The necessary condition and the sufficient condition on extremal values of functionals

2016 ◽  
pp. 29-41
Keyword(s):  
Necessary Condition ◽  
Sufficient Condition ◽  
Extremal Values

Separability criteria based on the Bloch representation of density matrices

2007 ◽  
Vol 7 (7) ◽  
pp. 624-638
Author(s):  
J. de Vicente
Keyword(s):  
Sufficient Conditions ◽  
Quantum Systems ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Pure Product ◽  
Separability Problem ◽  
Arbitrary Dimensions ◽  
Necessary And Sufficient ◽  
Bloch Representation

We study the separability of bipartite quantum systems in arbitrary dimensions using the Bloch representation of their density matrix. This approach enables us to find an alternative characterization of the separability problem, from which we derive a necessary condition and sufficient conditions for separability. For a certain class of states the necessary condition and a sufficient condition turn out to be equivalent, therefore yielding a necessary and sufficient condition. The proofs of the sufficient conditions are constructive, thus providing decompositions in pure product states for the states that satisfy them. We provide examples that show the ability of these conditions to detect entanglement. In particular, the necessary condition is proved to be strong enough to detect bound entangled states.

Rising or Constrained Power?

2010 ◽  
Author(s):  
Eswaran Sridharan
Keyword(s):  
Economic Growth ◽  
High Growth ◽  
The United States ◽  
Asia Pacific ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Economic Weight ◽  
Rising Power ◽  
Global Influence

This chapter analyses India’s prospects as a rising power by asking what kind of power India has the potential to be, given its military, economic, and institutional capacities and the economic and geostrategic constraints it faces. It argues that while sustained high growth is a necessary condition it is not a sufficient condition since economic growth does not necessarily convert smoothly into greater power. Due to such conversion problems India, like some other powers, might not be able to exercise commensurate regional, extra-regional, and global influence as might appear to follow from the revival of sustained high growth and increased economic weight. The more achievable and likely alternative is that of a coalitional or bridging power that can play the role of an effective partner in the security and other spheres to a range of powers, principally to the United States and in the Asia-Pacific and Indian Ocean regions.

scholarly journals Syarat Perlu dan Cukup untuk Keterbatasan Potensial Riesz di Ruang Morrey Klasik

2013 ◽  
Vol 14 (3) ◽  
pp. 227
Author(s):  
Mohammad Imam Utoyo ◽  
Basuki Widodo ◽  
Toto Nusantara ◽  
Suhariningsih Suhariningsih
Keyword(s):  
Characteristic Function ◽  
Morrey Space ◽  
Necessary Conditions ◽  
Riesz Potential ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Research Results ◽  
Necessary And Sufficient

This script was aimed to determine the necessary conditions for boundedness of Riesz potential in the classical Morrey space. If these results are combined with previous research results will be obtained the necessary and sufficient condition for boundedness of Riesz potential. This necessary condition is obtained through the use of characteristic function as one member of the classical Morrey space.

scholarly journals Spatio-temporal Dynamics of the Patterning of Arabidopsis Flower Meristem

2020 ◽  
Author(s):  
José Díaz ◽  
Elena R. Álvarez-Buylla
Keyword(s):  
Flower Development ◽  
Apical Meristem ◽  
Temporal Dynamics ◽  
Positional Information ◽  
Floral Organ ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Abc Model ◽  
Flower Meristem ◽  
Spatio Temporal

AbstractThe qualitative model presented in this work recovers the onset of the four fields that correspond to those of each floral organ whorl of Arabidopsis flower, suggesting a mechanism for the generation of the positional information required for the differential expression of the A, B and C identity genes according to the ABC model for organ determination during early stages of flower development. Our model integrates a previous model for the emergence of WUS pattern in the apical meristem, and shows that this pre-pattern is a necessary but not sufficient condition for the posterior information of the four fields predicted by the ABC model. Furthermore, our model predicts that LFY diffusion along the L1 layer of cells is not a necessary condition for the patterning of the floral meristem.

scholarly journals Generalized approximate boundary synchronization for a coupled system of wave equations

2020 ◽  
Author(s):  
Yanyan Wang
Keyword(s):  
Wave Equations ◽  
Dirichlet Boundary ◽  
Coupled System ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Exact Boundary ◽  
Boundary Controls ◽  
The Relationship ◽  
Exact Boundary Synchronization ◽  
Dirichlet Boundary Controls

In this paper, we consider the generalized approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls. We analyse the relationship between the generalized approximate boundary synchronization and the generalized exact boundary synchronization, give a sufficient condition to realize the generalized approximate boundary synchronization and a necessary condition in terms of Kalman’s matrix, and show the meaning of the number of total controls. Besides, by the generalized synchronization decomposition, we define the generalized approximately synchronizable state, and obtain its properties and a sufficient condition for it to be independent of applied boundary controls.

scholarly journals ON THE FINITENESS PROBLEM FOR AUTOMATON (SEMI)GROUPS

2012 ◽  
Vol 22 (06) ◽  
pp. 1250052 ◽  
Author(s):  
ALI AKHAVI ◽  
INES KLIMANN ◽  
SYLVAIN LOMBARDY ◽  
JEAN MAIRESSE ◽  
MATTHIEU PICANTIN
Keyword(s):  
Decision Problem ◽  
Decision Procedure ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Mealy Automata ◽  
Necessary And Sufficient ◽  
New Criteria

This paper addresses a decision problem highlighted by Grigorchuk, Nekrashevich, and Sushchanskiĭ, namely the finiteness problem for automaton (semi)groups. For semigroups, we give an effective sufficient but not necessary condition for finiteness and, for groups, an effective necessary but not sufficient condition. The efficiency of the new criteria is demonstrated by testing all Mealy automata with small stateset and alphabet. Finally, for groups, we provide a necessary and sufficient condition that does not directly lead to a decision procedure.

Criteria of Personal Identity

1975 ◽  
Vol 7 (1) ◽  
pp. 47-69
Author(s):  
Karl Ameriks
Keyword(s):  
Personal Identity ◽  
Primary Objective ◽  
Psychological Characteristics ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Bodily Continuity ◽  
Secondary Objective ◽  
The Way

The primary objective of this paper is to improve the defense of the thesis that(1) bodily continuity is the primary criterion of personal identity.This is to be done by establishing (in Parts Ill and IV) that there is a unique sense in which(2) bodily continuity is a necessary condition of personal identity.A secondary objective of the paper is to illustrate (in Part I) the way in which the value and validity of (2) has been obscured in recent defenses and criticisms of (1), which inappropriately interpret it in terms of the claim that(3) bodily continuity is a sufficient condition of personal identity.Since the truth of (3) will be denied, the defense of (1) will also involve arguing (in Part II) that the traditional alternative to the bodily criterion, namely the criterion of memory and psychological characteristics, not only is not a necessary condition of personal identity but also is not a sufficient condition.

The Logic of Competence

2016 ◽  
Author(s):  
Arthur Lupia
Keyword(s):  
Necessary Condition ◽  
Sufficient Condition ◽  
Logical Relationship ◽  
Improve Efficiency ◽  
Short Summary ◽  
Increasing Knowledge ◽  
Types Of Information

This chapter clarifies the logical relationship between the types of information an educator can convey and competence at the types of tasks that people perform. Here is a short summary of the chapter’s main claims: . . . Knowledge of many things does not require recalling “all the facts.” Competence at many tasks does not require knowing “everything.” Trying to give “all the facts” to an audience is usually a sign of inefficiency. Cues and non-declarative memories facilitate competence even when they leave people unable to answer certain fact-based questions about a task. The effectiveness of attempts to increase competence depends on what information is given to whom. For example, when the competence of a group is being measured, not all members of that group must have the same knowledge for the group as a whole to be competent. . . . To improve efficiency, educators should understand what kinds of information are most relevant for increasing knowledge and competence. To reach these conclusions, I rely heavily on two terms: necessity and sufficiency. These terms describe logical relationships between two or more items. Educators who understand these terms can better distinguish information that increases desired competences from information that has no such power. X is a necessary condition for Y if Y can happen only after X happens. For example, suppose that earning at least a million votes is a necessary condition for a candidate to win an election. Two implications follow. First, any candidate who does not earn at least a million votes cannot win the election (which means that there may be no winner). Second, any candidate who earns at least a million votes has the potential to win the election. X is a sufficient condition for Y if X happening means that Y must also happen. Suppose that earning a million votes is sufficient to win the election. Two implications follow. First, any candidate who earns at least a million votes wins the election (which implies that multiple candidates could win). Second, any candidate who does not earn at least a million votes retains the potential to win the election.

The necessary and sufficient condition for clustering of nodes based on the signs of connections in generalized signed networks

2019 ◽  
Vol 33 (10) ◽  
pp. 1950086
Author(s):  
Qi Wang ◽  
Yinhe Wang ◽  
Zilin Gao ◽  
Lili Zhang ◽  
Wenli Wang
Keyword(s):  
Clustering Algorithms ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Signed Networks ◽  
Clustering Problem ◽  
Structural Hole ◽  
Signed Network ◽  
Necessary And Sufficient ◽  
Sufficient And Necessary Condition

This paper investigates the clustering problem for the generalized signed networks. By rigorous derivations, a sufficient and necessary condition for clustering of the nodes in generalized signed networks is proposed in this paper. In order to obtain this condition, the concept of friends group is first introduced for the nodes based on their links’ sign. Then, the unprivileged network is also defined in this paper by employing the concepts of structural hole and broker. Compared with the existing clustering algorithms, the outstanding advantage in this paper is that only the positive or negative (especially, or zero) sign of the links is required regardless of their density or sparsity. We have proved mathematically that a generalized signed network is classifiable if and only if it is an unprivileged network. Finally, two examples associated with numerical simulations are proposed to generate the unprivileged networks.

Some Results on Matrices with Prescribed Diagonal Elements and Singular Values

1976 ◽  
Vol 19 (1) ◽  
pp. 89-92 ◽  
Author(s):  
Fuk-Yum Sing
Keyword(s):  
Singular Values ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Complex Matrix ◽  
Square Roots

Let A be an n × n complex matrix. The singular values of A are the non-negative square-roots of the eigenvalues of A*A. G. N. De Oliviera [4] gave a necessary condition for the existence of a matrix A with a1..., an as diagonal elements and α1,..., αn as singular values. We shall give another necessary condition which implies the above author’s condition and we show that this is also a sufficient condition for the case n =2.

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