rss_2.0Sciendo RSS Feed for Moroccan Journal of Pure and Applied Analysishttps://sciendo.com/journal/MJPAAhttps://www.sciendo.comhttps://sciendo-parsed.s3.eu-central-1.amazonaws.com/64724b86215d2f6c89dc33ae/cover-image.jpghttps://sciendo.com/journal/MJPAA140216https://sciendo.com/article/10.2478/mjpaa-2023-0019<abstract> <title style='display:none'>Abstract</title> <p>For a Bloch function <italic>f</italic> in the unit ball in ℂ<italic>ⁿ</italic>, we study the maximal locus of the Bloch norm of <italic>f</italic>; namely, the set <italic>L_f</italic> where the Bergman length of the gradient vector field of <italic>f</italic> attains its maximum. We prove that for <italic>n</italic> ≥<italic>,</italic> the set <italic>L_f</italic> consists of a finite union of real analytic sets with dimensions at most 2<italic>n −</italic> 2. This is not the case for <italic>n</italic> = 1 as was proved earlier by Cima and Wogen. We also give some rigidity properties of the set <italic>L_f</italic>. In particular, we give some sufficient criteria for constructing extreme functions in the Little Bloch ball.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00192023-06-07T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0014<abstract> <title style='display:none'>Abstract</title> <p>There exist two real valued periodic functions on the real line such that, for every <italic>x</italic> ∈ ℝ, <italic>f</italic>₁(<italic>x</italic>) + <italic>f</italic>₂(<italic>x</italic>) = <italic>x</italic>, but it is impossible to find two real valued periodic functions on the real line such that, for every <italic>x</italic> ∈ ℝ, <italic>f</italic>₁(<italic>x</italic>) + <italic>f</italic>₂(<italic>x</italic>) = <italic>x</italic>². The purpose of this note is to prove this result and also to study the possibility of decomposing more general polynomials into sum of periodic functions.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00142023-06-07T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0016<abstract> <title style='display:none'>Abstract</title> <p>In this paper, we study the cyclicity problem with respect to the forward shift operator <italic>S_b</italic> acting on the de Branges–Rovnyak space ℋ (<italic>b</italic>) associated to a function <italic>b</italic> in the closed unit ball of <italic>H</italic>^∞ and satisfying log(1− |<italic>b</italic>| ∈ <italic>L</italic>¹(𝕋). We present a characterisation of cyclic vectors for <italic>S_b</italic> when <italic>b</italic> is a rational function which is not a finite Blaschke product. This characterisation can be derived from the description, given in [22], of invariant subspaces of <italic>S_b</italic> in this case, but we provide here an elementary proof. We also study the situation where <italic>b</italic> has the form <italic>b</italic> = (1+ <italic>I</italic>)/2, where <italic>I</italic> is a non-constant inner function such that the associated model space <italic>K</italic>_I = ℋ (<italic>I</italic>) has an orthonormal basis of reproducing kernels.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00162023-06-07T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0012<abstract> <title style='display:none'>Abstract</title> <p>We consider a Nevanlinna–Pick interpolation problem on finite sequences of the unit disc, constrained by Beurling–Sobolev norms. We find sharp asymptotics of the corresponding interpolation quantities, thereby improving the known estimates. On our way we obtain a S. M. Nikolskii type inequality for rational functions whose poles lie outside of the unit disc. It shows that the embedding of the Hardy space <italic>H</italic>² into the Wiener algebra of absolutely convergent Fourier/Taylor series is invertible on the subset of rational functions of a given degree, whose poles remain at a given distance from the unit circle.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00122023-06-07T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0017<abstract> <title style='display:none'>Abstract</title> <p>The aim of these notes is to discuss the completeness of the dilated systems in a most general framework of an arbitrary sequence lattice <italic>X</italic>, including weighted <italic>ℓ</italic>^p spaces. In particular, general multiplicative and completely multiplicative sequences are treated. After the Fourier–Bohr transformation, we deal with the cyclicity property in function spaces on the corresponding infinite dimensional Reinhardt domain <inline-formula> <alternatives> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0017_ieq_001.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>𝔻</mml:mi></mml:mrow><mml:mi>X</mml:mi><mml:mo>∞</mml:mo></mml:msubsup></mml:mrow></mml:math> <tex-math>\mathbb{D}_X^\infty</tex-math> </alternatives> </inline-formula> . Functions with (weakly) dominating free term and (in particular) linearly factorable functions are considered. The most attention is paid to the cases of the polydiscs <inline-formula> <alternatives> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0017_ieq_002.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>𝔻</mml:mi></mml:mrow><mml:mi>X</mml:mi><mml:mo>∞</mml:mo></mml:msubsup><mml:mo>,</mml:mo><mml:mo>|</mml:mo><mml:msup><mml:mrow><mml:mi>ℂ</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>𝔻</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup></mml:mrow></mml:math> <tex-math>\mathbb{D}_X^\infty ,|{\mathbb{C}^N} = {\mathbb{D}^N}</tex-math> </alternatives> </inline-formula> and the <italic>ℓ</italic>^p-unit balls <inline-formula> <alternatives> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0017_ieq_003.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>𝔻</mml:mi></mml:mrow><mml:mi>X</mml:mi><mml:mo>∞</mml:mo></mml:msubsup><mml:mo>,</mml:mo><mml:mo>|</mml:mo><mml:msup><mml:mrow><mml:mi>ℂ</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>𝔹</mml:mi></mml:mrow><mml:mi>p</mml:mi><mml:mi>N</mml:mi></mml:msubsup></mml:mrow></mml:math> <tex-math>\mathbb{D}_X^\infty ,|{\mathbb{C}^N} = \mathbb{B}_p^N</tex-math> </alternatives> </inline-formula> , in particular to Dirichlet-type and Dirichlet–Drury–Arveson-type spaces and algebras, as <inline-formula> <alternatives> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0017_ieq_004.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>X</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mi>p</mml:mi></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>ℤ</mml:mi></mml:mrow><mml:mo>+</mml:mo><mml:mi>N</mml:mi></mml:msubsup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>α</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mi>s</mml:mi></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> <tex-math>X = {\ell ^p}\left( {_ + ^N,{{\left( {1 + \alpha } \right)}^s}} \right))</tex-math> </alternatives> </inline-formula>, <italic>s</italic> = (<italic>s</italic>₁, <italic>s</italic>₂, … ) and <inline-formula> <alternatives> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0017_ieq_005.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>X</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mi>p</mml:mi></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>ℤ</mml:mi></mml:mrow><mml:mo>+</mml:mo><mml:mi>N</mml:mi></mml:msubsup><mml:mo>,</mml:mo><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mi>α</mml:mi><mml:mo>!</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mo>|</mml:mo> <mml:mi>α</mml:mi> <mml:mo>|</mml:mo></mml:mrow><mml:mo>!</mml:mo></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mi>t</mml:mi></mml:msup><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mrow><mml:mo>|</mml:mo> <mml:mi>α</mml:mi> <mml:mo>|</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mi>s</mml:mi></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> <tex-math>X = {\ell ^p}\left( {\mathbb{Z}_ + ^N,\,\,{{\left( {{{\alpha !} \over {\left| \alpha \right|!}}} \right)}^t}{{\left( {1 + \left| \alpha \right|} \right)}^s}} \right)</tex-math> </alternatives> </inline-formula>, <italic>s,t</italic> ≥ 0, as well as to their infinite variables analogues. We priviledged the largest possible scale of spaces and the most elementary instruments used.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00172023-06-07T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0011<abstract> <title style='display:none'>Abstract</title> <p>This note is dedicated to recalling the virtues and the important contributions in mathematics of mohamed zarabi who passed a way on mid december 2021.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00112023-06-07T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0015<abstract> <title style='display:none'>Abstract</title> <p>Zarrabi proved in 1993 that if the spectrum of a contraction <italic>T</italic> on a Banach space is a countable subset of the unit circle 𝕋, and if <inline-formula> <alternatives> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0015_ieq_001.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mo>lim</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>→</mml:mo><mml:mo>+</mml:mo><mml:mo>∞</mml:mo></mml:mrow></mml:msub><mml:mfrac><mml:mrow><mml:mo>log</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mo>‖</mml:mo> <mml:mrow><mml:msup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:mrow> <mml:mo>‖</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:msqrt><mml:mi>n</mml:mi></mml:msqrt></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> <tex-math>{\lim _{n \to + \infty }}{{\log \left( {\left\| {{T^{ - n}}} \right\|} \right)} \over {\sqrt n }} = 0</tex-math> </alternatives> </inline-formula>, then <italic>T</italic> is an isometry, so that ‖<italic>Tⁿ</italic>‖ = 1 for every <italic>n</italic> ∈ ℤ. It is also known that if <italic>C</italic> is the usual triadic Cantor set then every contraction <italic>T</italic> on a Banach space such that <italic>Spec</italic>(<italic>T</italic> ) ⊂ 𝒞 satisfying <inline-formula> <alternatives> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0015_ieq_002.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>lim</mml:mo><mml:mi> </mml:mi><mml:mi>s</mml:mi><mml:mi>u</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>→</mml:mo><mml:mo>+</mml:mo><mml:mo>∞</mml:mo></mml:mrow></mml:msub><mml:mfrac><mml:mrow><mml:mo>log</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mo>‖</mml:mo> <mml:mrow><mml:msup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:mrow> <mml:mo>‖</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mi>α</mml:mi></mml:msup></mml:mrow></mml:mfrac><mml:mo>&lt;</mml:mo><mml:mo>+</mml:mo><mml:mo>∞</mml:mo></mml:mrow></mml:math> <tex-math>\lim \,su{p_{n \to + \infty }}{{\log \left( {\left\| {{T^{ - n}}} \right\|} \right)} \over {{n^\alpha }}} &lt; + \infty</tex-math> </alternatives> </inline-formula> for some <inline-formula> <alternatives> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0015_ieq_003.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>α</mml:mi><mml:mo>&lt;</mml:mo><mml:mfrac><mml:mrow><mml:mo>log</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>3</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mo>log</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi> </mml:mi><mml:mo>log</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>3</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mo>log</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mrow></mml:math> <tex-math>\alpha &lt; {{\log \left( 3 \right) - \log \left( 2 \right)} \over {2\,\log \left( 3 \right) - \log \left( 2 \right)}}</tex-math> </alternatives> </inline-formula> is an isometry.</p> <p>In the other direction an easy refinement of known results shows that if a closed <italic>E</italic> ⊂ 𝕋 is not a “strong <italic>AA</italic>⁺-set” then for every sequence (<italic>u_n)_n</italic>_≥1 of positive real numbers such that lim inf<italic>_n</italic>_→+∞<italic>u_n</italic> = + ∞ there exists a contraction <italic>T</italic> on some Banach space such that <italic>Spec</italic>(<italic>T</italic> )⊂ <italic>E</italic>, ‖<italic>T</italic>^{−<italic>n</italic>}‖ = <italic>O</italic>(<italic>u</italic>_n) as <italic>n</italic> → + ∞ and sup_n_≥1 ‖<italic>T</italic>⁻ⁿ‖ = + ∞.</p> <p>We show conversely that if <italic>E</italic> ⊂ 𝕋 is a strong <italic>AA</italic>⁺-set then there exists a nondecreasing unbounded sequence (<italic>u</italic>_n)_n_≥1 such that for every contraction <italic>T</italic> on a Banach space satsfying <italic>Spec</italic>(<italic>T</italic>) ⊂ <italic>E</italic> and ‖<italic>T</italic>⁻ⁿ ‖ = <italic>O</italic>(<italic>u</italic>_n) as <italic>n</italic> → + ∞ we have <italic>sup_n</italic>_&gt;0 ‖<italic>T</italic>⁻ⁿ ‖ ≤ <italic>K</italic>, where <italic>K &lt;</italic> + ∞ denotes the “<italic>AA</italic>⁺-constant” of <italic>E</italic> (closed countanble subsets of 𝕋 and the triadic Cantor set are strong <italic>AA</italic>⁺-sets of constant 1).</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00152023-06-07T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0018<abstract> <title style='display:none'>Abstract</title> <p>In this expository article, we discuss the evaluation and estimation of the operator norms of various functions of the Volterra operator.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00182023-06-07T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0013<abstract> <title style='display:none'>Abstract</title> <p>We extend Carleson’s formula to radially polynomially weighted Dirichlet spaces.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00132023-06-07T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0008<abstract> <title style='display:none'>Abstract</title> <p>New copulas, based on perturbation theory, are introduced to clarify a <italic>symmetrization</italic> procedure for asymmetric copulas. We give also some properties of the <italic>symmetrized</italic> copula mainly conservation of concordance. Finally, we examine some copulas with a prescribed symmetrized part. The start point of the treatment is the independence copula and the last one will be an arbitrary member of Farlie-Gumbel-Morgenstein family. By the way, we study topologically, the set of all symmetric copulas and give some of its classical and new properties.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00082023-02-01T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0007<abstract> <title style='display:none'>Abstract</title> <p>To have a more realistic model, in this paper, This manuscript is devoted to investigating a fractional-order mathematical model of Kouidere et al. That describes the dynamics of spread of African swine fever virus (ASFV). The aim of this work is to protect susceptible pigs from the virus, In our model, by including three controls which represent: the iron fencing and spraying pesticides and get rid.</p> <p>The aims of this paper is to reduce the number of infected pigs and ticks by using optimal control strategy and fractinal order derivation.</p> <p>Pontryagin’s maximal principle is used to describe optimal controls with Caputo time-fractional derivative and the optimal system is resolved in an iterative manner. Numerical simulations are presented based on the presented method. We finished tis article with a conclusion.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00072023-02-01T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0004<abstract> <title style='display:none'>Abstract</title> <p>In this work, we study the existence of at least one non decreasing sequence of nonnegative eigenvalues for the problem: <disp-formula> <alternatives> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0004_eq_001.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mo>{</mml:mo> <mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi></mml:mrow></mml:msup><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mi>λ</mml:mi><mml:mi>m</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mfrac><mml:mrow><mml:mo>∂</mml:mo><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>v</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mo>∂</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>u</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>v</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mo>…</mml:mo><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mo>∂</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>u</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>v</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi>o</mml:mi><mml:mi>n</mml:mi><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mo>∂</mml:mo><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow> </mml:mrow></mml:mrow></mml:math> <tex-math>\left\{ {\matrix{ {{\Delta ^{2p}}u = \lambda m\left( x \right)u\,\,\,in\,\,\,\Omega ,} \cr {{{\partial u} \over {\partial v}} = {{\partial \left( {\Delta u} \right)} \over {\partial v}} = \ldots = {{\partial \left( {{\Delta ^{2p - 1}}u} \right)} \over {\partial v}} = 0\,\,\,on\,\,\,\partial \Omega .} \cr } } \right.</tex-math> </alternatives> </disp-formula> Where Ω is a bounded domain in ℝ<italic>^N</italic> with smooth boundary ∂ Ω, <italic>p</italic> ∈ ℕ*, <italic>m</italic> ∈ <italic>L</italic>^∞ (Ω), and Δ²<italic>^p</italic><italic>u</italic> := Δ (Δ...( Δ<italic>u</italic>)), 2<italic>p</italic> times the operator Δ.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00042023-02-01T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0006<abstract> <title style='display:none'>Abstract</title> <p>In [Appl. Comput. Harmon. Anal., 46 (2019), 664673] O. Christensen and M. Hasannasab observed that assuming the existence of an operator <italic>T</italic> sending <italic>en</italic> to <italic>e</italic><italic>_n</italic>₊₁ for all <italic>n</italic> ∈ ℕ (where (<italic>e</italic><italic>_n</italic>)<italic>_n</italic>_∈ℕ is a sequence of vectors) guarantees that (<italic>e</italic><italic>_n</italic>)<italic>_n</italic>_∈ℕ is linearly independent if and only if dim<italic>{e</italic><italic>_n</italic><italic>}</italic><italic>_n</italic>_∈ℕ = ∞. In this article, we recover this result as a particular case of a general order-theory-based model-theoretic result. We then return to the context of vector spaces to show that, if we want to use a condition like <italic>T</italic>(<italic>e</italic><italic>_i</italic>) = <italic>e</italic><italic>_ϕ</italic>₍<italic>_i</italic>₎ for all <italic>i</italic> ∈ <italic>I</italic> where <italic>I</italic> is countable as a replacement of the previous one, the conclusion will only stay true if <italic>ϕ</italic> : <italic>I</italic> → <italic>I</italic> is conjugate to the successor function <italic>succ</italic> : <italic>n</italic> ↦<italic>n</italic> + 1 defined on ℕ. We finally prove a tentative generalization of the result, where we replace the condition <italic>T</italic>(<italic>e</italic><italic>_i</italic>) = <italic>e</italic><italic>_ϕ</italic>₍<italic>_i</italic>₎ for all <italic>i</italic> ∈ <italic>I</italic> where <italic>ϕ</italic> is conjugate to the successor function with a more sophisticated one, and to which we have not managed to find a new application yet.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00062023-02-01T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0001<abstract> <title style='display:none'>Abstract</title> <p>In this paper, we propose to compare three a posteriori error estimators namely equilibrated, star-based and residual based for the Poisson problem and the Stokes problem with lowest-order Crouzeix-Raviart finite element discretization. The numerical results are presented to compare the performance of the three estimators in an adaptive refinement strategy.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00012023-02-01T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0009<abstract> <title style='display:none'>Abstract</title> <p>We propose a novel diffusion process having a mean function equal to the Pareto probability density function up to a constant of proportionality. We examine the probabilistic properties of the proposed model. Then, referring to the problem of statistical inference, we describe the approach employed to tackle the issue of obtaining parameter estimates by maximizing the likelihood function based on discrete sampling. This estimation reduces to solving a set of complex equations, that is accomplished using the simulated annealing algorithm. A simulation study is also given to validate the methodology presented. Finally, using a real-world example of the Moroccan child mortality rate, we obtain the fits and forecasts by employing the suggested stochastic process and nonlinear regression model.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00092023-02-01T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0005<abstract> <title style='display:none'>Abstract</title> <p>In the history of option pricing, Black-Scholes model is one of the most significant models. In this paper, we present a new numerical strategy for valuing American option pricing problems governed by Black-Scholes model (BSM). Numerical computations are carried out to show the efficiency and robustness of the proposed method. We compare our numerical solution with the ones based on Finite Element Method (FEM) and the Enriched Finite Element Method (PUFEM). Our result shows the efficiency of the proposed strategy. In addition, that approach can be used to treat nonlinear evolutionary problems.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00052023-02-01T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0010<abstract> <title style='display:none'>Abstract</title> <p>In this work and in the context of PDE constrained optimization problems, we are interested in identification of a parameter in the diffusion equation proposed in [1]. We propose to identify this parameter automatically by a gradient descent algorithm to improve the restoration of a noisy image. Finally, we give numerical results to illustrate the performance of the automatic selection of this parameter and compare our numerical results with other image denoising approaches or algorithms.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00102023-02-01T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0002<abstract> <title style='display:none'>Abstract</title> <p>We study the existence and uniqueness of a bounded weak solution for a triply nonlinear thermistor problem in Sobolev spaces. Furthermore, we prove the existence of an absorbing set and, consequently, the universal attractor.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00022023-02-01T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0003<abstract> <title style='display:none'>Abstract</title> <p>The aim of this paper is to present the mathematical and numerical study of a nonlocal nonlinear model based on the variable exponent <italic>p</italic>(<italic>x</italic>)-Laplacian for removing Cauchy noise, which is a type of impulsive and non-Gaussian degradation. The proposed model benefits from the performance of the nonlocal approach to preserve small details and textures, and the efficiency of the variable exponent to reduce the execution time. To demonstrate the reliability of our proposed model, we provide some experimental denoising results and illustrate the comparison with some models from the literature.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00032023-02-01T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2022-0025<abstract> <title style='display:none'>Abstract</title> <p>In this paper, we introduce a new family of stochastic Lundqvist-Korf diffusion process, defined from a <italic>g</italic>-power of the Lundqvist-Korf diffusion process. First, we determine the probabilistic characteristics of the process, such as its analytic expression, the transition probability density function from the corresponding It ˆo stochastic differential equation and obtain the conditional and non-conditional mean functions. We then study the statistical inference in this process. The parameters of this process are estimated by using the maximum likelihood estimation method with discrete sampling, thus we obtain a nonlinear equation, which is achieved via the simulated annealing algorithm. Finally, the results of the paper are applied to simulated data.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2022-00252022-10-11T00:00:00.000+00:00en-us-1