Positive Function Research Articles

Metric theory for continued β-fractions

For a fixed real number β>1, let ϵ⁎(1,β)=(ϵ1⁎(1,β),ϵ2⁎(1,β),⋯) be the infinite β-expansion of 1. For any n≥1, we define ℓn(β)=sup⁡{j≥0:ϵn⁎(1,β)=⋯=ϵn+j⁎(1,β)=0} be the maximal length of consecutive 0's in ϵ⁎(1,β) starting from ϵn⁎(1,β). Write A0={β>1:{ℓn(β)}is bounded} and let φ:N→R+ be a positive function. For any fixed β∈A0, we show that the Lebesgue measure of the following setE(φ)={x∈[0,1):an(x)≥φ(n)for infinitely manyn∈N} is null or full according to the convergence or divergence of the series ∑n=1∞1φ(n) respectively, where an(x) denotes the n-th partial quotient of the continued β-fraction expansion of x. We further determine the Hausdorff dimension of the set E(φ). Such measure and dimension results generalize a classical theorem of Borel–Bernstein and a well-known result of Wang–Wu (2008) [28] on regular continued fractions.

On representations and topological aspects of positive maps on non-unital quasi *- algebras

In this paper we provide a representation of a certain class of C*-valued positive sesquilinear and linear maps on non-unital quasi *-algebras, thus extending the results from Bellomonte (GNS-construction for positive C∗-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*-$$\\end{document}valued sesquilinear maps on a quasi ∗-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$*-$$\\end{document}algebra, Mediterr. J. Math., 21 166 (22 pp) (2024)) to the case of non-unital quasi *-algebras. Also, we illustrate our results on the concrete examples of non-unital Banach quasi *-algebras, such as the standard Hilbert module over a commutative C*-algebra, Schatten p-ideals and noncommutative L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-spaces induced by a semifinite, nonfinite trace. As a consequence of our results, we obtain a representation of all bounded positive linear C*-valued maps on non-unital C*-algebras. We also deduce some norm inequalities for these maps. Finally, we consider a noncommutative L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-space equipped with the topology generated by a positive sesquilinear form and we construct a topologically transitive operator on this space.

Gibberellins Play an Essential Role in the Bud Growth of Petunia hybrida.

This study delves into the role of gibberellin (GA) in governing plant branch development, a process that remains incompletely understood. Through a combination of exogenous hormone treatment, gene expression analysis, and transgenic phenotype investigations, the impact of GA on petunia's branch development was explored. The results showed that GA3 alone did not directly induce axillary bud germination. However, paclobutrazol (PAC), an inhibitor of GA synthesis, effectively inhibited bud growth. Interestingly, the simultaneous application of GA3 and 6-BA significantly promoted bud growth in both intact and decapitated plants compared to using 6-BA alone. Moreover, this study observed a significant downregulation of GA synthesis genes, including GA20ox1, GA20ox2, GA20ox3, GA3ox1, and CPS1, alongside an upregulation of GA degradation genes such as GA2ox2, GA2ox4, and GA2ox8. The expression of GA signal transduction gene GID1 and GA response factor RGA was found to be upregulated. Notably, the PhGID1 gene, spanning 1029 bp and encoding 342 amino acids, exhibited higher expression in buds and the lowest expression in leaves. The overexpression of PhGID1 in Arabidopsis resulted in a noteworthy rise in the number of branches. This study highlights the crucial role of GA in bud germination and growth and the positive regulatory function of GA signaling in shoot branching processes.

A country’s culture and reporting of sustainability practices in energy industries: does a corporate sustainability committee matter?

This study investigates the impact of the national culture on the reporting of sustainability practices. Likewise, it explores the moderating effect of sustainability committees within firms on the nexus between a country’s cultural and sustainability reporting. The study uses a sample over a decade (2010–2019) from 18 countries. It employs six well-established cultural dimensions: power distance, Individualism, Masculinity, uncertainty avoidance, long-term orientation, and Indulgence. These dimensions are aggregated to create a composite cultural index score for each country, reflecting the nuanced interplay of cultural attributes. The sustainability reporting score is based on a firm Environment, Social, and Governance (ESG) scores. The results reveal that reporting sustainability practices is a positive function of our country’s cultural index score. This indicates that a firm dissimilates higher levels of sustainability information when it operates in a country with a higher power distance, lower Individualism, higher Masculinity, higher uncertainty avoidance, higher long-term orientation, and lower Indulgence and vice versa. Likewise, a sustainability committee moderates the relationship between the national culture and sustainability reporting, suggesting that a corporate sustainability committee can act as a mechanism for aligning and adapting sustainability reporting practices to the specific cultural context in which a firm operates. The results provide essential implications for firms and policymakers. Firms should be aware of how cultural values shape their sustainability reporting. Also, a specialized sustainability committee navigates cultural complexities and aligns initiatives with expectations. Moreover, Policymakers should consider cultural influences and develop guidelines that accommodate variations, leading to more effective sustainability reporting. This study contributes by examining environmentally sensitive industries (i.e., the energy industries), providing industry-specific analysis that deepens our understanding of how a country’s culture shapes the reporting of sustainability practices. Likewise, the study uses a holistic approach, exploring the multifaceted dimensions of the country’s culture and its consequences for reporting sustainability practices.

Positive Reinforced Generalized Time-Dependent Pólya Urns via Stochastic Approximation

Consider a generalized time-dependent Pólya urn process defined as follows. Let d∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\in \\mathbb {N}$$\\end{document} be the number of urns/colors. At each time n, we distribute σn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma _n$$\\end{document} balls randomly to the d urns, proportionally to f, where f is a valid reinforcement function. We consider a general class of positive reinforcement functions R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {R}$$\\end{document} assuming some monotonicity and growth condition. The class R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {R}$$\\end{document} includes convex functions and the classical case f(x)=xα\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(x)=x^{\\alpha }$$\\end{document}, α>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha >1$$\\end{document}. The novelty of the paper lies in extending stochastic approximation techniques to the d-dimensional case and proving that eventually the process will fixate at some random urn and the other urns will not receive any balls anymore.

Extensions of k-regulous functions from two-dimensional varieties

AbstractWe prove that a k-regulous function defined on a two-dimensional non-singular affine variety can be extended to an ambient variety. Additionally we derive some results concerning sums of squares of k-regulous functions; in particular we show that every positive semi-definite regular function on a non-singular affine variety can be written as a sum of squares of locally Lipschitz regulous functions.

Asymptotic lifting for completely positive maps

Let A and B be C⁎-algebras with A separable, let I be an ideal in B, and let ψ:A→B/I be a completely positive contractive linear map. We show that there is a continuous family Θt:A→B, for t∈[1,∞), of lifts of ψ that are asymptotically linear, asymptotically completely positive and asymptotically contractive. If ψ is of order zero, then Θt can be chosen to have this property asymptotically. If A and B carry continuous actions of a second countable locally compact group G such that I is G-invariant and ψ is equivariant, we show that the family Θt can be chosen to be asymptotically equivariant. If a linear completely positive lift for ψ exists, we can arrange that Θt is linear and completely positive for all t∈[1,∞). In the equivariant setting, if A, B and ψ are unital, we show that asymptotically linear unital lifts are only guaranteed to exist if G is amenable. This leads to a new characterization of amenability in terms of the existence of asymptotically equivariant unital sections for quotient maps.

On Łojasiewicz inequalities and the effective Putinar's Positivstellensatz

The representation of positive polynomials on a semi-algebraic set in terms of sums of squares is a central question in real algebraic geometry, which the Positivstellensatz answers. In this paper, we study the effective Putinar's Positivestellensatz on a compact basic semi-algebraic set S and provide a new proof and new improved bounds on the degree of the representation of positive polynomials. These new bounds involve a parameter ε measuring the non-vanishing of the positive function, the constant c and exponent L of a Łojasiewicz inequality for the semi-algebraic distance function associated to the inequalities g=(g1,…,gr) defining S. They are polynomial in c and ε−1 with an exponent depending only on L. We analyse in details the Łojasiewicz inequality when the defining inequalities g satisfy the Constraint Qualification Condition. We show that, in this case, the Łojasiewicz exponent L is 1 and we relate the Łojasiewicz constant c with the distance of g to the set of singular systems.

Positive psychological functioning, resilience and styles of coping as buffers against suicidal behaviours. A case-control study

BackgroundStudies in the literature mainly focus on understanding the risk factors for suicide, giving little relevance to protective variables. This study aimed at exploring the specific contribution of protective variables (resilience, coping and psychological well-being) in hospitalized suicide attempt (SA) makers. MethodsWe recruited 50 inpatients who made a SA before admission and 50 inpatients with no history of SA matched for DSM-5 diagnosis, gender and age. Protective variables were evaluated with: Brief COPE questionnaire, Dispositional Resilience Scale (DRS-15), Psychological Well-Being Scale (PWB-18). Psychopathological features and symptom severity were assessed with: Global Assessment of Functioning Scale (GAF), Rapid Dimensional Assessment Scale (SVARAD), Brief Psychiatric Rating Scale (BPRS), Clinical Global Impressions (CGI), Hamilton Depression Rating Scale (HDRS17). ResultsThe DRS-15 total score was significantly lower in SA makers. SA makers displayed significantly lower scores on the Engagement and Cognitive Restructuring subscales of the Brief COPE. On the PWB-18, the Self-Acceptance subscale score was lower in SA makers. LimitationsThe small sample size suggests the need for caution in interpreting the results. Matching was carried out by excluding diagnoses of personality disorders. ConclusionsPatients hospitalized following a SA are more often diagnosed with personality disorders, have deficit areas concerning resilience and coping, and lower psychological well-being compared to patients without a SA. When approaching a patient who has committed a SA, it may be useful to evaluate protective variables as well as risk factors, and encourage the development of adaptive coping mechanisms and positive self-evaluation through more dynamic therapeutic paths.

On Gabor frames generated by B-splines, totally positive functions, and Hermite functions

The frame set of a window ϕ∈L2(R) is the subset of all lattice parameters (α,β)∈R+2 such that G(ϕ,α,β)={e2πiβm⋅ϕ(⋅−αk):k,m∈Z} forms a frame for L2(R). In this paper, we investigate the frame set of B-splines, totally positive functions, and Hermite functions. We derive a sufficient condition for Gabor frames using the connection between sampling theory in shift-invariant spaces and Gabor analysis. As a consequence, we obtain a new frame region belonging to the frame set of B-splines and Hermite functions. For a class of functions that includes certain totally positive functions, we prove that for any choice of lattice parameters α,β>0 with αβ<1, there exists a γ>0 depending on αβ such that G(ϕ(γ⋅),α,β) forms a frame for L2(R). Our results give explicit frame bounds.

On positive solutions of the nonlocal Yamabe type equation in compact Riemannian manifold

In this paper we deal with a nonlocal curvature equation involving a critical nonlinearity and a lower order perturbation f on a compact Riemannian N-manifold (M,g) (with or without boundary):(−Δg)psu+a(x)up−1=K(x)ups⁎−1+f(x,u) in Mu>0 in Mu=0 on ∂M where (−Δg)ps is the fractional p-Laplacian on (M,g) for p>1 and the fractional exponent 0<s<1 with ps<N and ps⁎:=NpN−ps. Here a(x) is a non-trivial function in L∞(M) and K is a positive smooth function on M. Using critical point theory on the manifold, we prove the existence of a positive solution under certain assumptions.

Nonlipogenic ABCA1 Inducers (NLAI) for Alzheimer's Disease Validated in a Mouse Model Expressing Human APOE3/APOE4.

Therapeutics enhancing apolipoprotein (APOE) positive function are a priority, because APOE4 is the major genetic risk factor for Alzheimer's disease (AD). The function of APOE, the key constituent of lipoprotein particles that transport cholesterol and lipids in the brain, is dependent on lipidation by ABCA1, a cell-membrane cholesterol transporter. ABCA1 transcription is regulated by liver X receptors (LXR): agonists have been shown to increase ABCA1, often accompanied by unwanted lipogenesis and elevated triglycerides (TG). Therefore, nonlipogenic ABCA1-inducers (NLAI) are needed. Two rounds of optimization of an HTS hit, derived from a phenotypic screen, gave lead compound 39 that was validated and tested in E3/4FAD mice that express human APOE3/4 and five mutant APP and PSEN1 human transgenes. Treatment with 39 increased ABCA1 expression, enhanced APOE lipidation, and reversed multiple AD phenotypes, without increasing TG. This NLAI/LXR-agonist study is the first in a human APOE-expressing model with hallmark amyloid-β pathology.

Comment on “Asymptotic quantum algorithm for the Toeplitz systems”

In their recent paper [], Wan proposed a quantum algorithm to solve systems of linear equations Ax=b, where A is an n×n Toeplitz matrix generated by the Fourier coefficients of a continuous, 2π-periodic, and positive function f. For large enough n, they claimed that the algorithm is valid for all b. In this Comment we prove that, in general, this is not true. Published by the American Physical Society 2024

From the Choi formalism in infinite dimensions to unique decompositions of generators of completely positive dynamical semigroups

Given any separable complex Hilbert space, any trace-class operator [Formula: see text] which does not have purely imaginary trace, and any generator [Formula: see text] of a norm-continuous one-parameter semigroup of completely positive maps we prove that there exists a unique bounded operator [Formula: see text] and a unique completely positive map [Formula: see text] such that (i) [Formula: see text], (ii) the superoperator [Formula: see text] is trace class and has vanishing trace, and (iii) [Formula: see text] is a real number. Central to our proof is a modified version of the Choi formalism which relates completely positive maps to positive semi-definite operators. We characterize when this correspondence is injective and surjective, respectively, which in turn explains why the proof idea of our main result cannot extend to non-separable Hilbert spaces. In particular, we find examples of positive semi-definite operators which have empty pre-image under the Choi formalism as soon as the underlying Hilbert space is infinite-dimensional.

Almost All Quantum Channels Are Diagonalizable

We prove the statement “The collection of all elements of[Formula: see text] which have only simple eigenvalues is dense in[Formula: see text]” for different sets [Formula: see text], including: all quantum channels, the unital channels, the positive trace-preserving maps, all Lindbladians (gksl-generators), and all time-dependent Markovian channels. Therefore any element from each of these sets can always be approximated by diagonalizable elements of the same set to arbitrary precision.

The Privilege of Well-Being in an Increasingly Unequal Society

This article provides an overview of a model of psychological well-being put forth over 30 years ago. The intent was to advance new dimensions of positive functioning based on integration of clinical, developmental, existential, and humanistic thinking along with Aristotle’s writings about eudaimonia. The operationalization and validation of the model are briefly described, followed by an overview of scientific findings organized around (a) demographic and experiential predictors of well-being, (b) well-being as predictors of health and biomedical outcomes, (c) pathway studies that examine intervening processes (moderators, mediators), and (d) underlying mechanistic processes (neuroscience, genomics). Much prior work underscores the benefits of well-being, including for longevity. Widening socioeconomic inequality is, however, increasingly compromising the well-being of disadvantaged segments of the population. These problems have been exacerbated by recent historical stressors (Great Recession, COVID-19 pandemic). Cumulative hardships from these events and their implications for health are critical targets for future science and practice.

A Gradient Dynamics-Based Singularity Avoidance Method for Backstepping Control of Underactuated TORA Systems.

In this paper, a gradient dynamics-based control method is proposed to directly tackle the singularity problem in the backstepping control design of the TORA system. This method is founded upon the construction of an energy-like positive function, which includes an auxiliary variable in terms of the intermediate virtual control law. On this basis, a gradient dynamics is created to obtain a new virtual control command, which is capable of making the auxiliary variable gradually approach zero, thereby mitigating the issue of division by zero. The core innovation is the integration of the gradient dynamics into the recursive backstepping design to overcome the singularity problem and stabilize the system at the equilibrium quickly. In addition, it rigorously proves that all the signals in the closed-loop control system are uniformly ultimately bounded, and the tracking errors converge to a small neighborhood around zero through a Lyapunov-based stability analysis. Comparative simulations demonstrate that the proposed approach not only avoids the singularity issue, but also achieves a better transient performance over other methods.

Protopanaxadiol derivative: A plant origin of novel selective glucocorticoid receptor modulator with anti-inflammatory effect

Constant efforts have been made to move towards maintaining the positive anti-inflammatory functions of glucocorticoids (GCs) while minimizing side effects. The anti-inflammatory effect of GCs is mainly attributed to the inhibition of major inflammatory pathways such as NF-κB through GR transrepression, while its side effects are mainly mediated by transactivation. Here, we investigated the selective glucocorticoid receptor modulator (SGRM)-like properties of a plant-derived compound. In this study, glucocorticoid receptor (GR)-mediated alleviation of inflammation by SP-8 was investigated by a combination of in vitro, in silico, and in vivo approaches. Molecular docking and cellular thermal shift assay suggested that SP-8 bound stably to the active site of GR via hydrogen bonding and hydrophobic interactions. SP-8 activated GR, induced GR nuclear translocation, and inhibited NF-κB pathway activation. Furthermore, SP-8 did not up-regulate the gene and protein expression of PEPCK and TAT in HepG2 cells, and it did not induce fat deposition like GC and has little effect on bone metabolism. Interestingly, SP-8 upregulated GR protein expression and did not cause GR phosphorylation at Ser211 in RAW264.7 cells. This work proved that SP-8 dissociated characteristics of transrepression and transactivation can be separated. In addition, the in vitro and in vivo anti-inflammatory effects of SP-8 were confirmed in LPS-induced RAW 264.7 cells and in a mouse model of DSS-induced ulcerative colitis, respectively. In conclusion, SP-8 might serve as a potential SGRM and might hold great potential for therapeutic use in inflammatory diseases.

GhEXL3 participates in brassinosteroids regulation of fiber elongation in Gossypium hirsutum.

Cotton fiber (Gossypium hirsutum) serves as an ideal model for investigating the molecular mechanisms of plant cell elongation at the single-cell level. Brassinosteroids (BRs) play a crucial role in regulating plant growth and development. However, the mechanism by which BR influences cotton fiber elongation remains incompletely understood. In this study, we identified EXORDIUM-like (GhEXL3) through transcriptome analysis of fibers from BR-deficient cotton mutant pagoda 1 (pag1) and BRI1-EMS-SUPPRESSOR 1 (GhBES1.4, encoding a central transcription factor of BR signaling) overexpression cotton lines. Knockout of GhEXL3 using CRISPR/Cas9 was found to impede cotton fiber elongation, while its overexpression promoted fiber elongation, suggesting a positive regulatory function for GhEXL3 in fiber elongation. Furthermore, in vitro ovule culture experiments revealed that the overexpression of GhEXL3 partially counteracted the inhibitory effects of brassinazole (BRZ) on cotton fiber elongation, providing additional evidence of GhEXL3 involvement in BR signaling pathways. Moreover, our findings demonstrate that GhBES1.4 directly binds to the E-box (CACGTG) motif in the GhEXL3 promoter region and enhances its transcription. RNA-seq analysis revealed that overexpression of GhEXL3 upregulated the expression of EXPs, XTHs, and other genes associated with fiber cell elongation. Overall, our study contributes to understanding the mechanism by which BR regulates the elongation of cotton fibers through the direct modulation of GhEXL3 expression by GhBES1.4.

Intelligent cotter pins defect detection for electrified railway based on improved faster R-CNN and dilated convolution

The cotter pin (CP) is a vital fastener for the catenary support components (CSCs) of high-speed electrified railways. Due to the vibration and excitation caused by the passing of railway vehicles, some CPs may be broken or fallen off over time, which poses a significant safety hazard to the railway systems. Currently, the CP defect detection is primarily conducted by humans, which is inefficient and inconsistent. Therefore, there is an urgent need for automatic CP defect detection to ensure railway safety. However, this task is very challenging as it requires covering hundreds or thousands of miles in limited times when the railway stops running. To this end, we first design a traffic track intelligent imaging device to capture catenary images at various angles at high speed. Then, inspired by the success of deep learning-based object detection, we develop a CP detection model based on an improved Faster R-CNN with a multi-scale region proposal network (MS-RPN) and propose the positive sample adaptive loss function (PSALF) to enhance detection accuracy. Finally, we propose a module to recognize the CP defect based on dilated convolution. The experimental results show that our method can effectively detect the CP defect in the catenary image, achieving 99.05 % precision and 98.40 % recall rate on CP defect detection. Furthermore, CP detection method and CP defect detection are significantly faster than baseline method, with FPS improvements of 2.76 and 24.67, respectively, thus making it more suitable for real-time applications in railway systems.