• Journal of applied mathematics & informatics
• /
• v.31 no.3_4
• /
• pp.545-558
• /
• 2013

In this paper, the formulas for calculating the extremal ranks and inertias of the Hermitian least squares solutions to matrix equation AX = B are established. In particular, the necessary and sufficient conditions for the existences of the positive and nonnegative definite solutions to this matrix equation are given. Meanwhile, the least squares problem of the above matrix equation with Hermitian R-symmetric and R-skew symmetric constraints are also investigated.

• Journal of the Korean Mathematical Society
• /
• v.22 no.2
• /
• pp.181-189
• /
• 1985

• Communications of the Korean Mathematical Society
• /
• v.16 no.4
• /
• pp.679-690
• /
• 2001

Doubly stochastic matrices are n$\times$n nonnegative ma-trices whose row and column sums are all 1. Convex polytope $\Omega$$_{n}$ of doubly stochastic matrices and more generally (R,S), so called transportation polytopes, are important since they form the domains for the transportation problems. A theorem by Birkhoff classifies the extremal matrices of , $\Omega$$_{n}$ and extremal matrices of transporta-tion polytopes (R,S) were all classified combinatorially. In this article, we consider signed version of $\Omega$$_{n}$ and (R.S), obtain signed Birkhoff theorem; we define a new class of convex polytopes (R,S), calculate their dimensions, and classify their extremal matrices, Moreover, we suggest an algorithm to express a matrix in (R,S) as a convex combination of txtremal matrices. We also give an example that a polytope of signed matrices is used as a domain for a decision problem. In this context of finite reflection(Coxeter) group theory, our generalization may also be considered as a generalization from type $A_{*}$ n/ to type B$_{n}$ D$_{n}$. n/.

• Journal of the Korean Mathematical Society
• /
• v.60 no.5
• /
• pp.959-986
• /
• 2023

Zhai and Lin recently proved that if G is an n-vertex connected 𝜃(1, 2, r + 1)-free graph, then for odd r and n ⩾ 10r, or for even r and n ⩾ 7r, one has ${\rho}(G){\leq}{\sqrt{{\lfloor}{\frac{n^2}{4}}{\rfloor}}}$, and equality holds if and only if G is $K_{{\lceil}{\frac{n}{2}}{\rceil},{\lfloor}{\frac{n}{2}}{\rfloor}}$. In this paper, for large enough n, we prove a sharp upper bound for the spectral radius in an n-vertex H-free non-bipartite graph, where H is 𝜃(1, 2, 3) or 𝜃(1, 2, 4), and we characterize all the extremal graphs. Furthermore, for n ⩾ 137, we determine the maximum number of edges in an n-vertex 𝜃(1, 2, 4)-free non-bipartite graph and characterize the unique extremal graph.

• Journal of applied mathematics & informatics
• /
• v.5 no.1
• /
• pp.235-240
• /
• 1998

Given a Unimodular polynomial P of degree N$\geq$1, the exteremal problem for ${\gamma}$ =max{｜P(eit)｜:0 $\leq$t$\leq$2$\pi$} satisfies ${\gamma}$$\leq$C{{{{ SQRT { N+1} where C is a universal constant. Here we show that C < 2+{{{{ whenever N is fixed and P has the coefficients of a Rudin-Shapiro polynomial.

• Management Science and Financial Engineering
• /
• v.21 no.1
• /
• pp.11-17
• /
• 2015

Min-max stochastic optimization is an approach to address the distribution ambiguity of the underlying random variable. We present a unified approach to the problem which utilizes the theory of convex order on the random variables. First, we consider a general framework for the problem and give a condition under which the convex order can be utilized to transform the min-max optimization problem into a simple minimization problem. Then extremal distributions are presented for some interesting classes of distributions. Finally, applications to the single-period inventory control problems are given.

• Korean Journal of Mathematics
• /
• v.20 no.2
• /
• pp.247-254
• /
• 2012

The Klee-Quaife problem is finding the minimum order ${\mu}(d,c,v)$ of the $(d,c,v)$ graph, which is a $c$-vertex connected $v$-regular graph with diameter $d$. Many authors contributed finding ${\mu}(d,c,v)$ and they also enumerated and classied the graphs in several cases. This problem is naturally extended to the case of digraphs. So we are interested in the extended Klee-Quaife problem. In this paper, we deal with an equivalent problem, finding the maximum diameter of digraphs with given order, focused on 2-regular case. We show that the maximum diameter of strongly connected 2-regular digraphs with order $n$ is $n-3$, and classify the digraphs which have diameter $n-3$. All 15 nonisomorphic extremal digraphs are listed.

• Journal of applied mathematics & informatics
• /
• v.33 no.3_4
• /
• pp.365-377
• /
• 2015

For a (molecular) graph, the first and second Zagreb indices (M₁ and M₂) are two well-known topological indices, first introduced in 1972 by Gutman and Trinajstić. The first Zagreb index M₁ is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M₂ is equal to the sum of the products of the degrees of pairs of adjacent vertices. Let $K_{n_1,n_2}^{P}$ with n₁ $\leq$ n₂, n₁ + n₂ = n and p < n₁ be the set of bipartite graphs obtained by deleting p edges from complete bipartite graph K_n1,n2. In this paper, we determine sharp upper and lower bounds on Zagreb indices of graphs from $K_{n_1,n_2}^{P}$ and characterize the corresponding extremal graphs at which the upper and lower bounds on Zagreb indices are attained. As a corollary, we determine the extremal graph from $K_{n_1,n_2}^{P}$ with respect to Zagreb coindices. Moreover a problem has been proposed on the first and second Zagreb indices.

• Journal of the Chungcheong Mathematical Society
• /
• v.29 no.1
• /
• pp.137-150
• /
• 2016

We introduce a new approach that allows us to solve, algorithmically, the contractive completion problem. In this article, we provide concrete necessary and sufficient conditions for the existence of contractive completions of Hankel partial contractions of size $4{\times}4$ using a Moore-Penrose inverse of a matrix.

• Journal of the Korean Mathematical Society
• /
• v.57 no.3
• /
• pp.641-653
• /
• 2020

To extend the well-known extremal characterization of the geometric mean of two n × n positive definite matrices A and B, we solve the following problem: $${\max}\{X:X=X^*,\;\(\array{A&V&X\\V&B&W\\X&W&C}\){\geq}0\}$$. We find an explicit expression of the maximum value with respect to the matrix geometric mean of Schur complements.