Let R be a commutative ring (with 1) and let Z(R) be its set of zero-divisors. The zero-divisor graph Γ(R) has vertex set Z∗(R) = Z(R) \ {0} and for distinct x, y ∈ Z∗(R), the vertices x and y are adjacent if and only if xy = 0. In this paper, we consider the domination number and signed domination number on zero-divisor graph Γ(R) of commutative ring R such that for every 0 6= x ∈ Z∗(R), x 6= ...

For a family A and a set Z , denote {A ∈ A : A ∩ Z = ∅} by A(Z). For positive integers n and r , let Sn,r be the trivial compressed intersecting family {A ∈ ([n] r ) : 1 ∈ A}, where [n] := {1, . . . , n} and ([n] r ) := {A ⊂ [n] : |A| = r}. The following problem is considered: For r ≤ n/2, which sets Z ⊆ [n] have the property that |A(Z)| ≤ |Sn,r (Z)| for any compressed intersecting family A ⊂ (...

Here, 2 ≤ p < ∞, j : Z × R → R is a function which is measurable in z ∈ Z and locally Lipschitz in x ∈ R and ∂ j(z,x) is the Clarke subdifferential of j(z, ·). If f : Z × R → R is a measurable function which is in general discontinuous in the x ∈ R variable, for almost all z ∈ Z, all M > 0, and all |x| ≤ M, we have | f (z,x)| ≤ aM(z) with aM ∈ L1(Z) and we set j(z,x) = ∫x 0 f (z, r)dr, then j(z...

Let P := k [X,Y, Z] be a polynomial ring over an algebraic closed field k and (X, Y , Z, XY Z, XY Z ) · k [X,Y, Z] an (X,Y, Z)-primary ideal in P , (m, n, l, a, b, c, d, e, f are integers). The ideal Q=(X, Y , Z, XY Z, XY Z ) ·R is (X,Y, Z) · R-primary ideal in the local ring R = k [X,Y, Z](x,y,z). In this short note we give a formula for the calculation of Samuel multiplicity e0(Q,R) of the id...

For r ∈ (0, 1), let Z r = {x ∈ R 2 | dist(x, Z 2) > r/2} and τ r (x, v) = inf{t > 0 | x + tv ∈ ∂Z r }. Let Φ r (t) be the probability that τ r (x, v) ≥ t for x and v uniformly distributed in Z r and S 1 respectively. We prove in this paper that lim sup ǫ→0 +

Let Z = {z : Iz r < r} be a disk in the complex plane with the center ~" = mid Z E C and the radius r = rad Z > O, denoted shorter by parametric notation Z = {r r}. If f is a closed complex function then the complex-valued set f (Z) = { f ( z ) : z E Z} is closed. In general, the range f (Z) is not a disk, which is quite impractical in calculations. In order to remain in the realm of disks, it ...