Basis In Lp Research Articles

Power set of some quasinilpotent weighted shifts on ℓp

For a quasinilpotent operator T on a Banach space X, Douglas and Yang defined kx=limsupz→0ln⁡‖(z−T)−1x‖ln⁡‖(z−T)−1‖ for each nonzero vector x∈X, and call Λ(T)={kx:x≠0} the power set of T. They proved that the power set have a close link with T's lattice of hyperinvariant subspaces. In this paper, we compute the power set of some weighted shifts on ℓp for 1≤p<∞. The following results are obtained: (1) If T is an injective quasinilpotent forward unilateral weighted shift on ℓp(N), then Λ(T)={1} when ke0=1, where {en}n=0∞ be the canonical basis for ℓp(N); (2) There is a class of backward unilateral weighted shifts on ℓp(N) whose power set is [0,1]; (3) There exists a bilateral weighted shift on ℓp(Z) with power set [12,1].

Lipoprotein(a) Is Associated With the Progression and Vulnerability of New-Onset Carotid Atherosclerotic Plaque.

Although important progress has been made in understanding Lp(a) (lipoprotein[a])-mediated stroke risk, the contribution of Lp(a) to the progression of vulnerable plaque features associated with stroke risk remains unclear. This study aims to evaluate whether Lp(a) is associated with carotid plaque progression, new-onset plaque features, and plaque vulnerability in a prospective community-based cohort study. Baseline Lp(a) levels were measured using latex-enhanced turbidimetric immunoassay among 804 participants aged 45 to 74 years and free of cardiovascular disease in the Chinese Multi-provincial Cohort Study-Beijing project. Carotid atherosclerosis was measured twice by B-mode ultrasonography over a 10-year interval during the 2002 and 2012 surveys to assess the progression of total, vulnerable and stable plaques, and plaque vulnerability. The total plaque area and plaque vulnerability score were calculated. The median baseline Lp(a) level was 10.20 mg/dL (interquartile range, 6.20 to 17.18 mg/dL). Modified Poisson regression analysis showed that Lp(a) ≥50 mg/dL was significantly associated with 10-year progression of total carotid plaque (relative risk [RR], 1.41 [95% CI, 1.21-1.64]; E-value=2.17), vulnerable plaque (RR, 1.93 [95% CI, 1.54-2.41]), and stable plaque (RR, 1.51 [95% CI, 1.11-2.07]) compared with Lp(a) <50 mg/dL. Moreover, among participants without plaque at baseline, Lp(a) ≥50 mg/dL was related to an increased total plaque area (β=0.36 [95% CI, 0.06-0.65]; P=0.018) and increased plaque vulnerability score (β=0.30 [95% CI, 0.01-0.60]; P=0.045) in multivariable linear regression. Elevated Lp(a) levels were associated with 10-year carotid plaque progression and plaque vulnerability, providing a basis for Lp(a) as a treatment target for stroke prevention.

Lipoprotein (a), Inflammation, and Atherosclerosis

Growing evidence has shown that high levels of lipoprotein (a) (Lp(a)) and chronic inflammation may be responsible for the residual risk of cardiovascular events in patients managed with an optimal evidence-based approach. Clinical studies have demonstrated a correlation between higher Lp(a) levels and several atherosclerotic diseases including ischemic heart disease, stroke, and degenerative calcific aortic stenosis. The threshold value of Lp(a) serum concentrations associated with a significantly increased cardiovascular risk is >125 nmol/L (50 mg/dL). Current available lipid-lowering drugs have modest-to-no impact on Lp(a) levels. Chronic inflammation is a further condition potentially implicated in residual cardiovascular risk. Consistent evidence has shown an increased risk of cardiovascular events in patients with high sensitivity C reactive protein (>2 mg/dL), an inflammation biomarker. A number of anti-inflammatory drugs have been investigated in patients with or at risk of cardiovascular disease. Of these, canakinumab and colchicine have been found to be associated with cardiovascular risk reduction. Ongoing research aimed at improving risk stratification on the basis of Lp(a) and vessel inflammation assessment may help refine patient management. Furthermore, the identification of these conditions as cardiovascular risk factors has led to increased investigation into diagnostic and therapeutic strategies targeting them in order to reduce atherosclerotic cardiovascular disease burden.

How Genetic Variants in Children with Familial Hypercholesterolemia Not Only Guide Detection, but Also Treatment.

Familial hypercholesterolemia (FH) is a hereditary disorder that causes severely elevated low-density lipoprotein (LDL-C) levels, which leads to an increased risk for premature cardiovascular disease. A variety of genetic variants can cause FH, namely variants in the genes for the LDL receptor (LDLR), apolipoprotein B (APOB), proprotein convertase subtilisin/kexin type 9 (PCSK9), and/or LDL-receptor adaptor protein 1 (LDLRAP1). Variants can exist in a heterozygous form (HeFH) or the more severe homozygous form (HoFH). If affected individuals are diagnosed early (through screening), they benefit tremendously from early initiation of lipid-lowering therapy, such as statins, and cardiovascular imaging to detect possible atherosclerosis. Over the last years, due to intensive research on the genetic basis of LDL-C metabolism, novel, promising therapies have been developed to reduce LDL-C levels and subsequently reduce cardiovascular risk. Results from studies on therapies focused on inhibiting PCSK9, a protein responsible for degradation of the LDLR, are impressive. As the effect of PCSK9 inhibitors (PCSK9-i) is dependent of residual LDLR activity, this medication is less potent in patients without functional LDLR (e.g., null/null variant). Novel therapies that are expected to become available in the near future focused on inhibition of another major regulatory protein in lipid metabolism (angiopoietin-like 3 (ANGPTL3)) might dramatically reduce the frequency of apheresis in children with HoFH, independently of their residual LDLR. At present, another independent risk factor for premature cardiovascular disease, elevated levels of lipoprotein(a) (Lp(a)), cannot be effectively treated with medication. Further understanding of the genetic basis of Lp(a) metabolism, however, offers a possibility for the development of novel therapies.

Relationship between lipoprotein(a) and revascularization after percutaneous coronary intervention in type 2 diabetes mellitus patients with acute coronary syndrome

Background As a residual risk factor for coronary heart disease, lipoprotein(a) [Lp(a)] is associated with the occurrence of cardiovascular events after percutaneous coronary intervention (PCI). The revascularization rate after PCI is high among diabetic patients. However, the relationship between Lp(a) and revascularization after PCI in type 2 diabetes mellitus (T2DM) patients with acute coronary syndrome (ACS) remains unclear. Methods The investigation was a single-center, observational, retrospective cohort study. Patients with T2DM who were first diagnosed with ACS and underwent PCI were included in the study. As a result, 362 patients were enrolled and divided into three groups according to tertiles on basis of Lp(a) levels (11.48 mg/dL and 21.70 mg/dL). The incidence of major adverse cardiac events (MACEs), including cardiac death, revascularization due to myocardial ischemia, readmission due to angina, and nonfatal stroke, was evaluated. Subgroups were established according to the low-density lipoprotein cholesterol (LDL-C) level (70 mg/dL). Results During follow-up (median: 2.0 years), 69 MACEs occurred, and 76.81% of these patients underwent revascularization. The Lp(a) level in the MACE group was significantly higher than that in the non-MACE group (22.90 mg/dL vs. 14.10 mg/dL, p < .001). Kaplan–Meier analysis revealed that the incidence of adverse cardiovascular events was significantly higher in the high Lp(a) groups than in the low Lp(a) groups (p = .001), mainly because of the increased occurrence of revascularization irrespective of LDL-C level (<70 mg/dL; ≥70 mg/dL, both p < .05) rather than death, nonfatal stroke, or hospital readmission due to angina (both p > .05). The receiver operating characteristic (ROC) curve showed that the area under the curve (AUC) for Lp(a) in predicting the occurrence of MACE and revascularization were 0.664 and 0.668 respectively, both p < .05. Furthermore, multivariate Cox regression models indicated that Lp(a) was independently associated with revascularization [medium Lp(a) category: HR (95% CI): 2.988 (1.164–7.671), p = .023; high Lp(a) category: HR (95% CI): 4.937 (2.023–12.052), p < .001]. Conclusion Lp(a) was an independent predictor of revascularization in patients with ACS complicated with T2DM, regardless of LDL-C levels. This suggests that Lp(a) measurement may help identify high-risk diabetic patients with ACS.

Quasi-greedy bases in ℓp (0 < p < 1) are democratic

The list of known Banach spaces whose linear geometry determines the (nonlinear) democracy functions of their quasi-greedy bases to the extent that they end up being democratic, reduces to c0, ℓ2, and all separable L1-spaces. Oddly enough, these are the only Banach spaces that, when they have an unconditional basis, it is unique. Our aim in this paper is to study the connection between quasi-greediness and democracy of bases in non-locally convex spaces. We prove that all quasi-greedy bases in ℓp for 0<p<1 (which also has a unique unconditional basis) are democratic with fundamental function of the same order as (m1/p)m=1∞. The methods we develop allow us to obtain even more, namely that the same occurs in any separable Lp-space, 0<p<1, with the bounded approximation property.

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.

Duality problem for disjointly homogeneous rearrangement invariant spaces

Let 1≤p<∞. A Banach lattice E is said to be disjointly homogeneous (resp. p-disjointly homogeneous) if two arbitrary normalized disjoint sequences from E contain equivalent in E subsequences (resp. every normalized disjoint sequence contains a subsequence equivalent in E to the unit vector basis of lp). Answering a question raised in the paper [11], for each 1<p<∞, we construct a reflexive p-disjointly homogeneous rearrangement invariant space on [0,1] whose dual is not disjointly homogeneous. Employing methods from interpolation theory, we provide new examples of disjointly homogeneous rearrangement invariant spaces; in particular, we show that there is a Tsirelson type disjointly homogeneous rearrangement invariant space, which contains no subspace isomorphic to lp, 1≤p<∞, or c0.

Greedy algorithm with gaps

We generalize the well-known greedy approximation algorithm, by allowing gaps in the approximating sequence. We give examples of bases which are “quasi-greedy with gaps,” in spite of failing to be quasi-greedy in the usual sense. However, we also show that for some classical bases (such as the normalized Haar basis in L1, and the trigonometric basis in Lp for p≠2), the greedy algorithm may diverge, even if gaps are introduced into the approximating sequence.

The geometry of two-valued subsets of Lp -spaces

Abstract Let 𝓜(Ω, μ) denote the algebra of all scalar-valued measurable functions on a measure space (Ω, μ). Let B ⊂ 𝓜(Ω, μ) be a set of finitely supported measurable functions such that the essential range of each f ∈ B is a subset of {0,1}. The main result of this paper shows that for any p ∈ (0, ∞), B has strict p-negative type when viewed as a metric subspace of L p (Ω, μ) if and only if B is an affinely independent subset of 𝓜(Ω, μ) (when 𝓜(Ω, μ) is considered as a real vector space). It follows that every two-valued (Schauder) basis of L p (Ω, μ) has strict p-negative type. For instance, for each p ∈ (0, ∞), the system of Walsh functions in L p [0,1] is seen to have strict p-negative type. The techniques developed in this paper also provide a systematic way to construct, for any p ∈ (2, ∞), subsets of L p (Ω, μ) that have p-negative type but not q-negative type for any q &gt; p. Such sets preclude the existence of certain types of isometry into L p -spaces.

Haar bases on quasi-metric measure spaces, and dyadic structure theorems for function spaces on product spaces of homogeneous type

We give an explicit construction of Haar functions associated to a system of dyadic cubes in a geometrically doubling quasi-metric space equipped with a positive Borel measure, and show that these Haar functions form a basis for Lp. Next we focus on spaces X of homogeneous type in the sense of Coifman and Weiss, where we use these Haar functions to define a discrete square function, and hence to define dyadic versions of the function spaces H1(X) and BMO(X). In the setting of product spaces X˜=X1×⋯×Xn of homogeneous type, we show that the space BMO(X˜) of functions of bounded mean oscillation on X˜ can be written as the intersection of finitely many dyadic BMO spaces on X˜, and similarly for Ap(X˜), reverse-Hölder weights on X˜, and doubling weights on X˜. We also establish that the Hardy space H1(X˜) is a sum of finitely many dyadic Hardy spaces on X˜, and that the strong maximal function on X˜ is pointwise comparable to the sum of finitely many dyadic strong maximal functions. These dyadic structure theorems generalize, to product spaces of homogeneous type, the earlier Euclidean analogues for BMO and H1 due to Mei, and Li, Pipher and Ward.

Nonnegativity constraints for structured complete systems

We investigate pointwise nonnegativity as an obstruction to various types of structured completeness in L p ( R ) L^p(\mathbb {R}) . For example, we prove that if each element of the system { f n } n = 1 ∞ ⊂ L p ( R ) \{f_n\}_{n=1}^\infty \subset L^p(\mathbb {R}) is pointwise nonnegative, then { f n } n = 1 ∞ \{f_n\}_{n=1}^{\infty } cannot be an unconditional basis or unconditional quasibasis (unconditional Schauder frame) for L p ( R ) L^p(\mathbb {R}) . In particular, in L 2 ( R ) L^2(\mathbb {R}) this precludes the existence of nonnegative Riesz bases and frames. On the other hand, there exist pointwise nonnegative conditional quasibases in L p ( R ) L^p(\mathbb {R}) , and there also exist pointwise nonnegative exact systems and Markushevich bases in L p ( R ) L^p(\mathbb {R}) .

Polarization and propagation characteristics of switchable first-order azimuthally asymmetric beam generated in dual-mode fiber.

We report here the controlled generation of a linearly polarized first-order azimuthally asymmetric beam (F-AAB) in a dual-mode fiber (DMF) by appropriate superposition of selectively excited zeroth-order vector modes that are doughnut-shaped azimuthally symmetric beams (D-ASBs). We first demonstrate continually switching polarization mode structures having an identical two-lobe intensity profile (i.e., intra-F-AAB conversion). Then, under a distinct launching state, we generate mode structures progressively toggling between the doughnut-shaped profile and two-lobe pattern having dissimilar polarization orientations (i.e., F-AAB to D-ASB conversion). Interestingly, a decentralized elliptical Gaussian beam possessing homogenous spatial polarization is obtained by enhancing the contribution of the fundamental mode (HE₁₁/LP₀₁) in selectively excited F-AAB. A smoothly varying azimuth of the input beam in this situation resulted in redistribution of transverse energy procuring a unique and exciting unconventional two-grain T-polarized beam having mutually orthogonal state of polarization (SOP). All of the above three were achieved under a given set of launching conditions (tilt/offset) of a Gaussian mode (TEM₀₀) devised with changing SOP of the input beam. A strong modulation in the output beam characteristics was also observed with the variation in propagation distance (for a fixed input SOP) owing to the large difference in propagation constants of the participating modes (LP₀₁ and one of the F-AABs). Finally, this particular study led to a design for a low-cost highly sensitive strain measuring device based on tracking the centroid movement of the output intensity pattern. Each of our experimentally observed intensity/polarization distributions is theoretically mapped on a one-to-one basis considering a linear superposition of appropriately excited LP basis modes of the waveguide toward a complete understanding of the polarization and mode propagation in the dual-mode structure.

Unconditional structures of translates for L p (ℝ d )

We prove that a sequence (fi)i=1∞ of translates of a fixed f ∈ Lp(ℝ) cannot be an unconditional basis of Lp(ℝ) for any 1 ≤ p < ∞. In contrast to this, for every 2 < p < ∞, d ∈ ℕ and unbounded sequence (λn)n∈ℕ ⊂ ℝd we establish the existence of a function f ∈ Lp(ℝd) and sequence (gn*)n∈ℕ ⊂ Lp*(ℝd) such that \({({T_{{\lambda _n}}}f,g_n^*)_{n \in {\Bbb N}}}\) forms an unconditional Schauder frame for Lp(ℝd). In particular, there exists a Schauder frame of integer translates for Lp(ℝ) if (and only if) 2 < p < ∞.

Disjointly homogeneous rearrangement invariant spaces via interpolation

A Banach lattice E is called p-disjointly homogeneous, 1≤p≤∞, when every sequence of pairwise disjoint normalized elements in E has a subsequence equivalent to the unit vector basis of ℓp. Employing methods from interpolation theory, we clarify which r.i. spaces on [0,1] are p-disjointly homogeneous. In particular, for every 1<p<∞ and any increasing concave function φ on [0,1], which is not equivalent to neither 1 nor t, there exists a p-disjointly homogeneous r.i. space with the fundamental function φ. Moreover, it is shown that given 1<p<∞ and an increasing concave function φ with non-trivial dilation indices, there is a unique p-disjointly homogeneous space among all interpolation spaces between the Lorentz and Marcinkiewicz spaces associated with φ.

The fuzzy basis of the sequence space ℓP (ℱ)

Abstract Nanda [6] introduced and discussed ℓp (ℱ) space of sequences of fuzzy numbers, and various authors studied many different properties of the space ℓp (ℱ). In the sequel of these studies, we have shown in this paper that the set (Ek ) = (E 0, E 1, E 2,. . . , Ek , . . .) is Schauder basis for ℓp (ℱ) and dual space of the ℓp (ℱ) is ℓq (ℱ), where p –1 + q –1 = 1.

On the convergence of greedy algorithms for initial segments of the Haar basis

AbstractWe consider the X-Greedy Algorithm and the Dual Greedy Algorithm in a finite-dimensional Banach space with a strictly monotone basis as the dictionary. We show that when the dictionary is an initial segment of the Haar basis in Lp[0, 1] (1 &lt; p &lt; ∞) then the algorithms terminate after finitely many iterations and that the number of iterations is bounded by a function of the length of the initial segment. We also prove a more general result for a class of strictly monotone bases.

Genome-wide linkage analysis for identifying quantitative trait loci involved in the regulation of lipoprotein a (Lpa) levels

Lipoprotein Lp(a) levels are highly heritable and are associated with cardiovascular risk. We performed a genome-wide linkage analysis to delineate the genomic regions that influence the concentration of Lp(a) in families from the Genetic Analysis of Idiopathic Thrombophilia (GAIT) Project. Lp(a) levels were measured in 387 individuals belonging to 21 extended Spanish families. A total of 485 DNA microsatellite markers were genotyped to provide a 7.1 cM genetic map. The variance component linkage method was used to evaluate linkage and to detect quantitative trait loci (QTLs). The main QTL that showed strong evidence of linkage with Lp(a) levels was located at the structural gene for apo(a) on chromosome 6 (LOD score=13.8). Interestingly, another QTL influencing Lp(a) concentration was located on chromosome 2 with an LOD score of 2.01. This region contains several candidate genes. One of them is the tissue factor pathway inhibitor (TFPI), which has antithrombotic action and also has the ability to bind lipoproteins. However, quantitative trait association analyses performed with 12 SNPs in TFPI gene revealed no association with Lp(a) levels. Our study confirms previous results on the genetic basis of Lp(a) levels. In addition, we report a new QTL on chromosome 2 involved in the quantitative variation of Lp(a). These data should serve as the basis for further detection of candidate genes and to elucidate the relationship between the concentration of Lp(a) and cardiovascular risk.

Serum Lipoprotein(a) Level and Clinical Coronary Stenosis Progression in Patients With Myocardial Infarction Re-Revascularization Rate is High in Patients With High-Lp(a)

High serum lipoprotein(a) (Lp(a)) levels are associated with coronary artery disease. The serum Lp(a) levels of 130 patients with acute myocardial infarction (AMI) who underwent direct percutaneous coronary intervention were investigated. On the basis of Lp(a) level at 1 month after the onset of AMI, the patients were classified into 2 groups (high-Lp(a) (> or =30 mg/dl) and low-Lp(a) (< 30 mg/dl)) for evaluation of the clinical coronary stenosis progression (CCSP) rate. CCSP is defined as either target lesion revascularization (TLR) or new lesion revascularization (NLR). The CCSP rate was significantly higher in the high-Lp(a) group than in the low-Lp(a) group (65.8% vs 29.3%, p<0.01). In patients who had coronary stents in the acute phase (n=79), the CCSP and NLR rates were significantly higher in the high-Lp(a) group than in the low-Lp(a) group (45.0% vs 20.3%, p<0.05; 35.0% vs 6.8%, p<0.01), but there was no significant difference in TLR rate between the 2 groups (10.0% vs 13.6%, p=0.858). High serum Lp(a) level is a significant risk factor for CCSP, but does not influence restenosis after stenting.

Recovering an optimal LP basis from an optimal dual solution

Given a linear program, we describe an approach for crossing over from an optimal dual solution to an optimal basic primal solution. It consists in restricting the dual problem to a small box around the available optimal dual solution then, resolving the associated modified primal problem.