Convergence of hydrodynamic modes: insights from kinetic theory and holography

We study the mechanisms setting the radius of convergence of hydrodynamic dispersion relations in kinetic theory in the relaxation time approximation. This introduces a quali\-tatively new feature with respect to holography: a nonhydrodynamic sector represented by a branch cut in the retarded Green's function. In contrast with existing holographic examples, we find that the radius of convergence in the shear channel is set by a collision of the hydrodynamic pole with a branch point. In the sound channel it is set by a pole-pole collision on a non-principal sheet of the Green's function. More generally, we examine the consequences of the Implicit Function Theorem in hydrodynamics and give a prescription to determine a set of points that necessarily includes all complex singularities of the dispersion relation. This may be used as a practical tool to assist in determining the radius of convergence of hydrodynamic dispersion relations.

Collective modes of polarizable holographic media in magnetic fields

We consider a neutral holographic plasma with dynamical electromagnetic interactions in a finite external magnetic field. The Coulomb interactions are introduced via mixed boundary conditions for the Maxwell gauge field. The collective modes at finite wave-vector are analyzed in detail and compared to the magneto-hydrodynamics results valid only at small magnetic fields. Surprisingly, at large magnetic field, we observe the appearance of two plasmon-like modes whose corresponding effective plasma frequency grows with the magnetic field and is not supported by any background charge density. Finally, we identify a mode collision which allows us to study the radius of convergence of the linearized hydrodynamics expansion as a function of the external magnetic field. We find that the radius of convergence in momentum space, related to the diffusive transverse electromagnetic mode, increases quadratically with the strength of the magnetic field.

On some iterative methods with frozen derivatives for solving equations

We determine a radius of convergence for an efficient iterative method with frozen derivatives to solve Banach space defined equations. Our convergence analysis use \(\omega-\) continuity conditions only on the first derivative. Earlier studies have used hypotheses up to the seventh derivative, limiting the applicability of the method. Numerical examples complete the article.

Virtual corrections to gg → ZH in the high-energy and large-mt limits

We compute the next-to-leading order virtual corrections to the partonic cross-section of the processgg → ZH, in the high-energy and large-mtlimits. We use Padé approximants to increase the radius of convergence of the high-energy expansion in$$ {m}_t^2/s $$mt2/s,$$ {m}_t^2/t $$mt2/tand$$ {m}_t^2/u $$mt2/uand show that precise results can be obtained down to energies which are fairly close to the top quark pair threshold. We present results both for the form factors and the next-to-leading order virtual cross-section.

Wiman's type inequality for analytic and entire functions and $h$-measure of an exceptional sets

Let $\mathcal{E}_R$ be the class of analytic functions $f$ represented by power series of the form $f(z)=\sum\limits\limits_{n=0}^{+\infty}a_n z^n$ with the radius of convergence $R:=R(f)\in(0;+\infty].$ For $r\in [0, R)$ we denote the maximum modulus by $M_f(r)=\max\{|f(z)|\colon$ $ |z|=r\}$ and the maximal term of the series by $\mu_f(r)=\max\{|a_n| r^n\colon n\geq 0\}$. We also denote by $\mathcal{H}_R$, $R\leq +\infty$, the class of continuous positive functions, which increase on $[0;R)$ to $+\infty$, such that $h(r)\geq2$ for all $r\in (0,R)$ and $ \int^R_{r_{0}} h(r) d\ln r =+\infty $ for some $r_0\in(0,R)$. In particular, the following statements are proved. $1^0.$ If $h\in \mathcal{H}_R$ and $f\in \mathcal{E}_R,$ then for any $\delta>0$ there exist $E(\delta,f,h):=E\subset(0,R)$, $r_0 \in (0,R)$ such that $$ \forall\ r\in (r_0,R)\backslash E\colon\ M_f(r)\leq h(r) \mu_f(r) \big\{\ln h(r)\ln(h(r)\mu_f(r))\big\}^{1/2+\delta}$$ and $$\int\nolimits_E h(r) dr < +\infty. $$ $2^0.$ If we additionally assume that the function $f\in \mathcal{E}_R$ is unbounded, then $$ \ln M_f(r)\leq(1+o(1))\ln (h(r)\mu_f(r)) $$ holds as $r\to R$, $r\notin E$. Remark, that assertion $1^0$ at $h(r)\equiv \text{const}$ implies the classical Wiman-Valiron theorem for entire functions and at $h(r)\equiv 1/(1-r)$ theorem about the Kövari-type inequality for analytic functions in the unit disc. From statement $2^0$ in the case that $\ln h(r)=o(\ln\mu_f(r))$, $r\to R$, it follows that $ \ln M_f(r)=(1+o(1))\ln \mu_f(r) $ holds as $r\to R$, $r\notin E$.