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Equations $AX=Y$ and $Ax=y$ in Alg$\Cal L$
조영수,강주호,박동완 대한수학회 2006 대한수학회지 Vol.43 No.2
Let $\Cal L$ be a subspace lattice on a Hilbert space $\Cal H$ and $X$ and $Y$ be operators acting on a Hilbert space ${\Cal H}$. Let $P$ be the projection onto $\overline{{\Cal R} (X)}$, where ${\Cal R} X$ is the range of $X$. If $PE=EP$ for each $E\in {\Cal L}$, then there exists an operator $A$ in Alg$\Cal L$ such that $AX=Y$ if and only if $$\displaystyle\sup\{ {{\|E^\bot Yf\|} / {\|E^\bot Xf\|}} : f\in {\Cal H}, ~E\in {\Cal L} \}= K <\infty.$$ Moreover, if the necessary condition holds, then we may choose an operator $A$ such that $AX=Y$ and $\|A\| = K$. Let $x$ and $y$ be vectors in ${\Cal H}$ and let $P_x$ be the projection onto the singlely generated space by $x$. If $P_xE=EP_x$ for each $E\in {\Cal L}$, then the assertion that there exists an operator $A$ in Alg$\Cal L$ such that $Ax=y$ is equivalent to the condition $$K_0 : = \displaystyle\sup\{ {{\|E^\bot y\|} / {\|E^\bot x\|}} : ~E\in {\Cal L} \} <\infty.$$ Moreover, we may choose an operator $A$ such that $\|A\|=K_0$ whose norm is $K_0 $ under this case.
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Separable minimal surfaces and their limit behavior
김대환,Yuta Ogata 대한수학회 2024 대한수학회지 Vol.61 No.4
A separable minimal surface is represented by the form of $f(x) + g(y)+ h(z) = 0$, where $f$, $g$ and $h$ are real-valued functions of $x$, $y$ and $z$, respectively. We provide exact equations for separable minimal surfaces with elliptic functions that are singly, doubly and triply periodic minimal surfaces and completely classify all them. In particular, parameters in the separable minimal surfaces change the shape of the surfaces, such as fundamental periods and its limit behavior, within the form $f(x) + g(y)+ h(z) = 0$.
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On the existence of real quadratic fields with odd period of minimal type
Takanobu Eguchi,Yasuhiro Kishi 대한수학회 2024 대한수학회지 Vol.61 No.4
In this paper, under the ABC-conjecture, we show that there exist infinitely many real quadratic fields with odd period of minimal type.
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Realizing a Fake Projective Plane as a Degree 25 Surface in $\P^5$
Lev Borisov,Zachary Lihn 대한수학회 2024 대한수학회지 Vol.61 No.4
Fake projective planes are smooth complex surfaces of general type with Betti numbers equal to that of the usual projective plane. Recent explicit constructions of fake projective planes embed them via their bicanonical embedding in $\mathbb P^9$. In this paper, we study Keum's fake projective plane $(a=7, p=2, \{7\}, D_3 2_7)$ and use the equations of \cite{Borisov} to construct an embedding of fake projective plane in $\mathbb P^5$. We also simplify the 84 cubic equations defining the fake projective plane in $\mathbb P^9$.
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Floer mini-max theory, the Cerf diagram, and the spectral invariants
오용근 대한수학회 2009 대한수학회지 Vol.46 No.2
The author previously defined the spectral invariants, denoted by ρ(H;a), of a Hamiltonian function H as the mini-max value of the action functional A^H over the Novikov Floer cycles in the Floer homology class dual to the quantum cohomology class a. The spectrality axiom of the invariant ρ(H;a) states that the mini-max value is a critical value of the action functional A^H. The main purpose of the present paper is to prove this axiom for nondegenerate Hamiltonian functions in irrational symplectic manifolds (M, ω). We also prove that the spectral invariant function a : H ↦ ρ(H;a) can be pushed down to a continuous function defined on the universal (etale) covering space Ham(M, ω) of the group Ham(M, ω) of Hamiltonian diffeomorphisms on general (M, !). For a certain generic homotopy, which we call a Cerf homotopy H = {H^s}0≤s≤1 of Hamiltonians, the function ρ^a ․H : s ↦ ρ(H^s;a) is piecewise smooth away from a countable subset of [0, 1] for each non-zero quantum cohomology class a. The proof of this nondegenerate spectrality relies on several new ingredients in the chain level Floer theory, which have their own independent interest: a structure theorem on the Cerf bifurcation diagram of the critical values of the action functionals associated to a generic one-parameter family of Hamiltonian functions, a general structure theorem and the handle sliding lemma of Novikov Floer cycles over such a family and a family version of new transversality statements involving the Floer chain map, and many others. We call this chain level Floer theory as a whole the Floer mini-max theory.
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RESULT ON GENERALIZED DERIVATIONS WITH ENGEL CONDITIONS ON ONE-SIDED IDEALS
C¸aˇgri Demir,Nurcan Arga\c{c} 대한수학회 2010 대한수학회지 Vol.47 No.3
Let R be a non-commutative prime ring and I a non-zero left ideal of R. Let U be the left Utumi quotient ring of R and C be the center of U and k, m, n, r fixed positive integers. If there exists a generalized derivation g of R such that [g(xm)xn, xr]k = 0 for all x ∈ I,then there exists a ∈ U such that g(x) = xa for all x ∈ R except when R »= M2(GF(2)) and I[I, I] = 0.
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Cuiping Zhang,Jianlong Chen 대한수학회 2010 대한수학회지 Vol.47 No.3
For an endomorphism α of a ring R, we introduce the weak α-skew Armendariz rings which are a generalization of the α-skew Armendariz rings and the weak Armendariz rings, and investigate their properties. Moreover, we prove that a ring R is weak α-skew Armendariz if and only if for any n, the n × n upper triangular matrix ring Tn(R) is weak ¯α-skew Armendariz, where ¯α : Tn(R)→ Tn(R) is an extension of α. If R is reversible and α satisfies the condition that ab = 0 implies aα(b)=0 for any a, b ∈ R, then the ring R[x]/(xn) is weak ¯α-skew Armendariz,where (xn) is an ideal generated by xn, n is a positive integer and ¯α : R[x]/(xn)→ R[x]/(xn) is an extension of α. If α also satisfies the condition that αt = 1 for some positive integer t, the ring R[x] (resp,R[x; α]) is weak ¯α-skew (resp, weak) Armendariz, where ¯α : R[x] → R[x]is an extension of α.
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ON GENERALIZED (σ,ι)-DERIVATIONS Ⅱ
Raluca Mocanu,Hulya G. Inceboz 대한수학회 2010 대한수학회지 Vol.47 No.3
This paper continues a line investigation in [1]. Let A be a K-algebra and M an A/K-bimodule. In [5] Hamaguchi gave a necessary and sufficient condition for gDer(A,M) to be isomorphic to BDer(A,M). The main aim of this paper is to establish similar relationships for generalized (σ,ι)-derivations.
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Weak laws of large numbers for arrays under a condition of uniform integrability
성수학,Supranee Lisawadi,Andrei Volodin 대한수학회 2008 대한수학회지 Vol.45 No.1
For an array of dependent random variables satisfying a new notion of uniform integrability, weak laws of large numbers are obtained. Our results extend and sharpen the known results in the literature. For an array of dependent random variables satisfying a new notion of uniform integrability, weak laws of large numbers are obtained. Our results extend and sharpen the known results in the literature.
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Multigrid methods for 3$D$ $\HCurl$ problems with nonoverlapping domain decomposition smoothers
오덕순 대한수학회 2024 대한수학회지 Vol.61 No.4
We propose V--cycle multigrid methods for vector field problems arising from the lowest order hexahedral N\'{e}d\'{e}lec finite element. Since the conventional scalar smoothing techniques do not work well for the problems, a new type of smoothing method is necessary. We introduce new smoothers based on substructuring with nonoverlapping domain decomposition methods. We provide the convergence analysis and numerical experiments that support our theory.
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