Archimedean lattices

Anarchimedean lattice is a complete algebraic latticeL with the property that for each compact elementc∈L, the meet of all the maximal elements in the interval [0,c] is 0.L ishyper-archimedean if it is archimedean and for eachx∈L, [x, 1] is archimedean. The structure of these lattices is analysed from the point of view of their meet-irreducible elements. If the lattices are also Brouwer, then the existence of complements for the compact elements characterizes a particular class of hyper-archimedean lattices. The lattice ofl-ideals of an archimedean lattice ordered group is archimedean, and that of a hyper-archimedean lattice ordered group is hyper-archimedean. In the hyper-archimedean case those arising as lattices ofl-ideals are fully characterized.     

Bases of rearrangement invariant spaces

We prove that if E is a rearrangement-invariant space, then a boundedly complete basis exists in E, if and only if one of the following conditions holds: 1) E is maximal and E ≠ L ₁[0, 1]; 2) a certain (any) orthonormal system of functions from L _∞[0, 1], possessing the properties of the Schauder basis for the space of continuous on [0, 1] functions with the norm L _∞, represents a boundedly complete basis in E. As a corollary, we state the following assertion: Any (certain) orthonormal system of functions from L _∞[0, 1], possessing the properties of the Schauder basis for the space of continuous on [0, 1] functions with the norm L _∞, represents a spanning basis in a separable rearrangement-invariant space E, if and only if the adjoint space E* is separable. We prove that in any separable rearrangement-invariant space E the Haar system either forms an unconditional basis, or a strongly conditional one. The Haar system represents a strongly conditional basis in a separable rearrangement-invariant space, if and only if at least one of the Boyd indices of this space is trivial.     

Linear splicing and syntactic monoid

Splicing systems were introduced by Head in 1987 as a formal counterpart of a biological mechanism of DNA recombination under the action of restriction and ligase enzymes. Despite the intensive studies on linear splicing systems, some elementary questions about their computational power are still open. In particular, in this paper we face the problem of characterizing the proper subclass of regular languages which are generated by finite (Paun) linear splicing systems. We introduce here the class of marker languages L, i.e., regular languages with the form L=L₁[x]₁L₂, where L₁,L₂ are regular languages, [x] is a syntactic congruence class satisfying special conditions and [x]₁ is either equal to [x] or equal to [x]∪{1}, 1 being the empty word. Using classical properties of formal language theory, we give an algorithm which allows us to decide whether a regular language is a marker language. Furthermore, for each marker language L we exhibit a finite Paun linear splicing system and we prove that this system generates L.     

Extension theorems for linear lattices with positive algebraic basis

An ordered linear spaceV with positive wedgeK is said to satisfy extension property (E1) if for every subspaceL ₀ ofV such thatL ₀ ∩K is reproducing inL ₀, and every monotone linear functionalf ₀ defined onL ₀,f ₀ has a monotone linear extension to all ofV. A linear latticeX is said to satisfy extension property (E2) if for every sublatticeL ofX, and every linear functionalf defined onL which is a lattice homomorphism,f has an extensionf′ to all ofX which is also a linear functional and a lattice homomorphism. In this paper it is shown that a linear lattice with a positive algebraic basis has both extension property (E1) and (E2). In obtaining this result it is shown that the linear span of a lattice idealL and an extremal element not inL is again a lattice ideal. (HereX does not have to have a positive algebraic basis.) It is also shown that a linear lattice which possesses property (E2) must be linearly and lattice isomorphic to a functional lattice. An example is given of a function lattice which has property (E2) but does not have a positive algebraic basis. Yudin [12] has shown a reproducing cone in ann-dimensional linear lattice to be the intersection of exactlyn half-spaces. Here it is shown that the positive wedge in ann-dimensional archimedean ordered linear space satisfying the Riesz decomposition property must be the intersection ofn half-spaces, and hence the space must be a linear lattice with a positive algebraic basis. The proof differs from those given for the linear lattice case in that it uses no special techniques, only well known results from the theory of ordered linear space.     

Darstellung durch Spinorgeschlechter ternärer quadratischer formen

Let V be a regular ternary quadratic space over the algebraic number field F, L a lattice on V over the maximal order o of F. A number a, which is represented by all completions L_p, is not necessarily represented by L itself, but only by a lattice in the genus of L. We determine in which cases such a number is not represented by all spinor genera in the genus. Theorem 1 repeats the known [8,3] necessary conditions for this, which show that this behaviour is exceptional. They are sharpened in Theorem 2 to a necessary and sufficient condition in terms of certain groups Θ(L_p, a) (Definition 1). These groups are computed in Theorems 3 and 4 for nondyadic and 2-adip p. Some applications are given in the last section: We give new proofs (and in one case a correction) of results from [5] on the numbers represented by some genera of positive definite ternaries.     

A subclass of Ockham algebras

Here we introduce a subclass of the class of Ockham algebras ( L ; f ) for which L satisfies the property that for every x ∈ L , there exists n ≥ 0 such that fn ( x ) and fn+1 ( x ) are complementary. We characterize the structure of the lattice of congruences on such an algebra ( L ; f ). We show that the lattice of compact congruences on L is a dual Stone lattice, and in particular, that the lattice Con L of congruences on L is boolean if and only if L is finite boolean. We also show that L is congruence coherent if and only if it is boolean. Finally, we give a sufficient and necessary condition to have the subdirectly irreducible chains.     

Fourier series of half-range functions by smooth extension

This paper considers Fourier series approximations of one- and two-dimensional functions over the half-range, that is, over the sub-interval [0, L] of the interval [−L, L] in one-dimensional problems and over the sub-domain [0, L_x] × [0, L_y] of the domain [−L_x, L_x] × [−L_y, L_y] in two-dimensional problems. It is shown how to represent these functions using a Fourier series that employs a smooth extension. The purpose of the smooth extension is to improve the convergence characteristics otherwise obtained using the even and odd extensions. Significantly improved convergence characteristics are illustrated in one-dimensional and two-dimensional problems.     

Narrow and ℓ2-strictly singular operators from L p

In the first part of the paper we prove that for 2 < p, r < ∞ every operator T: L _p → ? _r is narrow. This completes the list of sequence and function Lebesgue spaces X with the property that every operator T : L _p → X is narrow. Next, using similar methods we prove that every ?₂-strictly singular operator from L _p , 1 < p < ∞, to any Banach space with an unconditional basis, is narrow, which partially answers a question of Plichko and Popov posed in 1990. A theorem of H. P. Rosenthal asserts that if an operator T from L ₁[0, 1] to itself satisfies the assumption that for each measurable set A ? [0, 1] the restriction \(T{|_{{L_1}(A)}}\) is not an isomorphic embedding, then T is narrow. (Here L ₁(A) = {x ∈ L ₁ : supp x ? A}.) Inspired by this result, in the last part of the paper, we find a sufficient condition, of a different flavor than being ?₂-strictly singular, for operators from L _p [0, 1] to itself, 1 < p < 2, to be narrow. We define a notion of a “gentle” growth of a function and we prove that for 1 < p < 2 every operator T from L _p to itself which, for every A ? [0, 1], sends a function of “gentle” growth supported on A to a function of arbitrarily small norm is narrow.     

Simple Lie algebras having extremal elements

Let L be a simple finite-dimensional Lie algebra of characteristic distinct from 2 and from 3. Suppose that L contains an extremal element that is not a sandwich, that is, an element x such that [x, [x, L]] is equal to the linear span of x in L. In this paper we prove that, with a single exception, L is generated by extremal elements. The result is known, at least for most characteristics, but the proofs in the literature are involved. The current proof closes a gap in a geometric proof that every simple Lie algebra containing no sandwiches (that is, ad-nilpotent elements of order 2) is in fact of classical type.     

The “Full Müntz Theorem” inL p[0, 1] for 0<p<∞

Denote by span {f ₁,f ₂, …} the collection of all finite linear combinations of the functionsf ₁,f ₂, … over ?. The principal result of the paper is the following. Theorem (Full Müntz Theorem in L_p(A) for p ∈ (0, ∞) and for compact sets A ? [0, 1] with positive lower density at 0). Let A ? [0, 1] be a compact set with positive lower density at 0. Let p ∈ (0, ∞). Suppose (λ _j ) _j=1 ^∞ is a sequence of distinct real numbers greater than ?(1/p). Then span {x ^λ1,x ^λ2,…} is dense in L_p(A) if and only if $\sum\limits_{j = 1}^\infty {\frac{{\lambda _j + \left( {1/p} \right)}}{{\left( {\lambda _j + \left( {1/p} \right)} \right)^2 + 1}} = \infty } $ . Moreover, if $\sum\limits_{j = 1}^\infty {\frac{{\lambda _j + \left( {1/p} \right)}}{{\left( {\lambda _j + \left( {1/p} \right)} \right)^2 + 1}} = \infty } $ , then every function from the L_p(A) closure of {x ^λ1,x ^λ2,…} can be represented as an analytic function on {z ∈ ? \ (?∞,0] : |z| < r_A} restricted to A ∩ (0, r_A) where $r_A : = \sup \left\{ {y \in \mathbb{R}:\backslash ( - \infty ,0]:\left| z \right|< r_A } \right\}$ (m(·) denotes the one-dimensional Lebesgue measure). This improves and extends earlier results of Müntz, Szász, Clarkson, Erdös, P. Borwein, Erdélyi, and Operstein. Related issues about the denseness of {x ^λ1,x ^λ2,…} are also considered.     

An Elemental Characterization of Orthogonal Ideals in Lie Algebras

In this paper, we prove that the ideals generated by two elements x, y in a nondegenerate Lie algebra L over a ring of scalars Φ with \({\frac 1 2, \frac 1 3}\) are orthogonal if and only if [x, [y, L]] = 0.     

Broken solutions of homogeneous variational problems

A real-valued function L on the tangent bundle of $\text{R}$ⁿ gives rise to variational problems as follows: for two points x₀, x₁ in $\text{R}$ⁿ and a time interval [0, T] to determine a curve γ: [O,T] → $\text{R}$ⁿ, connecting x₀ with x₁ which minimizes ∫₀^TL(γ(t), gg(t)) dt. We consider the associated Hamiltonian vectorfield on the cotangent bundle. If L is not convex on each fibre then the corresponding Hamiltonian vectorfield is not continuous. For homogeneous L and n = 2 restriction to an energy level gives an essentially three-dimensional vectorfield. In this case we list the possible discontinuities for generic L. Then we observe that there exits an open class of such variational problems, which admit no minimizing solution.     

A local solvability result for operators with characteristics having odd order multiplicity

It is shown that an operator L with the canonical form L = D_t^{2p + 1} + a(t, D_x) is locally solvable if and only if a(t, D_x) satisfies a Nirenberg-Treves-type condition.     

A maximum principle in Rn + 1

In this paper we establish maximum principles of the Cauchy problem for hyperbolic equations in $\text{R}$³ and $\text{R}$^{n + 1}(n ? 2). Our maximum principles generalize the results of Weinberger [5], and Sather [3, 4] for a class of equations such that the coefficients can be allowed to depend upon t, as well, in {x₁, x₂, t}-space and {x₁, x₂,…, x_n, t}-space. Throughout this paper, the influence of the work of Douglis [1] is apparent. See [2].     

Bipartite Q-polynomial distance-regular graphs and uniform posets

Let Γ denote a bipartite distance-regular graph with vertex set X and diameter D≥3. Fix x∈X and let L (resp., R) denote the corresponding lowering (resp., raising) matrix. We show that each Q-polynomial structure for Γ yields a certain linear dependency among RL ², LRL, L ² R, L. Define a partial order ≤ on X as follows. For y,z∈X let y≤z whenever ?(x,y)+?(y,z)=?(x,z), where ? denotes path-length distance. We determine whether the above linear dependency gives this poset a uniform or strongly uniform structure. We show that except for one special case a uniform structure is attained, and except for three special cases a strongly uniform structure is attained.     

On a multiplicative inequaluty for derived functions

В работе доказываетс я следующее неравенс тво. Пусть α₀, α₁, α₂, - произво льные неотрицательн ые числа α₀≠α₂. Тогда, еслиx(t) люб ая функция, для которой п роизводнаяx непреры вна и функция \(x^{a_0 } (\dot x)^{a_1 } (\ddot x)^{a_2 } \) принадлежи т пространствуL _∞[0, 1], то (*) $$\left\| x \right\|_{H^r [0,1]} \leqq c\left\| {x^{a_0 } (\dot x)^{a_1 } (\ddot x)^{a_2 } } \right\|L_{\infty [0,1]} ,$$ где ∥ · ∥H ^r [0,1] - норма в кл ас се функций на отрезке [0, 1], обладающих в простра нствеL _∞[0, 1] дробной производной гельдеровского типа порядкаr=(α₁+2α₂)/(α₀+α₁+α₂);с - конста нта, зависящая только от α₀, α₁, α₂. Это неравенство является точным в том смысле, что показател ьr есть максимальный, п ри котором неравенст во (*) имеет место с конечной конс тантойс. При α₀=α_? появляются логарифмические доб авки. Хорошо известно, что д ля непрерывной на [0, 1] фу нкции частные суммы Фурье п о тригонометрической системе равномерно с уммируются к ней методом (С, 1). И. Пра йс доказал, что для любой неограниченной последовательности целых положительных чисел {P _k} _k ^∞ =1 и Для любогоa∈[0, 1] существует непрерыв ная на [0, 1] функция, ряд Фурье которой по ортонормированно й мультипликативной системе (OHMC) не суммируется методом (С, 1) в точкеx=a. СССР, МОСКВА 103 055 УЛ. ОБРАЗЦОВА 15 МОСКОВСКИЙ ИНСТИТУТ ИНЖЕНЕРОВ ЖЕЛЕЗНОДО РОЖНОГО ТРАНСТПОРТА     

Generalized derivations as Jordan homomorphisms on lie ideals and right ideals

Let R be a prime ring, L a non-central Lie ideal of R and g a non-zero generalized derivation of R. If g acts as a Jordan homomorphism on L, then either g（x） = x for all x ∈ R, or char（R） = 2, R satisfies the standard identity s4（x1, x2, x3, x4）, L is commutative and u2 ∈ Z（R）, for any u C L. We also examine some consequences of this result related to generalized derivations which act as Jordan homomorphisms on the set [I, I], where I is a non-zero right ideal of R.     

Excess of a Lattice

 A pair (x,y) of elements in a lattice is mismatching if x is join-irreducible, y is meet-irreducible, and xy. The excess of a lattice L is defined by ex(L):=|L|− min{|V _x |+|I _y |:(x,y) is mismatching}, where V _x (I _y ) is the principal filter (ideal) generated by x(y), respectively. In the first half of this paper, it is shown that a lattice has excess zero (one) if and only if it is isomorphic to a Boolean lattice (one of a chain of length two, a diamond, and a pentagon), respectively. In the second half, we show that for each relatively complemented lattice which is not Boolean, its size is at most five times its excess. Moreover we determine extremal configurations. Received: September 2, 1999 Final version received: July 25, 2000 Acknowledgments. The author thanks the referee for constructive comments. Present address: Bureau of Human Resources, National Personnel Authority, 1-2-3 Kasumigaseki, Chiyoda-ku, Tokyo 100-8913, Japan. e-mail: abetet@jinji.go.jp     

Minkowski Length of 3D Lattice Polytopes

We study the Minkowski length L(P) of a lattice polytope P, which is defined to be the largest number of non-trivial primitive segments whose Minkowski sum lies in P. The Minkowski length represents the largest possible number of factors in a factorization of polynomials with exponent vectors in P, and shows up in lower bounds for the minimum distance of toric codes. In this paper we give a polytime algorithm for computing L(P) where P is a 3D lattice polytope. We next study 3D lattice polytopes of Minkowski length 1. In particular, we show that if Q, a subpolytope of P, is the Minkowski sum of L=L(P) lattice polytopes Q _i , each of Minkowski length 1, then the total number of interior lattice points of the polytopes Q ₁,??,Q _L is at most 4. Both results extend previously known results for lattice polygons. Our methods differ substantially from those used in the two-dimensional case.     

Hyperbolicity in median graphs

If X is a geodesic metric space and x ₁,x ₂,x ₃?∈?X, a geodesic triangle T?=?{x ₁,x ₂,x ₃} is the union of the three geodesics [x ₁ x ₂], [x ₂ x ₃] and [x ₃ x ₁] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by δ(X) the sharp hyperbolicity constant of X, i.e., ${\delta}(X)=\inf\{{\delta}\ge 0: \, X \, \text{is $\delta$-hyperbolic}\}. $ In this paper we study the hyperbolicity of median graphs and we also obtain some results about general hyperbolic graphs. In particular, we prove that a median graph is hyperbolic if and only if its bigons are thin.