proved that the closure of a {syndetic}-product recurrent point (i.e., weakly product recurrent point) has a dense minimal points; and a {piecewise syndetic}-product recurrent point is minimal. Results on product recurrence when the closure of an F-recurrent point has zero entropy are obtained.     

Finite embedding theorems for partial Latin squares,quasi-groups,and loops

In this paper we prove that a finite partial commutative (idempotent commutative) Latin square can be embedded in a finite commutative (idempotent commutative) Latin square. These results are then used to show that the loop varieties defined by any non-empty subset of the identities {x(xy) = y, (yx)x = y} and the quasi-group varieties defined by any non-empty subset of {x² = x, x(xy) = y, (yx)x = y}, except possibly {x(xy) = y, (yx)x = y}, have the strong finite embeddability property. It is then shown that the finitely presented algebras in these varities are residually finite, Hopfian, and have a solvable word problem.     

On a class of systems of rational second-order difference equations solvable in closed form

We show that the following nonlinear system of difference equations where parameters a,b,c,d and initial values x₋₁,x₀,y₋₁,y₀ are real numbers, is solvable in closed form, considerably generalizing some recent results. To do this, we use the method of transformation along with several tricks, transforming the system to some known solvable difference equations, by use of which we obtain some closed-form formulas for general solution to the system. The following five cases are considered separately: (1) c=0; (2) d=0; (3) a=0; (4) b=0; and (5) abcd≠0.