Type Operators Research Articles

Solutions for a flame propagation model in porous media based on Hamiltonian and regular perturbation methods

This article extends the exploration of solutions to the issue of flame propagation driven by pressure and temperature in porous media that we introduced in earlier papers. We continue to consider a p-Laplacian type operator as a mathematical formalism to model slow and fast diffusion effects, that can be given in the non-homogeneous propagation of flames. In addition, we introduce a forced convection to model any possible induced flow in the porous media. We depart from previously known models to further substantiate our driving equations. From a mathematical standpoint, our goal is to deepen in the understanding of the general behavior of solutions via analyzing their regularity, boundedness, and uniqueness. We explore stationary solutions through a Hamiltonian approach and employ a regular perturbation method. Subsequently, nonstationary solutions are derived using a singular exponential scaling and, once more, a regular perturbation approach.

On the Nodal Set of Solutions to a Class of Nonlocal Parabolic Equations

We investigate the local properties, including the nodal set and the nodal properties of solutions to the following parabolic problem of Muckenhoupt-Neumann type: { ∂ t u ¯ − y − a ∇ ⋅ ( y a ∇ u ¯ ) = 0 a m p ; in B 1 + × ( − 1 , 0 ) − ∂ y a u ¯ = q ( x , t ) u a m p ; on B 1 × { 0 } × ( − 1 , 0 ) , \begin{equation*} \begin {cases} \partial _t \overline {u} - y^{-a} \nabla \cdot (y^a \nabla \overline {u}) = 0 \quad &\text { in } \mathbb {B}_1^+ \times (-1,0) \\ -\partial _y^a \overline {u} = q(x,t)u \quad &\text { on } B_1 \times \{0\} \times (-1,0), \end{cases} \end{equation*} where a ∈ ( − 1 , 1 ) a\in (-1,1) is a fixed parameter, B 1 + ⊂ R N + 1 \mathbb {B}_1^+\subset \mathbb {R}^{N+1} is the upper unit half ball and B 1 B_1 is the unit ball in R N \mathbb {R}^N . Our main motivation comes from its relation with a class of nonlocal parabolic equations involving the fractional power of the heat operator H s u ( x , t ) = 1 | Γ ( − s ) | ∫ − ∞ t ∫ R N [ u ( x , t ) − u ( z , τ ) ] G N ( x − z , t − τ ) ( t − τ ) 1 + s d z d τ . \begin{equation*} H^su(x,t) = \frac {1}{|\Gamma (-s)|} \int _{-\infty }^t \int _{\mathbb {R}^N} \left [u(x,t) - u(z,\tau )\right ] \frac {G_N(x-z,t-\tau )}{(t-\tau )^{1+s}} dzd\tau . \end{equation*} We characterise the possible blow-ups and we examine the structure of the nodal set of solutions vanishing with a finite order. More precisely, we prove that the nodal set has at least parabolic Hausdorff codimension one in R N × R \mathbb {R}^N\times \mathbb {R} , and can be written as the union of a locally smooth part and a singular part, which turns out to possess remarkable stratification properties. Moreover, the asymptotic behaviour of general solutions near their nodal points is classified in terms of a class of explicit polynomials of Hermite and Laguerre type, obtained as eigenfunctions to an Ornstein-Uhlenbeck type operator. Our main results are obtained through a fine blow-up analysis which relies on the monotonicity of an Almgren-Poon type quotient and some new Liouville type results for parabolic equations, combined with more classical results including Federer’s reduction principle and the parabolic Whitney’s extension.

Quantum Sugawara operators in type A

The quantum Sugawara operators associated with a simple Lie algebra g are elements of the center of a completion of the quantum affine algebra Uq(gˆ) at the critical level. By the foundational work of Reshetikhin and Semenov-Tian-Shansky (1990), such operators occur as coefficients of a formal Laurent series ℓV(z) associated with every finite-dimensional representation V of the quantum affine algebra. As demonstrated by Ding and Etingof (1994), the quantum Sugawara operators generate all singular vectors in generic Verma modules over Uq(gˆ) at the critical level and give rise to a commuting family of transfer matrices. Furthermore, as observed by E. Frenkel and Reshetikhin (1999), the operators are closely related with the q-characters and q-deformed W-algebras via the Harish-Chandra homomorphism.We produce explicit quantum Sugawara operators for the quantum affine algebra of type A which are associated with primitive idempotents of the Hecke algebra and parameterized by Young diagrams. This opens a way to understand all the related objects via their explicit constructions. We consider one application by calculating the Harish-Chandra images of the quantum Sugawara operators. The operators act by scalar multiplication in the q-deformed Wakimoto modules and we calculate the eigenvalues by identifying them with the Harish-Chandra images.

Lattice Boltzmann simulation of ion transport during the charging process of porous electrodes in randomly reconstructed seawater desalination batteries

The economics and desalination results of desalination cells are superior to those of other desalination technologies due to their unique electrochemical mechanisms. Nevertheless, there has been a paucity of research into the potential mechanisms of pore-scale ion removal during the operation of this specific type of battery. In this paper, the relationship between electrode microstructure and cell performance is investigated by establishing a lattice Boltzmann model and a high-fidelity two-dimensional electrode structure. The results demonstrate that the random morphology and irregular agglomeration of particles result in significant differences in electrode performance. During electrode deionization, the ion embedding process is faster in particles close to the main flow. In addition, sodium ions are inserted faster in small particles than in large particles. If the particles comprising the electrode exhibit a localized uneven distribution, it is advantageous to have a greater density of particles in the vicinity of the flow channel. The larger the reaction interface per unit area, the faster the interpolation rate of the particle, while the larger the total area, the larger the interpolation capacity of the particle.

T3SS protein EsrC binds to the lacI-like operator of type 1 fimbrial operon to suppress adhesion of Edwardsiella piscicida.

Type 1 fimbria, the short hair-like appendage assembled on the bacterial surface, plays a pivotal role in adhesion and invasion in Edwardsiella piscicida. The type III secretion system (T3SS), another bacterial surface appendage, facilitates E. piscicida's replication in vivo by delivering effectors into host cells. Our previous research demonstrated that E. piscicida T3SS protein EseJ inhibits adhesion and invasion of E. piscicida by suppressing type 1 fimbria. However, how EseJ suppresses type 1 fimbria remains elusive. In this study, a lacI-like operator (nt -245 to -1 of fimA) upstream of type 1 fimbrial operon in E. piscicida was identified, and EseJ inhibits type 1 fimbria through the lacI-like operator. Moreover, through DNA pull-down and electrophoretic mobility shift assay, an AraC-type T3SS regulator, EsrC, was screened and verified to bind to nt -145 to -126 and nt -50 to -1 of fimA, suppressing type 1 fimbria. EseJ is almost abolished upon the depletion of EsrC. EsrC and EseJ impede type 1 fimbria expression. Intriguingly, nutrition and microbiota-derived indole activate type 1 fimbria through downregulating T3SS, alleviating EsrC or EseJ's inhibitory effect on lacI-like operator of type 1 fimbrial operon. By this study, it is revealed that upon entering the gastrointestinal tract, rich nutrients and indole downregulate T3SS and thereof upregulate type 1 fimbria, stimulating efficient adhesion and invasion; upon being internalized into epithelium, the limit in indole and nutrition switches on T3SS and thereof switches off type 1 fimbria, facilitating effector delivery to guarantee E. piscicida's survival/replication in vivo.IMPORTANCEIn this work, we identified the lacI-like operator of type 1 fimbrial operon in E. piscicida, which was suppressed by the repressors-T3SS protein EseJ and EsrC. We unveiled that E. piscicida upregulates type 1 fimbria upon sensing rich nutrition and the microbiota-derived indole, thereof promoting the adhesion of E. piscicida. The increase of indole and nutrition promotes type 1 fimbria by downregulating T3SS. The decrease in EseJ and EsrC alleviates their suppression on type 1 fimbria, and vice versa.

Weighted approximations by sampling type operators: recent and new results

In this paper, we collect some recent results on the approximation properties of generalized sampling operators and Kantorovich operators, focusing on pointwise and uniform convergence, rate of convergence, and Voronovskaya-type theorems in weighted spaces of functions. In the second part of the paper, we introduce a new generalization of sampling Durrmeyer operators including a special function $\rho$ which satisfies certain assumptions. For the family of newly constructed operators, we obtain pointwise convergence, uniform convergence and rate of convergence for functions belonging to weighted spaces of functions.

Convergence in distribution of the Bernstein–Durrmeyer kernel and pointwise convergence of a generalised operator for functions of bounded variation

We study the convergence of Bernstein type operators leading to two results. The first: The kernel Kn of the Bernstein–Durrmeyer operator at each point x∈(0,1) — that is Kn(x,t)dt — once standardised converges to the normal distribution. The second result computes the pointwise limit of a generalised Bernstein–Durrmeyer operator applied to — possibly discontinuous — functions f of bounded variation.

Quasi-triangular, factorizable Leibniz bialgebras and relative Rota–Baxter operators

Abstract We introduce the notion of quasi-triangular Leibniz bialgebras, which can be constructed from solutions of the classical Leibniz Yang–Baxter equation (CLYBE) whose skew-symmetric parts are invariant. In addition to triangular Leibniz bialgebras, quasi-triangular Leibniz bialgebras contain factorizable Leibniz bialgebras as another subclass, which lead to a factorization of the underlying Leibniz algebras. Relative Rota–Baxter operators with weights on Leibniz algebras are used to characterize solutions of the CLYBE whose skew-symmetric parts are invariant. On skew-symmetric quadratic Leibniz algebras, such operators correspond to Rota–Baxter type operators. Consequently, we introduce the notion of skew-symmetric quadratic Rota–Baxter Leibniz algebras, such that they give rise to triangular Leibniz bialgebras in the case of weight 0, while they are in one-to-one correspondence with factorizable Leibniz bialgebras in the case of nonzero weights.

ψ$$ \psi $$‐Bernstein–Kantorovich operators

In this article, we introduce a modified class of Bernstein–Kantorovich operators depending on an integrable function and investigate their approximation properties. By choosing an appropriate function , the order of approximation of our operators to a function is at least as good as the classical Bernstein–Kantorovich operators on the interval . We compared the operators defined in this study not only with Bernstein–Kantorovich operators but also with some other Bernstein–Kantorovich type operators. In this paper, we also study the results on the uniform convergence and rate of convergence of these operators in terms of the first‐ and second‐order moduli of continuity, and we prove that our operators have shape‐preserving properties. Finally, some numerical examples which support the results obtained in this paper are provided.

Refinements of Pólya-SzegŐ and Chebyshev type inequalities via different fractional integral operators

Various differential and integral operators have been introduced and applied for the generalization of several integral inequalities. The purpose of this article is to create a more generalized fractional integral operator of Saigo type. This operator will be used alongwith the existing Saigo type and Q-Saigo type fractional integral operators to establish extended and generalized versions of several inequalities, including Pólya-Szegő and Chebyshev type inequalities.

Existence of weak solutions of the fractional p(x,y)$$ p\left(x,y\right) $$‐Laplacian problem by topological degree

In this paper, we prove the existence result of weak solutions to a class fractional ‐Laplacian problems involving the nonlocal integro‐differential operator of elliptique type . The main tool used here is based on the topological degree theory combined with the theory fractional Sobolev spaces with variable exponent. Our results generalize and improve the existing results.

Preserving properties of some Szasz-Mirakyan type operators

For a family of Szasz-Mirakyan type operators we prove that they preserve convex-type functions and that a monotonicity property verified by Cheney and Sharma in the case Szasz-Mirakyan operators holds for the variation study here. We also verify that several modulus of continuity are preserved.

Uncertainty principles for the continuous wavelet transform associated with a Bessel type operator on the half line

This paper presents uncertainty principles pertaining to generalized wavelet transforms associated with a second-order differential operator on the half line, extending the concept of the Bessel operator. Specifically, we derive a Heisenberg-Pauli-Weyl type uncertainty principle, as well as other uncertainty relations involving sets of finite measure

Neural Network Approach to Cutting Tools Selection

Cutting tools selection is a key issue in today's research of computer aided process planning (CAPP). Traditionally, this task is carried out by process planners and knowledge base systems. Recently, process planners have started using newer artificial intelligent techniques, such as neural networks, fuzzy logic, intelligent agents, etc. to model cutting tools. In this study, the problem of cutting tools selection for milling and drilling operations is investigated. A neural network model is proposed to select the needed cutting tools and their geometry based feature type and machining operation. Feature type, feature condition, dimension ratio, feature taper and machining operation are presented to the network for each feature as input parameters. The network selects the required cutting tool. The advantage and effectiveness of the proposed model are verified through a several types of machining features

Approximation by parametrized logistic activated convolution type operators

Approximation by parametrized logistic activated convolution type operators

Charlier polynomial-based modified Kantorovich–Szász type operators and related approximation outcomes

Charlier polynomial-based modified Kantorovich–Szász type operators and related approximation outcomes

Construction of New Operators by Composition of Integral-Type Operators and Discrete Operators

In this paper, we propose some new positive linear approximation operators, which are obtained from a composition of certain integral type operators with certain discrete operators. It turns out that the new operators can be expressed in discrete form. We provide representations for their coefficients. Furthermore, we study their approximation properties and determine their moment generating functions, which may be useful in finding several other convergence results in different settings.

Generalized Szász-Mirakian Type Operators

In this paper, we are proposing certain modifications of Szász-Mirakian type operators and study their approximation properties. We also give a Voronovskaya type theorem for these operators. Keywords: Linear positive operators, Schurer type operator, Szász-Mirakian operators, Voronovskaya type theorem.

Pointwise and weighted estimates for Bernstein-Kantorovich type operators including beta function

Pointwise and weighted estimates for Bernstein-Kantorovich type operators including beta function

On a class of Hausdorff type operators on interval

The paper studies integral operators on the interval of real line which naturally arise in some problems in the theory of integral equations and mathematical physics. We study boundedness in weighted Lebesgue spaces: Sufficient and necessary conditions for boundedness are given. Also, special important particular cases of operators and spaces are considered as examples. We construct identity approximation within the framework of the class of operators under study, and two different approaches are given.