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Yaoming Shi

Publications and source records attributed to Yaoming Shi.

12 recordsLinked to original sources
arXiv2026

Construction of Finite Hilbert--P\'olya Matrices from Weil's Explicit Formula

Starting from the Riemann--$\Xi$ specialization of Weil's explicit formula, we construct finite real-symmetric Prime--Weil matrices $(\mathbf S)$ from pole, archimedean, and finite prime-power data. This construction is a finite-dimensional arithmetic model within the Hilbert--P\'olya program, which seeks a self-adjoint spectral realization of the nontrivial zeta-zero parameters. Their off-diagonal entries form a Loewner-type divided-difference matrix with a rank-two displacement identity. We formulate the spectral quotient as a Hermitian definite generalized eigenproblem on the fixed zero-mean contrast space. This realization is invariant under positive affine rescaling, avoids ground-vector normalization and an ill-conditioned oblique projector, and preserves the finite quotient spectrum. For a dimension-matched zero-side matrix built from $N$ distinct positive ordinates $\gamma_k$, rational interpolation gives the exact contrast-pencil spectrum $\{\pm\gamma_1,\ldots,\pm\gamma_N\}$ and positive-parity square spectrum $\{\gamma_1^2,\ldots,\gamma_N^2\}$. Since the ordinates are inputs, this is a reconstruction theorem. Assuming RH, the same interpolation vector proves $\lambda_{\min}(\mathbf S)\to0$. At $N=L=13$, Lemke's ground-state quotient and the contrast pencil agree numerically and reproduce the first three zeta ordinates to the reported precision. The remaining problem is a relative prime-to-zero perturbation theorem with uniform control of the compressed metric. No proof of RH is claimed.

math.GM↗
arXiv2026

A unified Boussinesq--Euler formulation and finite-time blow-up for a Hou--Luo type boundary-jet system

We derive a unified vorticity--stream formulation $(Bm)$ for two parity-reduced inviscid systems in the meridian plane: the 2D inviscid Boussinesq equations $(m=1)$ and the 3D axisymmetric Euler equations with swirl $(m=2)$. In the Boussinesq case we set $\Theta=\vartheta/r$ and write $\Theta=u^2$ only when a smooth square-root branch has been fixed; equivalently, one may keep the scalar variable $\Theta$ throughout. In the squared radial variable $q=r^2$, the two cases are encoded by the same parameterized system with $m=1,2$. At the boundary $q=1$, a Taylor expansion gives an exact boundary jet: the transport equations close on the boundary, while the elliptic relation also contains the next normal jet $\varphi_{qq}(x,1,t)$. If the boundary jet is closed by the first-order Taylor truncation $\varphi_{qq}(x,1,t)=0$, it reduces to a closed unified $(1+1)$D system $(Q0)$ with the local boundary velocity law $u=-(m+2)^{-1}\omega$. We prove finite-time blow-up for this closed Hou--Luo type model on a periodic interval by a Riccati argument in the spirit of Choi--Hou--Kiselev--Luo--\v{S}ver\'ak--Yao. The theorem is therefore a blow-up result for the closed boundary-jet model, not for the unrestricted Boussinesq or Euler systems.

math.AP↗
arXiv2026

Finite-time blow-up of two $(1+1)$D systems rigorously derived from the 3D axisymmetric Euler equations

We study two $(1+1)$-dimensional systems, denoted $(R0)$ and $(Z0)$, that are rigorously derived from the three-dimensional axisymmetric Euler equations in a signed polar formulation on the meridian plane. The main point of view in this revision is that these $(1+1)$D systems are not ad hoc model equations and not merely ``symmetry-axis reductions.'' Rather, they arise as exact symmetry-axis/apex restrictions of the full $(1+2)$D system~$(E2)$ obtained from 3D axisymmetric Euler, and they already contain the core finite-time singularity mechanism of the full problem. The rev3 geometry is based on the symmetry axes \[ \theta=0,\qquad \theta=\pm \frac{\pi}{2}, \] for which ridge flatness is preserved automatically by the evenness in $(r,z)$. Along these axes, and in particular at the apex $x^2=r^2+z^2=0$, the reduced dynamics closes exactly. This yields two rigorously derived $(1+1)$D systems: the horizontal-axis system $(R0)$ and the vertical-axis system $(Z0)$. The apex trace of these systems reduces further to a closed ODE of Constantin--Lax--Majda type, from which we obtain finite-time blow-up at the coordinate origin. The paper has three main outputs. First, it derives the signed-polar $(1+2)$D subsystem~$(E2)$ from the 3D axisymmetric Euler equations and identifies the exact $(1+1)$D systems $(R0)$ and $(Z0)$ carried by the symmetry axes. Second, it proves finite-time blow-up for the resulting apex dynamics and analyzes the associated convective axis reduction. Third, it derives the exact background--remainder equations and formulates a conditional nonlinear stability mechanism: if a compatible full background exists on $[0,T)$ with the coefficient bounds required by the weighted energy method, then the full solution inherits the same finite-time apex blow-up.

nlin.SI↗
arXiv2026

2D inviscid Boussinesq equations and 3D axisymmetric Euler equations: (1) A unification ($Em$), (2) Finite-time blow-up of two unified $(1+1)$D systems rigorously derived from ($Em$)

We derive a unified polar $(1+2)$D subsystem $(Em)$, with $m=1,2$, from the 2D inviscid Boussinesq and 3D axisymmetric Euler equations. On the symmetry axes $\theta=0,\pm\pi/2,\pi$, ridge flatness closes the dynamics and gives two exact unified $(1+1)$D reductions: the horizontal-axis system $(R0)$ and the vertical-axis system $(Z0)$. Their common apex trace is a Constantin--Lax--Majda type ODE that yields finite-time blow-up at $x=0$. Subsection~\ref{seq:vorticity-strain} connects this pointwise mechanism with Euler continuation theory: for any compatible axisymmetric realization, explicit apex blow-up forces divergence of the time-integrated $L^\infty$ norm of $\nabla\boldsymbol v$, so the singularity is detected by the strain criterion. Section~\ref{sec:R0-SS} strengthens the reduced mechanism by constructing regular apex-only self-similar profiles for the convective horizontal-axis equation $(R0)$; the resulting solution is bounded away from $x=0$, blows up at the apex, and satisfies the same strain-divergence condition. Finally, we derive the exact background--remainder equations and state a conditional nonlinear stability framework: if a compatible full background, weighted elliptic/coercive estimates, and a spectral gap exponent are available, then the apex blow-up transfers to the full solution. Thus the rigorous components are the derivation of $(Em)$, $(R0)$, and $(Z0)$, the apex blow-up and strain verification, the apex-only $(R0)$ self-similar construction, and the perturbative framework; the remaining open step is the unconditional construction and control of the full background away from the apex.

math.AP↗
arXiv2017

On the zeros of Riemann $Ξ(z)$ function

The Riemann $Ξ(z)$ function (even in $z$) admits a Fourier transform of an even kernel $Φ(t)=4e^{9t/2}θ''(e^{2t})+6e^{5t/2}θ'(e^{2t})$. Here $θ(x):=θ_3(0,ix)$ and $θ_3(0,z)$ is a Jacobi theta function, a modular form of weight $\frac{1}{2}$. (A) We discover a family of functions $\{Φ_n(t)\}_{n\geqslant 2}$ whose Fourier transform on compact support $(-\frac{1}{2}\log n, \frac{1}{2}\log n)$, $\{F(n,z)\}_{n\geqslant2}$, converges to $Ξ(z)$ uniformly in the critical strip $S_{1/2}:=\{|\Im(z)|< \frac{1}{2}\}$. (B) Based on this we then construct another family of functions $\{H(14,n,z)\}_{n\geqslant 2}$ and show that it uniformly converges to $Ξ(z)$ in the critical strip $S_{1/2}$. (C) Based on this we construct another family of functions $\{W(n,z)\}_{n\geqslant 8}:=\{H(14,n,2z/\log n)\}_{n\geqslant 8}$ and show that if all the zeros of $\{W(n,z)\}_{n\geqslant 8}$ in the critical strip $S_{1/2}$ are real, then all the zeros of $\{H(14,n,z)\}_{n\geqslant 8}$ in the critical strip $S_{1/2}$ are real. (D) We then show that $W(n,z)=U(n,z)-V(n,z)$ and $U(n,z^{1/2})$ and $V(n,z^{1/2})$ have only real, positive and simple zeros. And there exists a positive integer $N\geqslant 8$ such that for all $n\geqslant N$, the zeros of $U(n,x^{1/2})$ are strictly left-interlacing with those of $V(n,x^{1/2})$. Using an entire function equivalent to Hermite-Kakeya Theorem for polynomials we show that $W(n\geqslant N,z^{1/2})$ has only real, positive and simple zeros. Thus $W(n\geqslant N,z)$ have only real and imple zeros. (E) Using a corollary of Hurwitz's theorem in complex analysis we prove that $Ξ(z)$ has no zeros in $S_{1/2}\setminus\mathbb{R}$, i.e., $S_{1/2}\setminus \mathbb{R}$ is a zero-free region for $Ξ(z)$. Since all the zeros of $Ξ(z)$ are in $S_{1/2}$, all the zeros of $Ξ(z)$ are in $\mathbb{R}$, i.e., all the zeros of $Ξ(z)$ are real.

math.GM↗
arXiv2014

Real-rooted Pólya-like approximations to the Riemann Xi-function

The Riemann $Ξ(z)$ function admits a Fourier transform of a even kernel $Φ(t)$. The latter is related to the derivatives of Jacobi theta function $θ(z)$, a modular form of weight $1/2$. Pólya noticed that when $t$ goes to infinity, $e^t$ goes to $e^t+ e^{-t}=2\cosh t$. He then approximated the kernel $Φ(t)$ by $Φ_{P}(t)$ that contained only the leading term and with $\exp t,\exp(9t/4)$ replaced by $2\cosh t,2\cos(9t/4)$. This procedure captured almost all of the contribution from the tail part (i.e., $t\to\infty$) of the kernel $Φ(t)$. We realize that when $t$ goes to infinity and $0\leqslant b<1,c\in\R$, $\cosh t+c \cosh(bt)$ goes to $\cosh t$. Thus we improve Pólya's approximation by replacing $\cosh(9t/4)$ with $\cosh(9t/4)+b\sum_{k=0}^{m-1}b_k \cosh(9kt/(4m))$ and adjusting the parameters $b,b_k,m$ such that (A) the approximated kernel $Φ_{S}(b,b_k,m;t)$ goes to $Φ(t)$when $t$ goes to infinity;(B) $Φ_{S}(b,b_k,m;t)$ is identical to $Φ(t)$ at $t=0$; (C) the Fourier transform of $Φ_{S}(b,b_k,m;t)$,like in Pólya's case, has only real zeros. Since this procedure also captures almost all of the contribution from the head part (i.e., near $t=0$) of the kernel $Φ(t)$, we are able to anchor both ends of the kernel $Φ(t)$.

math.NT↗
arXiv2011

A cyclic cosmological model based on the f(ρ) modified theory of gravity

We consider FLRW cosmological models for perfect fluid (with rho as the energy density) in the frame work of the f(rho) modified theory of gravity [V. N. Tunyak, Russ. Phys. J. 21, 1221 (1978); J. R. Ray, L. L. Smalley, Phys. Rev. D. 26, 2615 (1982) ]. This theory, with total Lagrangian R-f(rho), can be considered as a cousin of the F(R) theory of gravity with total Lagrangian F(R)-rho. We can pick proper function forms f(rho) to achieve, as the F(R) theory does, the following 4 specific goals, (1) producing a non-singular cosmological model (Ricci scalar and Ricci tensor curvature are bounded); (2) explaining the cosmic early inflation and late acceleration in a unified fashion; (3) passing the solar system tests; (4) unifying the dark matter with dark energy. In addition we also achieve goal number (5): unify the regular matter/energy with dark matter/energy in a seamless fashion. The mathematics is simplified because in the f(rho) theory the leading terms in Einstein's equations are linear in second order derivative of metric wrt coordinates but in the F(R) theory the leading terms are linear in fourth order derivative of metric wrt coordinates.

physics.gen-ph↗
arXiv2004

Andreev reflection resonant tunneling through a precessing spin

We investigate Andreev reflection (AR) resonant tunneling through a precessing spin which is coupled to a normal metallic lead and a superconducting lead. The formula of the AR conductance at zero temperature is obtained as a function of chemical potential and azimuthal angle of the spin precessing by using the nonequilibrium Green function method. It is found that as the local spin precesses in a weak external magnetic field at Larmor frequency $ω_l$, the AR tunneling conductance exhibits an oscillation at the frequency $2ω_l$ alone. The amplitude of AR conductance oscillation enhances with spin-flip tunneling coupling increasing. The study also shows that spin-orbit interaction in tunneling barriers is crucial for the oscillations of AR conductance. The effect of spin-flip tunneling coupling caused by spin-orbit interaction and local spin precessing on resonant behavior of the AR conductance are examined.

cond-mat.mes-hall↗
arXiv2004

Spin-dependent Andreev reflection tunneling through a quantum dot with intradot spin-flip scattering

We study Andreev reflection (AR) tunneling through a quantum dot (QD) connected to a ferromagnet and a superconductor, in which the intradot spin-flip interaction is included. By using the nonequibrium-Green-function method, the formula of the linear AR conductance is derived at zero temperature. It is found that competition between the intradot spin-flip scattering and the tunneling coupling to the leads dominantes resonant behaviours of the AR conductance versus the gate voltage.A weak spin-flip scattering leads to a single peak resonance.However, with the spin-flip scattering strength increasing, the AR conductance will develop into a double peak resonannce implying a novel structure in the tunneling spectrum of the AR conductance. Besides, the effect of the spin-dependent tunneling couplings, the matching of Fermi velocity, and the spin polarization of the ferromagnet on the AR conductance is eximined in detail.

cond-mat.mes-hall↗
arXiv2001

A Lax Pair for the Dynamics of DNA Modeled as a Shearable and Extensible Elastic Rod: II. Discretization of the Arc Length

In previous work, the dynamics of the elastic rod was recast in a Lax pair formulation, with fiducial arc length s and time t as continuous independent variables. However, the solution of these equations cannot apply directly to a system where the fiducial arc length s is a discrete variable. In this paper, we show how to discretize the continuous s variable in a way that preserves the integrability of the original system. The t parameter is not discretized, so this algorithm will be especially useful for solutions of the s-discrete and t-continuous elastic rod problem, as may occur in problems where the polymeric structure of the DNA is made explicit.

nlin.SI↗
arXiv2001

A Lax Pair for the Dynamics of DNA Modeled as a Shearable and Extensible Elastic Rod

We introduce a spectral parameter into the geometrically exact Hamiltonian equations for the elastic rod in a way that creates a Lax pair. This assures integrability and permits application of the inverse scattering transform solution method. If the method can be carried through, the solution of the original problem is recovered by setting the spectral parameter to zero.

nlin.SI↗