scholarly journals Pathological Phenomena in Denjoy–Carleman Classes

2016 ◽  
Vol 68 (1) ◽  
pp. 88-108 ◽  
Author(s):  
Ethan Y. Jaffe
Keyword(s):  
Smooth Function ◽  
Convex Sequence ◽  
Carleman Class

AbstractLet M denote a Denjoy–Carleman class of ∞ functions (for a given logarithmically-convex sequence M = (Mn)). We construct: (1) a function in M((−1, 1)) that is nowhere in any smaller class; (2) a function on ℝ that is formally M at every point, but not in M (ℝ); (3) (under the assumption of quasianalyticity) a smooth function on ℝp (p ≥ 2) that is M on every M curve, but not inM (ℝp).

Periodic Solutions of Non-autonomous Allen–Cahn Equations Involving Fractional Laplacian

2020 ◽  
Vol 20 (3) ◽  
pp. 725-737 ◽  
Author(s):  
Zhenping Feng ◽  
Zhuoran Du
Keyword(s):  
Variational Methods ◽  
Periodic Solutions ◽  
Smooth Function ◽  
Fractional Laplacian ◽  
Type Inequality ◽  
Energy Functional ◽  
Large Integer ◽  
Existence Of Periodic Solutions ◽  
Double Well Potential

AbstractWe consider periodic solutions of the following problem associated with the fractional Laplacian: {(-\partial_{xx})^{s}u(x)+\partial_{u}F(x,u(x))=0} in {\mathbb{R}}. The smooth function {F(x,u)} is periodic about x and is a double-well potential with respect to u with wells at {+1} and -1 for any {x\in\mathbb{R}}. We prove the existence of periodic solutions whose periods are large integer multiples of the period of F about the variable x by using variational methods. An estimate of the energy functional, Hamiltonian identity and Modica-type inequality for periodic solutions are also established.

scholarly journals Consensus Based Platoon Algorithm for Velocity-Measurement-Absent Vehicles with Actuator Saturation

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Maode Yan ◽  
Ye Tang ◽  
Panpan Yang ◽  
Lei Zuo
Keyword(s):  
Adaptive Control ◽  
Control Strategy ◽  
Smooth Function ◽  
Velocity Measurement ◽  
Actuator Saturation ◽  
Sharp Corner ◽  
Control Approach ◽  
Input Signals ◽  
Vehicle Platoon ◽  
Novel Algorithm

We investigate the vehicle platoon problems, where the actuator saturation and absent velocity measurement are taken into consideration. Firstly, a novel algorithm, where a smooth function is introduced to deal with the sharp corner of the input signals, is proposed for a group of vehicles with actuator saturation by using the consensus theory. Secondly, by applying an auxiliary system for the followers to estimate the velocities, a control strategy for the vehicle platoon with actuator saturation and absent velocity measurement is designed via the adaptive control approach. Finally, numerical simulations are provided to illustrate the effectiveness of the proposed approaches.

An improved calibration curve between soil pH measured in waterand CaCl2

Soil Research ◽  
2002 ◽  
Vol 40 (8) ◽  
pp. 1399 ◽  
Author(s):  
B. L. Henderson ◽  
E. N. Bui
Keyword(s):  
Statistical Analysis ◽  
Water Resources ◽  
Soil Ph ◽  
Calibration Curve ◽  
Smooth Function ◽  
Degrees Of Freedom ◽  
Additive Model ◽  
Smoothing Spline ◽  
Look Up Table ◽  
6 Degrees Of Freedom

A new pH water to pH CaCl2 calibration curve was derived from data pooled from 2 National Land and Water Resources Audit projects. A total of 70465 observations with both pH in water and pH in CaCl2 were available for statistical analysis. An additive model for pH in CaCl2 was fitted from a smooth function of pH in water created by a smoothing spline with 6 degrees of freedom. This model appeared stable outside the range of the data and performed well (R2 = 96.2, s = 0.24). The additive model for conversion of pHw to pHCa is sigmoidal over the range of pH 2.5 to 10.5 and is similar in shape to earlier models. Using this new model, a look-up table for converting pHw to pHCa was created.

scholarly journals Dual pairs of sequence spaces

2001 ◽  
Vol 28 (1) ◽  
pp. 9-23 ◽  
Author(s):  
Johann Boos ◽  
Toivo Leiger
Keyword(s):  
Sequence Space ◽  
Dual Pair ◽  
General Concept ◽  
Sequence Spaces ◽  
Dual Pairs ◽  
The Third ◽  
Convex Sequence ◽  
Starting Point ◽  
Alpha Beta ◽  
Locally Convex

The paper aims to develop for sequence spacesEa general concept for reconciling certain results, for example inclusion theorems, concerning generalizations of the Köthe-Toeplitz dualsE×(×∈{α,β})combined with dualities(E,G),G⊂E×, and theSAK-property (weak sectional convergence). TakingEβ:={(yk)∈ω:=𝕜ℕ|(ykxk)∈cs}=:Ecs, wherecsdenotes the set of all summable sequences, as a starting point, then we get a general substitute ofEcsby replacingcsby any locally convex sequence spaceSwith sums∈S′(in particular, a sum space) as defined by Ruckle (1970). This idea provides a dual pair(E,ES)of sequence spaces and gives rise for a generalization of the solid topology and for the investigation of the continuity of quasi-matrix maps relative to topologies of the duality(E,Eβ). That research is the basis for general versions of three types of inclusion theorems: two of them are originally due to Bennett and Kalton (1973) and generalized by the authors (see Boos and Leiger (1993 and 1997)), and the third was done by Große-Erdmann (1992). Finally, the generalizations, carried out in this paper, are justified by four applications with results around different kinds of Köthe-Toeplitz duals and related section properties.

scholarly journals A Schwarz lemma for hyperbolic harmonic mappings in the unit ball

2021 ◽  
Vol 127 (3) ◽  
Author(s):  
Jiaolong Chen ◽  
David Kalaj
Keyword(s):  
Unit Ball ◽  
Explicit Form ◽  
Smooth Function ◽  
Harmonic Mapping ◽  
Sharp Inequality ◽  
Schwarz Lemma ◽  
Harmonic Mappings ◽  
Manuscripta Math ◽  
Sharp Constant ◽  
Mapping Theory

Assume that $p\in [1,\infty ]$ and $u=P_{h}[\phi ]$, where $\phi \in L^{p}(\mathbb{S}^{n-1},\mathbb{R}^n)$ and $u(0) = 0$. Then we obtain the sharp inequality $\lvert u(x) \rvert \le G_p(\lvert x \rvert )\lVert \phi \rVert_{L^{p}}$ for some smooth function $G_p$ vanishing at $0$. Moreover, we obtain an explicit form of the sharp constant $C_p$ in the inequality $\lVert Du(0)\rVert \le C_p\lVert \phi \rVert \le C_p\lVert \phi \rVert_{L^{p}}$. These two results generalize and extend some known results from the harmonic mapping theory (D. Kalaj, Complex Anal. Oper. Theory 12 (2018), 545–554, Theorem 2.1) and the hyperbolic harmonic theory (B. Burgeth, Manuscripta Math. 77 (1992), 283–291, Theorem 1).

scholarly journals Non-Smooth Geodesic Flows and Classical Mechanics

1969 ◽  
Vol 12 (2) ◽  
pp. 209-212 ◽  
Author(s):  
J. E. Marsden
Keyword(s):  
Phase Space ◽  
Kinetic Energy ◽  
Hamiltonian Systems ◽  
Configuration Space ◽  
Smooth Function ◽  
Cotangent Bundle ◽  
Classical Mechanics ◽  
Riemannian Metric ◽  
Geodesic Flows ◽  
Image Position

As is well known, there is an intimate connection between geodesic flows and Hamiltonian systems. In fact, if g is a Riemannian, or pseudo-Riemannian metric on a manifold M (we think of M as q-space or the configuration space), we may define a smooth function Tg on the cotangent bundle T*M (q-p-space, or the phase space). This function is the kinetic energy of q, and locally is given by

On the Smoothness of General Kernels

1966 ◽  
Vol 18 ◽  
pp. 443-448
Author(s):  
Jose Barros-Neto
Keyword(s):  
Smooth Function ◽  
General Distribution ◽  
Distribution Kernel ◽  
Set Of Points

In (3, §2), the writer and F. E. Browder stated briefly, without proof, some results concerning general distribution kernels. It is our aim here to prove and complete those results.The terminology and notations are introduced in §1.In §2 we define the notion of domain of dependence with respect to the kernel Kx,y (Definition 1) as well as the notion of smoothness of a distribution kernel at a point (Definition 2). Theorem 1 states that the set of points, where the distribution kernel is smooth, is open and the kernel is a smooth function in this set. Theorems 2 and 3 are the converse of Theorem 1.

Reduced Sobolev Inequalities

1988 ◽  
Vol 31 (2) ◽  
pp. 159-167 ◽  
Author(s):  
R. A. Adams
Keyword(s):  
Compact Support ◽  
Sobolev Inequality ◽  
Smooth Function ◽  
Sobolev Inequalities ◽  
Partial Derivatives ◽  
Derivatives Of

AbstractThe Sobolev inequality of order m asserts that if p ≧ 1, mp < n and 1/q = 1/p — m/n, then the Lq-norm of a smooth function with compact support in Rn is bounded by a constant times the sum of the Lp-norms of the partial derivatives of order m of that function. In this paper we show that that sum may be reduced to include only the completely mixed partial derivatives or order m, and in some circumstances even fewer partial derivatives.

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