A characterization of Schauder frames which are near-Schauder bases

A basic problem of interest in connection with the study of Schauder frames in Banach spaces is that of characterizing those Schauder frames which can essentially be regarded as Schauder bases. In this paper, we give a solution to this problem using the notion of the minimal-associated sequence spaces and the minimal-associated reconstruction operators for Schauder frames. We prove that a Schauder frame is a near-Schauder basis if and only if the kernel of the minimal-associated reconstruction operator contains no copy of $c_0$. In particular, a Schauder frame of a Banach space with no copy of $c_0$ is a near-Schauder basis if and only if the minimal-associated sequence space contains no copy of $c_0$. In these cases, the minimal-associated reconstruction operator has a finite dimensional kernel and the dimension of the kernel is exactly the excess of the near-Schauder basis. Using these results, we make related applications on Besselian frames and near-Riesz bases.Comment: 12 page

The Paulsen Problem, Continuous Operator Scaling, and Smoothed Analysis

The Paulsen problem is a basic open problem in operator theory: Given vectors $u_1, \ldots, u_n \in \mathbb R^d$ that are $\epsilon$-nearly satisfying the Parseval's condition and the equal norm condition, is it close to a set of vectors $v_1, \ldots, v_n \in \mathbb R^d$ that exactly satisfy the Parseval's condition and the equal norm condition? Given $u_1, \ldots, u_n$, the squared distance (to the set of exact solutions) is defined as $\inf_{v} \sum_{i=1}^n \| u_i - v_i \|_2^2$ where the infimum is over the set of exact solutions. Previous results show that the squared distance of any $\epsilon$-nearly solution is at most $O({\rm{poly}}(d,n,\epsilon))$ and there are $\epsilon$-nearly solutions with squared distance at least $\Omega(d\epsilon)$. The fundamental open question is whether the squared distance can be independent of the number of vectors $n$. We answer this question affirmatively by proving that the squared distance of any $\epsilon$-nearly solution is $O(d^{13/2} \epsilon)$. Our approach is based on a continuous version of the operator scaling algorithm and consists of two parts. First, we define a dynamical system based on operator scaling and use it to prove that the squared distance of any $\epsilon$-nearly solution is $O(d^2 n \epsilon)$. Then, we show that by randomly perturbing the input vectors, the dynamical system will converge faster and the squared distance of an $\epsilon$-nearly solution is $O(d^{5/2} \epsilon)$ when $n$ is large enough and $\epsilon$ is small enough. To analyze the convergence of the dynamical system, we develop some new techniques in lower bounding the operator capacity, a concept introduced by Gurvits to analyze the operator scaling algorithm.Comment: Added Subsection 1.4; Incorporated comments and fixed typos; Minor changes in various place

A Guide to Localized Frames and Applications to Galerkin-like Representations of Operators

This chapter offers a detailed survey on intrinsically localized frames and the corresponding matrix representation of operators. We re-investigate the properties of localized frames and the associated Banach spaces in full detail. We investigate the representation of operators using localized frames in a Galerkin-type scheme. We show how the boundedness and the invertibility of matrices and operators are linked and give some sufficient and necessary conditions for the boundedness of operators between the associated Banach spaces.Comment: 32 page

The Effect of Psyllium Hydrocolloid and Cholestyramine on Hepatic Bile Lipid Composition in Man

The effects of a mucoid - psyllium hydrocolloid - and an anion exchange polymercholestryamine - on the total cholesterol, total phospholipid, total bile salt, cholate, chenodeoxycholate, and deoxycholate concentrations of hepatic bile were determined in six post-cholecystectomy patients. Bile was obtained by drainage through an indwelling T-tube, which was clamped except during bile collection. Psyllium hydrocolloid treatment (12 gm/day) for 6 to 29 days had little or no effect on the cholesterol or phospholipid concentration of hepatic bile, but increased the total bile salt pool by gradually increasing the concentration of deoxycholate. Cholestyramine treatment (12 gm/day) for 8 to 12 days had no significant effect on cholesterol, phospholipid or total bile salt concentrations. There was a significant increase in the tri- to di-hydroxy bile salt ratio due to decreases in chenodeoxycholate and deoxycholate concentrations. The ratio of taurine to glycine conjugates decreased because of reductions in concentrations of taurine conjugates and compensating increases in glycine conjugates. The influence of these changes on bile micelle stability and cholesterol solubility is discussed. It is concluded that the changes effected by psyllium hydrocolloid may result in more stable bile micelles and greater cholesterol solubility. No definite conclusions can be reached with respect to cholestyramine\u27s effects

On R-duals and the duality principle in Gabor analysis

The concept of R-duals of a frame was introduced by Casazza, Kutyniok and Lammers in 2004, with the motivation to obtain a general version of the duality principle in Gabor analysis. For tight Gabor frames and Gabor Riesz bases the three authors were actually able to show that the duality principle is a special case of general results for R-duals. In this paper we introduce various alternative R-duals, with focus on what we call R-duals of type II and III. We show how they are related and provide characterizations of the R-duals of type II and III. In particular, we prove that for tight frames these classes coincide with the R-duals by Casazza et el., which is desirable in the sense that the motivating case of tight Gabor frames already is well covered by these R-duals. On the other hand, all the introduced types of R-duals generalize the duality principle for larger classes of Gabor frames than just the tight frames and the Riesz bases; in particular, the R-duals of type III cover the duality principle for all Gabor frames

A note on block sequences in Hilbert space

Block sequences with respect to frames in Hilbert spaces have been defined. Examples have been provided to show that a block sequence with respect to a given frame may not even be a Bessel sequence. Also, a necessary and sufficient condition under which a block sequence with respect to a frame is a frame has been given. Further, applications of block sequences to obtain Fusion frames and Fusion frame systems have been given. Finally, a problem has been posed and observed that an affirmative answer to this problem gives an affirmative answer to the Feichtinger Conjecture

Pinyon and Juniper Encroachment into Sagebrush Ecosystems Impacts Distribution and Survival of Greater Sage-Grouse

AbstractIn sagebrush (Artemisia spp.) ecosystems, encroachment of pinyon (Pinus spp.) and juniper (Juniperus spp.; hereafter, “pinyon-juniper”) trees has increased dramatically since European settlement. Understanding the impacts of this encroachment on behavioral decisions, distributions, and population dynamics of greater sage-grouse (Centrocercus urophasianus) and other sagebrush obligate species could help benefit sagebrush ecosystem management actions. We employed a novel two-stage Bayesian model that linked avoidance across different levels of pinyon-juniper cover to sage-grouse survival. Our analysis relied on extensive telemetry data collected across 6 yr and seven subpopulations within the Bi-State Distinct Population Segment (DPS), on the border of Nevada and California. The first model stage indicated avoidance behavior for all canopy cover classes on average, but individual grouse exhibited a high degree of heterogeneity in avoidance behavior of the lowest cover class (e.g., scattered isolated trees). The second stage modeled survival as a function of estimated avoidance parameters and indicated increased survival rates for individuals that exhibited avoidance of the lowest cover class. A post hoc frailty analysis revealed the greatest increase in hazard (i.e., mortality risk) occurred in areas with scattered isolated trees consisting of relatively high primary plant productivity. Collectively, these results provide clear evidence that local sage-grouse distributions and demographic rates are influenced by pinyon-juniper, especially in habitats with higher primary productivity but relatively low and seemingly benign tree cover. Such areas may function as ecological traps that convey attractive resources but adversely affect population vital rates. To increase sage-grouse survival, our model predictions support reducing actual pinyon-juniper cover as low as 1.5%, which is lower than the published target of 4.0%. These results may represent effects of pinyon-juniper cover in areas with similar ecological conditions to those of the Bi-State DPS, where populations occur at relatively high elevations and pinyon-juniper is abundant and widespread

LCA of hospital solid waste treatment alternatives in a developing country: The case of District Swat, Pakistan

Improper management of hospital waste leads to serious health and environmental issues, particularly in the case of developing countries, where, often, applied technologies are obsolete and there is a lack of compliance with respect to international best practices. The present study is designed to assess the environmental impacts of hospital waste management practices in Swath District, Pakistan. For this purpose, a life-cycle assessment (LCA) is applied for the estimation of different impacts of current and alternative hospital solid waste (HSW) treatment practices. Two scenarios are used to describe the current alternative practices (Scenario A and Scenario B), referring either to incineration or to direct landfilling of HSW without any sorting of collected materials. Conversely, Scenario C, which includes the use of pyrolysis and chemical disinfection, are considered as an up-to-date alternative, based on current international recommendations in this field. Prior to the analysis of impacts, due to the lack of available information, data were directly collected from both government and private hospitals in District Swat, involving measurements and a characterization of collected waste. In parallel, interviews were conducted, involving the hospitals' personnel. With respect to waste generation, government hospitals produce a larger amount of waste (74%) compared to private hospitals (24%). Poor regulatory indications and the absence of clear obligations for collection, disposal and management still represent a first obstacle to implement good practices. After defining the boundary of the system and the functional unit, according to standardized LCA practices, a life-cycle impact assessment (LCIA) was conducted, considering eight impact categories: human toxicity, freshwater eco-toxicity, marine aquatic eco-toxicity, terrestrial eco-toxicity, acidification potential, climate change, eutrophication and photochemical oxidation. The current practices (Scenario A and Scenario B) turned out to be the worst for all categories. In particular, the largest impact of all is recorded for human toxicity generated by incineration. In parallel, it must be considered that, currently, no recycling or reusing practices are implemented. Conversely, Scenario C (alternative up-to-date practices) would generate lower impacts. In detail, the highest value was recorded for marine aquatic ecotoxicity in relation to pyrolysis. Applying Scenario C, it would be possible to recover some materials, such as plastics, paper and sharps. In detail, considering the observed compositional characteristics, it would be possible to recover up to 78% of sharps and recycle 41% of plastic and paper from the general waste stream. Moreover, energy could be recovered from the pyrolysis process, generating a further benefit for the surrounding area. A lack of awareness, knowledge and infrastructures prevents the application of correct management practices, further degrading life and environmental conditions of this remote region of Pakistan. The huge difference in impacts between current practices and alternatives is demonstrated, showing a clear alternative for future management plans in this remote region and supporting future actions for local policymakers and hospital managers

Multipliers for p-Bessel sequences in Banach spaces

Multipliers have been recently introduced as operators for Bessel sequences and frames in Hilbert spaces. These operators are defined by a fixed multiplication pattern (the symbol) which is inserted between the analysis and synthesis operators. In this paper, we will generalize the concept of Bessel multipliers for p-Bessel and p-Riesz sequences in Banach spaces. It will be shown that bounded symbols lead to bounded operators. Symbols converging to zero induce compact operators. Furthermore, we will give sufficient conditions for multipliers to be nuclear operators. Finally, we will show the continuous dependency of the multipliers on their parameters.Comment: 17 page