Ultrapowers ofL 1(μ) and the subsequence splitting principle

We study the ultrapowers $L_1 (\mu )_\mathfrak{U} $ of aL ₁(μ) space, by describing the components of the well-known representation $L_1 (\mu )_\mathfrak{U} = L_1 (\mu _\mathfrak{U} ) \oplus _1 L_1 (\nu _\mathfrak{U} )$ , and we give a representation of the projection from $L_1 (\mu )_\mathfrak{U} $ onto $L_1 (\mu _\mathfrak{U} )$ . Moreover, the subsequence splitting principle forL ₁(μ) motivates the following question: if $\mathfrak{V}$ is an ultrafilter on ? and $[f_i ] \in L_1 (\mu )_\mathfrak{V} $ , is it possible to find a weakly convergent sequence (g _i ) ?L ₁(μ) following $\mathfrak{V}$ and a disjoint sequence (h _i ) ?L ₁(μ) such that [f _i ]=[g _i ]+[h _i ]? If $\mathfrak{V}$ is a selective ultrafilter, we find a positive answer by showing that $f = [f_i ] \in L_1 (\mu )_\mathfrak{V} $ belongs to $L_1 (\mu _{_\mathfrak{V} } )$ if and only if its representatives {f _i } are weakly convergent following $\mathfrak{V}$ and $f \in L_1 (\nu _\mathfrak{V} )$ if and only if it admits a representative consisting of pairwise disjoint functions. As a consequence, we obtain a new proof of the subsequence splitting principle. If $\mathfrak{V}$ is not a p-point then the above characterizations of $L_1 (\nu _{_\mathfrak{V} } )$ and $L_1 (\nu _{_\mathfrak{V} } )$ fail and the answer to the question is negative.     

On the best approximation of functions on the sphere in the metric ofL p (Sn), 1<p<∞

L_p, 1<p<, SⁿRⁿ⁺¹,n1, , , .     

Precipitous ideals and Σ 1 4 setssets

We prove under the assumption of the existence of a measurable, cardinal and precipitous ideal onw ¹ that every Σ ₁ ³ set is Lebesgue measurable, has the Baire property and is either countable or contians a perfect subset. We get similar results for Σ ₁ ⁴ sets, if we add the additional assumptions of C. H. and that carries a normal precipitous ideal.     

Multiwavelet packets and frame packets ofL 2(ℝ d )

The orthonormal basis generated by a wavelet ofL ²(ℝ) has poor frequency localization. To overcome this disadvantage Coifman, Meyer, and Wickerhauser constructed wavelet packets. We extend this concept to the higher dimensions where we consider arbitrary dilation matrices. The resulting basis ofL ²(ℝ ^d ) is called the multiwavelet packet basis. The concept of wavelet frame packet is also generalized to this setting. Further, we show how to construct various orthonormal bases ofL ²(ℝ ^d ) from the multiwavelet packets.     

Blaschke products satisfying the Carleson-Newman conditions and ideals of the algebra H∞

One obtains tests for an ideal of the algebra H to contain a Blaschke product satisfying the Carleson-Newman condition. If the generalized divisor of the ideal is bounded on the space of maximal ideals of H, then the ideal contains a function with bounded divisor.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 149, pp. 93–102, 1986.     

Lie ideals,Morita context and Generalized (α, β)-Derivations

A classical problem in ring theory is to study conditions under which a ring is forced to become commutative. Stimulated from Jacobson's famous result, several techniques are developed to achieve this goal. In the present note, we use a pair of rings, which are the ingredients of a Morita context, and obtain that if one of the ring is prime with the generalized (α, β)-derivations that satisfy certain conditions on the trace ideal of the ring, which by default is a Lie ideal, and the other ring is reduced, then the trace ideal of the reduced ring is contained in the center of the ring. As an outcome, in case of a semi-projective Morita context, the reduced ring becomes commutative.     

On translation and dilation invariant subspaces of L p (ℝn), 0 < p < 1

We prove that each translation and dilation invariant subspace X ⊂ L ^p (ℝⁿ), X ≠ L ^p (ℝⁿ), is contained in a maximal translation and dilation invariant subspace of L ^p (ℝⁿ). Moreover, we prove that the set of all maximal translation and dilation invariant subspaces of L ^p (ℝⁿ) has the power of continuum. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 345, 2007, pp. 5–21.     

One and two soliton solutions for the sinh–Gordon equation in (1+1), (2+1) and (3+1) dimensions

In this work we study the nonlinear sinh–Gordon equation in (1+1), (2+1) and (3+1) dimensions. We use the simplified form of Hirota’s method, to derive one and two soliton solutions for each equation.     

Estimates in the corona theorem and ideals ofH ∞ : A problem of T. Wolff

The main result of the paper is that there exist functionsf ₁,f ₂,f inH ^∞ satisfying the “corona condition”

冠图(G1 (o) G2)(o) G3的Wiener指数

给出了冠图(G1 (o)G2)(o) G3的Wiener指数W((G1(o)G2)(o) G3),得出了W((G1 (o)C2)(o)G3)与三个图的边数和顶点数有关,且与G1的结构有关,而与G2、G3的结构无关.     

Isomorphisms and projections for quotient-spaces of ℒ1-spaces by their reflexive subspaces

Let Z₁=X₁/E₁ and Z₂=X₂/E₂, where X₁ and X₂ are ₁-spaces, E₁cX₁, E₂c X₂. In this paper we study the following questions: 1) under what conditions are Z₁ and Z₂ isomorphic; 2) under what conditions is Z₁ isomorphic to a complemented subspace of Z₂. Some results: (a) if E₁ and E₂ are reflexive and Z₁. is isomorphic to Z₂, then one of the spaces E₁ E₂ is isomorphic to the product of the other by a finite-dimensional space; (b) if is a circle), E₁=H and E₂ is reflexive and X₂=Y^* for some Y, then it is impossible to imbed Z₁ in Z₂ as a complemented subspace.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 73, pp. 91–101, 1977.     

不存在秩为r(G)+2c(G)-1或r(G)-2c(G)+1的符号图

设Γ=(G,σ)是一个符号图,其中G是Γ的基图.设r(G,σ)是Γ的秩.[Linear Algebra Appl.,2018,538:166-186]和[Linear Multilinear Algebra,2019,67:2520-2539]分别证明了r(G)-2c(G)≤r(G,σ)≤r(G)+2c(G),其中,r(G)和c(G)分别是G的秩和圈空间维数.本文主要证明没有符号图的秩能够达到r(G)+2c(G)-1和r(G)-2c(G)+1,并且证明了存在无穷多个符号图的秩r(G,σ)=r(G)+2c(G)-s,其中s∈[0,4c(G)]且s≠1及4c(G)-1.     

Homomorphisms onC ∞ (E) andC ∞-bounding sets

We prove, for the class of real locally convex spacesE that are continuously and linearly injectable into somec ₀(), that every non-zero homomorphism on the algebraC (E) ofC -functions onE is given by a point evaluation at some point ofE. Furthermore, if every real-valuedC -function on the weak topology of a quasi-complete locally convex spaceE is bounded on a subsetA ofE, thenA is relatively weakly compact.     

Invariant subspaces and generalization of Nagaoka's theorem for the Hubbard model (U=∞)

The hubbard model (U=) on an arbitrary graph of sites in the presence of one hole in the system is considered. A sufficient condition for the absence of invariant subspaces of the space of states with fixed value of thez projection of the total spin that differ in the sets of possible spin configurations is found. A generalization of Nagaoka's results for bilobate graphs is given.Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 94, No. 1, pp. 160–164, January, 1993.     

从L~p(G,A)到L~p(G,X)的L~1(G,A)-模同态和不变算子(1

于树模 《数学年刊A辑(中文版)》1990,(6)

On complemented subspaces of the spaces (l p ) N ∩l q (l q )

Summary In this paper we study the complemented subspaces of the spaces (l^p)^N∩l^q(l^q), with 1≤p<q≤∞ or q=0, thereby showing that if (l^p)^N∩l^q(l^q)=F⊕G then either F or G contains a complemented copy of the whole space. Entrata in Redazione il 19 aprile 1999. Ricevuta versione finale il 4 febbraio 2000. Research supported by the Italian MURST.     

Characterization of G 2(q), where 2 < q ≡ −1(mod 3), by order components

We prove that the simple group G ₂(q), where 2 < q ≡ ?1(mod 3), is recognizable by the set of its order components. In other words, we prove that if G is a finite group with OC(G) = OC(G ₂(q)), then G ≌ G ₂(q).