Operators on the space of bounded strongly measurable functions

AbstractLet L(X,Y) stand for the space of all bounded linear operators between real Banach spaces (X,‖⋅‖X) and (Y,‖⋅‖Y), and let Σ be a σ-algebra of subsets of a non-empty set Ω. Let L∞(Σ,X) denote the Banach space of all bounded strongly Σ-measurable functions f:Ω→X equipped with the supremum norm ‖⋅‖. A bounded linear operator T from L∞(Σ,X) to a Banach space Y is said to be σ-smooth if ‖T(fn)‖Y→0 whenever ‖fn(ω)‖X→0 for all ω∈Ω and supn‖fn‖<∞. It is shown that if an operator measure m:Σ→L(X,Y) is variationally semi-regular (i.e., m˜(An)→0 as An↓∅, where m˜(A) stands for the semivariation of m on A∈Σ), then the corresponding integration operator Tm:L∞(Σ,X)→Y is σ-smooth. Conversely, it is proved that every σ-smooth operator T:L∞(Σ,X)→Y admits an integral representation with respect to its representing operator measure. We prove a Banach–Steinhaus type theorem for σ-smooth operators from L∞(Σ,X) to Y. In particular, we study the topological properties of the space L∞(Σ,X)c⁎ of all σ-smooth functionals on L∞(Σ,X). We prove a form of a generalized Nikodým convergence theorem and characterize relative σ(L∞(Σ,X)c⁎,L∞(Σ,X))-sequential compactness in L∞(Σ,X)c⁎. We derive a Grothendieck type theorem for L∞(Σ,X)c⁎. The relationships between different classes of linear operators on L∞(Σ,X) are established

A universal context-free grammar

In this report we show that, for each alphabet Σ, there exists a context-free grammar G which satisfies the property that for each context-free language L ⊆ Σ* a regular control set C can be found such that LC(G) = L

Non-equilibrium dynamics of dense gas under tight confinement

The force-driven Poiseuille flow of dense gases between two parallel plates is investigated through the numerical solution of the generalized Enskog equation for two-dimensional hard discs. We focus on the competing effects of the mean free path λ, the channel width L, and the disc diameter σ. For elastic collisions between hard discs, the normalised mass flow rate in the hydrodynamic limit increases with L/σ for a fixed Knudsen number (defined as Kn = λ/L), but is always smaller than that predicted by the Boltzmann equation. Also, for a fixed L/σ, the mass flow rate in the hydrodynamic flow regime is not a monotonically decreasing function of Kn but has a maximum when the solid fraction is about 0.3. Under ultra-tight confinement the famous Knudsen minimum disappears, and the mass flow rate increases with Kn, and is larger than that predicted by the Boltzmann equation in the free-molecular flow regime; for a fixed Kn, the smaller the L/σ, the larger the mass flow rate. In the transitional flow regime, however, the variation of the mass flow rate with L/σ is not monotonic for a fixed Kn: the minimum mass flow rate occurs at L/σ ≈ 2 ∼ 3. For inelastic collisions, the energy dissipation between the hard discs always enhances the mass flow rate. Anomalous slip velocity is also found, which decreases with increasing Knudsen number. The mechanism for these exotic behaviour is analysed

Time and tape complexity of pushdown automaton languages

An algorithm is presented which will determine whether any string w in Σ*, of length n, is contained in a language L ⊆ Σ* defined by a two-way nondeterministic pushdown automation. This algorithm requires time n3 when implemented on a random access computer. It requires n4 time and n2 tape when implemented on a multitape Turing machine.If the pushdown automaton is deterministic, the algorithm requires n2 time on a random access computer and n2 log n time on a multitape Turing machine

Implications of the kinematical structure of circumnuclear star-forming regions on their derived properties

We reviewthe results of high-dispersion spectroscopy of 17 circumnuclear star-forming regions (CNSFRs) in 3 nearby early spiral galaxies, NGC 2903, NGC 3310 and NGC 3351. We findthat single-Gaussian fitting to the Hβ and [O III]λ5007 Å line profiles results in velocity dispersions around 32 and 52 km s-1, respectively, while the IR Ca II triplet cross-correlation technique provides stellar velocity dispersion values close to 50 km s-1. Even though multiple kinematical components are present, the relation between gas velocity dispersion and Balmer emission line luminosity (L-σ relation) reproduces the correlation for disc giant HII regions albeit with a larger scatter. The scatter in the L-σ relation is considerably reduced when theoretical evolutionary corrections are applied suggesting that an age range is present in the sample of CNSFRs. To analyse the observed complex profiles, we performed multiple Gaussian component fits to the Hβ and [O III]λ5007 Å lines obtaining optimal fits with two Gaussians of different widths. These best fits indicate that the narrower component has average velocity dispersion close to 23 km s-1 while the broader component shows average values in the range 50-60 km s-1 for both lines, close to the observed stellar velocity dispersions. The fluxes of the broad and narrow Hβ components are similar. This is not the case for [O III]λ5007 Å for which the broad components have higher fluxes than the narrow ones, thus producing a clear segregation in their [OIII]/Hβ ratios. We suggest a possible scenario for understanding the behaviour of CNSFRs in the L-σ and σgas-σ* diagrams involving aninner gaseous disc responsible for the narrow component of the emission lines. Our main conclusion is that the presence of different kinematical components with similar total fluxes in the emission line spectrum of CNSFRs raises important doubts regarding the properties of the ionized gas derived from global line ratios obtained with low-resolution spectroscopy in star-forming regions in the central regions of early-type galaxies. Given the ubiquity of these star-forming systems, ionized gas analyses should be done preferably from high-dispersion spectra with high spatial resolution.Facultad de Ciencias Astronómicas y GeofísicasInstituto de Astrofísica de La Plat

On sparse languages L such that LL = Σ

AbstractA language L ∈ Σ∗ is said to be sparse if L contains a vanishingly small fraction of all possible strings of length n in Σ∗. C. Ponder asked if there exists a sparse language L such that LL = Σ∗. We answer this question in the affirmative. Several different constructions are provided, using ideas from probability theory, fractal geometry, and analytic number theory. We obtain languages that are optimally sparse, up to a constant factor. Finally, we consider the generalization Lj = Σ∗

On the separability of the restriction functor

Let G be a group, Λ = L σ∈G Λσ a strongly graded ring by G, H a subgroup of G and ΛH = L σ∈H Λσ. We give a necessary and sufficient condition for the ring Λ/ΛH to be separable, generalizing the corresponding result for the ring extension Λ/Λ1. As a consequence of this result we give a condition for Λ to be a hereditary order in case Λ is a strongly graded by finite group R-order in a separable K-algebra, for R a Dedekind domain with quotient field K

A non-ambiguous decomposition of regular languages and factorizing codes

AbstractGiven languages Z,L⊆Σ∗,Z is L-decomposable (finitely L-decomposable, resp.) if there exists a non-trivial pair of languages (finite languages, resp.) (A,B), such that Z=AL+B and the operations are non-ambiguous. We show that it is decidable whether Z is L-decomposable and whether Z is finitely L-decomposable, in the case Z and L are regular languages. The result in the case Z=L allows one to decide whether, given a finite language S⊆Σ∗, there exist finite languages C,P such that SC∗P=Σ∗ with non-ambiguous operations. This problem is related to Schützenberger's Factorization Conjecture on codes. We also construct an infinite family of factorizing codes

Inequalities for the q-permanent. II

AbstractFor a complex number q, the q-permanent of an n × n complex matrix A = ((aij)), written perq(A), is defined asperq(A)=∑σ∈Lnql(σ)φi=lnaiσ(i),where Jn is the symmetric group of degree n, and l(σ) the number of inversions of σ [i.e., the number of pairs, i, j such that 1 ⩽ i < j ⩽ n and σ(i) > σ(j)]. The function is of interest in that it includes both the determinant and the permanent as special cases. It is known that if A is positive semidefinite and if −1 ⩽ q ⩽ 1, then perq(A) ⩾ 0. We obtain results for the q-permanent, a few of which are generalizations of some results of Ando

О ПОЛИАДИЧЕСКОЙ ОПЕРАЦИИ СПЕЦИАЛЬНОГО ВИДА

In the article the authors continue to study the polyadic operation ηs, σ, k that was earlier defined at the Cartesian power Ak of the nth groupoid &lt; A, η &gt; by the substitution of σ ∈ Sk and the nth operation η. The special case of the polyadic operation ηs, σ, k is the lth operation [ ]l, σ, k that is defined by one of the authors for any integer k ≥ 2, l ≥ 2 and for any substitution of the σ set {1, …, k} at the Cartesian power Ak of the semigroup A. In turn, the special case of the lth operation [ ]l, σ, k consists of two polyadic operations by E. Post, one of which he defined at the Cartesian power of the symmetric group and the second – at the Cartesian power of the general linear group over the field of complex numbers. The properties of the operations ηs, σ, k are studied in the article. In particular, a new proof of the associativity of the polyadic operation ηs, σ, k was obtained. В статье продолжается изучение полиадической операции ηs, σ, k , которая была определена ранее на декартовой степени Ak n-арного группоида &lt; A, η &gt; с помощью подстановки σ ∈ Sk и n-арной операции η. Частным случаем полиадической операции ηs, σ, k является l-арная операция [ ]l, σ, k , которую один из авторов определил для любых целых k ≥ 2, l ≥ 2 и любой подстановки σ множества {1, …, k} на k-й декартовой степени Ak полугруппы A. В свою очередь, частными случаями l-арной операции [ ]l, σ, k являются две полиадические операции Э. Поста, одну из которых он определил на декартовой степени симметрической группы, вторую – на декартовой степени полной линейной группы над полем комплексных чисел. В статье приведены новые результаты об операции ηs, σ, k . В частности, получено новое доказательство ассоциативности этой полиадической операции.