Geometric conditions of differentiability for a regular locally Lipschitz function

1998 ◽  
Vol 14 (2) ◽  
pp. 209-222
Author(s):  
Shi Shuzhong ◽  
Wang Bingwu
Keyword(s):  
Lipschitz Function ◽  
Locally Lipschitz Function ◽  
Geometric Conditions ◽  
Locally Lipschitz

Strict lower semicontinuity of the level sets and invexity of a locally Lipschitz function

1995 ◽  
Vol 87 (3) ◽  
pp. 579-594 ◽  
Author(s):  
T. D. Phuong ◽  
P. H. Sach ◽  
N. D. Yen
Keyword(s):  
Lipschitz Function ◽  
Lower Semicontinuity ◽  
Level Sets ◽  
Locally Lipschitz Function ◽  
Locally Lipschitz

scholarly journals Relaxed submonotone mappings

2003 ◽  
Vol 2003 (1) ◽  
pp. 19-31 ◽  
Author(s):  
Tzanko Donchev ◽  
Pando Georgiev
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Lipschitz Function ◽  
Clarke Subdifferential ◽  
Monotone Mappings ◽  
Locally Lipschitz Function ◽  
Residual Subset ◽  
Locally Lipschitz ◽  
Whole Space ◽  
The Clarke Subdifferential

The notions ofrelaxed submonotoneandrelaxed monotonemappings in Banach spaces are introduced and many of their properties are investigated. For example, the Clarke subdifferential of a locally Lipschitz function in a separable Banach space is relaxed submonotone on a residual subset. For example, it is shown that this property need not be valid on the whole space. We prove, under certain hypotheses, the surjectivity of the relaxed monotone mappings.

scholarly journals Locally Lipschitz functions are generically pseudo-regular on separable Banach spaces

1993 ◽  
Vol 47 (2) ◽  
pp. 205-212 ◽  
Author(s):  
J.R. Giles ◽  
Scott Sciffer
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Lipschitz Function ◽  
Separable Banach Space ◽  
Locally Lipschitz Function ◽  
Locally Lipschitz Functions ◽  
Dini Derivative ◽  
Locally Lipschitz ◽  
Set Of Points ◽  
Clarke Derivative

For a locally Lipschitz function on a separable Banach space the set of points of Gâteaux differentiability is dense but not necessarily residual. However, the set of points where the upper Dini derivative and the Clarke derivative agree is residual. It follows immediately that the set of points of intermediate differentiability is also residual and the set of points where the function is Gâteaux but not strictly differentiable is of the first category.

BORNOLOGIES AND LOCALLY LIPSCHITZ FUNCTIONS

2014 ◽  
Vol 90 (2) ◽  
pp. 257-263 ◽  
Author(s):  
GERALD BEER ◽  
M. I. GARRIDO
Keyword(s):  
Metric Space ◽  
Lipschitz Function ◽  
Lipschitz Functions ◽  
Locally Lipschitz Function ◽  
Locally Lipschitz Functions ◽  
The Family ◽  
Locally Lipschitz

AbstractLet$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\langle X,d \rangle $be a metric space. We characterise the family of subsets of$X$on which each locally Lipschitz function defined on$X$is bounded, as well as the family of subsets on which each member of two different subfamilies consisting of uniformly locally Lipschitz functions is bounded. It suffices in each case to consider real-valued functions.

Optimality conditions for maximizing a locally Lipschitz function

Optimization ◽  
2005 ◽  
Vol 54 (4-5) ◽  
pp. 377-389 ◽  
Author(s):  
Joydeep Dutta
Keyword(s):  
Optimality Conditions ◽  
Lipschitz Function ◽  
Locally Lipschitz Function ◽  
Locally Lipschitz

Geometric Factor in Skeletal Reactions of 2-Methylpentane on Stepped Surfaces of Platinum Catalyst

1992 ◽  
Vol 57 (10) ◽  
pp. 2012-2020
Author(s):  
Vladimír Hejtmánek
Keyword(s):  
Active Sites ◽  
Platinum Catalyst ◽  
Point Of View ◽  
Geometric Factor ◽  
Published Data ◽  
Stepped Surfaces ◽  
Geometric Conditions ◽  
Surface Intermediates ◽  
Isomerization Mechanism

The role of geometric factor in the course of skeletal reactions (isomerization, hydrogenolysis) of 2-methylpentane on stepped (119), (557) and reconstructed R(557) surfaces of single crystals of platinum was evaluated with computer designed models. These calculations were compared with reported experimental data. It was found by analysis of geometric conditions that there are accessible active ensembles on double step of the reconstructed R(557) surface. In addition, these active sites are unsaturated in their coordination sphere and thus catalytically effective. This finding is consistent with published data, confirming higher catalytic activity of this surface. The various pathways of Bond Shift isomerization mechanism of 2-methylpentane from the point of view of steric demands of surface intermediates on differently located ensembles are discussed, too.

scholarly journals No Infimum Gap and Normality in Optimal Impulsive Control Under State Constraints

2021 ◽  
Author(s):  
Giovanni Fusco ◽  
Monica Motta
Keyword(s):  
Original Problem ◽  
State Constraints ◽  
Impulsive Control ◽  
Sufficient Conditions ◽  
Continuous Data ◽  
Lipschitz Continuous ◽  
Extended Sense ◽  
Unbounded Controls ◽  
Locally Lipschitz ◽  
Optimal Impulsive Control

AbstractIn this paper we consider an impulsive extension of an optimal control problem with unbounded controls, subject to endpoint and state constraints. We show that the existence of an extended-sense minimizer that is a normal extremal for a constrained Maximum Principle ensures that there is no gap between the infima of the original problem and of its extension. Furthermore, we translate such relation into verifiable sufficient conditions for normality in the form of constraint and endpoint qualifications. Links between existence of an infimum gap and normality in impulsive control have previously been explored for problems without state constraints. This paper establishes such links in the presence of state constraints and of an additional ordinary control, for locally Lipschitz continuous data.

scholarly journals Experimental Study on Dynamic Combustion Characteristics in Swirl-Stabilized Combustors

Energies ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 1609
Author(s):  
Donghyun Hwang ◽  
Kyubok Ahn
Keyword(s):  
Experimental Study ◽  
Heat Release ◽  
Dynamic Pressure ◽  
Combustion Instability ◽  
Flame Structure ◽  
Pressure Sensors ◽  
Necessary Condition ◽  
Rayleigh Criterion ◽  
Pressure Oscillations ◽  
Geometric Conditions

An experimental study was performed to investigate the combustion instability characteristics of swirl-stabilized combustors. A premixed gas composed of ethylene and air was burned under various flow and geometric conditions. Experiments were conducted by changing the inlet mean velocity, equivalence ratio, swirler vane angle, and combustor length. Two dynamic pressure sensors, a hot-wire anemometer, and a photomultiplier tube were installed to detect the pressure oscillations, velocity perturbations, and heat release fluctuations in the inlet and combustion chambers, respectively. An ICCD camera was used to capture the time-averaged flame structure. The objective was to understand the relationship between combustion instability and the Rayleigh criterion/the flame structure. When combustion instability occurred, the pressure oscillations were in-phase with the heat release oscillations. Even if the Rayleigh criterion between the pressure and heat release oscillations was satisfied, stable combustion with low pressure fluctuations was possible. This was explained by analyzing the dynamic flow and combustion data. The root-mean-square value of the heat release fluctuations was observed to predict the combustion instability region better than that of the inlet velocity fluctuations. The bifurcation of the flame structure was a necessary condition for combustion instability in this combustor. The results shed new insight into combustion instability in swirl-stabilized combustors.

scholarly journals Geometric conditions of a two-link-and-six muscle structure based on internal force stability

2020 ◽  
Vol 7 (1) ◽  
Author(s):  
Hiroaki Ochi ◽  
Hitoshi Kino ◽  
Kenji Tahara ◽  
Yuki Matsutani
Keyword(s):  
Internal Force ◽  
Muscle Structure ◽  
Geometric Conditions

Geometric conditions for G3 continuity of surfaces

1997 ◽  
Vol 14 (8) ◽  
pp. 719-729 ◽  
Author(s):  
Holger Theisel
Keyword(s):  
Geometric Conditions
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