In this paper, we consider a reaction infiltration problem consisting of a parabolic equation for the concentration, an elliptic equation for the pressure, and an ordinary differential equation for the porosity. Existence and uniqueness of a global classical solution is proved for bounded domains Ω⊂RN, under suitable boundary conditions.

A subalgebra U of a Lie algebra L over a field F is called modular* in L if U satisfies the dual of the modular identities in the lattice of subalgebras of L. Our aim is the study of the influence of the modular* identities in the structure of the algebra. First we prove that if the modular* conditions are imposed on an ideal of L then every element of L acts as an scalar on this ideal and if they are imposed on a non-ideal subalgebra U of L then the largest ideal of L contained in U also satisfies the modular* identities. We determine Lie algebras having a subalgebra which satisfies both the modular and modular* identities, provided that either L is solvable or char(F)≠ 2,3. As immediate consequences of this result we obtain that the existence of a co-atom satisfying the modular* identities in the lattice L(L) forces that the lattice L(L) is modular and that the modular* identities on any subalgebra U forces that U is quasi-abelian. In the case when L is supersolvable we obtain that the modular* conditions on any non-ideal of L are enough to guarantee that L(L) is modular. For arbitrary fields and any Lie algebra L, we prove that the modular* conditions on every co-atom of the lattice L(L) guarantee that L(L) is modular.

An inverse transversal of a regular semigroup S is an inverse subsemigroup So that contains precisely one inverse of each element of S. Here we consider the case where S is quasi-orthodox. We give natural characterisations of such semigroups and consider various properties of congruences.

A scalar multiple of the Kähler form of a Kähler manifold is called a Kähler magnetic field. We are focused on trajectories of charged particles under this action. As a variation of trajectories we define a magnetic Jacobi field. In this paper we discuss a comparison theorem on magnetic Jacobi fields, which corresponds to the Rauch's comparison theorem.

The group of symplectic transformations acts on the unit ball of a Hilbert space. The structure of the orbits has been determined by N. J. Young in [8]. We provide a new proof of this theorem; it is slightly simpler than the original one, and does not involve Brown–Douglas–Fillmore theory. Moreover, the steps followed hopefully throw some additional light on the subject. We rely heavily on previous work of Khatskevich, Shmulyan and Shulman ([5, 6, 7[); the proofs of the results used are included for completeness.

The Kervaire Conjecture is correct if it can be shown to hold for non-singular systems of equations of length 3. In this paper we prove it for the case of equations over a group G where each equation has the form axbx−1cy = 1 for a, b, c ∈ G.

In this note we propose an effective method based on the computation of a Gröbner basis of a left ideal to calculate the Gelfand-Kirillov dimension of modules.

In 1962 Gilbert Baumslag introduced the class of groups Gi, j for natural numbers i, j, defined by the presentations Gi, j = < a, b, t; a−1 = [bi, a] [bj, t] >. This class is of special interest since the groups are para-free, that is they share many properties with the free group F of rank 2.

Magnus and Chandler in their History of Combinatorial Group Theory mention the class Gi, j to demonstrate the difficulty of the isomorphism problem for torsion-free one-relator groups. They remark that as of 1980 there was no proof showing that any of the groups Gi, j are non-isomorphic. S. Liriano in 1993 using representations of Gi, j into PSL(2, pk), k ∈ ℕ, showed that G1,1 and G30,30 are non-isomorphic. In this paper we extend these results to prove that the isomorphism problem for Gi, 1, i ∈ ℕ is solvable, that is it can be decided algorithmically in finitely many steps whether or not an arbitrary one-relator group is isomorphic to Gi, 1. Further we show that Gi, 1 ≇ G1, 1 for all i > 1 and if i, k are primes then Gi, 1 ≅ Gk, 1 if and only if i = k.

A finite presentation F/N of a group G is called efficient if dF(N) = d(H2(G)) + d(F) – r(H1(G)). A finitely presented group is called efficient if it admits an efficient presentation. We show that a finitely presented group embeds into an efficient group.

Let X be a rearrangement invariant function space on [0,1] in which the Rademacher functions (rn) generate a subspace isomorphic to ℓ2. We consider the space Λ(R, X) of measurable functions f such that fg∈X for every function g=∑bnrn where (bn)∈ℓ2. We show that if X satisfies certain conditions on the fundamental function and on certain interpolation indices then the space Λ(R, X) is not order isomorphic to a rearrangement invariant space. The result includes the spaces Lp, q and certain classes of Orlicz and Lorentz spaces. We also study the cases X = Lexp and X = Lψ2 for ψ2) = exp(t2) – 1.

We characterize the test words of ℤm * ℤn. They are those elements not contained in a proper retract.

In this paper we generalise some of the results obtained in [1] for the n-dimensional real spaces ℓp(n) to the infinite dimensional real spaces ℓp. Let p >1 with p ≠ 2, and let x be a non-zero real sequence in ℓp. Let ε(x) denote the closed linear subspace spanned by the set of all those sequences in ℓp which are biorthogonal to x with respect to the unique semi-inner-product on ℓp consistent with the norm on ℓp. In this paper we show that codim ε(x)=1 unless either x has exactly two non-zero coordinates which are equal in modulus, or x has exactly three non-zero coordinates α, β, γ with |α| ≥ |β| ≥ |γ| and |α|p > |β|p + |γ|p. In these exceptional cases codim ε(x) = 2. We show that is a linear subspace if, and only if, x has either at most two non-zero coordinates or x has exactly three non-zero coordinates which satisfy the inequalities stated above.

Simple characterizations of a pseudosphere or a plane in Minkowski 3-space by the existence of pseudocircles are given.

We study the initial value problem to the Davey-Stewartson system for the elliptic-hyperbolic case in the usual Sobolev space. We prove local existence and uniqueness H5/2 with a condition such that the L2 norm of the data is sufficiently small.

We calculate the box-dimension for a class of nowhere differentiable curves defined by Markov attractors of certain iterated function systems of affine maps.

Suppose λ is a positive number, and let , x∈Rd, denote the d-dimensional Gaussian. Basic theory of cardinal interpolation asserts the existence of a unique function , x∈Rd, satisfying the interpolatory conditions , k∈Zd, and decaying exponentially for large argument. In particular, the Gaussian cardinal-interpolation operator , given by , x∈Rd, , is a well-defīned linear map from ℓ2(Zd) into L2(Rd). It is shown here that its associated operator-norm is , implying, in particular, that is contractive. Some sidelights are also presented.

In analogy with a construction from function theory, we herein define right, left, and two-sided Bourgain algebras associated with an operator algebra A. These algebras are defined initially in Banach space terms, using the weak-* topology on A, and our main result is to give a completely algebraic characterization of them in the case where A is a nest algebra. Specifically, if A = alg N is a nest algebra, we show that each of the Bourgain algebras defined has the form A + K ∩ B, where B is the nest algebra corresponding to a certain subnest of N. We also characterize algebraically the second-order (and higher) Bourgain algebras of a nest algebra, showing for instance that the second-order two-sided Bourgain algebra coincides with the two-sided Bourgain algebra itself in this case.

On pages 338 and 339 in his first notebook, Ramanujan records eighteen values for a certain product of theta-functions depending on two integral parameters m and n. When (m, n) = 1, it can be seen that each of these values is a unit. The purpose of this paper is to establish each of these eighteen values and to prove that under certain general conditions this product is indeed a unit. Lastly, we prove that certain quotients of theta-functions are algebraic integers.

For R ∈ {bv0, c0, ℓ∞} a multiplier of FK spaces, the classical sectional convergence theorems permit the reduction of R to any of its dense barrelled subspaces as a simple consequence of the Closed Graph Theorem. (Cf. the Bachelis/Rosenthal reduction of R = ℓ∞ to its dense barrelled subspace m0.) A natural modern setting permits the reduction of R to any of the larger class of dense βφ subspaces. Bennett and Kalton's FK setting remarkably reduced R = ℓ∞ to any of its dense subspaces. This extreme reduction also obtains in the modern βφ setting since, surprisingly, every dense subspace of ℓ∞ is a βφ subspace. Moreover, the standard results, including the Bennett/Kalton reduction, easily follow from their βφ versions and the Closed Graph Theorem. Our two supporting papers find relevant “Non-barrelled dense βφ subspaces” and study “Generalized sectional convergence and multipliers”. Here we specialize the βφ approach to ordinary, particularly unconditional, sectional convergence.

Let K be a complete field with respect to a discrete valuation and let L be a finite Galois extension of K. If the residue field extension is separable then the different of L/K can be expressed in terms of the ramification groups by a well-known formula of Hilbert. We will identify the necessary correction term in the general case, and we give inequalities for ramification groups of subextensions L′/K in terms of those of L/K. A question of Krasner in this context is settled with a counterexample. These ramification phenomena can be related to the structure of the module of differentials of the extension of valuation rings. For the case that [L: K] = p2, where p is the residue characteristic, this module is shown to determine the correction term in Hilbert's formula.