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The commuting graph of an arbitrary ring R, denoted by ( R), is a graph whose vertices are all non-central elements of R, and two distinct vertices a and b are adjacent if and only if ab = ba. In this paper, we investigate the connectivity, the diameter, the maximum degree and the minimum degree of the commuting graph of group ring ZnQ8. The main result is that ( ZnQ8) is connected if and only if n is not a prime. If ( ZnQ8) is connected, then diam(ZnQ8)= 3, while ( ZnQ8) is disconnected then every connected component of ( ZnQ8) must be a complete graph with a same size. Further, we obtain the degree of every vertex in ( ZnQ8), the maximum degree and the minimum degree of ( ZnQ8).

In 2003, Marques-Smith and Sullivan described the join Ω of thenatural order' and thecontainment order' on P ( X), the semigroup under composition of all partial transformations of a set X. And, in 2004, Pinto and Sullivan described all automorphisms of PS( q), the partial Baer-Levi semigroup consisting of all injective 2 P ( X) such that jX n X j = q, where @0 q j Xj. In this paper, we describe the group of automorphisms of R( q), the largest regular subsemigroup of PS( q). In 2010, we studied some properties of and on PS( q). Here, we characterize the meet and join under those orders for elements of R( q) and PS( q). In addition, since does not equal Ω on I( X), the symmetric inverse semigroup on X, we formulate an algebraic version of

In the present paper we describe a class of Gorenstein, �nite and at morphism ϱ : X ! Y of degree 6 of algebraic varieties, called Anglo{American covers. We prove a general Bertini theorem for them and we give some evidence that the cover ϱ : A ! P2 associated general (1 ; 3){polarized abelian surface is Anglo{American.

In this paper we give asymptotic formulas for the number of l-cyclic extensions of the rational function eld k = F q(T ) with pre- scribed l-class numbers inside some cyclotomic function elds, and den- sity results for l-cyclic extensions of k with certain properties on the ideal class groups.

Assume G is a finite p-group and i is a fixed positive integer. In this paper, finite p-groups G with for all nonnormal subgroups H are classified up to isomorphism. As a corollary, this answer Problem 116(i) proposed by Y. Berkovich in his book "Groups of Prime Power Order Vol. I" in 2008.

Let {, } be a sequence of i.i.d. nondegenerate random variables which is in the domain of attraction of the normal law with mean zero and possibly infinite variance. Denote , and . Then for d > -1, we showed that under some regularity conditions, holds in this paper, where If g denotes the indicator function.

Let R be a ring and M a right R-module, S = (M). The module M is called almost principally small injective (or APS-injective for short) if, for any a J(R), there exists an S-submodule of M such that (a) = Ma as left S-modules. If is a APS-injective module, then we call R a right APS-injective ring. We develop, in this paper, APS-injective rings as a generalization of PS-injective rings and AP-injective rings. Many examples of APS-injective rings are listed. We also extend some results on PS-injective rings and AP-injective rings to APS-injective rings.

Silverman (14) proved a height inequality for a jointly regular family of rational maps and the author (10) improved it for a jointly reg- ular pair. In this paper, we provide the same improvement for a jointly regular family: let h : P n ! R be the logarithmic absolute height on the projective space, let r(f) be the D-ratio of a rational map f which is dened in (10) and let ff1;:::;fk j fl : A n ! A ng be a nite set of poly- nomial maps which is dened over a number eld K. If the intersection of the indeterminacy loci of f1;:::;fk is empty, then there is a constant C such that k ∑ l=1 1 degfl h ( fl(P) ) > ( 1 + 1 r ) f(P) C for all P 2 A n

The main object of this paper is to prove several inclusion re-lations associated with ( j; )-neighborhoods of various subclasses de nedby Salagean operator by making use of the familiar concept of neigh-borhoods of analytic functions. Special cases of some of these inclusionrelations are shown to yield known results. 1. IntroductionLet T ( j ) denote the class of functions of the form:(1.1) f ( z ) = z ∑ 1k = j +1 a k z k ( a k 0; j 2 N = f 1 ; 2 ;:::g )which are analytic in the open unit disc U = fz : jzj < 1 g . Following ([7] and[10]), we de ne the ( j; )-neighborhood of a function f ( z ) 2 T ( j ) by(1.2) N j; ( f ) =8

In [6, 8], we showed that any ideal of is generated by at most two polynomials of the `standard' forms when l is even. The purpose of this paper is to find the `standard' generators of the cyclic codes over of length a multiple of p, namely the ideals of with an integer l which is a multiple of p. We also find an explicit description of their duals in terms of the generators when a = 2.

Cho and Kim (4) and Kim (6) introduced the concept of the competition index of a digraph. Cho and Kim (4) and Akelbek and Kirk- land (1) also studied the upper bound of competition indices of primitive digraphs. In this paper, we study the upper bound of competition in- dices of strongly connected digraphs. We also study the relation between competition index and ordinary index for a symmetric strongly connected digraph.

The ff bracket skein module K t ( M ) of a 3-manifold M becomes an algebra for t = 1. We prove that this algebra has nonon-trivial nilpotent elements for M being the exterior of the twist knotin 3-sphere and, therefore, it is isomorphic to the SL 2 (C)-character ringof the fundamental group of M . Our proof is based on some propertiesof Chebyshev polynomials. 1. IntroductionIn this paper, we show a property of the ff bracket skein module(KBSM for short) by using polynomials S n ( z ) ( n 2 Z) in an indeterminate z de ned by the following recursive relation: S n +2 ( z ) = zS n +1 ( z ) S n ( z ) ; S 1 ( z ) = z; S 0 ( z ) = 1 : The polynomial S n ( z ) can be transformed into the Chebyshev polynomial of thesecond type U n ( z ) de ned by U n +2 ( z ) = 2 zU n +1 ( z ) U n ( z ) ; U 1 ( z ) = 2 z; U 0 ( z ) = 1 : Indeed, U n ( z ) = S n (2 z ) holds for any n 0.In Theorem 2.1 of [3], Bullock and LoFaro found that the KBSM K t ( E K m )of the exterior E K m of an m-twist knot K

In this paper, we dene an Lp analytic generalized Fourier- Feynman transform and a convolution product of functionals in a Ba- nach algebra F ( Ca;b(0 ;T )) which is called the Fresnel type class, and in more general class FA1;A2 of functionals dened on general function space Ca;b(0 ;T ) rather than on classical Wiener space. Also we obtain some relationships between the Lp analytic generalized Fourier-Feynman transform and convolution product for functionals in F ( Ca;b(0 ;T )) and in FA1;A2 .

In this paper, we establish a Marcinkiewicz-Zygmund type strong law for sequences of blockwise and pairwise m-dependent random variables. The sharpness of the results is illustrated by an example.

In this paper, the connection of the inverse transversal with the adequate transversal is explored. It is proved that if S is an abundant semigroup with an adequate transversal , then S is regular if and only if is an inverse semigroup. It is also shown that adequate transversals of a regular semigroup are just its inverse transversals. By means of a quasi-adequate semigroup and a right normal band, we construct an abundant semigroup containing a quasi-ideal S-adequate transversal and conversely, every such a semigroup can be constructed in this manner. It is simpler than the construction of Guo and Shum [9] through an SQ-system and the construction of El-Qallali [5] by W(E, S).

We improve some inequalities of Ostrowski-like type and further generalize them.Key words: Inequality, Error, Integral, Taylor, Ostrowski2000 MSC: 26D10, 41A55, 65D301. IntroductionIn 1938, Ostrowski [8] proved the following interesting integral inequality which has received con-siderable attention from many researchers.Theorem 1 (See [8]). Let f : [a,b] → R be continuous on [a,b] and differentiable on (a,b) whosederivative function f 0 : (a,b) → R is bounded on (a,b), i.e., kf 0 k ∞ = sup t∈(a,b) |f 0 (t)| < ∞. ThenZb f(x)−1b−a af(t)dt ∞5 14+x− a+b22 (b−a) 2 !(b−a)kf 0 k (1)for all x ∈ [a,b].This inequality gives an upper bound for the approximation of the integral average 1b−a R ba f(t)dtby the value f(x) at point x ∈ [a,b]. The first generalization of Ostrowskis inequality was given byG.V. Milovanovi´c and J.E. Peˇcari´c in [7]. However, note that estimate (1) can be applied only if f 0 is bounded. In the first part of this paper, we will improve (1) by assuming f 0 ∈ L p

In the present paper sufficient conditions, in terms of hyper- geometric inequalities, are found so that the Hohlov operator preserves a certain subclass of close-to-convex functions (denoted by R ( A;B)) and transforms the classes consisting of k-uniformly convex functions, k-starlike functions and univalent starlike functions into R ( A;B).

In the present paper we introduce the lower autocentral seriesof autocommutator subgroups of a given group. Following our previouswork on the subject in 2009, it is shown that every nite abelian groupis isomorphic with n th -term of the lower autocentral series of some niteabelian group. 1. IntroductionLet A = Aut( G ) denote the group of automorphisms of a given group G . Forany element g 2 G and 2 A the element [ g; ] = g 1 g is an autocommutator of g and : We de ne the autocommutator of higher weight inductively asfollows:[ g; 1 ; 2 ;:::; i ] = [[ g; 1 ; 2 ;:::; i 1 ] ; i ]for all 1 ; 2 ;:::; i 2 A: So the autocommutator subgroup of weight i + 1 is de ned in the followingway: K i ( G ) = [ G;A;:::;A | {z } i -times ] = ⟨ [ g; 1 ; 2 ;:::; i ] j g 2 G; 1 ; 2 ;:::; i 2 A⟩: Clearly K i ( G ) is a characteristic subgroup of G for all i 1. Therefore, oneobtains a descending chain of autocommutator subgroups of G as follows: G K 1 ( G ) K 2 ( G ) K i ( G ) ; which we may call it the

Let M be a compact orientable closed 3-manifold, and F a non-separating incompressible closed surface in M. Let M ' = M (F), where (F) is an open regular neighborhood of F in M. In the paper, we give a lower bound of genus of self-amalgamation of minimal Heegaard splitting V ' ( SW ' of M ' under some conditions on the distance of the Heegaard splitting.

Let {, } be a sequence of asymptotically almost negatively associated random variables and . In the paper, we get the precise results of Hjek-Rnyi type inequalities for the partial sums of asymptotically almost negatively associated sequence, which generalize and improve the results of Theorem 2.4-Theorem 2.6 in Ko et al. ([4]). In addition, the large deviation of for sequence of asymptotically almost negatively associated random variables is studied. At last, the Marcinkiewicz type strong law of large numbers is given.

The main purpose of this paper is using the elementary andanalytic methods to study the properties of the 2 k -th power mean valueof the generalized quadratic Gauss sums, and give two exact mean valueformulae for k = 3 and 4. 1. IntroductionLet q 2 be an integer, ˜ denotes a Dirichlet character modulo q . For anyinteger n , we de ne the generalized quadratic Gauss sums G ( n;˜ ; q ) as follows: G ( n;˜ ; q ) =∑ qa =1 ˜ ( a ) e ( na 2 q ) ; where e ( y ) = e 2 ˇiy . This sum is important, because it is a generalization ofthe classical quadratic Gauss sums G ( n;q ), which is de ned by G ( n ; q ) =∑ qa =1 e ( na 2 q ) : About the properties of G ( n;˜ ; q ), some authors had studied it, and obtainedmany interesting results. For example, for any integer n with ( n;q ) = 1, fromthe general result of Cochrane and Zheng [2] we can deduce that jG ( n;˜ ; q ) j 2 ! ( q ) q 12 ; where ! ( q ) denotes the number of all distinct prime divisors of q . The casewhere q is a prime is due to Weil [4]. Zhang [5] proved that for any odd prime

In this paper we study some properties of rank one perturba- tions of the unilateral shift operators T = S+u v. In particular, we give some criteria for eigenvalues of T. Also we characterize some conditions for T to be hyponormal.

Let f be a function which assigns a positive integer f(v) to each vertex v V (G), let r, s and t be non-negative integers. An f-coloring of G is an edge-coloring of G such that each vertex v V (G) has at most f(v) incident edges colored with the same color. The minimum number of colors needed to f-color G is called the f-chromatic index of G and denoted by (G). An [r, s, t; f]-coloring of a graph G is a mapping c from V(G) E(G) to the color set C = {0, 1, ; k - 1} such that |c() - c( )| r for every two adjacent vertices and , |c( - c()| s and f() for all V (G), C where denotes the number of -edges incident with the vertex and , are edges which are incident with but colored with different colors, |c()-c()| t for all pairs of incident vertices and edges. The minimum k such that G has an [r, s, t; f]-coloring with k colors is defined as the [r, s, t; f]-chromatic number and denoted by (G). In this paper, we present some general bounds for [r, s, t; f]-coloring firstly. After that, we obtain some important properties under the restriction min{r, s, t} = 0 or min{r, s, t} = 1. Finally, we present some problems for further research.

The notion of the generalized Fibonacci matrix F (a;b;s) n of type s, whose nonzero elements are generalized Fibonacci numbers, is introduced in the paper (23). Regular case s = 0 is investigated in (23). In the present article we consider singular case s = 1. Pseudoinverse of the generalized Fibonacci matrix F (a;b; 1) n is derived. Correlations between the matrix F (a;b; 1) n and the Pascal matrices are considered. Some combinatorial identities involving generalized Fibonacci numbers are derived. A class of test matrices for computing the Moore-Penrose inverse is presented in the last section.

In this paper, a pair of Wolfe type nondifferentiable sec- ond order symmetric minimax mixed integer dual problems is formu- lated. Symmetric and self-duality theorems are established under 1- bonvexity/ 2-boncavity assumptions. Several known results are obtained as special cases. Examples of such primal and dual problems are also given.

In this paper, we flnd the maximum exponent of D £ E, the cartesian product of two digraphs D and E on n, n + 2 vertices, respectively for an even integer n ‚ 4. We also characterize the extremal cases.

Nakaoka and Oda ([1] and [2]) introduced the notion of max-imal open sets and minimal closed sets in topological spaces. In thispaper, we introduce new classes of sets called maximal θ -open sets, min-imal θ -closed sets, θ -semi maximal open and θ -semi minimal closed andinvestigate some of their fundamental properties. 1. Introduction and preliminariesGeneralized open sets play a very important role in General Topology andthey are now the research topics of many topologists worldwide. Indeed asignificant theme in General Topology and Real Analysis concerns the variouslymodified forms of continuity, separation axioms etc by utilizing generalizedopen sets. One of the most well-known notions and also an inspiration sourceis the notion of θ -open sets introduced by N. V. Veli˘cko [3] in 1968. Since thecollection of θ -open sets in a topological space ( X,τ ) forms a topology τ θ on X then the union of two θ -open sets is of course θ -open. Moreover τ = τ θ if andonly if ( X,τ ) is regular.F. Nakaoka and N. Oda in [1] and [2] introduced the notion of maximal opensets and minimal closed sets. The purpose of the present paper is to introducethe concept of a new class of open sets called maximal

In this paper we propose new primal-dual interior point meth- ods (IPMs) for P⁄(•) linear complementarity problems (LCPs) and an- alyze the iteration complexity of the algorithm. New search directions and proximity measures are defined based on a class of kernel functions, ˆ(t) = t 2 i1 2 i R t 1 e q 1 i1 d», q ‚ 1. If a strictly feasible starting point

Let D be an integrally closed domain with quotient field K, ⁄ be a star operation on D, X,Y be indeterminates over D, N⁄ = {f 2 D(X)| (cD(f)) ⁄ = D} and R = D(X)N⁄. Let b be the b-operation on R, and let ⁄c be the star operation on D defined by I ⁄c = (ID(X)N⁄) b \K. Finally, let Kr(R,b) (resp., Kr(D,⁄c)) be the Kronecker function ring of R (resp., D) with respect to Y (resp., X,Y ). In this paper, we show that Kr(R,b) µ Kr(D,⁄c) and Kr(R,b) is a kfr with respect to K(Y ) and X in the notion of (2). We also prove that Kr(R,b) = Kr(D,⁄c) if and only if D is a P⁄MD. As a corollary, we have that if D is not a P⁄MD, then Kr(R,b) is an example of a kfr with respect to K(Y ) and X but not a Kronecker function ring with respect to K(Y ) and X.

In this paper, strongly cotorsion (torsion-free) modules are studied and strongly cotorsion (torsion-free) dimension is introduced. It is shown that every module has a special SCn-preenvelope and an ST Fn- cover for any n 2 N based on some results of cotorsion pairs from (9). Some characterizations of strongly cotorsion (torsion-free) dimension of a module are given.

For convex subsets X of a topological vector space E, we show that a KKM principle implies a Fan-Browder type fixed point theo- rem and that this theorem implies generalized forms of the Sion minimax theorem. The von Neumann-Sion minimax theorem is fundamental in convex analysis and in game theory. von Neumann (8) proved his theorem for simplexes by reducing the problem to the 1-dimensional cases. Sion's generalization (7) was proved by the aid of Helly's theorem and the KKM theorem due to Knaster, Kuratowski, and Mazurkiewicz (5). In a recent paper, Kindler (4) proved Sion's theorem by applying the 1-dimensional KKM theorem (i.e., every interval in R is connected), the 1-dimensional Helly theorem (i.e., any family of pairwise intersecting compact intervals in R has nonempty intersection), and Zorn's lemma (or other method). In this short note, for convex subsets X of a topological vector space E, we show that a KKM principle implies a Fan-Browder type fixed point theorem and that this theorem implies a generalized form of the Sion minimax theorem. Definition. If a multimap G : X( X satisfies

Let Z n be the Cartesian product of the set of integers Z and let (Z,T) and (Z n ,T n ) be the Khalimsky line topology on Z and the Khalimsky product topology onZ n , respectively. Then for a set X ‰Z n , consider the subspace (X,T n X) induced from (Z n ,T n ). Considering a k-adjacency on (X,T n X), we call it a (computer topological) space with k-adjacency and use the notation (X,k,Tn X ) := Xn,k. In this paper we

P. M. Cohn called a ring R reversible if whenever ab = 0, then ba = 0 for a,b 2 R. Commutative rings and reduced rings are reversible. In this paper, we extend the reversible condition of a ring as follows: Let R be a ring and fi an endomorphism of R, we say that R is right (resp., left) fi-shifting if whenever afi(b) = 0 (resp., fi(a)b = 0) for a,b 2 R, bfi(a) = 0 (resp., fi(b)a = 0); and the ring R is called fi-shifting if it is both left and right fi-shifting. We investigate characterizations of fi- shifting rings and their related properties, including the trivial extension, Jordan extension and Dorroh extension. In particular, it is shown that for an automorphism fi of a ring R, R is right (resp., left) fi-shifting if and only if Q(R) is right (resp., left) ¯ fi-shifting, whenever there exists the classical right quotient ring Q(R) of R.

In this paper we give a new characterization of real hyper- surfaces of type B, that is, a tube over a totally geodesicQP n in complex two-plane Grassmannians G2(C m+2 ), where m = 2n, with the Reeb vec- tor » belonging to the distribution D, where D denotes a subdistribution in the tangent space TxM such that TxM = D'D? for any point x 2 M and D? = Span{»1,»2,»3 }.

In this paper, new q-analogs of Genocchi numbers and poly- nomials are defined. Some important arithmetic and combinatoric rela- tions are given, in particular, connections with q-Bernoulli numbers and polynomials are obtained.

In (6) and (7), we introduced graph von Neumann algebras which are the (groupoid) crossed product algebras of von Neumann al- gebras and graph groupoids via groupoid actions. We showed that such crossed product algebras have the graph-depending amalgamated reduced free probabilistic properties. In this paper, we will consider a scalar- valued W ⁄ -probability on a given graph von Neumann algebra. We show that a diagonal graph W ⁄ -probability space (as a scalar-valued W ⁄ - probability space) and a graph W ⁄ -probability space (as an amalgamated W ⁄ -probability space) are compatible. By this compatibility, we can find the relation between amalgamated free distributions and scalar-valued free distributions on a graph von Neumann algebra. Under this compat- ibility, we observe the scalar-valued freeness on a graph von Neumann algebra.

Let ae be an endomorphism and I a ae-ideal of a ring R. Pear- son and Stephenson called I a ae-semiprime ideal if whenever A is an ideal of R and m is an integer such that Aae t (A) µ I for all t ‚ m, then A µ I, where ae is an automorphism, and Hong et al. called I a ae-rigid ideal if aae(a) 2 I implies a 2 I for a 2 R. Notice that R is called a ae-semiprime ring (resp., a ae-rigid ring) if the zero ideal of R is a ae-semiprime ideal (resp., a ae-rigid ideal). Every ae-rigid ideal is a ae-semiprime ideal for an automorphism ae, but the converse does not hold, in general. We, in this paper, introduce the quasi ae-rigidness of ideals and rings for an automor- phism ae which is in between the ae-rigidness and the ae-semiprimeness, and study their related properties. A number of connections between the quasi ae-rigidness of a ring R and one of the Ore extension R(x;ae,-) of R are also investigated. In particular, R is a (principally) quasi-Baer ring if and only if R(x;ae,-) is a (principally) quasi-Baer ring, when R is a quasi ae-rigid ring. 1. Definitions Let ae be an endomorphism of a ring R, the additive map - : R ! R is called a ae-derivation if -(ab) = -(a)b+ae(a)-(b) for any a,b 2 R. For a ring R with an endomorphism ae of R and a ae-derivation -, the Ore extension R(x; ae, -) of R is the ring obtained by giving the polynomial ring over R with new multiplication: xr = ae(r)x+-(r) for all r 2 R. If - = 0, we write R(x;ae) for R(x;ae,0) and it is called the skew polynomial ring (or, an Ore extension of endomorphism type); while R((x;ae)) is called a skew power series ring. An endomorphism ae of a ring R is called rigid (17) if aae(a) = 0 implies a = 0 for a 2 R. A ring R is called a ae-rigid ring (9) if there exists a rigid endomorphism ae of R. The Ore extension R(x;ae,-) of R is reduced (i.e., it has no nonzero nilpotent elements) and ae is a monomorphism if and only if R is a ae-rigid ring if and only if R(x;ae) is reduced by (9, Proposition 5) and (10, Proposition 3), respectively. Hence, ae-rigid rings are reduced rings, but there exists an endomorphism ae of a commutative reduced ring which is not a ae-rigid

In this paper we will prove that the groups Dp+1(2) and Dp+1(3), where p is an odd prime number, are uniquely determined by their sets of order components. A main consequence of our result is the validity of Thompson's conjecture for the groups Dp+1(2) and Dp+1(3).

is the meaningful smallest product of three chains of each size 2n+1 since is a 1-element poset. The linear discrepancy of the product of three chains is found as . But the case of the product of three chains is not known yet. In this paper, we determine ld as a case to determine the linear discrepancy of the product of three chains of each size 2n + 1.

In this paper, we prove the generalized Hyers-Ulam stability of bi-homomorphisms in -ternary algebras and of bi-derivations on -ternary algebras for the following bi-additive functional equation f(x + y, z - w) + f(x - y, z + w) = 2f(x, z) - 2f(y, w). This is applied to investigate bi-isomorphisms between -ternary algebras.

If f is M-harmonic and integrable with respect to a weighted radial measure "fi over the unit ball Bn of C n , then R Bn (f - ˆ) d"fi = f(ˆ(0)) for every ˆ 2 Aut(Bn). Equivalently f is fixed by the weighted Berezin transform; Tfif = f. In this paper, we show that if a function f defined on Bn satisfies R(f - `) 2 L 1 (Bn) for every ` 2 Aut(Bn) and Sf = rf for some |r| = 1, where S is any convex combination of the iterations of Tfi 0 s, then f is M-harmonic.

In this paper, we give topological properties of collection of prime ideals in 2-primal near-rings. We show that Spec(N), the spectrum of prime ideals, is a compact space, and Max(N), the maximal ideals of N, forms a compact T1-subspace. We also study the zero-divisor graph iI(R) with respect to the completely semiprime ideal I of N. We show that iP(R), where P is a prime radical of N, is a connected graph with diameter less than or equal to 3. We characterize all cycles in the graph iP(R).

Given two integral self-reciprocal polynomials having the sa-me modulus at a point z 0 on the unit circle, we show that the minimalpolynomial of z 0 is also self-reciprocal and it divides an explicit integralself-reciprocal polynomial. Moreover, for any two integral self-reciprocalpolynomials, we give a sufficient condition for the existence of a point z 0 on the unit circle such that the two polynomials have the same modulusat z 0 . 1. Introduction and statement of resultsThroughout this paper, U denotes the unit circle and n is a positive integer.A polynomial P ( z ) = a n z n + a n− 1 z n− 1 + ··· + a 0 is said to be a self-reciprocalpolynomial of degree n if it satisfies a n 6= 0 and P ( z ) = z n P (1 /z ). Thusthe zeros of a self-reciprocal polynomial either lie on the unit circle or aresymmetricwith respect to U . There havebeen a number of interestingproblems(for example [2]) about the distribution of zeros of self-reciprocal polynomials.Also the minimal polynomial of an algebraic number

The main purpose of this paper is to deal with the unique- ness of meromorphic functions sharing sets concerning small functions. We obtain two main theorems which improve and extend strongly some results due to R. Nevanlinna, Li-Qiao, Yao, Yi, Thai-Tan, and Cao-Yi. It is well known that two nonconstant polynomials f and g over an algebraic closed field of characteristic zero are identical if there exist two distinct values a and b such that f(x) = a if and only if g(x) = a and f(x) = b if and only if g(x) = b. In 1926, R. Nevanlinna (3) proved his famous five-value theorem that for two nonconstant meromorphic functions f and g on the whole complex plane C, if they have the same inverse images (ignoring multiplicities) for five distinct values, then f(z) · g(z). After this very work, the uniqueness of meromorphic functions with shared values onC attracted many investigations (for references, see (10)). It is very interesting to consider distinct small functions instead of distinct complex numbers on C. Over the last few years, there were several generaliza- tions of Nevanlinna's five-value theorem. To state some of these results, we must introduce some notions. Let h be a nonzero holomorphic function on C, expanding h as h(z) = P1 i=0 bi(ziz0) i around z0, then we define "h(z0) := min{i : bi 6 0}. Let k and M be positive integers or +1. We set " M h (z) = min{M,"h(z)}

In this paper, we shall show that for any entire function f, the function of the form f m (f n i 1)f 0 has no non-zero finite Picard value for all positive integers m, n 2 N possibly except for the special case m = n = 1. Furthermore, we shall also show that for any two non- constant meromorphic functions f and g, if f m (f n i1)f 0 and g m (g n i1)g 0 share the value 1 weakly, then f · g provided that m and n satisfy some conditions. In particular, if f and g are entire, then the restrictions on m and n could be greatly reduced.

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A Roman k-dominating function on G is a function f : V (G) ! {0,1,2} such that every vertex u for which f(u) = 0 is adjacent to at least k vertices v1,v2,...,vk with f(vi) = 2 for i = 1,2,...,k. The weight of a Roman k-dominating function is the value f(V (G)) = u2V (G) f(u). The minimum weight of a Roman k-dominating function on a graph G is called the Roman k-domination number ∞kR(G) of G. Note that the Roman 1-domination number ∞1R(G) is the usual Roman domination number ∞R(G). In this paper, we investigate the properties of the Roman k-domination number. Some of our results extend these one given by Cockayne, Dreyer Jr., S. M. Hedetniemi, and S. T. Hedetniemi (2) in 2004 for the Roman domination number. 1. Terminology and introduction We consider finite, undirected and simple graphs G with vertex set V (G) and edge set E(G). The number of vertices |V (G)| of a graph G is called the order of G and is denoted by n = n(G). The open neighborhood N(v) = NG(v) of a vertex v consists of the vertices adjacent to v and d(v) = dG(v) = |N(v)| is the degree of v. The closed neigh- borhood of a vertex v is defined by N(v) = NG(v) = N(v)({v}. The maximum degree of a graph G is denoted by ¢(G) = ¢. For a subset S µ V (G), we define N(S) = NG(S) = S v2S N(v), N(S) = NG(S) = N(S) ( S, and G(S) is the subgraph induced by S. The complement of a graph G is denoted by G. If !(G) is the number of components of G and m(G) = |E(G)|, then c(G) = m(G) i n(G) + !(G) is the well-known cyclomatic number of G. A graph is a cactus graph if all its cycles are edge-disjoint. We write Kn for the complete graph of order n, and Kp,q for the complete bipartite graph with bipartition X,Y such that |X| = p and |Y | = q.

In this paper, we are concerned with the following eigenvalue problems of m-point boundary value problem for p-Laplacian dynamic equation on time scales 'p(u ¢ (t)) r + ‚h(t)f(u(t)) = 0, t 2 (0,T), u(0) = 0, 'p u ¢ (T) = mi2 i=1 ai'p u ¢ (»i) , where 'p(u) = |u| pi2 u,p > 1 and ‚ > 0 is a real parameter. Under certain assumptions, some new results on existence of one or two posi- tive solution and nonexistence are obtained for ‚ evaluated in dierent intervals. Our work develop and improve many known results in the literature even for the continual case. In doing so the usual restriction that f0 = limu!0+ f(u)/'p(u) and f1 = limu!1 f(u)/'p(u) exist is removed. As an applications, an example is given to illustrate the main results obtained.

It is well-known that if a convex hyperbolic polygon is con-structed as a fundamental domain for a subgroup of the modular group,then its translates by the group elements form a locally finite tessellationand its side-pairing transformations form a system of generators for thegroup. Such hyperbolically convex polygons can be obtained by usingDirichlet’s and Ford’s polygon constructions. Another method of obtain-ing a fundamental domain for subgroups of the modular group is throughthe use of a right coset decomposition and we call such domains standardfundamental domains. In this paper we give subgroups of the modulargroup which do not have hyperbolically convex standard fundamentaldomain containing only inequivalent cusps. 1. IntroductionLet Γ(1) denote the inhomogeneous modular group acting on the upper halfplane Hin the usual manner: τ 7−→ M aτ + bcτ + d, a,b,c,d ∈ Z , ad−bc = 1 . We shall denote the above mapping by its matrix form M = a bc d . Note that M and −M define the same mapping in Γ(1). The modular groupΓ(1) is generated by the transformations (see [6]

In this paper, we give a recursive formula for the Jones poly- nomial of a 2-bridge knot or link with Conway normal form C(i2n1, 2n2, i2n3,...,(i1) r 2nr) in terms of n1,n2,...,nr. As applications, we also give a recursive formula for the Jones polynomial of a 3-periodic link L (3) with rational quotient L = C(2,n1,i2,n2,..., nr,(i1) r 2) for any nonzero integers n1,n2,...,nr and give a formula for the span of the Jones polynomial of L(3) in terms of n1,n2,...,nr with ni 6 ±1 for all i = 1,2,...,r.

There are three standard weight functions on a linear code viz. Hamming weight, Lee weight, and Euclidean weight. Euclidean weight function is useful in connection with the lattice constructions (2) where the minimum norm of vectors in the lattice is related to the min- imum Euclidean weight of the code. In this paper, we obtain an upper bound over the number of parity check digits for Euclidean weight codes detecting and correcting burst errors.