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This paper contains the proof of general results on the calculation of the norms of monotone operators acting from one ideal space to another under matching convexity and concavity properties of the operator and the norms in ideal spaces.

The paper is devoted to the problem of exact calculation of the norms in ideal spaces for monotone operators on the cones of functions with monotonicity properties. We implement a general approach to this problem that covers many concrete variants of monotone operators in ideal spaces and different monotonicity conditions for functions. As applications, we calculate the norms of some integral operators on the cones, associate norms over some cones in Lebesgue spaces, the norms of the dilation operator and embedding operators on weighted Lorentz spaces with general weights. Under some more general conditions, we present order sharp estimates for Hardy-type operators on the cones.

Исследуются свойства пространств обобщенной гладкости, таких как пространства Кальдерона, которые включают в себя классические пространства Никольского-Бесова и многие их обобщения, и описываются дифференциальные свойства обобщенных бесселевых потенциалов, которые включают в себя классические потенциалы Бесселя и пространства Соболева. Ядра потенциалов могут иметь нестепенные особенности в окрестности начала координат. С помощью точных по порядку оценок модулей непрерывности потенциалов устанавливаются критерии вложений потенциалов в пространства Кальдерона и описываются оптимальные пространства для таких вложений.

We investigate the properties of spaces with generalized smoothness, such as Calderon spaces, that include the classical Nikolskii–Besov spaces and many of their generalizations, and describe differential properties of generalized Bessel potentials that include classical Bessel potentials and Sobolev spaces. The kernels of potentials may have non-power singularities at the origin. Using order-sharp estimates for the moduli of continuity of potentials, we establish criteria for the embeddings of potentials into Calderon spaces and describe the optimal spaces for such embeddings.

Background/objectives The European League Against Rheumatism (EULAR)/American College of Rheumatology (ACR) 2019 classification criteria for systemic lupus erythematosus system showed high specificity, while attaining also high sensitivity. We hereby analysed the performance of the individual criteria items and their contribution to the overall performance of the criteria. Methods We combined the EULAR/ACR derivation and validation cohorts for a total of 1197 systemic lupus erythematosus (SLE) and n=1074 non-SLE patients with a variety of conditions mimicking SLE, such as other autoimmune diseases, and calculated the sensitivity and specificity for antinuclear antibodies (ANA) and the 23 specific criteria items. We also tested performance omitting the EULAR/ACR criteria attribution rule, which defines that items are only counted if not more likely explained by a cause other than SLE. Results Positive ANA, the new entry criterion, was 99.5% sensitive, but only 19.4% specific, against a non-SLE population that included other inflammatory rheumatic, infectious, malignant and metabolic diseases. The specific criteria items were highly variable in sensitivity (from 0.42% for delirium and 1.84% for psychosis to 75.6% for antibodies to double-stranded DNA), but their specificity was uniformly high, with low C3 or C4 (83.0%) and leucopenia 80% for all items, explaining the higher overall specificity of the criteria set. © Author(s) (or their employer(s)) 2021. No commercial re-use. See rights and permissions. Published by BMJ.

BACKGROUND For high-volume assays, optimizing throughput reduces test cost and turn-around time. One approach for liquid chromatography-tandem mass spectrometry (LC-MS/MS) assays is sample multiplexing, wherein the analyte of interest is derivatized in different specimens with reagents of different molecular weight (differential mass tagging). Specimens can then be combined and simultaneously analyzed within a single injection to improve throughput. Here we developed and validated a quantitative, sample-multiplexed LC-MS/MS assay for serum total testosterone (TT) based on this approach. METHODS For the sample-multiplexed assay, calibrators, controls, and patient specimens were first extracted separately. After mass tagging with either methoxyamine or hydroxylamine, they were combined and injected into the LC-MS/MS system. To evaluate assay performance, we determined limit of quantification (LOQ), linearity, recovery, and imprecision. A method-comparison study was also performed, comparing the new assay with the standard LC-MS/MS assay in 1574 patient specimens. RESULTS The method was linear from 2.5 to 2000 ng/dL, with accuracies from 93% to 104% for both derivatives. An LOQ of 1.0 ng/dL was achieved. Intra-assay and total CVs across 4 quality control concentrations were less than 10%. The assay demonstrated good agreement (Deming regression, 1.03x + 6.07) with the standard LC-MS/MS assay for the patient specimens tested (TT, 3 to 4862 ng/dL). CONCLUSION Sample multiplexing by differential mass tagging of TT increases LC-MS/MS throughput 2-fold without compromising analytical accuracy and sensitivity.

Objectives The European League Against Rheumatism (EULAR)/American College of Rheumatology (ACR) 2019 Classification Criteria for systemic lupus erythematosus (SLE) have been validated with high sensitivity and specificity. We evaluated the performance of the new criteria with regard to disease duration, sex and race/ethnicity, and compared its performance against the Systemic Lupus International Collaborating Clinics (SLICC) 2012 and ACR 1982/1997 criteria. Methods Twenty-one SLE centres from 16 countries submitted SLE cases and mimicking controls to form the validation cohort. The sensitivity and specificity of the EULAR/ACR 2019, SLICC 2012 and ACR 1982/1997 criteria were evaluated. Results The cohort consisted of female (n=1098), male (n=172), Asian (n=118), black (n=68), Hispanic (n=124) and white (n=941) patients; with an SLE duration of 1 to

Order covering and order equivalence for cones of functions withmonotonicity properties related to decreasing rearrangements of generalized Bessel and Riesz potentials are considered.

Compound ion distributions, fi(v), have been measured with high-time resolution by NASA's Magnetospheric Multi-Scale Mission (MMS) and have been found in reconnection simulations. A compound distribution, fi(v), consisting, for example, of essentially disjoint pieces will be called a multi-beam distribution and modeled as a sum of “beams,” fi(v) = f1(v) + ⋯ + fN(v). Velocity moments of fi(v) are taken beam by beam and summed. Such multi-beam moments of fi(v) have advantages over the customary standard velocity moments of fi(v), for which there is only one mean flow velocity. For example, the standard thermal energy moment of a pair of equal and opposite cold particle beams is non-zero even though each beam has zero thermal energy. We therefore call this thermal energy pseudothermal. By contrast, a multi-beam moment of two or more beams has no pseudothermal energy. We develop three different ways of approximating a compound ion velocity distribution, fi(v), as a sum of beams and finding multi-beam moments for both a compound fi(v) measured by MMS in the dayside magnetosphere during reconnection and a compound fi(v) found in a particle-in-cell simulation of magnetotail reconnection. The three methods are (i) a visual method in which the velocity centroid of each beam is estimated and the beam densities are determined self-consistently, (ii) a k-means method in which particles in a particle representation of fi(v) are sorted into a minimum energy configuration of N (= k) clusters, and (iii) a nonlinear least squares method based on a fit to a sum of N kappa functions. Multi-beam energy moments are calculated and compared with standard moments for the thermal energy density, pressure tensor, thermal energy flux (heat plus enthalpy fluxes), bulk kinetic energy density, ram pressure, and bulk kinetic energy flux. Applying this new formalism to real data demonstrates in detail how multi-beam techniques provide new insights into the energetics of observed space plasmas.

We establish criteria for the validity of modular inequalities for the Hardy operator on the cone Ω of nonnegative decreasing functions from weighted Orlicz spaces with general weight. The result is based on the theorem on the reduction of modular inequalities for positively homogeneous operators on the cone Ω, which enables passing to modular inequalities for modified operators on the cone of all nonnegative functions from an Orlicz space. It is shown that, for the Hardy operator, the modified operator is a generalized Hardy operator. This enables us to establish explicit criteria for the validity of modular inequalities.

Modular inequalities and inequalities for the norms of Hardy-type operators on the cone Ω of positive functions and on the cone of positive decreasing functions with common weight and common Young function in a weighted Orlicz space are considered. A reduction theorem for the norm of the Hardy operator on the cone Ω is obtained. It is shown that this norm is equivalent to the norm of a modified operator on the cone of all positive functions in the space under consideration. It is proved that the modified operator is a generalized Hardy-type operator. The equivalence of modular inequalities on the cone Ω and modified modular inequalities on the cone of all positive functions in the Orlicz space is shown. A criterion for the validity of such inequalities in general Orlicz spaces is obtained and refined for weighted Lebesgue spaces.

Modular and norm inequalities are considered for positively homogeneous operators on the cone of all nonnegative functions and on the cone Ω of nonnegative decreasing functions from the weighted Orlicz space with a general weight and a general Young function. A reduction theorem is obtained for the norm of an operator on Ω. This norm is shown to be equivalent to the norm of a modified operator on the cone of all nonnegative functions in the above Orlicz space. A similar theorem is obtained for modular inequalities. The results are based on the application of the principle of duality, which gives a description of the associated Orlicz norm for Ω. We also establish the equivalence of modular inequalities on the cone Ω and modified modular inequalities on the cone of all nonnegative functions in Orlicz space. In the general situation, the forms of these answers are substantially different from the descriptions obtained earlier by P. Drabek, A. Kufner, and H. Heinig under the assumption that the Young function and its complementary function satisfy Δ2-conditions.

Objective To develop new classification criteria for systemic lupus erythematosus (SLE) jointly supported by the European League Against Rheumatism (EULAR) and the American College of Rheumatology (ACR). Methods This international initiative had four phases. (1) Evaluation of antinuclear antibody (ANA) as an entry criterion through systematic review and meta-regression of the literature and criteria generation through an international Delphi exercise, an early patient cohort and a patient survey. (2) Criteria reduction by Delphi and nominal group technique exercises. (3) Criteria definition and weighting based on criterion performance and on results of a multi-criteria decision analysis. (4) Refinement of weights and threshold scores in a new derivation cohort of 1001 subjects and validation compared with previous criteria in a new validation cohort of 1270 subjects. Results The 2019 EULAR/ACR classification criteria for SLE include positive ANA at least once as obligatory entry criterion; followed by additive weighted criteria grouped in seven clinical (constitutional, haematological, neuropsychiatric, mucocutaneous, serosal, musculoskeletal, renal) and three immunological (antiphospholipid antibodies, complement proteins, SLE-specific antibodies) domains, and weighted from 2 to 10. Patients accumulating ≥10 points are classified. In the validation cohort, the new criteria had a sensitivity of 96.1% and specificity of 93.4%, compared with 82.8% sensitivity and 93.4% specificity of the ACR 1997 and 96.7% sensitivity and 83.7% specificity of the Systemic Lupus International Collaborating Clinics 2012 criteria. Conclusion These new classification criteria were developed using rigorous methodology with multidisciplinary and international input, and have excellent sensitivity and specificity. Use of ANA entry criterion, hierarchically clustered and weighted criteria reflect current thinking about SLE and provide an improved foundation for SLE research. © Author(s) (or their employer(s)) 2019. No commercial re-use. See rights and permissions. Published by BMJ.

Objective. Given the complexity and heterogeneity of systemic lupus erythematosus (SLE), high-performing classification criteria are critical to advancing research and clinical care. A collaborative effort by the European League Against Rheumatism and the American College of Rheumatology was undertaken to generate candidate criteria, and then to reduce them to a smaller set. The objective of the current study was to select a set of criteria that maximizes the likelihood of accurate classification of SLE, particularly early disease. Methods. An independent panel of international SLE experts and the SLE classification criteria steering committee (conducting SLE research in Canada, Mexico, United States, Austria, Germany, Greece, France, Italy, and Spain) ranked 43 candidate criteria. A consensus meeting using nominal group technique (NGT) was conducted to reduce the list of criteria for consideration. Results. The expert panel NGT exercise reduced the candidate criteria for SLE classification from 43 to 21. The panel distinguished potential "entry criteria," which would be required for classification, from potential "additive criteria." Potential entry criteria were antinuclear antibody (ANA) = 1:80 (HEp-2 immunofluorescence), and low C3 and/or low C4. The use of low complement as an entry criterion was considered potentially useful in cases with negative ANA. Potential additive criteria included lupus nephritis by renal biopsy, autoantibodies, cytopenias, acute and chronic cutaneous lupus, alopecia, arthritis, serositis, oral mucosal lesions, central nervous system manifestations, and fever. Conclusion. The NGT exercise resulted in 21 candidate SLE classification criteria. The next phases of SLE classification criteria development will require refinement of criteria definitions, evaluation of the ability to cluster criteria into domains, and evaluation of weighting of criteria. Copyright © 2019. All rights reserved.

In the paper we consider general properties of operators acting from rearrangement invariant spaces into generalized Morrey-type spaces. We extract wide class of operators that preserve non-negativity and monotonicity of functions and prove two-sided estimates for their norms. As corollaries we obtain corresponding results for operators of embedding and for Hardy–Littlewood maximal operators. For operators commuting with a shift operator the results are extended to the case of global Morrey-type spaces. As an application of these approaches we establish a criterion of the embedding for a weighted Lorentz space into a Morrey-type space.

A monotone operator P mapping the Orlicz–Lorentz class to an ideal space is considered. The Orlicz–Lorentz class is the cone of measurable functions on R + =(0, ∞) whose decreasing rearrangements with respect to the Lebesgue measure on R + belong to the weighted Orlicz space L Φ,ν. Reduction theorems are proved, which make it possible to reduce estimates of the norm of the operator P: ΛΦ,ν →Y to those of the norm of its restriction to the cone of nonnegative step functions in L Φ,ν. The application of these results to the identity operator from ΛΦ,ν to the weighted Lebesgue space Y = L 1(R +; g) gives exact descriptions of associated norms for ΛΦ,ν.

We obtain formulas for the generalized functional norm associated with the two-weight integral quasi-norm. We describe a minimal generalized Banach function space containing a given quasi-Banach space defined by the two-weight integral quasi-norm.

The restriction of a monotone operator P to the cone Ω of nonnegative decreasing functions from a weighted Orlicz space Lφ,v without additional a priori assumptions on the properties of theOrlicz function φ and the weight function v is considered. An order-sharp two-sided estimate of the norm of this restriction is established by using a specially constructed discretization procedure. Similar estimates are also obtained for monotone operators over the corresponding Orlicz–Lorentz spaces Λφ,v. As applications, descriptions of associated spaces for the cone Ω and the Orlicz–Lorentz space are obtained. These new results are of current interest in the theory of such spaces.

We consider the monotone operator P, which maps Orlicz-Lorentz class \( \varLambda _{\varPhi , v} \) into some ideal space \(Y=Y(R_+ )\). Orlicz-Lorentz class is determined as the cone of Lebesgue-measurable functions on \(R_+ = ( 0, \infty )\) having the decreasing rearrangements that belong to weighted Orlicz space \( L_{\varPhi , v} \) under some general assumptions concerning properties of functions \( \varPhi \) and v. We prove the reduction theorems allowing reducing the estimates of the norm of operator \(P: \varLambda _{\varPhi , v} \rightarrow Y\) to the estimates for its restriction on some cone of nonnegative step-functions in \( L_{\varPhi , v} \). Application of these results to identical operator mapping \( \varLambda _{\varPhi , v} \) into the weighted Lebesgue space \(Y=L_{ 1} ( R_+ ; g )\) gives the sharp description of the associate space for \( \varLambda _{\varPhi , v} \). The main results of this paper were announced in [20]. They develop the results of our paper [19] related to the case of N-functions.

The paper is devoted to Bessel potentials constructed by convolutions of Bessel–McDonald kernelswith functions from the basic rearrangement invariantspace. If the criterion is fulfilled for the embedding ofpotentials into the space of bounded continuous functions, we state the equivalent description for the conesof moduli of continuity of potentials in the uniformnorm. This makes it possible to obtain the criterion forthe embedding of potentials into the Calderon space.In the case of Bessel potentials constructed over thebasic weighted Lorentz space, we explicitly describethe optimal Calderon space for such an embedding.1. DESCRIPTION OF THE CONE OF CONTINUITY MODULI FOR POTENTIALSLet

We study the problem of constructing a minimal Banach function space containing a given cone of nonnegative measurable functions. For the associate function norm of the norm of an optimal space, we obtain general formulas and specify them in the case of a cone defined by an integral representation. We also consider the similar problem of constructing an optimal rearrangement invariant space and compare the descriptions obtained.

It is proved that the boundedness of the maximal operator M from a Lebesgue space Lp1(R n) to a general local Morrey-type space LMp2θ ,w(R n) is equivalent to the boundedness of the embedding operator from Lp1 (R n) to LMp2θ ,w(R n) and in its turn to the boundedness of the Hardy operator from L p1 p2 (0,∞) to the weighted Lebesgue space L θ p2 ,v(0,∞) for a certain weight function v determined by the functional parameter w . This allows obtaining necessary and sufficient conditions on the function w ensuring the boundedness of M from Lp1(R n) to LMp2θ ,w(R n) for any 0 1 . These conditions with p1 = p2 = 1 are necessary and sufficient for the boundedness of M from L1(R) to the weak local Morreytype space WLM1θ ,w(R) . Mathematics subject classification (2010): 42B20, 42B25, 42B35.

We study generalized Bessel potentials constructed by means of convolutions of functions with kernels that generalize the classical Bessel-Macdonald kernels. In contrast to the classical case, nonpower singularities of kernels in a neighborhood of the origin are admitted. The integral properties of functions are characterized in terms of decreasing rearrangements. The differential properties of potentials are described by the kth-order moduli of continuity in the uniform norm. An order-sharp upper estimate is established for the modulus of continuity of a potential. Such estimates play an important role in the theory of function spaces. They allow one to establish sharp embedding theorems for potentials, find majorants of the moduli of continuity, and estimate the approximation numbers of embedding operators.

In this paper we study spaces of Bessel potentials in n -dimensional Euclidean spaces. They are constructed on the basis of a rearrangement-invariant space (RIS) by using convolutions with Bessel–MacDonald kernels. Specifically, the treatment covers spaces of classical Bessel potentials. We establish two-sided estimates for the corresponding modulus of smoothness of order k ∈ N , ω k ( f ; t ) , and determine their continuity envelope functions. This result is then applied to estimate the approximation numbers of some embeddings.

In the theory of function spaces it is an important problem to describe the differential properties for the classical Bessel and Riesz potentials as well as for their generalizations. Bessel potentials are determined by the convolutions of functions with Bessel-MacDonald kernels Gα. In this paper we characterize the integral properties of functions by their decreasing rearrangements. The differential properties of potentials are characterized by their modulus of continuity of order k in the uniform norm. Estimates of such type were obtained by A. Gogatishvili, J. Neves, and B. Opic in the case k > α. Here, we remove this restriction and obtain the results for all values k ∈ N. We find order-sharp estimates from above for moduli of continuity and construct the examples confirming the sharpness. On the base of these results we obtain the order-sharp estimates for continuity envelope function in the space of potentials, and give estimates for the approximation numbers of the embedding operator.