Convex Type Research Articles

Multiplicity of nonnegative solutions to a weighted indefinite elliptic equation of the concave–convex type

Multiplicity of nonnegative solutions to a weighted indefinite elliptic equation of the concave–convex type

Bounds of Different Integral Operators in Tensorial Hilbert and Variable Exponent Function Spaces

In dynamical systems, Hilbert spaces provide a useful framework for analyzing and solving problems because they are able to handle infinitely dimensional spaces. Many dynamical systems are described by linear operators acting on a Hilbert space. Understanding the spectrum, eigenvalues, and eigenvectors of these operators is crucial. Functional analysis typically involves the use of tensors to represent multilinear mappings between Hilbert spaces, which can result in inequality in tensor Hilbert spaces. In this paper, we study two types of function spaces and use convex and harmonic convex mappings to establish various operator inequalities and their bounds. In the first part of the article, we develop the operator Hermite–Hadamard and upper and lower bounds for weighted discrete Jensen-type inequalities in Hilbert spaces using some relational properties and arithmetic operations from the tensor analysis. Furthermore, we use the Riemann–Liouville fractional integral and develop several new identities which are used in operator Milne-type inequalities to develop several new bounds using different types of generalized mappings, including differentiable, quasi-convex, and convex mappings. Furthermore, some examples and consequences for logarithm and exponential functions are also provided. Furthermore, we provide an interesting example of a physics dynamical model for harmonic mean. Lastly, we develop Hermite–Hadamard inequality in variable exponent function spaces, specifically in mixed norm function space (lq(·)(Lp(·))). Moreover, it was developed using classical Lebesgue space (Lp) space, in which the exponent is constant. This inequality not only refines Jensen and triangular inequality in the norm sense, but we also impose specific conditions on exponent functions to show whether this inequality holds true or not.

Artificial Intelligence Measurement of Preoperative Radiographs in Adolescent Idiopathic Scoliosis Based on Multiple-View Semantic Segmentation.

Cross-sectional study. Imaging classification of adolescent idiopathic scoliosis (AIS) is directly related to the surgical strategy, but the artificial classification is complex and depends on doctors' experience. This study investigated deep learning-based automated classification methods (DL group) for AIS and validated the consistency of machine classification and manual classification (M group). A total of 506 cases (81 males and 425 females) and 1812 AIS full spine images in the anteroposterior (AP), lateral (LAT), left bending (LB) and right bending (RB) positions were retrospectively used for training. The mean age was 13.6 ± 1.8. The mean maximum Cobb angle was 46.8 ± 12.0. U-Net semantic segmentation neural network technology and deep learning methods were used to automatically segment and establish the alignment relationship between multiple views of the spine, and to extract spinal features such as the Cobb angle. The type of each test case was automatically calculated according to Lenke's rule. An additional 107 cases of adolescent idiopathic scoliosis imaging were prospectively used for testing. The consistency of the DL group and M group was compared. Automatic vertebral body segmentation and recognition, multi-view alignment of the spine and automatic Cobb angle measurement were implemented. Compare to the M group, the consistency of the DL group was significantly higher in 3 aspects: type of lateral convexity (0.989 vs 0.566), lumbar curvature modifier (0.932 vs 0.738), and sagittal plane modifier (0.987 vs 0.522). Deep learning enables automated Cobb angle measurement and automated Lenke classification of idiopathic scoliosis whole spine radiographs with higher consistency than manual measurement classification.

Hyers–Ulam Stability of 2D-Convex Mappings and Some Related New Hermite–Hadamard, Pachpatte, and Fejér Type Integral Inequalities Using Novel Fractional Integral Operators via Totally Interval-Order Relations with Open Problem

The aim of this paper is to introduce a new type of two-dimensional convexity by using total-order relations. In the first part of this paper, we examine the Hyers–Ulam stability of two-dimensional convex mappings by using the sandwich theorem. Our next step involves the development of Hermite–Hadamard inequality, including its weighted and product forms, by using a novel type of fractional operator having non-singular kernels. Moreover, we develop several nontrivial examples and remarks to demonstrate the validity of our main results. Finally, we examine approximate convex mappings and have left an open problem regarding the best optimal constants for two-dimensional approximate convexity.

Elliptic equations with critical and supercritical growth at the boundary

In this study, we consider two classes of elliptic problems with nonlinear boundary conditions of concave–convex type. In the first problem, we obtain two nonzero and nonnegative solutions when the nonlinear term exhibits critical growth. In the second problem, we obtain infinitely many solutions (with no prescribed sign) by assuming that the nonlinearity is even and subcritical near the origin but with no growth condition at infinity.

ON MULTIPLICATIVE (s,P)-CONVEXITY AND RELATED FRACTIONAL INEQUALITIES WITHIN MULTIPLICATIVE CALCULUS

In this paper, we propose a fresh conception about convexity, known as the multiplicative [Formula: see text]-convexity. Along with this direction, we research the properties of such type of convexity. In the framework of multiplicative fractional Riemann–Liouville integrals and under the [Formula: see text]differentiable [Formula: see text]-convexity, we investigate the multiplicative fractional inequalities, including the Hermite–Hadamard- and Newton-type inequalities. To further verify the validity of our primary outcomes, we give a few numerical examples. As applications, we proffer a number of inequalities of multiplicative type in special means as well.

A generalization of convexity via an implicit inequality

<abstract><p>We unified several kinds of convexity by introducing the class $ \mathcal{A}_{\zeta, w}([0, 1]\times I^2) $ of $ (\zeta, w) $-admissible functions $ F: [0, 1]\times I\times I\to \mathbb{R} $. Namely, we proved that most types of convexity from the literature generate functions $ F\in \mathcal{A}_{\zeta, w}([0, 1]\times I^2) $ for some $ \zeta\in C([0, 1]) $ and $ w\in C^1(I) $ with $ w(I)\subset I $ and $ w' > 0 $. We also studied some properties of $ (\zeta, w) $-admissible functions and established some integral inequalities that unify various Hermite-Hadamard-type inequalities from the literature.</p></abstract>

Novel fractional inequalities measured by Prabhakar fuzzy fractional operators pertaining to fuzzy convexities and preinvexities

<abstract><p>In this article, we implemented the idea of a fuzzy interval-valued function with the well-known generalized fuzzy fractional operators, associated with different types of convexities and preinvexities. We developed the Prabhakar fuzzy fractional operators using the fuzzy interval-valued function. We presented the novel extensions of Hermite-Hadamard fuzzy-type and trapezoidal fuzzy-type inequalities, based on the $ h $-Godunova-Levin convex and $ h $-Godunova preinvex fuzzy interval-valued functions.</p></abstract>

The optical possibilities of gelatin

Contact lenses have traditionally been manufactured using a variety of plastics: polymers for soft lenses and acrylic or polymethyl methacrylate for hard lenses. There is emerging interest in the use of gelatin for lenses to increase malleability, water solubility, and sustainability as a non-plastic. A predicted difficulty in the use of gelatin is the reduced refractive index (RI). We looked at how acrylic or gelatin changes the RI and focal length in order to see how gelatin lenses would have to be altered for more severe prescriptions. We tested the refractive rays of gelatin and acrylic with both convex and concave lens types to investigate which produced smaller angles of refraction and correspondingly greater focal length. The average RI of the biconvex and biconcave acrylic without Pam, acrylic with Pam, and gelatin were 1.47, 1.496 and 1.36, respectively. Gelatin’s lesser angles of refraction produced a longer focal length. The lower RI of the gelatin (0.11–0.136 less than acrylic) directly correlated with its lesser angles of refraction, which produced an overall greater focal length (3.51– 4.38 more than acrylic). Gelatin needs to be thickened, increased in curvature, or mixed with crosslinkers to strengthen the lens. Lens makers will face similar challenges with the switch to gelatin lenses, so they may need to incorporate a larger percentage of solid gelatin concentrate or a plastic alloy in the gelatin to increase the RI, decrease the focal length, and improve the stability of the gelatin.

On $ \psi $-convex functions and related inequalities

<abstract><p>We introduce the class of $ \psi $-convex functions $ f:[0, \infty)\to \mathbb{R} $, where $ \psi\in C([0, 1]) $ satisfies $ \psi\geq 0 $ and $ \psi(0)\neq \psi(1) $. This class includes several types of convex functions introduced in previous works. We first study some properties of such functions. Next, we establish a double Hermite-Hadamard-type inequality involving $ \psi $-convex functions and a Simpson-type inequality for functions $ f\in C^1([0, \infty)) $ such that $ |f'| $ is $ \psi $-convex. Our obtained results are new and recover several existing results from the literature.</p></abstract>

Anatomical Variations of The Diaphragma Sellae and Pituitary Stalk in Human Adult Cadavers in Myanmar

Objective: The middle cerebral fossa contains the sellar area. The diaphragma sellae, a dural fold, covers the sella turcica, which is located in the bony depression of the sphenoid bone. One of the most significant endocrine glands in the body, the pituitary gland’s stalk protrudes from the base of the brain through the diaphragma sellae’s primary hole.Material and Methods: This study was a cross-sectional experimental study and carried out on 30 human adult cadavers (20 males and 10 females) in order to describe the anatomical variations of diaphragma sellae and pituitary stalk.Results: The shapes of the diaphragma sellae were 17 flat types (56.7%), 12 concave types (40%) and 1 convex type (3.3%). The shapes of the central opening of the diaphragma sellae were 21 round types (70%), 5 coronal elliptical types (16.7%), and 4 sagittal elliptical types (13.3%). The mean anteroposterior diameter of the central opening of the diaphragma sellae was 6.92±1.47 mm and the transverse diameter was 7.06±1.46 mm. The mean anteroposterior diameter of the pituitary stalk was 1.83±0.41 mm and the transverse diameter of the pituitary stalk was 2.00±0.33 mm. The locations of the pituitary stalk were 20 posterior types (66.7%), 5 central types (16.7%), and 5 anterior types (16.7%).Conclusion: Neurosurgeons, maxillofacial surgeons, and clinical radiologists may find the current study of the diphragama sellae and sellar region helpful in contemplating the diseases and operational techniques in this area.

Counter-examples to Breckner-convexity

In this paper, we examine convexity type inequalities. Let D be a nonempty convex subset of a linear space, c>0 and α:D-D→R be a given even function. The inequality f((x+y)/2) ≤ c f(x) + c f(y) + α(x-y) (x,y € D) is the focus of our examinations. We will construct an example to show that for c=1, this Jensen type inequality does not imply the convexity of the function. Then, we compare this inequality with Hermite–Hadamard type inequalities.

Are methods to quantify osseous exposure in orthopedic surgery reliable?

BackgroundOur study examined if there were any limitations when using various measurement techniques in the literature to quantify osseous exposure. Additionally, we also examined if surface contour had any influence on obtained measurements, which no previous study has attempted. Materials and methodsThree methods used to quantify osseous exposure area were identified, one in which involves manually applying mesh over exposure area. The other two use digital image capture software (ImageJ, Bethesda, MD). We simulated flat, convex, and mixed surface types using synthetic bone analogs. We assessed the degree of variability between mean values using an ANOVA or Kruskal–Wallis equality of populations rank test. Cronbach's alpha test of internal reliability was used to assess the internal reliability of measurement technique. ResultsANOVA test for difference in measurement techniques on all three surface types was statistically significant (p < 0.05). Cronbach's alpha test of internal reliability for each technique on the convex surface did not obtain adequate significance (alpha >0.70). Only the mesh method obtained adequate alpha value for significance when applied to the flat and mixed surface types. DiscussionEach of the three measurement techniques tested demonstrated poor internal reliability. We suggest taking care when comparing studies that use different quantification techniques when calculating osseous exposure for different surgical approaches. Future studies should explore alternative methods of osseous exposure quantification.

A convexity in R^2 with river metric

In this paper we consider the space R2 with the river metric d* and different types of convexity of this space. We define W-convex structure in (R2, d*) and we give the complete characterization of the convex sets in this space. We consider some measures of noncompactness and we give the moduli of noncompactness for considered measures on this space.

Various Convexities and Some Relevant Properties of Consumer Preference Relations

Abstract The concept of convexity plays an important role in the study of economics and consumer theory. For the most part, such studies have been conducted on the assumption that consumer preferences are a binary relation that is complete, reflexive and transitive on the set X of consumption choices. However, each consumer is a biological being with multidimensional physiological needs so that possible consumptions from different dimensions cannot be compared by using preferences. By removing that unrealistic assumption, this paper examines how the various concepts of convex preferences and relevant properties can be re-established. We derive a series of 10 formal propositions and construct 6 examples to show that (a) a weighted combination of two possible consumptions is not necessarily comparable with any of the consumptions; (b) not every convergent sequence of a consumer’s preferred consumptions asymptotically preserves that consumer’s preference preordering; (c) not all preferences satisfy either positive multiplicativity or additive conservation; (d) three types of preference convexities – weak convexity, convexity and strong convexity – can all be introduced into general convex spaces. This paper concludes with some research topics of expected significance for future works.

Fgh-Convex Functions and Entropy Bounds

In this paper, we introduce an universal definition (fgh-convex) that results in several types of convexity. Particular cases of the fgh-convex are for instance the harmonically convex, geometrically convex, GA-convex, log-convex, and several others. Also, we obtain some useful inequalities such as Jensen, generalization of Jensen Hermite-Hadamard, Mercer inequalities. Moreover, with the use of these inequalities, we obtained bounds for Shannon’s entropy and Kapur’s entropy. Finally, we found an application of the obtained inequalities in means.

Not All Osteophytes Are Located on the Right Side of the Vertebrae in Diffuse Idiopathic Skeletal Hyperostosis: A Quantitative Analysis in Relation to the Position of Aorta.

Diffuse idiopathic skeletal hyperostosis (DISH) is characterized by osteophytes in the anterior vertebrae, and the presence of aorta may have an impact on their formation. However, the anatomical positional relationship between the aorta and osteophytes in patients with DISH remains controversial. This study aimed to evaluate the position of osteophytes in relation to aorta in DISH, and the influence of aortic pulsation on the formation of osteophytes from the perspective of morphology. We conducted a retrospective review of 101 patients diagnosed with DISH and symptomatic lumbar spinal stenosis between June 2018 and December 2021. A total of 637 segments with heterotopic ossification in DISH were used for quantitative measurements on CT scans. The Cartesian coordinate system was built up on the axial CT scans to reflect the relative position between aorta and osteophytes. Osteophytes were divided into adjacent aorta group (AD group) and non-adjacent aorta group (N-AD group). In terms of the morphology, osteophytes in the AD group were further divided into convex, flat, and concave types. The relative position between aorta and osteophytes, and the aorta-osteophyte distance and morphology of osteophytes were compared. Univariate analysis of variance was performed for multiple groups, and two independent-samples t-tests were used for two groups. From T5 to L4, aorta gradually descended from left side to middle of vertebrae, and osteophytes gradually shifted from right side of vertebrae (T5-T10) to bilateral sides (T11-L4). Of 637 osteophytes in DISH, 60.1% (383/637) were in AD group, including convex type 0.6% (4/637), flat type 34.7% (221/637), and concave type 24.8% (158/637). The N-AD group accounted for 39.9% (254/637). Flat osteophytes were concentrated in T5-T12, while concave osteophytes in T11-L4. Overall, the aorta-osteophyte distance of concave type was significantly smaller than that of flat type. Osteophytes are not always located on the right side of vertebrae, but move with the position of the descending aorta. Furthermore, the morphology of osteophytes varies by vertebral segment in DISH, which is related to aorta descending anteriorly in the spine.

New Perspectives of Symmetry Conferred by q-Hermite-Hadamard Type Integral Inequalities

The main goal of this work is to provide quantum parametrized Hermite-Hadamard like type integral inequalities for functions whose second quantum derivatives in absolute values follow different type of convexities. A new quantum integral identity is derived for twice quantum differentiable functions, which is used as a key element in our demonstrations along with several basic inequalities such as: power mean inequality, and Holder’s inequality. The symmetry of the Hermite-Hadamard type inequalities is stressed by the different types of convexities. Several special cases of the parameter are chosen to illustrate the investigated results. Four examples are presented.

Stability of Delay Hopfield Neural Networks with Generalized Riemann-Liouville Type Fractional Derivative.

The general delay Hopfield neural network is studied. We consider the case of time-varying delay, continuously distributed delays, time-varying coefficients, and a special type of a Riemann-Liouville fractional derivative (GRLFD) with an exponential kernel. The kernels of the fractional integral and the fractional derivative in this paper are Sonine kernels and satisfy the first and the second fundamental theorems in calculus. The presence of delays and GRLFD in the model require a special type of initial condition. The applied GRLFD also requires a special definition of the equilibrium of the model. A constant equilibrium of the model is defined. An inequality for Lyapunov type of convex functions with the applied GRLFD is proved. It is combined with the Razumikhin method to study stability properties of the equilibrium of the model. As a partial case we apply quadratic Lyapunov functions. We prove some comparison results for Lyapunov function connected deeply with the applied GRLFD and use them to obtain exponential bounds of the solutions. These bounds are satisfied for intervals excluding the initial time. Also, the convergence of any solution of the model to the equilibrium at infinity is proved. An example illustrating the importance of our theoretical results is also included.

2-point left Radau-type inequalities via s-convexity

Abstract Convexity is a fundamental concept in analysis. Over the past few decades, many significant error bounds have been established for various quadrature rules using different types of convexity. This paper focuses on the Gauss–Radau quadrature formula. Initially, we introduce a novel identity related to 2-point left Radau-type rule. Next, we derive several integral inequalities for functions whose first derivatives are s-convex in the second sense. Finally, we present applications to special means to demonstrate the effectiveness of our results.