‎The first variable Zagreb index of graph $G$ is defined as‎ ‎begin{eqnarray*}‎ ‎M_{1,lambda}(G)=sum_{vin V(G)}d(v)^{2lambda}‎, ‎end{eqnarray*}‎ ‎where $lambda$ is a real number and $d(v)$ is the degree of‎ ‎vertex $v$‎. ‎In this paper‎, ‎some upper and lower bounds for the distribution function and expected value of this index in random increasing trees (rec...

In this paper, an attempt is made to include division by zero in ordinary arithmetic. Counting or measuring done get the value of multiples, powers, quotients, sums and differences zeros. Zero divided taken be equal one. There are many infinities zeroes. The biggest all cannot imagined smallest also. An define a mathematical structure from set real numbers, zeros infinities. Some fallacies have...

Let G(z) be a real entire function of order less than 2 with only real zeros. Then we classify certain distributions functions F such that the convolution (G ∗ dF )(z) = ∫∞ −∞G(z − is) dF (s) has only real zeros.

Based on a new explicit upper bound for the number of zeros of exponential polynomials in a horizontal strip, we obtain a uniform upper bound for the number of zeros of solutions to an ordinary differential equation near its Fuchsian singular point, provided that any two distinct characteristic exponents at this point have distinct real parts. The latter result implies that a Fuchsian different...

It is well known that a random cosine polynomial $${V_n}\left(x \right) = \sum\nolimits_{j 0}^n {{a_j}\cos \left({jx} \right)} $$ , x ∈ (0, 2π), with the coefficients being independent and identically distributed (i.i.d.) real-valued standard Gaussian variables (asymptotically) has $$2n/\sqrt 3 expected real roots. On other hand, out of many ways to construct dependent polynomial, one force be ...