Understanding and Fluency

Turn Over

Philosophy and psychology appeal to a sense of understanding, typically a feeling invoked to explain people’s choices. ‘Understanding’ seems loosely associated with properties like transparency (things we understand we can also introspect), or voluntary (cognitive) control (things we understand we can turn over in our mind). Research on attention and memory shows that many candidate cases of understanding lack properties like transparency and voluntary control. In fact, ‘understanding’ may denote an unprincipled stew of states, processes, capacities, and goals that are only occasionally present when philosophers, and ordinary folks, apply the term or concept. A unified account of understanding might be valuable, but understanding isn’t a natural kind or defined by a set of necessary and sufficient conditions. Any unity we find in understanding comes not from the involvement of common mechanisms across diverse cases, but rather of messy cognitive activities in the common goal of pursuing the truth.

A Note on Normal Attraction to a Stable Law

Canadian Mathematical Bulletin ◽

10.4153/cmb-1971-081-5 ◽

1971 ◽

Vol 14 (3) ◽

pp. 451-452

Author(s):

M. V. Menon ◽

V. Seshadri

Keyword(s):

Distribution Function ◽

Sufficient Conditions ◽

Characteristic Exponent ◽

Random Variables ◽

Stable Law ◽

Common Distribution ◽

Normal Attraction ◽

Common Distribution Function ◽

The Common ◽

Let X1, X2, …, be a sequence of independent and identically distributed random variables, with the common distribution function F(x). The sequence is said to be normally attracted to a stable law V with characteristic exponent α, if for some an (converges in distribution to V). Necessary and sufficient conditions for normal attraction are known (cf [1, p. 181]).

Maximal and Minimal Ranks of the Common Solution of Some Linear Matrix Equations over an Arbitrary Division Ring with Applications

Algebra Colloquium ◽

10.1142/s1005386709000297 ◽

2009 ◽

Vol 16 (02) ◽

pp. 293-308 ◽

Cited By ~ 13

Author(s):

Qingwen Wang ◽

Guangjing Song ◽

Xin Liu

Keyword(s):

Division Ring ◽

Sufficient Conditions ◽

Linear Matrix ◽

Matrix Equations ◽

Common Solution ◽

Special Cases ◽

The Common ◽

Linear Matrix Equations

We establish the formulas of the maximal and minimal ranks of the common solution of certain linear matrix equations A1X = C1, XB2 = C2, A3XB3 = C3 and A4XB4 = C4 over an arbitrary division ring. Corresponding results in some special cases are given. As an application, necessary and sufficient conditions for the invariance of the rank of the common solution mentioned above are presented. Some previously known results can be regarded as special cases of our results.

On Sums of Progressions of Positive Integers

The Mathematical Gazette ◽

10.1017/s0025557200207109 ◽

1970 ◽

Vol 54 (388) ◽

pp. 113-115

Author(s):

R. L. Goodstein

Keyword(s):

Positive Integer ◽

Arithmetic Progression ◽

Sufficient Conditions ◽

The Common ◽

Positive Integers

We consider the problem of finding necessary and sufficient conditions for a positive integer to be the sum of an arithmetic progression of positive integers with a given common difference, starting with the case when the common difference is unity.

The common Re-nonnegative definite and Re-positive definite solutions to the matrix equations $ A_1XA_1^\ast = C_1 $ and $ A_2XA_2^\ast = C_2 $

AIMS Mathematics ◽

10.3934/math.2022026 ◽

2021 ◽

Vol 7 (1) ◽

pp. 384-397

Author(s):

Yinlan Chen ◽

◽

Lina Liu

Keyword(s):

Sufficient Conditions ◽

Positive Definite ◽

Linear Matrix ◽

Matrix Equations ◽

The Matrix ◽

The Common ◽

Nonnegative Definite ◽

Linear Matrix Equations

<abstract><p>In this paper, we consider the common Re-nonnegative definite (Re-nnd) and Re-positive definite (Re-pd) solutions to a pair of linear matrix equations $ A_1XA_1^\ast = C_1, \ A_2XA_2^\ast = C_2 $ and present some necessary and sufficient conditions for their solvability as well as the explicit expressions for the general common Re-nnd and Re-pd solutions when the consistent conditions are satisfied.</p></abstract>

Two remarks on amalgams of locally finite groups

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700003750 ◽

1987 ◽

Vol 36 (3) ◽

pp. 461-468 ◽

Cited By ~ 1

Author(s):

Berthold J. Maier

Keyword(s):

Finite Group ◽

Finite Index ◽

Finite Groups ◽

Sufficient Conditions ◽

Locally Finite Group ◽

Locally Finite ◽

Locally Finite Groups ◽

The Common ◽

We construct non amalgamation bases in the class of locally finite groups, and we present necessary and sufficient conditions for the embeddability of an amalgam into a locally finite group in the case that the common subgroup has finite index in both constituents.

A Common Value Experimentation with Multiarmed Bandits

Mathematical Problems in Engineering ◽

10.1155/2018/4791590 ◽

2018 ◽

Vol 2018 ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

Xiujuan Gao ◽

Hao Liang ◽

Tong Wang

Keyword(s):

Sufficient Conditions ◽

Second Derivative ◽

Value Functions ◽

Perfect Equilibrium ◽

Common Value ◽

A Value ◽

Markov Perfect ◽

Identical Preference ◽

The Common ◽

We study a value common experimentation with multiarmed bandits and give an application about the experimentation. The second derivative of value functions at cutoffs is investigated when an agent switches action with multiarmed bandits. If consumers have identical preference which is unknown and purchase products from only two sellers among multiple sellers, we obtain the necessary and sufficient conditions about the common experimentation. The Markov perfect equilibrium and the socially effective allocation in K-armed markets are discussed.

DECOMPOSITION OF MULTIUNIVERSE FUZZY TRUTH FUNCTIONS

International Journal of Uncertainty Fuzziness and Knowledge-Based Systems ◽

10.1142/s0218488501001095 ◽

2001 ◽

Vol 09 (05) ◽

pp. 623-643

Author(s):

DERYA ALTUNAY ◽

TURHAN ÇİFTÇİBAŞI

Keyword(s):

Propositional Logic ◽

Sufficient Conditions ◽

Fuzzy Relation ◽

Fuzzy Relations ◽

Decomposition Problem ◽

The Common ◽

The Given

This paper focuses on the decomposition problem of fuzzy relations using the concepts of multiuniverse fuzzy propositional logic. Given two fuzzy propositions in different universes, it is always possible to construct a fuzzy relation in the common universe through a prescribed combination. However, the converse is not so obvious, if possible at all. In other words, given a fuzzy relation, how would we know if it really represents a certain relationship between some fuzzy propositions? It is important to recognize whether the given fuzzy relation is a meaningful representation of information according to certain criteria applicable to some fuzzy propositions that constitute the fuzzy relation itself. Two basic structures of decomposition are investigated. Necessary and sufficient conditions for decomposition of multiuniverse fuzzy truth functions in terms of one-universe truth functions are presented. An algorithm for decomposition is proposed.

The Common Solution of Some Matrix Equations

Algebra Colloquium ◽

10.1142/s1005386716000092 ◽

2016 ◽

Vol 23 (01) ◽

pp. 71-81 ◽

Cited By ~ 8

Author(s):

Li Wang ◽

Qingwen Wang ◽

Zhuoheng He

Keyword(s):

General Solution ◽

Sufficient Conditions ◽

Linear Matrix ◽

Matrix Equations ◽

Existence Of A Solution ◽

Solvability Conditions ◽

Common Solution ◽

The Common ◽

Linear Matrix Equations

In this paper we investigate the system of linear matrix equations A1X=C1, YB2=C2, A3XB3=C3, A4YB4=C4, BX+YC=A. We present some necessary and sufficient conditions for the existence of a solution to this system and give an expression of the general solution to the system when the solvability conditions are satisfied.

Similarity transformation of matrices to one common canonical form and its applications to 2D linear systems

International Journal of Applied Mathematics and Computer Science ◽

10.2478/v10006-010-0037-z ◽

2010 ◽

Vol 20 (3) ◽

pp. 507-512 ◽

Cited By ~ 2

Author(s):

Tadeusz Kaczorek

Keyword(s):

Linear Systems ◽

Canonical Form ◽

Linear Transformation ◽

Similarity Transformation ◽

Sufficient Conditions ◽

Closed Loop System ◽

2D System ◽

The Common ◽

Similarity transformation of matrices to one common canonical form and its applications to 2D linear systemsThe notion of a common canonical form for a sequence of square matrices is introduced. Necessary and sufficient conditions for the existence of a similarity transformation reducing the sequence of matrices to the common canonical form are established. It is shown that (i) using a suitable state vector linear transformation it is possible to decompose a linear 2D system into two linear 2D subsystems such that the dynamics of the second subsystem are independent of those of the first one, (ii) the reduced 2D system is positive if and only if the linear transformation matrix is monomial. Necessary and sufficient conditions are established for the existence of a gain matrix such that the matrices of the closed-loop system can be reduced to the common canonical form.

Hermitian Solutions to a Quaternion Matrix Equation

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.50-51.391 ◽

2011 ◽

Vol 50-51 ◽

pp. 391-395

Author(s):

Ning Li ◽

Jing Jiang ◽

Wen Feng Wang

Keyword(s):

Matrix Equation ◽

Quaternion Algebra ◽

Sufficient Conditions ◽

Quaternion Matrix ◽

Quaternion Matrix Equation ◽

The Common ◽

In this paper, we consider Hermitian and skew-Hermitian solutions to a certain matrix equation over quaternion algebra H. Necessary and sufficient conditions are obtained for the quaternion matrix equation to have Hermitian and skew-Hermitian solutions, and the expressions of such solutions are also given. As an application, the common skew-Hermitian g-inverse of quaternion matrix A and B is considered.