Ill-posedness of waterline integral of time domain free surface Green function for surface piercing body advancing at dynamic speed

In the linear time domain computation of a floating body advancing at a dynamic speed, the source formulation for the velocity potential of the hydrodynamic problem is commonly used so that the velocity potential is expressed as the integral of time domain free surface sources distributed on the two-dimensional wetted body surface and the one-dimensional waterline, which is the intersection of the wetted body surface and the mean free water surface. A time domain free surface source is corresponding to the time domain free surface Green function associated with a suitable source strength, which is to be solved from body boundary condition and normal velocity boundary integral equation of the source formulation. The normal velocity boundary integral equation contains an integral of the normal derivative of the time domain free surface Green function on the waterline. It is shown that the waterline integral is ill-posed. Thus the source strength of velocity potential is not obtainable

Steady-state bifurcation analysis of a strong nonlinear atmospheric vorticity equation

The quasi-geostrophic equation or the Euler equation with dissipation studied in the present paper is a simplified form of the atmospheric circulation model introduced by Charney and DeVore [J. Atmos. Sci. 36(1979), 1205-1216] on the existence of multiple steady states to the understanding of the persistence of atmospheric blocking. The fluid motion defined by the equation is driven by a zonal thermal forcing and an Ekman friction forcing measured by $\kappa>0$. It is proved that the steady-state solution is unique for $\kappa >1$ while multiple steady-state solutions exist for $\kappa<\kappa_{crit}$ with respect to critical value $\kappa_{crit}<1$. Without involvement of viscosity, the equation has strong nonlinearity as its nonlinear part contains the highest order derivative term. Steady-state bifurcation analysis is essentially based on the compactness, which can be simply obtained for semi-linear equations such as the Navier-Stokes equations but is not available for the quasi-geostrophic equation in the Euler formulation. Therefore the Lagrangian formulation of the equation is employed to gain the required compactness.Comment: 20 pages, 0 figures, 30 reference

New formulation of the finite depth free surface Green function

For a pulsating free surface source in a three-dimensional finite depth fluid domain, the Green function of the source presented by John [F. John, On the motion of floating bodies II. Simple harmonic motions, Communs. Pure Appl. Math. 3 (1950) 45-101] is superposed as the Rankine source potential, an image source potential and a wave integral in the infinite domain $(0, \infty)$. When the source point together with a field point is on the free surface, John's integral and its gradient are not convergent since the integration $\int^\infty_\kappa$ of the corresponding integrands does not tend to zero in a uniform manner as $\kappa$ tends to $\infty$. Thus evaluation of the Green function is not based on direct integration of the wave integral but is obtained by approximation expansions in earlier investigations. In the present study, five images of the source with respect to the free surface mirror and the water bed mirror in relation to the image method are employed to reformulate the wave integral. Therefore the free surface Green function of the source is decomposed into the Rankine potential, the five image source potentials and a new wave integral, of which the integrand is approximated by a smooth and rapidly decaying function. The gradient of the Green function is further formulated so that the same integration stability with the wave integral is demonstrated. The significance of the present research is that the improved wave integration of the Green function and its gradient becomes convergent. Therefore evaluation of the Green function is obtained through the integration of the integrand in a straightforward manner. The application of the scheme to a floating body or a submerged body motion in regular waves shows that the approximation is sufficiently accurate to compute linear wave loads in practice.Comment: 24 pages, 7 figure

Instability of the Kolmogorov flow in a wall-bounded domain

In the magnetohydrodynamics (MHD) experiment performed by Bondarenko and his co-workers in 1979, the Kolmogorov flow loses stability and transits into a secondary steady state flow at the Reynolds number $R=O(10^3)$. This problem is modelled as a MHD flow bounded between lateral walls under slip wall boundary condition. The existence of the secondary steady state flow is now proved. The theoretical solution has a very good agreement with the flow measured in laboratory experiment at $R=O(10^3)$. Further transition of the secondary flow is observed numerically. Especially, well developed turbulence arises at $R=O(10^4)$

On the number of representations of n as a linear combination of four triangular numbers II

Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb N$ let $N(a,b,c,d;n)$ be the number of representations of $n$ by $ax^2+by^2+cz^2+dw^2$, and let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x-1)/2+by(y-1)/2+cz(z-1)/2 +dw(w-1)/2$ $(x,y,z,w\in\Bbb Z$). In this paper we reveal the connections between $t(a,b,c,d;n)$ and $N(a,b,c,d;n)$. Suppose $a,n\in\Bbb N$ and $2\nmid a$. We show that $$t(a,b,c,d;n)=\frac 23N(a,b,c,d;8n+a+b+c+d)-2N(a,b,c,d;2n+(a+b+c+d)/4)$$ for $(a,b,c,d)= (a,a,2a,8m),\ (a,3a,8k+2,8m+6),\ (a,3a,8m+4,8m+4)\ (n\equiv m+\frac{a-1}2 \pmod 2)$ and $(a,3a,16k+4,16m+4)\ (n\equiv \frac{a-1}2\pmod 2)$. We also obtain explicit formulas for $t(a,b,c,d;n)$ in the cases $(a,b,c,d)=(1,1,2,8),\ (1,1,2,16),(1,2,3,6),\ (1,3,4,12),\ (1,1,$$3,4),\ (1,1,5,5),\ (1,5,5,5),\ (1,3,3,12),\ (1,1,1,12),\ (1,1,3,12)$and$(1,3,3,4)$.Comment: 22 page

Global large solutions and incompressible limit for the compressible flow of liquid crystals

The present paper is dedicated to the global large solutions and incompressible limit for the compressible flow of liquid crystals under the assumption on almost constant density and large volume viscosity. The result is based on Fourier analysis and involved so-called critical Besov norm.Comment: 19 page

Global large solutions and incompressible limit for the compressible Navier-Stokes equations

The present paper is dedicated to the global large solutions and incompressible limit for the compressible Navier-Stokes system in $\mathbb{R}^d$ with $d\ge 2$. We aim at extending the work by Danchin and Mucha (Adv. Math., 320, 904--925, 2017) in $L^2$ structure to that in a critical $L^p$ framework. The result implies the existence of global large solutions initially from large highly oscillating velocity fields.Comment: The final version for publis

On the number of representations of n as a linear combination of four triangular numbers

Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb N$ let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x-1)/2+by(y-1)/2+cz(z-1)/2 +dw(w-1)/2$ $(x,y,z,w\in\Bbb Z$). In this paper we obtain explicit formulas for $t(a,b,c,d;n)$ in the cases $(a,b,c,d)=(1,2,2,4),\ (1,2,4,4),\ (1,1,4,4),\ (1,4,4,4)$, $(1,3,9,9),\ (1,1,3,9)$,$(1,3,3,9)$,$(1,1,9,9),\ (1,9,9,9)$and$(1,1,1,9).$Comment: 18 page

Fermion Masses and Flavor Mixing in A Supersymmetric SO(10) Model

we study fermion masses and flavor mixing in a supersymmetric SO(10) model, where $\mathbf{10}$, $\mathbf{120}$ and $\mathbf{\bar{126}}$ Higgs multiplets have Yukawa couplings with matter multiplets and give masses to quarks and leptons through the breaking chain of a Pati-Salam group. This brings about that, at the GUT energy scale, the lepton mass matrices are related to the quark ones via several breaking parameters, and the small neutrino masses arise from a Type II see-saw mechanism. When evolving renormalization group equations for the fermion mass matrices from the GUT scale to the electroweak scale, in a specific parameter scenario, we show that the model can elegantly accommodate all observed values of masses and mixing for the quarks and leptons, especially, it's predictions for the bi-large mixing in the leptonic sector are very well in agreement with the current neutrino experimental data.Comment: LaTeX2e, 14 pages, 2 figure

Entanglement entropy in quasi-symmetric multi-qubit states

We generalize the symmetric multi-qubit states to their q-analogs, whose basis vectors are identified with the q-Dicke states. We study the entanglement entropy in these states and find that entanglement is extruded towards certain regions of the system due to the inhomogeneity aroused by q-deformation. We also calculate entanglement entropy in ground states of a related q-deformed Lipkin-Meshkov-Glick model and show that the singularities of entanglement can correctly signify the quantum phase transition points for different strengths of q-deformation.Comment: 11 pages, 2 figure