Positive Part Research Articles

Approximation of the second eigenvalue of the [formula omitted]-Laplace operator in symmetric domains

A novel approach to approximate the second eigenfunction and the second eigenvalue of the p-Laplace operator is suggested for some symmetric domains. In the case of the Dirichlet boundary condition, the algorithm has the restriction that the positive and the negative part of the second eigenfunction have equal Lp-norm, however, in the case of the Neumann boundary condition, the algorithm does not have such restriction so we implement it to perform bi-clustering. Some interesting estimates for the iterative method are obtained. At the end of the paper, we present various examples and computational tests, which support that the suggested iterative algorithm is well-defined and effective.

Vaginal Epithelial Cell Membrane-Based Phototherapeutic Decoy Confers a "Three-in-One" Strategy to Treat against Intravaginal Infection of Candida albicans.

Phototherapy is an effective strategy to control Candida albicans (C. albicans) infection without raising the concern of drug resistance. Despite its effectiveness, a higher dose of phototherapeutic power is required for C. albicans elimination compared to bacteria that have to be used, which is readily accompanied by off-target heat and toxic singlet oxygen to damage normal cells, thus limiting its usefulness for antifungal applications. Here to overcome this, we develop a "three-in-one" biomimetic nanoplatform consisting of an oxygen-dissolved perfluorocarbon camouflaged by a photosensitizer-loaded vaginal epithelial cell membrane. With a cell membrane coating, the nanoplatform is capable of specifically binding with C. albicans at the superficial or deep vaginal epithelium, thereby centering the phototherapeutic agents on C. albicans. Meanwhile, the cell membrane coating endows the nanoplatform to competitively protect healthy cells from candidalysin-medicated cytotoxicity. Upon candidalysin sequestration, pore-forming on the surface of the nanoplatform accelerates release of the preloaded photosensitizer and oxygen, resulting in enhanced phototherapeutic power for improved anti-C. albicans efficacy under near-infrared irradiation. In an intravaginal C. albicans-infected murine model, treatment with the nanoplatform leads to a significantly decreased C. albicans burden, particularly when leveraging candidalysin for further elevated phototherapy and C. albicans inhibition. Also, the same trends hold true when using the nanoplatform to treat the clinical C. albicans isolates. Overall, this biomimetic nanoplatform can target and bind with C. albicans and simultaneously neutralize the candidalysin and then transform such toxins that are always considered a positive part in driving C. albicans infection with the power of enhancing phototherapy for improved anti-C. albicans efficacy.

Classification of simple locally finite modules over the affine-Virasoro algebras

In [12], the authors classified a class of simple modules over untwised affine Kac-Moody Lie algebras, on these modules each weight vector of the positive parts of affine Kac-Moody Lie algebras acts locally finitely. In this paper, for all affine-Virasoro algebras we also classify all simple modules with the similar property. We determine that there are precisely three classes of simple modules on which each weight vector of the positive part of any affine-Virasoro algebra acts locally finitely: simple highest weight or Whittaker Virasoro algebra modules, simple highest weight or Whittaker affine Lie algebra modules, and simple highest weight or Whittaker affine-Virasoro algebra modules which are neither simple Virasoro algebra modules nor simple affine Lie algebra modules. We also obtain three equivalent conditions to characterize such simple modules over affine-Virasoro algebras.

Strain orthogonal decomposition implemented within the phase field method with interfacial damage to model fracture in multi-phase heterogeneous materials

In recent times, the phase field method is a robust simulation tool that can predict crack formation and propagation in structures. In brittle and quasi-brittle materials, the strain tensor is decomposed into negative and positive parts corresponding to the compression and tension behaviors when the structure is loaded. In the previous studies related to the phase field method, the strain tensor decompositions do not satisfy the orthogonal condition of these two parts, even if the elastic stiffness tensor is isotropic. This problem leads to inaccuracies in the mechanical behavior of the materials. In this paper, the strain tensor orthogonal condition is applied to the bulk strain tensor parts representing the internal strain of the component phases. This orthogonal condition is implemented into the phase field method with interfacial damage to simulate crack propagation in a sample containing multi-phase heterogeneous materials. Furthermore, the shape and area fraction of these phases are recognized by a supplemental sub-function which was studied in our recent paper from image formats. Through several examples, the obtained results are compared with the relevant numerical ones to demonstrate the correctness and effectiveness of the proposed method.

ON THE INTEGRAL FORM OF RANK 1 KAC–MOODY ALGEBRAS

In this paper we shall prove that the ℤ-subalgebra generated by the divided powers of the Drinfeld generators xr±\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {x}_r^{\\pm } $$\\end{document} (r ∈ ℤ) of the Kac–Moody algebra of type A22\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\ extrm{A}}_2^{(2)} $$\\end{document} is an integral form (strictly smaller than Mitzman’s; see [Mi]) of the enveloping algebra, we shall exhibit a basis generalizing the one provided in [G] for the untwisted affine Kac–Moody algebras and we shall determine explicitly the commutation relations. Moreover, we prove that both in the untwisted and in the twisted case the positive (respectively negative) imaginary part of the integral form is an algebra of polynomials over ℤ.

Truncated Moments for Heavy-Tailed and Related Distribution Classes

Suppose that ξ+ is the positive part of a random variable defined on the probability space (Ω,F,P) with the distribution function Fξ. When the moment Eξ+p of order p>0 is finite, then the truncated moment F¯ξ,p(x)=min1,Eξp1I{ξ>x}, defined for all x⩾0, is the survival function or, in other words, the distribution tail of the distribution function Fξ,p. In this paper, we examine which regularity properties transfer from the distribution function Fξ to the distribution function Fξ,p and which properties transfer from the function Fξ,p to the function Fξ. The construction of the distribution function Fξ,p describes the truncated moment transformation of the initial distribution function Fξ. Our results show that the subclasses of heavy-tailed distributions, such as regularly varying, dominatedly varying, consistently varying and long-tailed distribution classes, are closed under this truncated moment transformation. We also show that exponential-like-tailed and generalized long-tailed distribution classes, which contain both heavy- and light-tailed distributions, are also closed under the truncated moment transformation. On the other hand, we demonstrate that regularly varying and exponential-like-tailed distribution classes also admit inverse transformation closures, i.e., from the condition that Fξ,p belongs to one of these classes, it follows that Fξ also belongs to the corresponding class. In general, the obtained results complement the known closure properties of distribution regularity classes.

Initial Coefficients Upper Bounds for Certain Subclasses of Bi-Prestarlike Functions

In this article, we introduce and study the behavior of the modules of the first two coefficients for the classes NΣ(γ,λ,δ,μ;α) and NΣ*(γ,λ,δ,μ;β) of normalized holomorphic and bi-univalent functions that are connected with the prestarlike functions. We determine the upper bounds for the initial Taylor–Maclaurin coefficients |a2| and |a3| for the functions of each of these families, and we also point out some special cases and consequences of our main results. The study of these classes is closely connected with those of Ruscheweyh who in 1977 introduced the classes of prestarlike functions of order μ using a convolution operator and the proofs of our results are based on the well-known Carathédory’s inequality for the functions with real positive part in the open unit disk. Our results generalize a few of the earlier ones obtained by Li and Wang, Murugusundaramoorthy et al., Brannan and Taha, and could be useful for those that work with the geometric function theory of one-variable functions.

D-SVM over networked systems with non-ideal linking conditions

This paper considers distributed optimization algorithms, with application in binary classification via distributed support vector machines (D-SVM) over multi-agent networks subject to some link nonlinearities. The agents solve a consensus-constraint distributed optimization cooperatively via continuous-time dynamics, while the links are subject to strongly sign-preserving odd nonlinear conditions. Logarithmic quantization and clipping (saturation) are two examples of such nonlinearities. In contrast to existing literature that mostly considers ideal links and perfect information exchange over linear channels, we show how general sector-bounded models affect the convergence to the optimizer (i.e. the SVM classifier) over dynamic balanced directed networks. In general, any odd sector-bounded nonlinear mapping can be applied to our dynamics. The main challenge is to show that the proposed system dynamics always have one zero eigenvalue (associated with the consensus) and the other eigenvalues all have negative real parts. This is done by recalling arguments from matrix perturbation theory. Then, the solution is shown to converge to the agreement state under certain conditions. For example, the gradient tracking (GT) step size is tighter than the linear case by factors related to the upper/lower sector bounds. To the best of our knowledge, no existing work in distributed optimization and learning literature considers non-ideal link conditions.

Hankel Determinant and Toeplitz Determinant on the Class of Bazileviˇc Functions Related to the Lemniscate Bernoulli

In this papers, we investigate the Hankel determinant and Toeplitz determinant for the class Bazileviˇc Function B1(α, δ) related to the Bernoulli Lemniscate function on the unit disk D = {z : |z| < 1} and obtain the upper bounds of the determinant H2(1), H2(2), T2(1), and investigate H2(1) using coefficients invers function. We used lemma from Charateodory-Toeplitz and Libera about sharp inequalities for functions with positive real part.

The compact presentation for the alternating central extension of the q-Onsager algebra

The q-Onsager algebra Oq is defined by two generators and two relations, called the q-Dolan/Grady relations. We investigate the alternating central extension Oq of Oq. The algebra Oq was introduced by Baseilhac and Koizumi, who called it the current algebra of Oq. Recently Baseilhac and Shigechi gave a presentation of Oq by generators and relations. The presentation is attractive, but the multitude of generators and relations makes the presentation unwieldy. In this paper we obtain a presentation of Oq that involves a subset of the original set of generators and a very manageable set of relations. We call this presentation the compact presentation of Oq. This presentation resembles the compact presentation of the alternating central extension for the positive part of Uq(slˆ2).

Massively parallel computation of tropical varieties, their positive part, and tropical Grassmannians

We present a massively parallel framework for computing tropicalizations of algebraic varieties which can make use of symmetries using the workflow management system GPI-Space and the computer algebra system Singular. We determine the tropical Grassmannian TGr0(3,8). Our implementation works efficiently on up to 840 cores, computing the 14763 orbits of maximal cones under the canonical S8-action in about 20 minutes.Relying on our result, we show that the Gröbner structure of TGr0(3,8) refines the 16-dimensional skeleton of the coarsest fan structure of the Dressian Dr(3,8), except for 23 orbits of special cones, for which we construct explicit obstructions to the realizability of their tropical linear spaces. Moreover, we propose algorithms for identifying maximal-dimensional cones which belong to positive tropicalizations of algebraic varieties. We compute the positive Grassmannian TGr+(3,8) and compare it to the cluster complex of the classical Grassmannian Gr(3,8).

Transient Electromagnetic Smoke Ring Due to a Grounded-Wire Source

The concept of a “smoke ring” in electromagnetic fields can be used to describe the spatial distribution and diffusion of electromagnetic fields with either frequency or time. This provides an intuitive basis for the qualitative interpretation of artificial-source electromagnetic measurement results. The existing discussions regarding this field of study have mainly focused on the smoke rings of a magnetic dipole or loop source. In the present work, smoke rings of other commonly used grounded-wire sources in the time domain are investigated. The results show that the smoke ring of grounded-wire sources is more complex than that of magnetic sources. A grounded-wire source can generate an induced current in both horizontal and vertical directions. The horizontal induced current includes positive and negative parts. Both parts of the horizontal induced current maximum diffuse downward perpendicular to the surface, but the diffusion velocity of the negative part is faster, approximately 5.6 times that of the positive part. The vertical induced current maximum diffuses at 45° from the grounded surface with a velocity approximately 5 times that of the positive horizontal induced current. The smoke ring of the horizontal induced current closely resembles a basin, while that of the vertical induced current is more similar to two separate cylinders. Finally, the imaging depth based on horizontal and vertical induced currents is obtained by the statistical relationship between the induced current maximums and the diffusion depth, which are equal to 1.4 times and 1.25 times the diffusion depth, respectively.

Blow-up criteria of the simplified Ericksen–Leslie system

In this paper, we establish scaling invariant blow-up criteria for a classical solution to the simplified Ericksen–Leslie system in terms of the positive part of the second eigenvalue of the strain matrix and orientation field in mixed-norm Lebesgue spaces. Our result may be also regarded as an extension or improvement of the corresponding results obatined by Neustupa and Penel (Trends in Partial Differential Equations of Mathematical Physics, pp. 197–212, 2005), Miller (Arch. Ration. Mech. Anal. 235(1):99–139, 2020) and Huang and Wang (Commun. Partial Differ. Equ. 37(5):875–884, 2012).

On Some Classes of Estimators Derived from the Positive Part of James–Stein Estimator

This work consists of developing shrinkage estimation strategies for the multivariate normal mean when the covariance matrix is diagonal and known. The domination of the positive part of James–Stein estimator (PPJSE) over James–Stein estimator (JSE) relative to the balanced loss function (BLF) is analytically proved. We introduce a new class of shrinkage estimators which ameliorate the PPJSE, and then we construct a series of polynomial shrinkage estimators which improve the PPJSE; also, any estimator of this series can be ameliorated by adding to it a new term of higher degree. We end this paper by simulation studies which confirm the performance of the suggested estimators.

Associated Water Dynamics Might Be a Key Factor Affecting Protein Stability in the Crowded Milieu.

Over the past 20 years, the most studied and debated aspect of macromolecular crowding is how it affects protein stability. Traditionally, it is explained by a delicate balance between the stabilizing entropic effect and the stabilizing or destabilizing enthalpic effect. However, this traditional crowding theory cannot explain experimental observations like (i) negative entropic effect and (ii) entropy-enthalpy compensation. Herein, we provide experimental evidence that associated water dynamics plays a crucial role in controlling protein stability in the crowded milieu for the first time. We have correlated the modulation of associated water dynamics with the overall stability and its individual components. We showed that rigid associated water would stabilize the protein through entropy but destabilize it through enthalpy. In contrast, flexible associated water destabilizes the protein through entropy but stabilizes through enthalpy. Consideration of entropic and enthalpic modulation through crowder-induced distortion of associated water successfully explains the negative entropic part and entropy-enthalpy compensation. Furthermore, we argued that the relationship between the associated water structure and protein stability should be better understood by individual entropic and enthalpic components instead of the overall stability. Although a huge effort is necessary to generalize the mechanism, this report provides a unique way of understanding the relationship between protein stability and associated water dynamics, which might be a generic phenomenon and should trigger much research in this area.

Radius of starlikeness through subordination

"A normalized function $f$ on the open unit disc is starlike (or convex) univalent if the associated function $zf'(z)/f(z)$ (or $1+zf''(z)/f'(z)$) is a function with positive real part. The radius of starlikeness or convexity is usually obtained by using the estimates for functions with positive real part. Using subordination, we examine the radius of various starlikeness, in particular, radii of Janowski starlikeness and starlikeness of order $\beta$, for the function $f$ when the function $f$ is either convex or $(zf'(z)+\alpha z^2f''(z))/f(z)$ lies in the right-half plane. Radii of starlikeness associated with lemniscate of Bernoulli and exponential functions are also considered."

Antimicrobial Resistance (AMR) in post COVID era – ‘Second Pandemic’ Slowly Moving Across the World

Antimicrobial Resistance (AMR) in post COVID era – ‘Second Pandemic’ Slowly Moving Across the World

Sharp bounds of logarithmic coefficient problems for functions with respect to symmetric points

The logarithmic coefficients play an important role for different estimates in the theory of univalent functions.Due to the significance of the recent studies about the logarithmic coefficients, the problem of obtaining the sharp bounds for the second Hankel determinant of these coefficients, that is $H_{2,1}(F_f/2)$ was paid attention. We recall that if $f$ and $F$ are two analytic functions in $\mathbb{D}$, the function $f$ is subordinate to $F$, written $f(z)\prec F(z)$, if there exists an analytic function $\omega$ in $\mathbb{D}$ with $\omega(0)=0$ and $|\omega(z)|<1$, such that $f(z)=F\left(\omega(z)\right)$ for all $z\in\mathbb{D}$. It is well-known that if $F$ is univalent in $\mathbb{D}$, then $f(z)\prec F(z)$ if and only if $f(0)=F(0)$ and $f(\mathbb{D})\subset F(\mathbb{D})$.A function $f\in\mathcal{A}$ is starlike with respect to symmetric points in $\mathbb{D}$ iffor every $r$ close to $1,$ $r < 1$ and every $z_0$ on $|z| = r$ the angular velocity of $f(z)$about $f(-z_0)$ is positive at $z = z_0$ as $z$ traverses the circle $|z| = r$ in the positivedirection. In the current study, we obtain the sharp bounds of the second Hankel determinant of the logarithmic coefficients for families $\mathcal{S}_s^*(\psi)$ and $\mathcal{C}_s(\psi)$ where were defined by the concept subordination and $\psi$ is considered univalent in $\mathbb{D}$ with positive real part in $\mathbb{D}$ and satisfies the condition $\psi(0)=1$. Note that $f\in \mathcal{S}_s^*(\psi)$ if\[\dfrac{2zf^\prime(z)}{f(z)-f(-z)}\prec\psi(z),\quad z\in\mathbb{D}\]and $f\in \mathcal{C}_s(\psi)$ if\[\dfrac{2(zf^\prime(z))^\prime}{f^\prime(z)+f^\prime(-z)}\prec\psi(z),\quad z\in\mathbb{D}.\]It is worthwhile mentioning that the given bounds in this paper extend and develop some related recent results in the literature. In addition, the results given in these theorems can be used for determining the upper bound of $\left\vert H_{2,1}(F_f/2)\right\vert$ for other popular families.

Optimum parameters of variant tuned mass damper using equal eigenvalue criterion

AbstractIn this study, optimum parameters of a variant tuned mass damper (V‐TMD) wherein, damper is connected to the ground are obtained using equal eigenvalue (EEV) criterion. In EEV, TMD parameters are obtained by equating eigenvalues of two modes. By equating eigenvalues, the modal damping ratios (i.e., negative real part of the eigenvalue) become equal and maximum in two modes. This optimum condition is achieved by finding the roots of a sixth order polynomial. One of the novel findings is regarding the existence of two sets of optimum parameters, out of which, one set matches exactly with one of the earlier study. Regions in which, single and two sets of optimum parameters exist, are clearly brought out. Performance of V‐TMD is demonstrated for harmonic as well as earthquake excitation. It is noted that displacement as well as acceleration of the main system reduces due to V‐TMD. Another significant finding is that the real part of eigenvalues of V‐TMD is larger than that of classical tuned mass damper (C‐TMD), which leads to faster decay of earthquake response and lower peak response. The optimum parameters obtained in this study, pertain to linear SDOF and V‐TMD.

One positive part of copper ion on biocatalysts: CNPs@Cu-Trypsin-assisted aqueous extraction technique for simultaneous exploration of edible insect oil and protein

Edible insects are rich in oil and protein resources, and has it become increasingly important to develop them reasonably when resources are increasingly scarce. However, due to the challenges in recovering enzymes in the enzyme-assisted aqueous extraction technique and the remaining organic solvents in the extraction process based on organic solvent, it is difficult to achieve simultaneous utilization of oil and protein from edible insects. Hence, an improved enzyme-assisted aqueous extraction process based on the assistance of chitosan cross-linked Fe3O4 nanoparticles (CNPs) immobilized enzymes and divalent copper ions chelation, CNPs@Cu-Trypsin-assisted aqueous extraction technique, was first proposed. The hydrolysis activity was significantly increased by 6.16 times, and the value of Kcat/Km was increased by 1.7 times for the results of the formation of the hydrophilic microenvironment. Furthermore, the extraction ratio of silkworm pupae oil extracted via CNPs@Cu-Trypsin-assisted aqueous extraction technique could be increased to 93%, and the oil extraction rate could be maintained above 65% after 10 times reused. Improvement of enzyme-assisted aqueous extraction under the addition of divalent copper ions could achieve an ideal effect of oil extracted from silkworm pupae with less time. Additionally, the quality of silkworm pupae oil was significantly improved and the silkworm pupae protein hydrolysate also showed improved biological activity. Therefore, the enzyme-assisted aqueous extraction enhancement by trypsin immobilization and divalent copper ion chelation is a promising strategy for the green and sustainable exploration of edible insect oil and protein without the use of organic solvents.