فهرست مطالب
Bulletin of Iranian Mathematical Society
Volume:40 Issue: 6, 2014
- تاریخ انتشار: 1393/10/03
- تعداد عناوین: 17
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Pages 1347-1372For singularities $fin K[[x_{1},ldots,x_{n}]]$ over an algebraically closed field $K$ of arbitrary characteristic, we introduce the finite $mathcal{S}-$determinacy under $mathca {S}-$equivalence, where $mathcal{S}=mathcal{R}_{mathcal{G}},~mathcal{R}_{mathca {A}}, ~mathcal{K}_{mathcal{G}},~mathcal{K}_{mathcal{A}}$. It is proved that the finite $mathcal{R}_{mathcal{G}}(mathcal{K}_{mathcal{G}})-$determinacy is equivalent to the finiteness of the relative $mathcal{G}-$Milnor ($mathcal{G}-$Tjurina) number and the finite $mathcal{R}_{mathcal{A}}(mathcal{K}_{mathcal{A}})-$determinacy is equivalent to the finiteness of the relative $mathcal{A}-$Milnor ($mathcal{A}-$Tjurina) number. Moreover, some estimates are provided on the degree of the $mathcal{S} $determinacy in positive characteristic.Keywords: Finite $mathcal{R}, {mathcal{G}}~(mathcal{R}, {mathcal{A}}), $determinacyý, ýfinite $mathcal{K}, {mathcal{G}}~(mathcal{K}, {mathcal{A}}), $ determinacyý, ýthe relative $mathcal{G}(mathcal{A}), $Milnor numberý, ýrelative $mathcal{G}(mathcal{A}), $ Tjurina numberý
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Pages 1373-1385In this paper, we study translation invariant surfaces in the 3-dimensional Heisenberg group $rm Nil_3$. In particular, we completely classify translation invariant surfaces in $rm Nil_3$ whose position vector $x$ satisfies the equation $Delta x = Ax$, where $Delta$ is the Laplacian operator of the surface and $A$ is a $3 times 3$-real matrix.Keywords: Heisenberg groupý, ýfinite type surfaceý, ýinvariantý ýsurfaceý
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Pages 1387-1401A $mu$-way $(v,k,t)$ $trade$ of volume $m$ consists of $mu$ disjoint collections $T_1$, $T_2, dots T_{mu}$, each of $m$ blocks, such that for every $t$-subset of $v$-set $V$ the number of blocks containing this t-subset is the same in each $T_i (1leq ileq mu)$. In other words any pair of collections ${T_i,T_j}$, $1leq iIn this paper we investigate the existence of $mu$-way $(v,k,t)$ trades and prove the existence of: (i)~3-way $(v,k,1)$ trades (Steiner trades) of each volume $m,mgeq2$. (ii) 3-way $(v,k,2)$ trades of each volume $m,mgeq6$ except possibly $m=7$. We establish the non-existence of 3-way $(v,3,2)$ trade of volume 7. It is shown that the volume of a 3-way $(v,k,2)$ Steiner trade is at least $2k$ for $kgeq4$. Also the spectrum of 3-way $(v,k,2)$ Steiner trades for $k=3$ and 4 are specified.Keywords: $mu$, way $(v, k, t)$ tradeý, ý3, way $(v, 2)$ tradeý, ýone, solelyý
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Pages 1403-1411Universally prestarlike functions of order $alphaleq 1$ in the slit domain $Lambda=mathbb{C}setminus [1,infty)$ have been recently introduced by S. Ruscheweyh.This notion generalizes the corresponding one for functions in the unit disk $Delta$ (and other circular domains in $mathbb{C}$). In this paper, we obtain the Fekete-Szegö coefficient functional for transforms of such functions.Keywords: restarlike functionsý, ýuniversallyý ýprestarlike functionsý, ýFekete, Szeg{o}ý ýfinctionalý
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Pages 1413-1431Let $G$ be a finite group which is not a cyclic $p$-group, $p$ a prime number. We define an undirected simple graph $Delta(G)$ whose vertices are the proper subgroups of $G$, which are not contained in the Frattini subgroup of $G$ and two vertices $H$ and $K$ are joined by an edge if and only if $G=langle H, Krangle$. In this paper we classify finite groups with planar graph. %For this, by Kuratowski's Theorem, we have to study subdivisions %of the Kuratowski graphs $K_{3, 3}$ and $K_5$ in the graph $Delta(G)$. Our result shows that only few groups have planar graphs.Keywords: Graph on groupý, ýplannar graphý, ýfinite groupý
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Pages 1433-1439In this paper, we consider a general integral operator $G_n(z).$ The main object of the present paper is to study some properties of this integral operator on the classes $mathcal{S}^{*}(alpha),$ $mathcal{K}(alpha),$ $mathcal{M}(beta),$ $mathcal{N}(beta $ and $mathcal{KD}(mu,beta).$Keywords: Analytic functionsý, ýintegral operatorý, ýstarlike functionsý, ýconvex functionsý
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Pages 1441-1451Let $R$ be a domain with quotiont field $K$, and let $N$ be a submodule of an $R$-module $M$. We say that $N$ is powerful (strongly primary) if $x,yin K$ and $xyMsubseteq N$, then $xin R$ or $yin R$ ($xMsubseteq N$ or $y^nMsubseteq N$ for some $ngeq1$). We show that a submodule with either of these properties is comparable to every prime submodule of $M$, also we show that an $R$-module $M$ admits a powerful submodule if and only if it admits a strongly primary submodule. Finally we study finitely generated torsion free modules over domain each of whose prime submodules are strongly primary.Keywords: Prime submoduleý, ýstrongly prime submoduleý, primary submoduleý, ýpower submoduleý
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Pages 1453-1468We show that the character space of the vector-valued Lipschitz algebra $Lip^{alpha}(X, E)$ of order $alpha$ is homeomorphic to the cartesian product $Xtimes M_E$ in the product topology, where $X$ is a compact metric space and $E$ is a unital commutative Banach algebra. We also characterize the form of each character on $Lip^{alpha}(X, E)$. By appealing to the injective tensor product, we then identify the character space of the vector-valued polynomial Lipschitz algebra $Lip_P^{alpha}(X, E)$, generated by the polynomials on the compact space $Xsubseteq Bbb{C}^{n}$. It is also shown that $Lip_P^{alpha}(X, E)$ is the injective tensor product $Lip_P^{alpha}(X)widehat{otimes}_epsilon E$. Finally, we characterize the form of each character on $Lip_{P}^{alpha}(X, E)$.Keywords: Vector, valued Lipschitz algebras, character space, injective tensor product, polynomial approximation
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Pages 1469-1478In the present paper, we introduce the concept of generalized multivalued $F$ -contraction mappings and give a fixed point result, which is a proper generalization of some multivalued fixed point theorems including Nadlers.Keywords: Fixed point, Multivalued map, generalized F, contraction
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Pages 1479-1489For any $kin mathbb{N}$, the $k$-subdivision of graph $G$ is a simple graph $G^{frac{1 {k}}$, which is constructed by replacing each edge of $G$ with a path of length $k$. In [Moharram N. Iradmusa, On colorings of graph fractional powers, Discrete Math., (310) 2010, No. 10-11, 1551-1556] the $m$th power of the $n$-subdivision of $G$ has been introduced as a fractional power of $G$, denoted by $G^{frac{m}{n}}$. In this regard, we investigate domination number and independent domination number of fractional powers of graphs.Keywords: Domination number_Subdivision of a graph_Power of a graph
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Pages 1491-1504Let $f$ be a proper $k$-coloring of a connected graph $G$ and $Pi=(V_1,V_2,ldots,V_k)$ be an ordered partition of $V(G)$ into the resulting color classes. For a vertex $v$ of $G$, the color code of $v$ with respect to $Pi$ is defined to be the ordered $k$-tuple $c_{{}_Pi}(v)=(d(v,V_1),d(v,V_2),ldots,d(v,V_k))$, where $d(v,V_i)=min{d(v,x):~xin V_i}, 1leq ileq k$. If distinct vertices have distinct color codes, then $f$ is called a locating coloring. The minimum number of colors needed in a locating coloring of $G$ is the locating chromatic number of $G$, denoted by $Cchi_{{}_L}(G)$. In this paper, we study the locating chromatic number of the join of graphs. We show that when $G_1$ and $G_2$ are two connected graphs with diameter at most two, then $Cchi_{{}_L}(G_1vee G_2)=Cchi_{{}_L}(G_1)+Cchi {{}_L}(G_2)$, where $G_1vee G_2$ is the join of $G_1$ and $G_2$. Also, we determine the locating chromatic number of the join of paths, cycles and complete multipartite graphs.Keywords: Locating coloringý, ýlocating chromatic numberý, ýfaný, ýwheelý, ýjoiný
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Pages 1505-1514In this paper we give a characterization of unmixed tripartite graphs under certain conditions which is a generalization of a result of Villarreal on bipartite graphs. For bipartite graphs two different characterizations were given by Ravindra and Villarreal. We show that these two characterizations imply each other.Keywords: Well, covered graphý, ýunmixed graphý, ýperfect matchingý
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Pages 1515-1526In this paper, we mainly investigate how the generalized metrizability properties of the remainders affect the metrizability of rectifiable spaces, and how the character of the remainders affects the character and the size of a rectifiable space. Some results in [A. V. Arhangelskii and J. Van Mill, On topological groups with a first-countable remainder, Topology Proc. 42 (2013) 157--163.] and [F. C. Lin, C. Liu, S. Lin, A note on rectifiable spaces, Topology Appl. 159 (2012), no. 8, 2090--2101.] are improved, respectively.Keywords: Rectifiable spaceý, ýsymmetrizable spaceý, ýcharacterý
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Pages 1527-1538We present a characterization of Arens regular semigroup algebras $ell^1(S)$, for a large class of semigroups. Mainly, we show that if the set of idempotents of an inverse semigroup $S$ is finite, then $ell^1(S)$ is Arens regular if and only if $S$ is finite.Keywords: Arens regularityý, ýcompletely simple semigroupý, ýinverseý ýsemigroupý, ýleft (right) groupý, ýweaklyý
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Pages 1539-1551In this paper we obtain coefficient characterization, extreme points and distortion bounds for the classes of harmonic $p-$valent functions defined by certain modified operator. Some of our results improve and generalize previously known results.Keywords: Analytic functionsý, ýharmonic functionsý, ýextreme pointsý, ýdistortion boundsý
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Pages 1553-1571In this paper, the maximal dissipative extensions of a symmetric singular 1D discrete Hamiltonian operator with maximal deficiency indices (2,2) (in limit-circle cases at ±∞) and acting in the Hilbert space ℓ_{Ω}²(Z;C²) (Z:={0,±1,±2,...}) are considered. We consider two classes dissipative operators with separated boundary conditions both at -∞ and ∞. For each of these cases we establish a selfadjoint dilation of the dissipative operator and construct the incoming and outgoing spectral representations that makes it possible to determine the scattering function (matrix) of the dilation. Further a functional model of the dissipative operator and its characteristic function in terms of the Weyl function of a selfadjoint operator are constructed. Finally we show that the system of root vectors of the dissipative operators are complete in the Hilbert space ℓ_{Ω}²(Z;C²).Keywords: Discrete Hamiltonian system, dissipative operator, selfadjoint dilation, characteristic function, completeness
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Pages 1573-1585Let $G$ be a finite group. A subset $X$ of $G$ is a set of pairwise non-commuting elements if any two distinct elements of $X$ do not commute. In this paper we determine the maximum size of these subsets in any finite non-abelian metacyclic $2$-group and in any finite non-abelian $p$-group with an abelian maximal subgroup.Keywords: Metacyclic $p$, groupý, powerful 2, groupý, coveringý, pairwiseý ýnon, commuting elementsý