• Journal of the Korean Mathematical Society
• /
• v.43 no.2
• /
• pp.323-356
• /
• 2006

Let X and Y be vector spaces. It is shown that a mapping $f\;:\;X{\rightarrow}Y$ satisfies the functional equation $$mn_{mn-2}C_{k-2}f(\frac {x_1+...+x_{mn}} {mn})$$ $(\ddagger)\;+mn_{mn-2}C_{k-1}\;\sum\limits_{i=1}^n\;f(\frac {x_{mi-m+1}+...+x_{mi}} {m}) =k\;{\sum\limits_{1{\leq}i_1<... if and only if the mapping $f : X{\rightarrow}Y$ is additive, and we prove the Cauchy-Rassias stability of the functional equation $(\ddagger)$ in Banach modules over a unital $C^*-algebra$. Let A and B be unital $C^*-algebra$ or Lie $JC^*-algebra$. As an application, we show that every almost homomorphism h : $A{\rightarrow}B$ of A into B is a homomorphism when $h(2^d{\mu}y) = h(2^d{\mu})h(y)\;or\;h(2^d{\mu}\;o\;y)=h(2^d{\mu})\;o\;h(y)$ for all unitaries ${\mu}{\in}A,\;all\;y{\in}A$, and d = 0,1,2,..., and that every almost linear almost multiplicative mapping $h:\;A{\rightarrow}B$ is a homomorphism when h(2x)=2h(x) for all $x{\in}A$. Moreover, we prove the Cauchy-Rassias stability of homomorphisms in $C^*-algebras$ or in Lie $JC^*-algebras$, and of Lie $JC^*-algebra$ derivations in Lie $JC^*-algebras$.

• Korean Journal of Mathematics
• /
• v.22 no.1
• /
• pp.91-121
• /
• 2014

Let X and Y be vector spaces. It is shown that a mapping $f:X{\rightarrow}Y$ satisfies the functional equation ${\ddag}$ $$2df(\frac{x_1+{\sum}_{j=2}^{2d}(-1)^jx_j}{2d})-2df(\frac{x_1+{\sum}_{j=2}^{2d}(-1)^{j+1}x_j}{2d})=2\sum_{j=2}^{2d}(-1)^jf(x_j)$$ if and only if the mapping $f:X{\rightarrow}Y$ is additive, and prove the Cauchy-Rassias stability of the functional equation (${\ddag}$) in Banach modules over a unital $C^*$-algebra, and in Poisson Banach modules over a unital Poisson $C^*$-algebra. Let $\mathcal{A}$ and $\mathcal{B}$ be unital $C^*$-algebras, Poisson $C^*$-algebras, Poisson $JC^*$-algebras or Lie $JC^*$-algebras. As an application, we show that every almost homomorphism $h:\mathcal{A}{\rightarrow}\mathcal{B}$ of $\mathcal{A}$ into $\mathcal{B}$ is a homomorphism when $h(d^nuy)=h(d^nu)h(y)$ or $h(d^nu{\circ}y)=h(d^nu){\circ}h(y)$ for all unitaries $u{\in}\mathcal{A}$, all $y{\in}\mathcal{A}$, and n = 0, 1, 2, ${\cdots}$. Moreover, we prove the Cauchy-Rassias stability of homomorphisms in $C^*$-algebras, Poisson $C^*$-algebras, Poisson $JC^*$-algebras or Lie $JC^*$-algebras, and of Lie $JC^*$-algebra derivations in Lie $JC^*$-algebras.

• Korean Journal of Child Studies
• /
• v.6 no.1
• /
• pp.17-40
• /
• 1985

This study investigated Korean children's early acquisition of negatives and focused on four research questions: 1) processing of negative variations; 2) the nature of negatives when negatives are completely acquired in Korean (in which meaning and form are matched in one to one mapping); 3) the validity of Bellugi's negative acquisition model in Korean; and 4) the cause of child's erroneous sentence production: limited ability or regularity in children's cognition. The language data of the five subjects (age span; 1.1 - 3.11) were collected by their parents in the natural setting of the home. The results showed that 1) the pivot form, was processed in many ways from a simple to a complicated form, such as <(X+X')+N> <(x+x')+N,Y> <(x+x') N,(y+y')>. It appeared that the children used a simple negative format to reach a one-step advanced negative format. 2) Korean negatives are divided into range of negation in the negative sentence (part or whole), strength of negation (absolute or general), functions of meaning (negation, absences, refusal, prohibition, impossibility). All five children acquired negative sentences in all functions and the complete range after 3 years of age. 3) In spite of the differences in age level, Bellugi's four stage model was in evidence; that is, Korean children's negative acquisition was almost identical with Bellugi's tour stage model in deep structure. 4) Analyses of children's error sentences showed that the sentences with errors were made not because of the children's limitation in cognitive ability but because of the strict application of regularity of rules from the original grammars. Consequently, the children produced negative sentences using two rules: the rule of additive complexity (from simple to complex) and the rule of division (from one to several).