We consider the number of zeros of the integral $I(h) = oint_{Gamma_h} omega$ of real polynomial form $omega$ of degree not greater than $n$ over a family of vanishing cycles on curves $Gamma_h:$ $y^2+3x^2-x^6=h$, where the integral is considered as a function of the parameter $h$. We prove that the number of zeros of $I(h)$, for $0 < h < 2$, is bounded above by $2[frac{n-1}{2}]+1$.

we consider the number of zeros of the integral $i(h) = oint_{gamma_h} omega$ of real polynomial form $omega$ of degree not greater than $n$ over a family of vanishing cycles on curves $gamma_h:$ $y^2+3x^2-x^6=h$, where the integral is considered as a function of the parameter $h$. we prove that the number of zeros of $i(h)$, for $0 < h < 2$, is bounded above by $2[frac{n-1}{2}]+1$.

Understanding vector fields resulting from large scientific simulations is an important and often difficult task. Integral curves, curves that are tangential to a vector field at each point, are a powerful visualization method in this context. Application of integral curve-based visualization to very large vector field data represents a significant challenge due to the non-local and data-depend...

‎In the present paper‎, ‎we are going to use geometric and topological concepts‎, ‎entities and properties of the‎ ‎integral curves of linear vector fields‎, ‎and the theory of differential equations‎, ‎to establish a representation for some groups on $R^{n} (ngeq 1)$‎. ‎Among other things‎, ‎we investigate the surjectivity and faithfulness of the representation‎. At the end‎, ‎we give some app...

in this paper, the dual area vector of a closed dual spherical curve is kinematically generatedand the dual steineer vector of a motion are extensively studied by the methods of differential geometry.jacobi’s theorems, known for real curves, are investigated for closed dual curves. the closed trajectorysurfaces generated by an oriented line are fixed in a moving rigid body in ir3 , in which the...