On the expressibility of piecewise-linear continuous functions as the difference of two piecewise-linear convex functions

1986 ◽  
pp. 118-134 ◽  
Author(s):  
D. Melzer
Keyword(s):  
Convex Functions ◽  
Piecewise Linear ◽  
Continuous Functions ◽  
The Difference

Successive coefficients of close-to-convex functions

2020 ◽  
Vol 32 (5) ◽  
pp. 1131-1141 ◽  
Author(s):  
Paweł Zaprawa
Keyword(s):  
Real Axis ◽  
Convex Functions ◽  
Positive Direction ◽  
The Real ◽  
Coefficient Problems ◽  
The Difference

AbstractIn this paper we discuss coefficient problems for functions in the class {{\mathcal{C}}_{0}(k)}. This family is a subset of {{\mathcal{C}}}, the class of close-to-convex functions, consisting of functions which are convex in the positive direction of the real axis. Our main aim is to find some bounds of the difference of successive coefficients depending on the fixed second coefficient. Under this assumption we also estimate {|a_{n+1}|-|a_{n}|} and {|a_{n}|}. Moreover, it is proved that {\operatorname{Re}\{a_{n}\}\geq 0} for all {f\in{\mathcal{C}}_{0}(k)}.

Refinements of the majorization-type inequalities via green and fink identities and related results

2018 ◽  
Vol 68 (4) ◽  
pp. 773-788 ◽  
Author(s):  
Sadia Khalid ◽  
Josip Pečarić ◽  
Ana Vukelić
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Convex Functions ◽  
Type Inequality ◽  
Cauchy Type ◽  
Linear Functionals ◽  
Ostrowski’S Inequality ◽  
New Family ◽  
Exponentially Convex Functions ◽  
The Difference

Abstract In this work, the Green’s function of order two is used together with Fink’s approach in Ostrowski’s inequality to represent the difference between the sides of the Sherman’s inequality. Čebyšev, Grüss and Ostrowski-type inequalities are used to obtain several bounds of the presented Sherman-type inequality. Further, we construct a new family of exponentially convex functions and Cauchy-type means by looking to the linear functionals associated with the obtained inequalities.

CONTINUOUS PIECEWISE LINEAR FUNCTIONS

2005 ◽  
Vol 10 (1) ◽  
pp. 77-99 ◽  
Author(s):  
CHARALAMBOS D. ALIPRANTIS ◽  
DAVID HARRIS ◽  
RABEE TOURKY
Keyword(s):  
Piecewise Linear ◽  
Continuous Functions ◽  
Linear Functions ◽  
Dimensional Euclidean Space ◽  
Piecewise Linear Functions ◽  
Space Of Continuous Functions ◽  
One Dimensional ◽  
Mathematical Background ◽  
Linear Regressions ◽  
Special Case

The paper studies the function space of continuous piecewise linear functions in the space of continuous functions on them-dimensional Euclidean space. It also studies the special case of one dimensional continuous piecewise linear functions. The study is based on the theory of Riesz spaces that has many applications in economics. The work also provides the mathematical background to its sister paper Aliprantis, Harris, and Tourky (2006), in which we estimate multivariate continuous piecewise linear regressions by means of Riesz estimators, that is, by estimators of the the Boolean formwhereX=(X1,X2, …,Xm) is some random vector, {Ej}j∈Jis a finite family of finite sets.

Automatic techniques for economic computer computation of continuous functions

SIMULATION ◽  
1968 ◽  
Vol 11 (1) ◽  
pp. 37-48 ◽  
Keyword(s):  
High Speed ◽  
Piecewise Linear ◽  
Continuous Functions ◽  
Computational Error ◽  
Mathematical Function ◽  
Chemical Plants ◽  
Approximation Techniques ◽  
Allowable Error ◽  
The Cost ◽  
Error Envelope

Methods are presented for the automatic preparation of functions of one or more variables for economical calculation by high-speed digital computers. The cost of calculation is considered according to the factors of number of functions, complexity, requirements for precision, and the frequency with which functions are to be calculated. Contrary to classic approaches, con sideration is not given to minimizing computational error for its own sake. On the contrary, the maximum allowable error may be sought in order to minimize computational costs. In this respect, each function is represented by an error envelope that specifies the required limits of computational precision. It is the error envelope rather than the function itself which is dealt with. The approximation techniques dealt with in this paper are limited to piecewise linear ap proximation of functions of one or two independent variables. Projects requiring the maintaining and computation of large quantities of continuous functions are fre quently to be found in industry and research; for example, in the simulation of real-time processes— aircraft flight and flight trainer simulations, simula tion for control and regulation of continuous pro cesses as in chemical plants, weather calculations, radiation studies, etc. In addition, computer service centers, providing computational services to many users, may extend the range and effectiveness of their mathematical function program library by the use of the economical com putational methods of this paper.

scholarly journals Continuous functions of bounded nth variation

1969 ◽  
Vol 16 (3) ◽  
pp. 205-214
Author(s):  
Gavin Brown
Keyword(s):  
Continuous Function ◽  
Positive Integer ◽  
Decomposition Theorem ◽  
Continuous Functions ◽  
Unit Interval ◽  
Jordan Decomposition ◽  
The Real ◽  
Elementary Construction ◽  
The Difference

Let n be a positive integer. We give an elementary construction for the nth variation, Vn(f), of a real valued continuous function f and prove an analogue of the classical Jordan decomposition theorem. In fact, let C[0, 1] denote the real valued continuous functions on the closed unit interval, let An denote the semi-algebra of non-negative functions in C[0, 1] whose first n differences are non-negative, and let Sn denote the difference algebra An - An. We show that Sn is precisely that subset of C[0, 1] on which Vn(f)<∞. (Theorem 1).

scholarly journals Well-posedness of the difference schemes of the high order of accuracy for elliptic equations

2006 ◽  
Vol 2006 ◽  
pp. 1-12 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Pavel E. Sobolevskiĭ
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Continuous Functions ◽  
High Order ◽  
Difference Schemes ◽  
Well Posedness ◽  
Smooth Functions ◽  
Order Of Accuracy ◽  
High Order Of Accuracy ◽  
The Difference

It is well known the differential equation−u″(t)+Au(t)=f(t)(−∞<t<∞)in a general Banach spaceEwith the positive operatorAis ill-posed in the Banach spaceC(E)=C((−∞,∞),E)of the bounded continuous functionsϕ(t)defined on the whole real line with norm‖ϕ‖C(E)=sup⁡−∞<t<∞‖ϕ(t)‖E. In the present paper we consider the high order of accuracy two-step difference schemes generated by an exact difference scheme or by Taylor's decomposition on three points for the approximate solutions of this differential equation. The well-posedness of these difference schemes in the difference analogy of the smooth functions is obtained. The exact almost coercive inequality for solutions inC(τ,E)of these difference schemes is established.

Simple anti-swing feedback control for a gantry crane

Robotica ◽  
2003 ◽  
Vol 21 (6) ◽  
pp. 655-666 ◽  
Author(s):  
Yannick Aoustin ◽  
Alexander Formal'sky
Keyword(s):  
Optimal Control ◽  
Piecewise Linear ◽  
Continuous Functions ◽  
Time Optimal Control ◽  
Gantry Crane ◽  
Control Law ◽  
Reference Trajectory ◽  
Controlling Parameter ◽  
Input Signals ◽  
Time Optimal

We propose a simple quasi time optimal control law for a gantry crane with a payload. The force applied to the trolley is a controlling parameter. The control law consists of two parts: a feedforward term and a trolley position and velocity feedback term.Initially, we synthesize the feedforward term and the corresponding reference trajectory by computing the time optimal control for the system mass center. The computed optimal control is a discontinuous function of time with several switching time instants. Undesirable large vibrations due to the payload sway appear under this control. Therefore, we transform this control, replacing its jumps by the piecewise linear continuous functions. The computed feedforward term and the reference trajectory are used as input signals of the PD-controller.

scholarly journals Piecewise linear discretization of Symbolic Implicit Monte Carlo radiation transport in the difference formulation

2006 ◽  
Vol 220 (1) ◽  
pp. 471-497 ◽  
Author(s):  
Eugene D. Brooks ◽  
Abraham Szőke ◽  
Jayson D.L. Peterson
Keyword(s):  
Monte Carlo ◽  
Radiation Transport ◽  
Piecewise Linear ◽  
The Difference
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