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Pole skipping is a recently discovered subtle effect in the thermal energy density retarded two point function at a special point in the complex $(\omega,p)$ planes. We propose that pole skipping is determined by the stress tensor contribution to many-body chaos, and the special point is at $(\omega,p)_\text{p.s.}= i \lambda^{(T)}(1,1/u_B^{(T)})$, where $\lambda^{(T)}=2\pi/\beta$ and $u_B^{(T)}$ are the stress tensor contributions to the Lyapunov exponent and the butterfly velocity respectively. While this proposal is consistent with previous studies conducted for maximally chaotic theories, where the stress tensor dominates chaos, it clarifies that one cannot use pole skipping to extract the Lyapunov exponent of a theory, which obeys $\lambda\leq \lambda^{(T)}$. On the other hand, in a large class of strongly coupled but non-maximally chaotic theories $u_B^{(T)}$ is the true butterfly velocity and we conjecture that $u_B\leq u_B^{(T)}$ is a universal bound. While it remains a challenge to explain pole skipping in a general framework, we provide a stringent test of our proposal in the large-$q$ limit of the SYK chain, where we determine $\lambda,\, u_B,$ and the energy density two point function in closed form for all values of the coupling, interpolating between the free and maximally chaotic limits. Since such an explicit expression for a thermal correlator is one of a kind, we take the opportunity to analyze many of its properties: the coupling dependence of the diffusion constant, the dispersion relations of poles, and the convergence properties of all order hydrodynamics., Comment: 39 pages, 13 figures

In this paper, we consider the following Schrödinger equation (0.1) − Δ u − μ u | x | 2 + V ( x ) u = K ( x ) | u | 2 ∗ − 2 u + f ( x , u ) , x ∈ R N , u ∈ H 1 ( R N ) , where N ⩾ 4, 0 ⩽ μ < μ ‾, μ ‾ = ( N − 2 ) 2 4 , V is periodic in x, K and f are asymptotically periodic in x, we take advantage of the generalized Nehari manifold approach developed by Szulkin and Weth to look for the ground state solution of (0.1).

This paper presents the optimality of decay estimate of solutions to the initial value problem of 1D Schrödinger equations containing a long-range dissipative nonlinearity, i.e., λ | u | 2 u. Our aim is to obtain the two results. One asserts that, if the L 2 -norm of a global solution, with an initial datum in the weighted Sobolev space, decays at the rate more rapid than ( log t ) − 1 / 2 , then it must be a trivial solution. The other asserts that there exists a solution decaying just at the rate of ( log t ) − 1 / 2 in L 2 .

We consider a Bessel process X of dimension δ ∈ ( 0 , 2 ) having 0 as a slowly reflecting (sticky) boundary point with a stickiness parameter 1 ∕ μ ∈ ( 0 , ∞ ) . We show that (i) the process X can be characterised through its square Y = X 2 as a unique weak solution to the SDE system d Y t = δ I ( Y t > 0 ) d t + 2 Y t d B t I ( Y t = 0 ) d t = 1 2 μ d l t 0 ( Y ) where B is a standard Brownian motion and l 0 ( Y ) is a diffusion local time process of Y at 0, and (ii) the transition density function of X can be expressed in the closed form as a convolution integral involving a Mittag-Leffler function and a modified Bessel function of the second kind. Appearance of the Mittag-Leffler function is novel in this context. We determine a (sticky) boundary condition at zero that characterises the transition density function of X as a unique solution to the Kolmogorov backward/forward equation of X . We also show that the convolution integral can be characterised as a unique solution to the generalised Abel equation of the second kind. Letting μ ↓ 0 (absorption) and μ ↑ ∞ (instantaneous reflection) the closed-form expression for the transition density function of X reduces to the ones found by Feller (1951) and Molchanov (1967) respectively.

We will present a consistent description of Hamiltonian dynamics on the ``symplectic extended phase space'' that is analogous to that of a time-\underline{in}dependent Hamiltonian system on the conventional symplectic phase space. The extended Hamiltonian $H_{1}$ and the pertaining extended symplectic structure that establish the proper canonical extension of a conventional Hamiltonian $H$ will be derived from a generalized formulation of Hamilton's variational principle. The extended canonical transformation theory then naturally permits transformations that also map the time scales of original and destination system, while preserving the extended Hamiltonian $H_{1}$, and hence the form of the canonical equations derived from $H_{1}$. The Lorentz transformation, as well as time scaling transformations in celestial mechanics, will be shown to represent particular canonical transformations in the symplectic extended phase space. Furthermore, the generalized canonical transformation approach allows to directly map explicitly time-dependent Hamiltonians into time-independent ones. An ``extended'' generating function that defines transformations of this kind will be presented for the time-dependent damped harmonic oscillator and for a general class of explicitly time-dependent potentials. In the appendix, we will reestablish the proper form of the extended Hamiltonian $H_{1}$ by means of a Legendre transformation of the extended Lagrangian $L_{1}$., 24 pages

A three-dimensional slowly rotating black hole solution in the presence of negative cosmological constant in the Einstein-power-Maxwell theory is studied. It is shown that in the small rotation limit the electric field, diagonal metric function and thermodynamic properties in the small rotation limit are the same as for static case, whereas the small rotation gives in addition a non-diagonal metric function and magnetic field which are also small. For these functions cased by rotation of black hole it was obtained exact integral solution and analytic asymptotic solution.

In this paper, we study the dynamics of a three-species food chain model with two delays duo to fear. First we discuss the permanence of this delayed model, then we discuss dynamics under six cases: (1) τ 1 = τ 2 = 0 , (2) τ 1 > 0 , τ 2 = 0 , (3) τ 2 > 0 , τ 1 = 0 , (4) τ 1 = τ 2 > 0 , (5) τ 1 ∈ ( 0 , τ 10 ) , τ 2 > 0 , (6) τ 2 ∈ ( 0 , τ 20 ) , τ 1 > 0 . We shall study the Hopf bifurcation about case (5) by center manifold theorem and normal form. At last we shall give simulations which reveal the influence of delays and fear on the dynamics of this predator–prey system.

In this work we study a non-flat Friedmann-Robertson-Walker universe filled with a pressure-less dark matter (DM) and Barrow holographic dark energy (BHDE) whose IR cutoff is the apparent horizon. Among various DE models, (BHDE) model shows the dynamical enthusiasm to discuss the universe's transition phase. According to the new research, the universe transitioned smoothly from a decelerated to an accelerated period of expansion in the recent past. We exhibit that the development of $q$ relies upon the type of spatial curvature. Here we study the equation of state (EoS) parameter for the BHDE model to determine the cosmological evolution for the non-flat universe. The (EoS) parameter and the deceleration parameter (DP) shows a satisfactory behaviour, it does not cross the the phantom line. We also plot the statefinder diagram to characterize the properties of the BHDE model by taking distinct values of barrow exponent $\triangle$. Moreover, we likewise noticed the BHDE model in the $(\omega_{D}-\omega_{D}^{'})$ plane, which can furnish us with a valuable, powerful finding to the mathematical determination of the statefinder. In the statefinder trajectory, this model was found to be able to reach the $\Lambda CDM$ fixed point., Comment: 19 pages, 21 figures

In this article we derive the expression of \textit{renormalized energies} for unit-valued harmonic maps defined on a smooth bounded domain in \(\mathbb{R}^2\) whose boundary has several connected components. The notion of renormalized energies was introduced by Bethuel-Brezis-H\'elein in order to describe the position of limiting Ginzburg-Landau vortices in simply connected domains. We show here, how a non-trivial topology of the domain modifies the expression of the renormalized energies. We treat the case of Dirichlet boundary conditions and Neumann boundary conditions as well.

The completeness of the system of eigenfunctions of the complex Schrodinger operator L c = − d 2 / d x 2 + c x 2 / 3 on the semi-axis in L 2 ( R + ) with Dirichlet boundary conditions is proved for all c: | arg ⁡ c | π / 2 + θ 0 , where π / 10 θ 0 π / 2 is determined as the only solution of a certain transcendental equation.

In this paper we consider singularly perturbed nonlinear Schrödinger equations with electromagnetic potentials and involving continuous nonlinearities with subcritical, critical or supercritical growth. By means of suitable variational techniques, truncation arguments and Lusternik–Schnirelman theory, we relate the number of nontrivial complex-valued solutions with the topology of the set where the electric potential attains its minimum value.

We obtain the growth of Sobolev norms of the solution to the Maxwell–Dirac equations in R 1 + 1 by applying elementary techniques. In particular, we estimate L ∞ bound of the solution by making use of a local energy conservation. The similar idea can be applied to the Dirac–Klein–Gordon equations.

In the present paper, the three dimensional (3D) settlement process of two side-by-side free-falling spheres in a water tank was numerically investigated using coupled Smoothed Particle Hydrodynamics (SPH) - Discrete Element Method (DEM) method. The maximum particle Reynolds number, calculated based on the terminal velocity and sphere diameter, is R e m = 2 . 4 × 1 0 4 . The accuracy of the model was firstly validated by simulating the single sphere water-entry, settlement of two circular cylinders arranged in series and two identical spheres settlement side by side problems, respectively. The simulation results exhibited a satisfactory agreement with the available numerical and experimental data in the literatures. Then, the settlement process of two spheres with different sizes and transverse distances were simulated. In particular, the motion patterns of two interacting spheres with different initial gaps and diameter ratios, including motion trajectory, vertical falling distance and flow field, were systematically investigated. The results demonstrate that when initial gap S 0 ∗ S 0 ∗ > 0.75, the interaction between the two spheres are negligible. In addition, the interaction between particles would slow down the settling velocity and affect the terminal settling distance.

This paper presents a numerical investigation of the laminar flow and pressure drop characteristics of planar Tee joints, a canonical flow of interest for the thermal-hydraulic design of oil-immersed power transformer windings. After formulating the problem in nondimensional form, the steady, constant property flow in planar 90 ∘ Tee joints is computed numerically by integrating the Navier–Stokes equations with fully developed upstream and downstream boundary conditions. The analysis assumes a straight-through configuration in which the straight duct holds flow in the same direction before and after the junction, whereas the flow in the side branch can either divide from the incoming flow or combine with it. The analysis starts with the description of the flow patterns that emerge in the dividing and combining cases for all mass split ratios, 0 ≤ m 1 ≤ 1 , and a wide range of straight duct to side branch width ratios, 1 ≤ α ≤ 3 , and Reynolds numbers of the common branch, 0 < Re3 ≤ 200, values that are representative of the cooling oil flow in oil-immersed transformer winding. Flow maps for planar Tee joints are then presented, showing the existence of different regions in the (Re3, m 1 )-plane that exhibit different number and location of recirculation zones. Pressure distributions and secondary loss coefficients are then computed and analyzed, providing a numerical database that is used to develop new local pressure loss correlations for planar Tee joints in an accompanying paper.

The Dunkl--Dirac operator is a deformation of the Dirac operator by means of Dunkl derivatives. We investigate the symmetry algebra generated by the elements supercommuting with the Dunkl--Dirac operator and its dual symbol. This symmetry algebra is realised inside the tensor product of a Clifford algebra and a rational Cherednik algebra associated with a reflection group or root system. For reducible root systems of rank three, we determine all the irreducible finite-dimensional representations and conditions for unitarity. Polynomial solutions of the Dunkl--Dirac equation are given as a realisation of one family of such irreducible unitary representations., Comment: v3 40p. Final version accepted in J. Algebra. See v2 for proof of Thm 4.1

We consider an Klein–Gordon relativistic equation with a boundary dissipation of fractional derivative type. We study of stability of the system using semigroups theory and classical theorems over asymptotic behavior.

In this paper, by using variational methods, we study the existence and concentration of ground state solutions for the following fractional Schrödinger equation ( − Δ ) α u + V ( x ) u = A ( ϵ x ) f ( u ) , x ∈ R N , where α ∈ ( 0 , 1 ), ϵ is a positive parameter, N > 2 α, ( − Δ ) α stands for the fractional Laplacian, f is a continuous function with subcritical growth, V ∈ C ( R N , R ) is a Z N -periodic function and A ∈ C ( R N , R ) satisfies some appropriate assumptions.

Vortex rings are turbulent coherent structures that are ubiquitous in various application domains ranging from gas turbines to aquatic locomotion. The vortex rings are highly sensitive to a variety of external governing parameters. In this paper, we have quantitatively investigated the topology and formation mechanisms of such rings under the effect of externally modulated temperature, pressure, and amount of fluid injection. Vortex characteristics like translational velocity, peak vorticity, and circulation are analytically predicted along with rigorous experimental corroboration. The experiments are carried out on high viscous fluids with circulation-based vortex Reynolds number varied in the range of 8–100.​ The rings under distinct conditions are generated from a solenoid valve-controlled pipe by varying the externally governed parameters. We have shown that these parameters can be collapsed into a universal variable to provide insights into the vortex roll-up characteristics and variation of energy distribution towards translation and circulation for a given vortex ring.

This note is concerned with the global well-posedness of nonlinear Schrödinger equations in the continuum with spatially homogeneous random initial data.

In this paper, we study the existence of a positive ground state solution for a class of generalized quasilinear Schrödinger equations with asymptotically periodic potential. By the variational method, a positive ground state solution is obtained. Compared with the existing results, our results improve and generalize some existing related results.

In this paper we discuss the model of fractional oscillator where the inertial and restoring force terms maintains their usual expression but the damping term involves a fractional derivative of Caputo type, the so called fractional Kelvin-Voigt oscillator. The transient solution of this model is given in terms of the so called bivariate Mittag-Leffler function, while the steady-state solution in response to a sinusoidal force involves a 4-variate Mittag-Leffler function. We give numerical examples comparing the solutions for different values of the order α of the fractional derivative (0

We study Boltzmann-Fermi-Dirac equation when quantum effects are taken into account in dilute gas dynamics. By employing new estimates on trilinear terms of collision kernels, we prove the global existence of the classical solution to Boltzmann-Fermi-Dirac equation near equilibrium. Furthermore, the limit from Boltzmann-Fermi-Dirac equation to incompressible Navier-Stokes-Fourier equations is justified rigorously. The corresponding formal analysis was given in the thesis of Zakrevskiy \cite{Zakrevskiy}, Comment: 32 pages

Laminar cross-flow over a circular cylinder with smooth as well as longitudinally grooved surfaces was investigated numerically for low freestream Reynolds number ranging from 50 to 300. Most of the currently published works on similar flow configurations considered high Reynolds number in the turbulent regime. The computations were performed by the ANSYS Fluent code. The numerical results were validated against the experimental results. The triangular V-shaped, the U-shaped, and Rectangular shaped grooves, with unit aspect ratio; along with the smooth cylinder, are considered. The predicted flow-field for the grooved cylinder cases are analyzed and compared with that for the smooth cylinder. The aerodynamic loads, time-averaged velocities, rms velocity, vorticity, and pressure coefficient plots around the smooth and grooved cylinders were compared to explore the effect of the shape of the grooves on flow dynamics over the cylinder. The flow physics around the cylinder is significantly affected by the shape and size of the grooves, and their circumferential distribution. It is found that the U-grooves significantly decrease the mean drag coefficient (around 13% for Re = 200 and 10% for Re = 300 compared to the smooth cylinder case). The grooves decreased viscous drag significantly (up to 30%) compared to pressure drag. The grooves on the cylinder also delayed the separation and reduced the extent of the recirculation zone in the cylinder wake due to the presence of recirculating flow within the individual grooves. The conclusion is that the properly designed longitudinal grooves on the cylinder in a cross-flow can be used advantageously for reducing aerodynamic loads like drag and lift force in laminar flow as well.

The particle-laden turbulent flow over a two-dimensional dune bathymetry is studied by means of Large Eddy Simulation (LES). The 4-way coupling regime for particle–particle and particle-flow interactions is applied and an appropriate empirical formula is used to estimate the bedload transport. The flow conditions correspond to river flow in laboratory scale at a Reynolds number equal to 17 , 500 with respect to the maximum water depth and bulk velocity. The dune-shaped bed is introduced within a Cartesian grid by means of the Immersed Boundary Method. A particle cloud consisting of spherical particles is released within the carrier fluid under the influence of the Stokes drag and gravity forces. Three different types of particles are examined: acrylic, glass and steel particles. The particle volume fraction is significant and a hard-sphere model is deployed to account for particle–particle collisions in a 4-way coupling regime. Closure parameters for the collision model have been adopted for each type of the examined particles, based on experimental works in literature. Results focus on the effect of the particle inertia on their kinematics, the velocity field and the shear stress along the dune bed which modifies bedload transport. Using the numerical simulations presented, it can be observed that when larger particle-to-fluid density ratios were examined, larger slip velocities between the fluid and particle phases were recorded.

The quasi-steady translational motion of an incompressible couple stress fluid confined between two eccentric spherical surfaces is investigated. It is assumed that the motion is generated by allowing the internal sphere to translate with a constant velocity. The Stokesian assumption of low Reynolds numbers is considered so that the nonlinear terms of the equation of motion are neglected. The superposition principle is employed to construct the general solution of the governing equations in the presence of spherical surfaces using two spherical systems of coordinates relative to the centers of the two spherical surfaces. The no-slip boundary conditions on the spherical boundaries are applied. In addition, the conditions of vanishing couple stresses on the bounding surfaces are imposed. The boundary collocation technique is utilized numerically to satisfy the boundary conditions on the spherical surfaces. The drag force experienced by the fluid flow on the spherical particle is obtained and represented numerically through table and graphs. The numerical results show that the values of the drag force increases monotonically with the increase in the radii ratio. It increases also with the increase of the couple stress viscosity parameter. The obtained results are consistence with the results available in the literature for viscous fluids.

The paper considers the existence of one- or two-hump solutions of a singularly perturbed nonlinear Schrodinger (NLS) equation, which is the standard NLS equation with a third order perturbation. In particular, this equation appears in the field of nonlinear optics, where it is used to describe pulses in optical fibers near a zero dispersion wavelength. It has been shown formally and numerically that the perturbed NLS equation has one- or multi-hump solutions with small oscillations at infinity, called generalized one- or multi-hump solutions. The main purpose of the paper is to provide the first rigorous proof of the existence of generalized one- or two-hump solutions of the singularly perturbed NLS equation. The several invariant properties of the equation, i.e., the translational invariance, the gauge invariance and the reversibility property, are essential to obtain enough free constants to prove the existence. The ideas and methods presented here may be applicable to show existence of generalized 2 k -hump solutions of the equation.

We collect some interesting results for equations driven by the fractional relativistic Schrodinger operator ( − Δ + m 2 ) s with s ∈ ( 0 , 1 ) and m > 0 . More precisely, for the linear theory, we prove Holder-Schauder-Zygmund regularity results and a Kato's inequality. For the nonlinear theory, we obtain L ∞ -regularity, exponential decay, a Pohozaev-type identity, and a symmetry result for solutions of certain nonlinear fractional problems.

Gupta et al. (2020) had obtained a nontrivial solution for the Chaplygin gas equations and the generalized Riemann problem for the Chaplygin gas equations was solved by them through a rarefaction wave. In this paper, we explain that the existence of a rarefaction wave is not possible and that the solution obtained in Gupta et al. (2020) is also in error. In fact, in this paper, we solved the generalized Riemann problem through a characteristic shock(s). For several classes of non-constant and smooth initial data, the solution to the generalized Riemann problem is presented.

The steady low-Reynolds-number rotation of a spherical soft particle (a hard core coated with a permeable porous layer) in a viscous fluid within a nonconcentric spherical cavity about their common diameter is semi-analytically studied. To solve the Stokes and Brinkman equations for the fluid velocity, a solution is constituted by the general solutions in two spherical coordinate systems originated from the particle and cavity centers and the boundary conditions are satisfied by a collocation technique. Numerical results of the hydrodynamic torque on the soft sphere are obtained as a function of the core-to-particle radius ratio, particle-to-cavity radius ratio, relative center-to-center distance of the particle and cavity, and ratio of the particle radius to the permeation length in the porous layer over the entire ranges. The effect of the cavity on the torque of a rotating soft particle is weaker than that of a corresponding hard particle (or soft one with lower permeability or thinner thickness of its porous layer). While the normalized torque of a soft sphere in general is an increasing function of the particle-to-cavity radius ratio, a weak minimum of it (surprisingly, less than the value of an unconfined particle) may occur for a particle with a small to mediate core-to-particle radius ratio and a high permeability inside a nonconcentric cavity at a moderate value of the particle-to-cavity radius ratio. Also, this torque in general is an increasing function of the eccentricity of the particle location, but it may decrease slightly with an increase in the eccentricity for a particle with a small to mediate core-to-particle radius ratio and a high permeability.

In an earlier paper, Cruz-Uribe, Rodney and Rosta proved an equivalence between weighted Poincaré inequalities and the existence of weak solutions to a family of Neumann problems related to a degenerate $ p $-Laplacian. Here we prove a similar equivalence between Poincaré inequalities in variable exponent spaces and solutions to a degenerate $ {p(\cdot)} $-Laplacian, a non-linear elliptic equation with nonstandard growth conditions.

In the framework of $f(T)$-gravity theory, classical and quantum cosmology has been studied in the present work for FLRW space-time model. The Noether symmetry, a point-like symmetry of the Lagrangian is used to the physical system and a specific functional form of $f(T)$ is determined. A point transformation in the 2D augmented space restricts one of the variable to be cyclic so that the Lagrangian as well as the field equations are simplified so that they are solvable. Lastly for quantum cosmology, the WD equation is constructed and possible solution has been evaluated., 15 pages, 2 figures

Conventional wisdom holds that the von Neumann entropy corresponds to thermodynamic entropy, but Meir Hemmo and Orly Shenker have recently argued against this view by attacking von Neumann’s argument. I argue that Hemmo and Shenker’s arguments fail because of several misunderstandings about statistical-mechanical and thermodynamic domains of applicability, about the nature of mixed states, and about the role of approximations in physics. As a result, their arguments fail in all cases: in the single-particle case, the finite-particles case, and the infinite-particles case.

The enigmatic nonlocal quantum correlation that was famously derided by Einstein as "spooky action at a distance" has now been experimentally demonstrated to be authentic. The quantum entanglement and nonlocal correlations emerged as inevitable consequences of John Bell's epochal paper on Bell's inequality. However, in spite of some extraordinary applications as well as attempts to explain the reason for quantum nonlocality, a satisfactory account of how Nature accomplishes this astounding phenomenon is yet to emerge. A cogent mechanism for the occurrence of this incredible event is presented in terms of a plausible quantum mechanical Einstein-Rosen bridge., Comment: 7 pages

In this paper, we investigate the Bohr phenomenon for the class of analytic functions defined on the simply connected domain \(\Omega_{\gamma}=\bigg\{z\in\mathbb{C} \colon \bigg|z+\frac{\gamma}{1-\gamma}\bigg

In the case of favorable pressure gradient, Oleinik obtained the global-in-x solutions to the steady Prandtl equations with low regularity (see Oleinik and Samokhin [9] , P.21, Theorem 2.1.1). Due to the degeneracy of the equation near the boundary, the question of higher regularity of Oleinik's solutions remains open. See the local-in-x higher regularity established by Guo and Iyer [5] . In this paper, we prove that Oleinik's solutions are smooth up to the boundary y = 0 for any x > 0 , using further maximum principle techniques. Moreover, since Oleinik only assumed low regularity on the data prescribed at x = 0 , our result implies instant smoothness (in the steady case, x = 0 is often considered as initial time).

In this paper, we study some spectral properties of the one-dimensional Hahn-Dirac boundary-value problem, such as formally self-adjointness, the case that the eigenvalues are real, orthogonality of eigenfunctions, Green’s function, the existence of a countable sequence of eigenvalues, eigenfunctions forming an orthonormal basis of L2ω,q((ω0, a); E).

Let η ( s ) = ∑ n = 1 ∞ ( − 1 ) n + 1 n − s be the alternating zeta function. For a real number τ we define certain complex numbers b M , m ( τ ) and consider finite Dirichlet series υ M ( τ , s ) = ∑ m = 1 M b M , m ( τ ) m − s and η N ( τ , s ) = ∑ M = 1 N υ M ( τ , s ) . Computations demonstrate some remarkable properties of these finite Dirichlet series, but nothing was supported by a proof so far. First, numerical data show that η N ( τ , s ) approximates η ( s ) with high accuracy for s in the vicinity of 1 / 2 + i τ ; this allows one to surmise that (*) η ( s ) = ∑ M = 1 ∞ υ M ( τ , s ) . Moreover, it looks that lim M → ∞ ⁡ m υ M ( τ , 1 − σ + i t ) ‾ m υ M ( τ , σ + i t ) = η ( σ + i t ) m η ( 1 − σ + i t ) ‾ m ; in other words, the individual summands in expected expansion ( ⁎ ) satisfy with an increasing accuracy a counterpart of the classical functional equation. Let ϒ M ( τ , σ + i t ) = υ M ( τ , σ + i t ) / η ( σ + i t ) . When M, τ, and either σ or t are fixed, and the fourth parameter varies, the plot of ϒ M ( τ , σ + i t ) on the complex plane contains numerous almost ideally circular arcs with geometrical parameters closely related to the non-trivial zeros of the zeta function.

We study the volume of compact Riemannian manifolds which are Einstein with respect to a metric connection with (parallel) skew-torsion. We provide a result for the sign of the first variation of the volume in terms of the corresponding scalar curvature. This generalizes a result of M. Ville, related with the first variation of the volume on a compact Einstein manifold., 7 pages, to appear in "Communications in Mathematics"

In this paper we construct unstable shocks in the context of 2D isentropic compressible Euler in azimuthal symmetry. More specifically, we construct initial data that when viewed in self-similar coordinates, converges asymptotically to the unstable $C^{\frac15}$ self-similar solution to the Burgers' equation. Moreover, we show the behavior is stable in $C^8$ modulo a two dimensional linear subspace. Under the azimuthal symmetry assumption, one cannot impose additional symmetry assumptions in order to isolate the corresponding manifold of initial data leading to stability: rather, we rely on modulation variable techniques in conjunction with a Newton scheme., 68 pages, accepted version

We improve previous results on dispersive decay for 1D Klein–Gordon equation. We develop a novel approach, which allows us to establish the decay in more strong norms and weaken the assumption on the potential.

We construct a distorted Fourier transformation associated with the multi-dimensional quantum walk. In order to avoid the complication of notations, almost all of our arguments are restricted to two dimensional quantum walks (2DQWs) without loss of generality. The distorted Fourier transformation characterizes generalized eigenfunctions of the time evolution operator of the QW. The 2DQW which will be considered in this paper has an anisotropy due to the definition of the shift operator for the free QW. Then we define an anisotropic Banach space as a modified Agmon-H\"{o}rmander's $\mathcal{B}^*$ space and we derive the asymptotic behavior at infinity of generalized eigenfunctions in these spaces. The scattering matrix appears in the asymptotic expansion of generalized eigenfunctions.