Unconditional Basis Research Articles

Fast and accurate structure probability estimation for simultaneous alignment and folding of RNAs with Markov chains

MotivationSimultaneous alignment and folding (SA&F) of RNAs is the indispensable gold standard for inferring the structure of non-coding RNAs and their general analysis. The original algorithm, proposed by Sankoff, solves the theoretical problem exactly with a complexity of O(n^6) in the full energy model. Over the last two decades, several variants and improvements of the Sankoff algorithm have been proposed to reduce its extreme complexity by proposing simplified energy models or imposing restrictions on the predicted alignments.ResultsHere, we introduce a novel variant of Sankoff’s algorithm that reconciles the simplifications of PMcomp, namely moving from the full energy model to a simpler base pair-based model, with the accuracy of the loop-based full energy model. Instead of estimating pseudo-energies from unconditional base pair probabilities, our model calculates energies from conditional base pair probabilities that allow to accurately capture structure probabilities, which obey a conditional dependency. This model gives rise to the fast and highly accurate novel algorithm Pankov (Probabilistic Sankoff-like simultaneous alignment and folding of RNAs inspired by Markov chains).ConclusionsPankov benefits from the speed-up of excluding unreliable base-pairing without compromising the loop-based free energy model of the Sankoff’s algorithm. We show that Pankov outperforms its predecessors LocARNA and SPARSE in folding quality and is faster than LocARNA.

On certain subspaces of $${\ell _p}$$ for $${0<p\le 1}$$ and their applications to conditional quasi-greedy bases in p-Banach spaces

We construct for each $$0<p\le 1$$ an infinite collection of subspaces of $$\ell _p$$ that extend the example of Lindenstrauss (Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 12, 539–542, 1964) of a subspace of $$\ell _{1}$$ with no unconditional basis. The structure of this new class of p-Banach spaces is analyzed and some applications to the general theory of $${\mathcal {L}}_{p}$$ -spaces for $$0<p<1$$ are provided. The introduction of these spaces serves the purpose to develop the theory of conditional quasi-greedy bases in p-Banach spaces for $$p<1$$ . Among the topics we consider are the existence of infinitely many conditional quasi-greedy bases in the spaces $$\ell _{p}$$ for $$p\le 1$$ and the careful examination of the conditionality constants of the “natural basis” of these spaces.

Equivalent Norms in Hilbert Spaces with Unconditional Bases of Reproducing Kernels

In a Hilbert space H with an unconditional basis of reproducing kernels, we study the existence of a weighted integral norm with respect to an absolutely continuous measure, which is equivalent to the original H-norm. If the space H is defined via weighted integrals, the problem can be interpreted as restoring the original structure.

Russian projects in tractor engineering intended for increasing the efficiency of agricultural production

Technical and technological upgrading of agricultural production is a necessary factor in maintaining the competitiveness of an agricultural producer. According to experts, the lag in technological development as compared to the level of developed countries is a generation in production process, and from two to three generations in machinery for most agricultural producers. To ensure a decent position of the Russian Federation on the global food market, it is necessary to close this gap in the shortest possible time. Therefore, the timely introduction of the latest scientific and technical projects in production is currently an unconditional basis for its further development. Technical and technological upgrading has a direct impact on the efficiency of agricultural production and the sustainability of agriculture in general. The experience of these agricultural producers who have upgraded their own production shows that agriculture is becoming a high-tech and highly efficient industry. The upgrading of agricultural machinery is a continuous process of introducing innovations into individual components and systems that increase the power, functionality and efficiency of the machinery as a whole. The article analyzes a number of projects of domestic machine builders that are competitive with respect to foreign models.

Analytic P-ideals and Banach spaces

We study the interplay between Banach space theory and theory of analytic P-ideals. Applying the observation that, up to isomorphism, all Banach spaces with unconditional bases can be constructed in a way very similar to the construction of analytic P-ideals from submeasures, we point out numerous symmetries between the two theories. Also, we investigate a special case, the interactions between combinatorics of families of finite sets, topological properties of the “Schreier type” Banach spaces associated to these families, and the complexity of ideals generated by the canonical bases in these spaces. Among other results, we present some new examples of Banach spaces and analytic P-ideals, a new characterization of precompact families and its applications to enhance Pták's Lemma and Mazur's Lemma.

On two long standing open problems on $L_p$-spaces

The present note was written during the preparation of the talk at the International Conference dedicated to 70-th anniversary of Professor O. Lopushansky, September 16-19, 2019, Ivano-Frankivsk (Ukraine). We focus on two long standing open problems. The first one, due to Lindenstrauss and Rosenthal (1969), asks of whether every complemented infinite dimensional subspace of $L_1$ is isomorphic to either $L_1$ or $\ell_1$. The second problem was posed by Enflo and Rosenthal in 1973: does there exist a nonseparable space $L_p(\mu)$ with finite atomless $\mu$ and $1&lt;p&lt;\infty$, $p \neq 2$ having an unconditional basis? We analyze partial results and discuss on some natural ideas to solve these problems.

Closed ideals of operators on the Tsirelson and Schreier spaces

Let B(X) denote the Banach algebra of bounded operators on X, where X is either Tsirelson's Banach space or the Schreier space of order n for some n∈N. We show that the lattice of closed ideals of B(X) has a very rich structure; in particular B(X) contains at least continuum many maximal ideals.Our approach is to study the closed ideals generated by the basis projections. Indeed, the unit vector basis is an unconditional basis for each of the above spaces, so there is a basis projection PN∈B(X) corresponding to each non-empty subset N of N. A closed ideal of B(X) is spatial if it is generated by PN for some N. We can now state our main conclusions as follows:•the family of spatial ideals lying strictly between the ideal of compact operators and B(X) is non-empty and has no minimal or maximal elements;•for each pair I⫋J of spatial ideals, there is a family {ΓL:L∈Δ}, where the index set Δ has the cardinality of the continuum, such that ΓL is an uncountable chain of spatial ideals, ⋃ΓL is a closed ideal that is not spatial, andI⫋L⫋JandL+M‾=J whenever L,M∈Δ are distinct and L∈ΓL, M∈ΓM.

Corrigendum to the paper ``The universal Banach space with a $K$-suppression unconditional basis''

Corrigendum to the paper ``The universal Banach space with a $K$-suppression unconditional basis''

Functional continuity of topological algebras with orthonormal bases

(i) Every complex [Formula: see text]-algebra with an identity and with an orthonormal basis is functionally continuous; (ii) Every complex complete LMC algebra with an orthogonal basis is functionally continuous; (iii) Every complex sequentially complete locally convex algebra with an unconditional orthonormal basis and with an element [Formula: see text] for which [Formula: see text]th coefficient functional value tends to infinity as [Formula: see text] tends to infinity is functionally continuous. These results are proved and an example is provided for non-extendability of these results. A representation for positive linear functionals on a sequentially complete locally convex algebra with an unconditional orthonormal basis, with an identity, and with an element [Formula: see text] mentioned in (iii) is obtained. All results are obtained only for commutative algebras.

Unconditional bases of wavelets in local fields

We provide sufficient conditions on the wavelets on a local field K for the wavelet system to form an unconditional basis for the Hardy space H1(K) and the Lebesgue spaces Lp(K), 1 < p < ∞.

Some Remarks on Schauder Bases in Lipschitz Free Spaces

We show that the basis constant of every retractional Schauder basis on the Free space of a graph circle increases with the radius. As a consequence, there exists a uniformly discrete subset $M\subseteq\R^2$ such that $\F(M)$ does not have a retractional Schauder basis. Furthermore, we show that for any net $ N\subseteq\R^n$, $n\geq 2$, there is no retractional unconditional basis on the Free space $\mathcal F(N)$.

WAVELET CHARACTERIZATIONS OF WEIGHTED HERZ SPACES

We characterize the homogeneous weighted Herz space ˙ K α,p q (w1, w2) and the non-homogeneous weighted Herz space K α,p q (w1, w2) using wavelets in C 1 (R n ) with compact support. Applying the characterizations, we prove that the wavelet basis forms an unconditional basis in ˙ K α,p q (w1, w2) and in K α,p q (w1, w2) .

Smooth norms in dense subspaces of Banach spaces

In the first part of our paper, we show that ℓ∞ has a dense linear subspace which admits an equivalent real analytic norm. As a corollary, every separable Banach space, as well as ℓ1(c), also has a dense linear subspace which admits an analytic renorming. By contrast, no dense subspace of c0(ω1) admits an analytic norm. In the second part, we prove (solving in particular an open problem of Guirao, Montesinos, and Zizler in [7]) that every Banach space with a long unconditional Schauder basis contains a dense subspace that admits a C∞-smooth norm. Finally, we prove that there is a proper dense subspace of ℓ∞c(ω1) that admits no Gâteaux smooth norm. (Here, ℓ∞c(ω1) denotes the Banach space of real-valued, bounded, and countably supported functions on ω1.)

On the complete separation of asymptotic structures in Banach spaces

Let (ei)i denote the unit vector basis of ℓp, 1≤p<∞, or c0. We construct a reflexive Banach space with an unconditional basis that admits (ei)i as a uniformly unique spreading model while it has no subspace with a unique asymptotic model, and hence it has no asymptotic-ℓp or c0 subspace. This solves a problem of E. Odell. We also construct a space with a unique ℓ1 spreading model and no subspace with a uniformly unique ℓ1 spreading model. These results are achieved with the utilization of a new version of the method of saturation under constraints that uses sequences of functionals with increasing weights.

Universal decomposed Banach spaces

Let {mathcal {B}} be a class of finite-dimensional Banach spaces. A {mathcal {B}}-decomposed Banach space is a Banach space X endowed with a family {mathcal {B}}_Xsubset {mathcal {B}} of subspaces of X such that each xin X can be uniquely written as the sum of an unconditionally convergent series sum _{Bin {mathcal {B}}_X}x_B for some (x_B)_{Bin {mathcal {B}}_X}in prod _{Bin {mathcal {B}}_X}B. For every Bin {mathcal {B}}_X let mathrm {pr}_B:Xrightarrow B denote the coordinate projection. Let Csubset [-1,1] be a closed convex set with Ccdot Csubset C. The C-decomposition constant K_C of a {mathcal {B}}-decomposed Banach space (X,{mathcal {B}}_X) is the smallest number K_C such that for every function alpha :{mathcal {F}}rightarrow C from a finite subset {mathcal {F}}subset {mathcal {B}}_X the operator T_alpha =sum _{Bin {mathcal {F}}}alpha (B)cdot mathrm {pr}_B has norm Vert T_alpha Vert le K_C. By varvec{{mathcal {B}}}_C we denote the class of {mathcal {B}}-decomposed Banach spaces with C-decomposition constant K_Cle 1. Using the technique of Fraïssé theory, we construct a rational {mathcal {B}}-decomposed Banach space mathbb {U}_Cin varvec{{mathcal {B}}}_C which contains an almost isometric copy of each {mathcal {B}}-decomposed Banach space Xin varvec{{mathcal {B}}}_C. If {mathcal {B}} is the class of all 1-dimensional (resp. finite-dimensional) Banach spaces, then mathbb {U}_{C} is isomorphic to the complementably universal Banach space for the class of Banach spaces with an unconditional (f.d.) basis, constructed by Pełczyński (and Wojtaszczyk).

Geometry of radial Hilbert spaces with unconditional bases of reproducing kernels

Geometry of radial Hilbert spaces with unconditional bases of reproducing kernels

Algebras of Diagonal Operators of the Form Scalar-Plus-Compact Are Calkin Algebras

For every Banach space X with a Schauder basis, consider the Banach algebra RI⊕Kdiag(X) of all diagonal operators that are of the form λI+K. We prove that RI⊕Kdiag(X) is a Calkin algebra, that is, there exists a Banach space YX such that the Calkin algebra of YX is isomorphic as a Banach algebra to RI⊕Kdiag(X). Among other applications of this theorem, we obtain that certain hereditarily indecomposable spaces and the James spaces Jp and their duals endowed with natural multiplications are Calkin algebras; that all nonreflexive Banach spaces with unconditional bases are isomorphic as Banach spaces to Calkin algebras; and that sums of reflexive spaces with unconditional bases with certain James–Tsirelson type spaces are isomorphic as Banach spaces to Calkin algebras.

Spectra of a discrete Sturm-Liouville problem with eigenparameter-dependent boundary conditions in Pontryagin space

In this paper, we consider the spectra of a discrete Sturm-Liouville problem with two eigenparameter-dependent boundary conditions. Different from the previous results, the operator corresponding to the problem here is not self-adjoint in the corresponding Hilbert space, but J-self-adjoint in a new Pontryagin space under a new J-metric. No matter what, the existence, simplicity and interlacing properties of real eigenvalues, nonreal eigenvalues and the oscillation properties of eigenfunctions are obtained in this paper. Meanwhile, we also obtain the existence of a pair of nonreal eigenvalues and an unconditional basis of the Pontryagin space under the new J-metric.

Regular Ordinary Differential Operators with Involution

The main results of the paper are related to the study of differential operators of the form $$Ly = {y^{\left( n \right)}}\left( { - x} \right) + \sum\limits_{k = 1}^n {pk\left( x \right){y^{\left( {n - k} \right)}}\left( { - x} \right) + } \sum\limits_{k = 1}^n {{q_k}\left( x \right){y^{\left( {n - k} \right)}}} \left( x \right),\,x \in \left[ { - 1,1} \right],$$ with boundary conditions of general form concentrated at the endpoints of a closed interval. Two equivalent definitions of the regularity of boundary conditions for the operator L are given, and a theorem on the unconditional basis property with brackets of the generalized eigenfunctions of the operator L in the case of regular boundary conditions is proved.

Embeddability of ℓp and bases in Lipschitz free p-spaces for 0 < p ≤ 1

Our goal in this paper is to continue the study initiated by the authors in [4] of the geometry of the Lipschitz free p-spaces over quasimetric spaces for 0<p≤1, denoted Fp(M). Here we develop new techniques to show that, by analogy with the case p=1, the space ℓp embeds isomorphically in Fp(M) for 0<p<1. Going further we see that despite the fact that, unlike the case p=1, this embedding need not be complemented in general, complementability of ℓp in a Lipschitz free p-space can still be attained by imposing certain natural restrictions to M. As a by-product of our discussion on bases in Fp([0,1]), we obtain examples of p-Banach spaces for p<1 that are not based on a trivial modification of Banach spaces, which possess a basis but fail to have an unconditional basis.