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Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 59
Author(s):  
Pedro González-Rodelas ◽  
Hasan M. H. Idais ◽  
Mohammed Yasin ◽  
Miguel Pasadas

Function interpolation and approximation are classical problems of vital importance in many science/engineering areas and communities. In this paper, we propose a powerful methodology for the optimal placement of centers, when approximating or interpolating a curve or surface to a data set, using a base of functions of radial type. In fact, we chose a radial basis function under tension (RBFT), depending on a positive parameter, that also provides a convenient way to control the behavior of the corresponding interpolation or approximation method. We, therefore, propose a new technique, based on multi-objective genetic algorithms, to optimize both the number of centers of the base of radial functions and their optimal placement. To achieve this goal, we use a methodology based on an appropriate modification of a non-dominated genetic classification algorithm (of type NSGA-II). In our approach, the additional goal of maintaining the number of centers as small as possible was also taken into consideration. The good behavior and efficiency of the algorithm presented were tested using different experimental results, at least for functions of one independent variable.


2021 ◽  
Vol 10 (12) ◽  
pp. 3533-3548
Author(s):  
H. Desalegn ◽  
T. Abdi ◽  
J.B. Mijena

In this paper we discuss the following problem with additive noise, \[\begin{cases} \frac{\partial^{\beta} u(t,x) }{\partial t}=-(-\triangle)^{\frac{\alpha}{2}} u(t,x)+b(u(t,x))+\sigma\dot{W}(t,x),~~t>0, \\u(0,x)=u_{0}(x),\end{cases},\] where $\alpha \in(0,2) $ and $ \beta \in (0,1)$, the fractional time derivative is in the sense of Caputo, $-(-\Delta)^{\frac{\alpha}{2}}$ is the fractional Laplacian, $\sigma$ is a positive parameter, $\dot{W}$ is a space-time white noise, $u_0(x)$ is assumed to be non-negative, continuous and bounded. We study first the equation on $[0,\,1]$ with homogeneous Drichlet boundary condition and show that the solution of the equation blows up in finite time if and only if $b$ satisfies the Osgood condition, \[ \int_{c}^{\infty} \frac{ds}{b(s)} <\infty \] for some constant $c>0$. We then consider the same equation on the whole line and show that the above Osgood condition is satisfied whenever the solution of the equation blows up.


Author(s):  
Nan Deng

Aims/ Objectives: We discuss the existence and nonexistence of nontrivial nonnegative doubly periodic solutions for nonlinear telegraph equations                                                                  utt-uxx+cut+a(t,x)u=λf (t,x,u) , where c > 0 is a constant, λ > 0 is a positive parameter, a ∈ C(R2,R+), f ∈ C(R2 × R+,R+), and a, f are 2π-periodic in t and x. The proof is based on a known xed point theorem due to Schauder. In previous articles, a single telegraph equation or telegraph system have been widely studied, but there is relatively little research on nonlinear telegraph equations with a parameter and the nonlinearities are nonnegative. We would like do some research on this topic. We give new conclusions on the existence and nonexistence of nontrivial nonnegative doubly periodic solutions for nonlinear telegraph equations under sublinear assumptions. Study Design: Study on the existence and nonexistence of nontrivial nonnegative doubly periodic solutions. Place and Duration of Study: School of Applied Science, Beijing Information Science & Technology University, September 2020 to present.Methodology: We prove the existence and nonexistence of nontrivial nonnegative doubly periodic solutions by the results of Schauder's xed point theorem. Results: We give new conclusions of existence and nonexistence of nontrivial nonnegative doubly periodic solutions for the equations. Conclusion: We prove the existence and nonexistence of nontrivial nonnegative doubly periodic solutions for nonlinear telegraph equations                                                                 utt − uxx + cut + a(t, x)u = λf (t, x, u), and give new conclusions.


2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Antonio Iannizzotto ◽  
Roberto Livrea

AbstractWe consider a Dirichlet type problem for a nonlinear, nonlocal equation driven by the degenerate fractional p-Laplacian, whose reaction combines a sublinear term depending on a positive parameter and an asymmetric perturbation (superlinear at positive infinity, at most linear at negative infinity). By means of critical point theory and Morse theory, we prove that, for small enough values of the parameter, such problem admits at least four nontrivial solutions: two positive, one negative, and one nodal. As a tool, we prove a Brezis-Oswald type comparison result.


Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1874
Author(s):  
Denis I. Borisov

We consider a general second order self-adjoint elliptic operator on an arbitrary metric graph, to which a small graph is glued. This small graph is obtained via rescaling a given fixed graph γ by a small positive parameter ε. The coefficients in the differential expression are varying, and they, as well as the matrices in the boundary conditions, can also depend on ε and we assume that this dependence is analytic. We introduce a special operator on a certain extension of the graph γ and assume that this operator has no embedded eigenvalues at the threshold of its essential spectrum. It is known that under such assumption the perturbed operator converges to a certain limiting operator. Our main results establish the convergence of the spectrum of the perturbed operator to that of the limiting operator. The convergence of the spectral projectors is proved as well. We show that the eigenvalues of the perturbed operator converging to limiting discrete eigenvalues are analytic in ε and the same is true for the associated perturbed eigenfunctions. We provide an effective recurrent algorithm for determining all coefficients in the Taylor series for the perturbed eigenvalues and eigenfunctions.


Author(s):  
Zhiqian He ◽  
Liangying Miao

Abstract In this paper, we study the number of classical positive radial solutions for Dirichlet problems of type (P) − d i v ∇ u 1 − | ∇ u | 2 = λ f ( u )   in B 1 , u = 0                     on ∂ B 1 , $$\left\{\begin{aligned}\hfill & -\mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{\nabla u}{\sqrt{1-\vert \nabla u{\vert }^{2}}}\right)=\lambda f(u)\quad \text{in}\enspace {B}_{1},\hfill \\ \hfill & u=0\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \enspace \enspace \enspace \enspace \enspace \enspace \enspace \enspace \enspace \enspace \text{on}\enspace \partial {B}_{1},\enspace \hfill \end{aligned}\right.$$ where λ is a positive parameter, B 1 = { x ∈ R N : | x | < 1 } ${B}_{1}=\left\{x\in {\mathbb{R}}^{N}:\vert x\vert {< }1\right\}$ , f : [0, ∞) → [0, ∞) is a continuous function. Using the fixed point index in a cone, we prove the results on both uniqueness and multiplicity of positive radial solutions of (P).


2021 ◽  
Vol 36 (Supplement_1) ◽  
Author(s):  
A Volodarsky-Perel ◽  
M Zajicek ◽  
D Shai ◽  
H Raanani ◽  
N Gruber ◽  
...  

Abstract Study question What is the predictive value of ovarian reserve evaluation in patients with non-iatrogenic primary ovarian insufficiency (NIPOI) for follicle detection in ovarian tissue harvested for cryopreservation? Summary answer Ovarian tissue cryopreservation (OTCP) should be considered if patients present at least one of the following parameters: detectable AMH, FSH≤20mIU/ml, detection of ≥ 1 antral follicle. What is known already In pre-pubertal girls suffering from NIPOI, which majorly has a genetic etiology, fertility preservation using OTCP is commonly practiced. When OTCP was performed in an unselected group of children and adolescents with NIPOI, only 26% of them had follicles in ovarian tissue while 74% did not benefit from the surgery. The role of preoperative evaluation of anti-müllerian hormone (AMH) serum level, follicular stimulating hormone (FSH) serum level, and trans-abdominal ultrasound for the antral follicle count to predict the detection of primordial follicles in the harvested ovarian tissue is unclear. Study design, size, duration We conducted a retrospective analysis of all patients ≤ 18 years old who were referred for fertility preservation counseling due to NIPOI at a single tertiary hospital between 2010 and 2020. If initial evaluation suggested a diminished ovarian reserve and at least one positive parameter indicating a follicular activity (AMH &gt; 0.16ng/ml, FSH ≤ 20mIU/ml, detection of ≥ 1 antral follicle by transabdominal sonography), OTCP was offered. Patients with 46XY gonadal dysgenesis were excluded. Participants/materials, setting, methods OTCP was performed laparoscopically in all cases. A fresh sample of cortical tissue was fixed in buffered formaldehyde for histological analysis. The rest of the ovarian tissue was cut into small cuboidal slices 1–2 mm in thickness and cryopreserved. After the serial sections, the histological slides were evaluated for the presence of follicles by a certified pathologist. Follicles were counted and categorized as primordial, primary, and secondary. Main results and the role of chance During the study period, 39 patients with suspected NIPOI were referred to the fertility preservation center. Thirty-seven patients included in the study were diagnosed with Turner’s syndrome (n = 28), Galactosemia (n = 3), Blepharophimosis-Ptosis-Epicanthus Inversus syndrome (n = 1), and idiopathic NIPOI (n = 6). Of 28 patients with Turner’s syndrome, 6 had 45X monosomy, 15 had mosaicism and 7 had structural anomalies in X-chromosome. One patient with gonadal dysgenesis and one with the presence of Y-chromosome in 20% of somatic cells were excluded from the study. OTCP was conducted in 14 patients with at least one positive parameter suggesting ovarian function. No complications of the surgical procedure or the anesthesia were observed. Primordial follicles were found in all patients with two or three positive parameters (100%) and in three of six cases with one positive parameter (50%). In total, of the 14 patients who underwent OTCP with at least one positive parameter, 11 (79%) had primordial follicles at biopsy (mean 23.9, range 2–47). This study demonstrates a positive predictive value of 79% for the detection of primordial follicles in patients who had at least one positive parameter of ovarian reserve evaluation. If two or three parameters were positive, the positive predictive value increased to 100%. Limitations, reasons for caution This study did not examine the negative predictive value of our protocol as OTCP was not recommended in the absence of positive parameters. The future fertility potential of cryopreserved tissue in the population with NIPOI is unclear and should be discovered in further studies. Wider implications of the findings: We suggest the evaluation of ovarian reserve by antral follicles count, AMH, and FSH serum levels prior to OTCP in patients with NIPOI. By recommendation of OTCP only if ≥ 1 parameter suggesting the ovarian function is positive, unnecessary procedures can be avoided. Trial registration number Not applicable


Author(s):  
Claudianor O. Alves ◽  
Prashanta Garain ◽  
Vicenţiu D. Rădulescu

AbstractThis paper is concerned with the qualitative analysis of solutions to the following class of quasilinear problems $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{\Phi }u=f(x,u) &{}\quad \text {in } \Omega ,\\ u=0 &{}\quad \text {on }\partial \Omega , \end{array} \right. \end{aligned}$$ - Δ Φ u = f ( x , u ) in Ω , u = 0 on ∂ Ω , where $$\Delta _{\Phi }u=\mathrm{div}\,(\varphi (x,|\nabla u|)\nabla u)$$ Δ Φ u = div ( φ ( x , | ∇ u | ) ∇ u ) and $$\Phi (x,t)=\int _{0}^{|t|}\varphi (x,s)s\,ds$$ Φ ( x , t ) = ∫ 0 | t | φ ( x , s ) s d s is a generalized N-function. We assume that $$\Omega \subset {\mathbb {R}}^N$$ Ω ⊂ R N is a smooth bounded domain that contains two open regions $$\Omega _N,\Omega _p$$ Ω N , Ω p with $${\overline{\Omega }}_N \cap {\overline{\Omega }}_p=\emptyset $$ Ω ¯ N ∩ Ω ¯ p = ∅ . The features of this paper are that $$-\Delta _{\Phi }u$$ - Δ Φ u behaves like $$-\Delta _N u $$ - Δ N u on $$\Omega _N$$ Ω N and $$-\Delta _p u $$ - Δ p u on $$\Omega _p$$ Ω p , and that the growth of $$f:\Omega \times {\mathbb {R}} \rightarrow {\mathbb {R}}$$ f : Ω × R → R is like that of $$e^{\alpha |t|^{\frac{N}{N-1}}}$$ e α | t | N N - 1 on $$\Omega _N$$ Ω N and as $$|t|^{p^{*}-2}t$$ | t | p ∗ - 2 t on $$\Omega _p$$ Ω p when |t| is large enough. The main result establishes the existence of solutions in a suitable Musielak–Sobolev space in the case of high perturbations with respect to the values of a positive parameter.


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