Norm inequalities involving derivatives

1978 ◽  
Vol 82 (1-2) ◽  
pp. 51-70 ◽  
Author(s):  
W. D. Evans ◽  
A. Zettl
Keyword(s):  
General Class ◽  
New Method ◽  
Elementary Operator ◽  
Best Constant ◽  
New Approach ◽  
Theoretic Proof ◽  
General Class Of Functions ◽  
Image Position ◽  
Class Of Functions

SynopisisA new method for studying inequalities of the type ‖y(r)‖2<ε‖Sk−ry(k)‖2 + K(ε)‖S−ry‖2 and ‖y′‖2≦ Kp(S)‖Sy″‖ ‖S−1y‖ is presented here. With this new approach we obtain new and far reaching extensions of previously known inequalities of this sort as well as simpler proofs of the known cases. In addition we obtain an inequality of type ‖Sy′‖<ε‖(Sy′)′‖ + K(ε)‖y‖ for a general class of functions S. Also we give an elementary operator-theoretic proof of Everitt's characterization of the best constant as well as all cases of equality for

Heteroepitaxy and Characterization of Low-Dislocation-Density GaN on Periodically Grooved Substrates

2000 ◽  
Vol 639 ◽  
Author(s):  
T. Detchprohm ◽  
M. Yano ◽  
R. Nakamura ◽  
S. Sano ◽  
S. Mochiduki ◽  
...  
Keyword(s):  
Dislocation Density ◽  
New Method ◽  
Growth Technique ◽  
New Approach ◽  
Density Area ◽  
Low Dislocation Density ◽  
Dislocation Densities ◽  
Movpe Growth

ABSTRACTWe have developed a new method to prepare low-dislocation-density GaN by using periodically grooved substrates in a conventional MOVPE growth technique. This new approach was demonstrated for GaN grown on periodically grooved α-Al2O3(0001), 6H-SiC(0001)Si and Si(111) substrates. Dislocation densities were 2×107 cm−2 in low-dislocation-density area.

scholarly journals Concentration of the error between a function and its polynomial of best uniform approximation

1998 ◽  
Vol 41 (3) ◽  
pp. 447-463 ◽  
Author(s):  
Maurice Hasson
Keyword(s):  
Uniform Approximation ◽  
Algebraic Polynomial ◽  
General Class ◽  
Supremum Norm ◽  
Best Uniform Approximation ◽  
Class A ◽  
Image Position ◽  
Class Of Functions

Let f be a continuous real valued function defined on [−1, 1] and let En(f) denote the degree of best uniform approximation to f by algebraic polynomial of degree at most n. The supremum norm on [a, b] is denoted by ∥.∥[a, b] and the polynomial of degree n of best uniform approximation is denoted by Pn. We find a class of functions f such that there exists a fixed a ∈(−1, 1) with the following propertyfor some positive constants C and N independent of n. Moreover the sequence is optimal in the sense that if is replaced by then the above inequality need not hold no matter how small C > 0 is chosen.We also find another, more general class a functions f for whichinfinitely often.

A note on the age-dependent branching process with immigration

1975 ◽  
Vol 12 (01) ◽  
pp. 130-134 ◽  
Author(s):  
Norman Kaplan
Keyword(s):  
General Class ◽  
Branching Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Direct Generalization ◽  
Age Dependent ◽  
General Class Of Functions ◽  
Branching Process With Immigration ◽  
Necessary And Sufficient ◽  
Class Of Functions

Let {Z(t)}t0be an age-dependent branching process with immigration. For a general class of functions Φ(x), a necessary and sufficient condition is given for whenE{Φ (Z(t))} &lt;∞. This result is a direct generalization of a theorem proven for the branching process without immigration.

The completeness of the first-order functional calculus

1949 ◽  
Vol 14 (3) ◽  
pp. 159-166 ◽  
Author(s):  
Leon Henkin
Keyword(s):  
Functional Calculus ◽  
Propositional Calculus ◽  
Higher Order ◽  
New Method ◽  
Completeness Proof ◽  
New Approach ◽  
First Order ◽  
Formal Systems ◽  
The Subject ◽  
Image Position

Although several proofs have been published showing the completeness of the propositional calculus (cf. Quine (1)), for the first-order functional calculus only the original completeness proof of Gödel (2) and a variant due to Hilbert and Bernays have appeared. Aside from novelty and the fact that it requires less formal development of the system from the axioms, the new method of proof which is the subject of this paper possesses two advantages. In the first place an important property of formal systems which is associated with completeness can now be generalized to systems containing a non-denumerable infinity of primitive symbols. While this is not of especial interest when formal systems are considered as logics—i.e., as means for analyzing the structure of languages—it leads to interesting applications in the field of abstract algebra. In the second place the proof suggests a new approach to the problem of completeness for functional calculi of higher order. Both of these matters will be taken up in future papers.The system with which we shall deal here will contain as primitive symbolsand certain sets of symbols as follows:(i) propositional symbols (some of which may be classed as variables, others as constants), and among which the symbol “f” above is to be included as a constant;(ii) for each number n = 1, 2, … a set of functional symbols of degree n (which again may be separated into variables and constants); and(iii) individual symbols among which variables must be distinguished from constants. The set of variables must be infinite.

scholarly journals Hydropyrolysis: Implications for Radiocarbon Pretreatment and Characterization of Black Carbon

Radiocarbon ◽  
2010 ◽  
Vol 52 (3) ◽  
pp. 1336-1350 ◽  
Author(s):  
P L Ascough ◽  
M I Bird ◽  
W Meredith ◽  
R E Wood ◽  
C E Snape ◽  
...  
Keyword(s):  
Black Carbon ◽  
Radiocarbon Dating ◽  
Elevated Temperatures ◽  
New Method ◽  
Humic Material ◽  
New Approach ◽  
Pyrogenic Carbon ◽  
Organic Components ◽  
Diagenetic Alteration

Charcoal is the result of natural and anthropogenic burning events, when biomass is exposed to elevated temperatures under conditions of restricted oxygen. This process produces a range of materials, collectively known as pyrogenic carbon, the most inert fraction of which is known as black carbon (BC). BC degrades extremely slowly and is resistant to diagenetic alteration involving the addition of exogenous carbon, making it a useful target substance for radiocarbon dating particularly of more ancient samples, where contamination issues are critical. We present results of tests using a new method for the quantification and isolation of BC, known as hydropyrolysis (hypy). Results show controlled reductive removal of non-BC organic components in charcoal samples, including lignocellulosic and humic material. The process is reproducible and rapid, making hypy a promising new approach not only for isolation of purified BC for 14C measurement but also in quantification of different labile and resistant sample C fractions.

scholarly journals Fairness criteria for allocating scarce resources

2020 ◽  
Vol 14 (6) ◽  
pp. 1533-1541
Author(s):  
Bismark Singh
Keyword(s):  
Optimization Model ◽  
General Class ◽  
Fair Allocation ◽  
Scarce Resources ◽  
Multiple Resources ◽  
Multiple Users ◽  
Special Cases ◽  
General Class Of Functions ◽  
Class Of Functions

Abstract We develop an optimization model to provide a fair allocation of multiple resources to multiple users. All resources might not be suitable to all users. We develop a notion of fairness, and then provide a general class of functions achieving it. Next, we develop more restricted notions of fairness—special cases of which exist in literature. Finally, we distinguish between scarce and abundant resources, and show that if a resource is abundant, all users seeking it achieve the maximum possible coverage.

On the “Preconditioning” Function Used in Planewave DFT Calculations and its Generalization

2015 ◽  
Vol 18 (1) ◽  
pp. 167-179 ◽  
Author(s):  
Yunkai Zhou ◽  
James R. Chelikowsky ◽  
Xingyu Gao ◽  
Aihui Zhou
Keyword(s):  
Dft Calculations ◽  
General Formula ◽  
Significant Role ◽  
General Class ◽  
Order Approximation ◽  
Higher Order ◽  
Approximation Accuracy ◽  
General Class Of Functions ◽  
Class Of Functions

AbstractThe Teter, Payne, and Allan “preconditioning” function plays a significant role in planewave DFT calculations. This function is often called the TPA preconditioner. We present a detailed study of this “preconditioning” function. We develop a general formula that can readily generate a class of “preconditioning” functions. These functions have higher order approximation accuracy and fulfill the two essential “preconditioning” purposes as required in planewave DFT calculations. Our general class of functions are expected to have applications in other areas.

scholarly journals ON THE NUMBER OF LIMIT CYCLES IN PIECEWISE-LINEAR LIÉNARD SYSTEMS

2005 ◽  
Vol 15 (04) ◽  
pp. 1417-1422 ◽  
Author(s):  
A. TONNELIER
Keyword(s):  
Limit Cycles ◽  
General Class ◽  
Piecewise Linear ◽  
Liénard System ◽  
Liénard Systems ◽  
Lienard System ◽  
General Class Of Functions ◽  
Class Of Functions

In a previous paper [Tonnelier, 2002] we conjectured that a Liénard system of the form ẋ = p(x) - y, ẏ = x where p is piecewise linear on n + 1 intervals has up to 2n limit cycles. We construct here a general class of functions p satisfying this conjecture. Limit cycles are obtained from the bifurcation of the linear center.

On a problem of Chowla and some related problems

1936 ◽  
Vol 32 (4) ◽  
pp. 530-540 ◽  
Author(s):  
Paul Erdös
Keyword(s):  
General Class ◽  
Number Of Divisors ◽  
General Class Of Functions ◽  
Class Of Functions

Let d(m) denote the number of divisors of the integer m. Chowla has conjectured that the integers for which d(m + 1) > d(m) have density ½. In this paper I prove and generalize this conjecture. I prove in § 1 a corresponding result for a general class of functions f(m), and in § 2 the result for d(m) which is not included among the f(m). I employ the method used in my paper: “On the density of some sequences of numbers.”

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