The behavior at infinity of a solution to a Sobolev type system

2007 ◽  
Vol 48 (5) ◽  
pp. 784-797
Author(s):  
L. N. Bondar
Keyword(s):  
Type System ◽  
Sobolev Type

scholarly journals A multi-species chemotaxis system: Lyapunov functionals, duality, critical mass

2017 ◽  
Vol 29 (3) ◽  
pp. 515-542 ◽  
Author(s):  
N. I. KAVALLARIS ◽  
T. RICCIARDI ◽  
G. ZECCA
Keyword(s):  
Total Population ◽  
Critical Mass ◽  
Type System ◽  
Type Inequality ◽  
Chemical Substance ◽  
Sobolev Type ◽  
Lyapunov Functionals ◽  
Single Chemical ◽  
Chemotaxis System ◽  
Sobolev Type Inequality

We introduce a multi-species chemotaxis type system admitting an arbitrarily large number of population species, all of which are attracted versus repelled by a single chemical substance. The production versus destruction rates of the chemotactic substance by the species is described by a probability measure. For such a model, we investigate the variational structures, in particular, we prove the existence of Lyapunov functionals, we establish duality properties as well as a logarithmic Hardy–Littlewood–Sobolev type inequality for the associated free energy. The latter inequality provides the optimal critical value for the conserved total population mass.

Symmetry and Regularity of Solutions to the Weighted Hardy–Sobolev Type System

2016 ◽  
Vol 16 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Lu Chen ◽  
Zhao Liu ◽  
Guozhen Lu
Keyword(s):  
Lipschitz Continuity ◽  
Type System ◽  
Regularity Of Solutions ◽  
Sobolev Type ◽  
Sobolev Inequalities ◽  
Plane Method ◽  
Integral Forms ◽  
Special Cases ◽  
Formula Group ◽  
Moving Plane

AbstractHardy–Littlewood–Sobolev inequalities and the Hardy–Sobolev type system play an important role in analysis and PDEs. In this paper, we consider the very general weighted Hardy–Sobolev type system$u(x)=\int_{\mathbb{R}^{n}}\frac{1}{|x|^{\tau}|x-y|^{n-\alpha}|y|^{t}}f_{1}(u(y% ),v(y))dy,\quad v(x)=\int_{\mathbb{R}^{n}}\frac{1}{|x|^{t}|x-y|^{n-\alpha}|y|^% {\tau}}f_{2}(u(y),v(y))dy,$where$\displaystyle f_{1}(u(y),v(y))=\lambda_{1}u^{p_{1}}(y)+\mu_{1}v^{q_{1}}(y)+% \gamma_{1}u^{\alpha_{1}}(y)v^{\beta_{1}}(y),$$\displaystyle f_{2}(u(y),v(y))=\lambda_{2}u^{p_{2}}(y)+\mu_{2}v^{q_{2}}(y)+% \gamma_{2}u^{\alpha_{2}}(y)v^{\beta_{2}}(y).$Only the special cases when ${\gamma_{1}=\gamma_{2}=0}$ and one of ${\lambda_{i}}$ and ${\mu_{i}}$ is zero (for both ${i=1}$ and ${i=2}$) have been considered in the literature. We establish the integrability of the solutions to the above Hardy–Sobolev type system and the ${C^{\infty}}$ regularity of solutions to this system away from the origin, which improves significantly the Lipschitz continuity in most works in the literature. Moreover, we also use the moving plane method of [8] in integral forms developed in [6] to prove that each pair ${(u,v)}$ of positive solutions of the above integral system is radially symmetric and strictly decreasing about the origin.

Ultrastructural characteristics of sporidia of Ustilago

1989 ◽  
Vol 47 ◽  
pp. 786-787
Author(s):  
Patricia N. Hackney
Keyword(s):  
Electron Microscope ◽  
Type System ◽  
Tube Formation ◽  
High Voltage Electron Microscopy ◽  
Voltage Electron ◽  
University Of Wisconsin ◽  
Transmission Electron ◽  
Ustilago Hordei ◽  
Silene Alba ◽  
The University

Ustilago hordei and Ustilago violacea are yeast-like basidiomycete pathogens ofHordeum vulgare and Silene alba respectively. The mating type system in both species of Ustilago is bipolar, with alleles, A,a, (U.hordei) and a1, a2 (U.violacea) at a single locus. Haploid sporidia maintain the asexual phase by budding, while the sexual phase is initiated by conjugation tube formation between the mating types during budding and conjugation.For observation of budding, sporidia were prepared by culturing the four types on YEG (yeast extract glucose) broth for 24 hours. After centrifugation at 5000g cells were either left unmated or mated in a1/a2,A/a combinations. The sporidia were then mixed 1:1 with 4% agar and the resulting 1mm cubes fixed in 8% gluteraldehyde and post fixed in osmium tetroxide. After dehydration and embedding cubes were thin sectioned with a LKB ultratome and photographed in a Zeiss 9s transmission electron microscope or in an AE1 electron microscope of MK11 1MEV at the High Voltage Electron Microscopy Center of the University of Wisconsin-Madison.

Thermal comfort conditions in a classroom conditioned by a split-type system

2017 ◽  
Author(s):  
Rogério Vilain ◽  
Marcelo Pereira ◽  
Nathan Mendes ◽  
katia cordeiro ◽  
anastacio da silva junior
Keyword(s):  
Thermal Comfort ◽  
Type System

Modelling of impregnation of γ-alumina with cobalt and molybdenum salts. CoCl2-(NH4)2MoO4-γ-Al2O3 (aluminate type) system

1987 ◽  
Vol 52 (3) ◽  
pp. 663-671 ◽  
Author(s):  
Jiří Hanika ◽  
Vladimír Janoušek ◽  
Karel Sporka
Keyword(s):  
Effective Diffusion ◽  
Type System ◽  
Catalyst Particle ◽  
Ammonium Molybdate ◽  
Active Components ◽  
Model Set ◽  
Cobalt Dichloride ◽  
Set Up ◽  
Electron Microanalysis ◽  
Kinetics Of

Adsorption data for the impregnation of alumina with an aqueous solution of cobalt dichloride and ammonium molybdate were treated in terms of the Langmuir adsorption isotherm and compared with a mathematical model set up to describe the kinetics of simultaneous impregnation of a support by two components. The effective diffusion coefficients of the two components at 25 °C in a cylindrical particle of alumina were obtained. The validity of the model used was verified qualitatively by comparing the numerical results with the experimental time dependent concentration profiles of the active components in a catalyst particle, measured by electron microanalysis technique.

A Lightweight Formalism for Reference Lifetimes and Borrowing in Rust

2021 ◽  
Vol 43 (1) ◽  
pp. 1-73
Author(s):  
David J. Pearce
Keyword(s):  
Memory Management ◽  
Programming Model ◽  
Type System ◽  
Control Flow ◽  
Two Phase ◽  
Type Checking ◽  
Strong Focus ◽  
Reference Implementation ◽  
Two Phases ◽  
Complex Features

Rust is a relatively new programming language that has gained significant traction since its v1.0 release in 2015. Rust aims to be a systems language that competes with C/C++. A claimed advantage of Rust is a strong focus on memory safety without garbage collection. This is primarily achieved through two concepts, namely, reference lifetimes and borrowing . Both of these are well-known ideas stemming from the literature on region-based memory management and linearity / uniqueness . Rust brings both of these ideas together to form a coherent programming model. Furthermore, Rust has a strong focus on stack-allocated data and, like C/C++ but unlike Java, permits references to local variables. Type checking in Rust can be viewed as a two-phase process: First, a traditional type checker operates in a flow-insensitive fashion; second, a borrow checker enforces an ownership invariant using a flow-sensitive analysis. In this article, we present a lightweight formalism that captures these two phases using a flow-sensitive type system that enforces “ type and borrow safety .” In particular, programs that are type and borrow safe will not attempt to dereference dangling pointers. Our calculus core captures many aspects of Rust, including copy- and move-semantics, mutable borrowing, reborrowing, partial moves, and lifetimes. In particular, it remains sufficiently lightweight to be easily digested and understood and, we argue, still captures the salient aspects of reference lifetimes and borrowing. Furthermore, extensions to the core can easily add more complex features (e.g., control-flow, tuples, method invocation). We provide a soundness proof to verify our key claims of the calculus. We also provide a reference implementation in Java with which we have model checked our calculus using over 500B input programs. We have also fuzz tested the Rust compiler using our calculus against 2B programs and, to date, found one confirmed compiler bug and several other possible issues.

scholarly journals B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces

2020 ◽  
Vol 18 (1) ◽  
pp. 715-730
Author(s):  
Javanshir J. Hasanov ◽  
Rabil Ayazoglu ◽  
Simten Bayrakci
Keyword(s):  
Differential Operator ◽  
Singular Integral ◽  
Riesz Potential ◽  
Type Theorem ◽  
Integral Operators ◽  
Morrey Spaces ◽  
Singular Integral Operators ◽  
Sobolev Type ◽  
Riesz Potentials ◽  
Bessel Differential Operator

Abstract In this article, we consider the Laplace-Bessel differential operator {\Delta }_{{B}_{k,n}}=\mathop{\sum }\limits_{i=1}^{k}\left(\frac{{\partial }^{2}}{\partial {x}_{i}^{2}}+\frac{{\gamma }_{i}}{{x}_{i}}\frac{\partial }{\partial {x}_{i}}\right)+\mathop{\sum }\limits_{i=k+1}^{n}\frac{{\partial }^{2}}{\partial {x}_{i}^{2}},{\gamma }_{1}\gt 0,\ldots ,{\gamma }_{k}\gt 0. Furthermore, we define B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials associated with the Laplace-Bessel differential operator. Moreover, we also obtain the boundedness of the B-maximal commutator {M}_{b,\gamma } and the commutator {[}b,{A}_{\gamma }] of the B-singular integral operator and Hardy-Littlewood-Sobolev-type theorem for the commutator {[}b,{I}_{\alpha ,\gamma }] of the B-Riesz potential on B-Morrey spaces {L}_{p,\lambda ,\gamma } , when b\in {\text{BMO}}_{\gamma } .

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