scholarly journals A criterion for differentiability

1956 ◽  
Vol 40 ◽  
pp. 8-11
Author(s):  
A. M. Macbeath
Keyword(s):  
Real Number ◽  
Number System ◽  
The Other ◽  
Alternative Methods ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Explicit Mention ◽  
Necessary And Sufficient ◽  
Real Number System

There are two alternative methods of defining the concept of “convergence” of a sequence, one involving explicit mention of the limit, the other (Cauchy's condition) giving a necessary and sufficient condition in terms of the elements of the sequence only. The two definitions are equivalent, because of the property of completeness of the real number system.

scholarly journals Equipartitioning and balancing points of polygons

Pythagoras ◽  
2010 ◽  
Vol 0 (71) ◽  
Author(s):  
Shunmugam Pillay ◽  
Poobhalan Pillay
Keyword(s):  
General Theorem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Sufficient Condition ◽  
Centre Of Mass ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

The centre of mass G of a triangle has the property that the rays to the vertices from G sweep out triangles having equal areas. We show that such points, termed equipartitioning points in this paper, need not exist in other polygons. A necessary and sufficient condition for a quadrilateral to have an equipartitioning point is that one of its diagonals bisects the other. The general theorem, namely, necessary and sufficient conditions for equipartitioning points for arbitrary polygons to exist, is also stated and proved. When this happens, they are in general, distinct from the centre of mass. In parallelograms, and only in them, do the two points coincide.

Orthogonal Fractional Factorial Plans

1980 ◽  
Vol 29 (3-4) ◽  
pp. 143-160 ◽  
Author(s):  
Rahul Mukerjee
Keyword(s):  
Broad Class ◽  
Existence Results ◽  
Fractional Factorial ◽  
The Other ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Equal Frequency ◽  
Other Hand ◽  
Necessary And Sufficient ◽  
Generally True

This paper shows that the criterion of proportional frequency for (unblocked) orthogonal fractional factorial plans, as suggested by some previous authors, is not generally true. On the other hand, the criterion of equal frequency has been established as a necessary and sufficient condition in the general case. Some other properties of orthogonal fractional factorial plans have been investigated. A necessary and sufficient condition for designs involving two or more blocks has also been presented. A broad class of non-existence results follow.

Uncertainty Regarding Interpretation of the “Negligence Rule” and Its Implications for the Efficiency of Outcomes

2016 ◽  
Vol 7 (2) ◽  
Author(s):  
Satish K. Jain
Keyword(s):  
Subjective Probability ◽  
The Other ◽  
Sufficient Condition ◽  
Subjective Probabilities ◽  
Necessary And Sufficient Condition ◽  
Liability Rule ◽  
Important Conclusion ◽  
Standard Version ◽  
Necessary And Sufficient ◽  
Negligence Rule

AbstractThere are two ways that the negligence rule is interpreted. Under one interpretation a negligent injurer is liable for the entire harm to the victim; and under the other interpretation a negligent injurer is liable only for that part of the harm which can be ascribed to his negligence. Both these versions are efficient. However, if there is uncertainty regarding whether the court will be employing the full liability version or the incremental liability version for determining the liability of a negligent injurer, notwithstanding the fact that both the versions are efficient, inefficiency is possible. It is shown in the paper that a necessary and sufficient condition for efficiency in all cases is that the subjective probability with which the injurer expects the standard version to be employed must be greater than or equal to the subjective probability with which the victim expects the standard version to be employed. For the subset of applications without complementarities in the cares of the two parties and which are such that the total social costs are minimized at a unique care-configuration, it is shown that efficiency obtains regardless of the subjective probabilities with which the parties expect the two versions. One very important conclusion that emerges from the analysis of this paper is that when courts employ more than one liability rule, even if all the employed rules are efficient, the efficiency of all outcomes cannot be taken for granted merely on the ground of the efficiency of the employed rules.

scholarly journals The cognitive process of extended trans students in understanding the real number system

2020 ◽  
Vol 1470 ◽  
pp. 012070
Author(s):  
Dewi Herawaty ◽  
Wahyu Widada ◽  
Iran Sairan ◽  
Fizi Herdian ◽  
Khathibul U Z Nugroho ◽  
...  
Keyword(s):  
Real Number ◽  
Cognitive Process ◽  
Number System ◽  
The Real ◽  
Real Number System

The Inverse Eigenvalue Problem for Symmetric Doubly Arrow Matrices

2013 ◽  
Vol 444-445 ◽  
pp. 625-627
Author(s):  
Kan Ming Wang ◽  
Zhi Bing Liu ◽  
Xu Yun Fei
Keyword(s):  
Eigenvalue Problem ◽  
Special Kind ◽  
Inverse Eigenvalue Problem ◽  
Symmetric Matrices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Inverse Eigenvalue ◽  
Necessary And Sufficient

In this paper we present a special kind of real symmetric matrices: the real symmetric doubly arrow matrices. That is, matrices which look like two arrow matrices, forward and backward, with heads against each other at the station, . We study a kind of inverse eigenvalue problem and give a necessary and sufficient condition for the existence of such matrices.

The Structure of the Real Number System.

1964 ◽  
Vol 71 (9) ◽  
pp. 1061
Author(s):  
J. B. Roberts ◽  
L. W. Cohen ◽  
G. Ehrlich
Keyword(s):  
Real Number ◽  
Number System ◽  
The Real ◽  
Real Number System
Sign in / Sign up

Export Citation Format

Share Document