We derive a formula for the number of pre-images under non-degenerate harmonic mapping $f$, using argument principle. This reveals connection between and caustics. Our results allow to deduce $f$ geometrically every non-caustic point. approximately locate points near Moreover, we apply our prove that $k = n, n+1, \ldots, n^2$ there exists polynomial degree $n$ with $k$ zeros.

Abstract This article investigates the long-wave anti-plane shear motion in a symmetric three-layered laminate composed of pre-stressed compressible elastic layers. The layers are perfectly bonded, while traction-free and fixed boundary conditions considered on outer faces laminate. In both cases, dispersion relation is obtained terms anti-symmetric decompositions. Numerical results an asymptot...

A detailed biomechanical model of the low-back musculature that predicts muscle-force distribution in response to external loading is presented. The paper shows that the class of loading tasks that involve an erect posture and an arbitrary load placed on the upper limbs can be described as a loading plane whose axes are the flexion and lateral bending moments. Under these conditions, the indivi...

Accumulating evidence documents the initiation of diverse physiologic and biochemical response subsequent to an oral glucose load. However, significant gaps in knowledge exist in the understanding of consequences of glucose load during pregnancy, a state of insulin resistance. Using high dimensional protein arrays, we conducted a pilot proof-of-concept and feasibility study to investigate profi...

In this paper, buckling and free vibration analysis of a circular tapered nanoplate subjected to in-plane forces were studied. The linear variation of the plate thickness was considered in radial direction. Nonlocal elasticity theory was employed to capture size-dependent effects. The Raleigh-Ritz method and differential transform method were utilized to obtain the frequency equations for simpl...

In this paper, homotopy perturbation and modified Lindstedt-Poincare methods are employed for nonlinear free vibrational analysis of simply supported and double-clamped beams subjected to axial loads. Mid-plane stretching effect has also been accounted in the model. Galerkin's decomposition technique is implemented to convert the dimensionless equation of the motion to nonlinear ordinary differ...

Pre-eclampsia and late-onset Alzheimer's disease (LOAD) share no clinical features. In contrast to these clinical dissimilarities, striking parallels exist between the (epi)genetic features associated with pre-eclampsia and LOAD for the genes located on 10q22. The parallels in identity between the 10q22 genes involved and active in the organs (placenta, brain) primarily affected in the respecti...

For the optimum design of Piled Raft Foundations (PRF), the concept of settlement reducer piles has been applied. However, when these piles are connected to the raft, a high stress may develop in the relatively small number of piles. Therefore, an alternative approach is to disconnect the piles from the raft forming a new system called a non-connected or disconnected piled-raft foundation (NCPR...