Reverse Mathematics (RM) is a program in the foundations of mathematics where the aim is to find the minimal axioms needed to prove a given theorem of ordinary mathematics. Generally, the minimal axioms are equivalent to the theorem at hand, assuming a weak logical system called the base theory. Moreover, many theorems are either provable in the base theory or equivalent to one of four logical systems, together called the Big Five. For instance, the Weierstrass approximation theorem, i.e., that a continuous function can be approximated uniformly by a sequence of polynomials, has been classified in RM as being equivalent to weak König’s lemma, the second Big Five system. In this paper, we study approximation theorems for discontinuous functions via Bernstein polynomials from the literature. We obtain many equivalences between the latter and weak König’s lemma. We also show that slight variations of these approximation theorems fall far outside of the Big Five but fit in the recently developed RM of new ‘big’ systems, namely the uncountability of ${\mathbb R}$, the enumeration principle for countable sets, the pigeon-hole principle for measure, and the Baire category theorem.

In this work we study features of inertia-gravity wave turbulence in the rotating shallow water equations. On examining the dynamics of waves with varying rotation rates, we find that the turbulent cascade of waves is strongest at low rotation rates, forming a $k^{-2}$ energy spectrum, and a rich distribution of shocks in physical space. At high rotation rates, the forward cascade of waves weakens along with a steeper energy spectra and vanishing of shocks in physical space. The wave cascade is seen to be scale-local, resulting in a noticeable time interval for energy to get transferred from domain scale to dissipative scale. Furthermore, we find that the vortical flow has a non-negligible effect on the wave cascade, especially at high rotation rates. The vortical flow assists in the forward cascade of waves and shock formation at high rotation rates, while the waves by themselves in the absence of the vortical flow lack a forward cascade and shock formation at such high rotation rates. On investigating the physical space structures in the vortical flow and their connections to the wave cascade, we find that strain-dominant regions, that are located around the boundaries of coherent vortices, are the physical space regions that contribute majorly to the forward cascade of waves. Our results in general highlight intriguing features of dispersive inertia-gravity wave turbulence that are qualitatively similar to those seen in three-dimensional homogeneous isotropic turbulence and are beyond the predictions of asymptotic resonant wave interaction theory.

This paper presents advances towards the data-based control of periodic oscillator flows, from their fully developed regime to their equilibrium stabilized in closed loop, with linear time-invariant (LTI) controllers. The proposed approach directly builds upon the iterative method of Leclercq et al. (J. Fluid Mech., vol. 868, 2019, pp. 26–65) and provides several improvements for an efficient online implementation, aimed at being applicable in experiments. First, we use input–output data to construct an LTI mean transfer functions of the flow. The model is subsequently used for the design of an LTI controller with linear quadratic Gaussian synthesis, which is practical to automate online. Then, using the controller in a feedback loop, the flow shifts in phase space and oscillations are damped. The procedure is repeated until equilibrium is reached, by stacking controllers and performing balanced truncation to deal with the increasing order of the compound controller. In this article, we illustrate the method for the classic flow past a cylinder at Reynolds number $Re=100$. Care has been taken such that the method may be fully automated and hopefully used as a valuable tool in a forthcoming experiment.

The results of an experimental investigation of smooth-body adverse pressure gradient (APG) turbulent boundary layer flow separation and reattachment over a two-dimensional ramp are presented. These results are part of a larger archival smooth-body flow separation data set acquired in partnership with NASA Langley Research Center and archived on the NASA Turbulence Modeling Resource website. The experimental geometry provides initial canonical turbulent boundary layer growth under nominally zero pressure gradient conditions prior to encountering a smooth, two-dimensional, backward facing ramp geometry onto which a streamwise APG that is fully adjustable is imposed. Detailed surface and off-surface flow field measurements are used to fully characterize the smooth-body APG turbulent boundary layer separation and reattachment at multiple spanwise locations over the ramp geometry. Unsteady aspects of the flow separation are characterized. It is shown that the first and second spatial derivatives of the streamwise static surface pressure profile are sufficient to determine key detachment and reattachment locations. The imposed streamwise APG gives rise to inflectional mean velocity profiles and the associated formation of an embedded shear layer, which is shown to play a dominant role in the subsequent flow development. Similarity scaling is developed for both the mean velocity and turbulent stresses that is found to provide self-similar collapse of profiles for different regions of the ramp flow. Despite the highly non-equilibrium flow environment, a new similarity scaling proved capable of providing self-similar turbulent stress profiles over the full streamwise extent of flow separation and downstream reattachment.

Given a positive integer m, let $\mathbb {Z}_m$ be the set of residue classes mod m. For $A\subseteq \mathbb {Z}_m$ and $n\in \mathbb {Z}_m$, let $\sigma _A(n)$ be the number of solutions to the equation $n=x+y$ with $x,y\in A$. Let $\mathcal {H}_m$ be the set of subsets $A\subseteq \mathbb {Z}_m$ such that $\sigma _A(n)\geq 1$ for all $n\in \mathbb {Z}_m$. Let

$$ \begin{align*} \ell_m=\min\limits_{A\in \mathcal{H}_m}\bigg\lbrace m^{-1}\sum_{n\in \mathbb{Z}_m}\sigma_A(n)\bigg\rbrace. \end{align*} $$

Ding and Zhao [‘A new upper bound on Ruzsa’s numbers on the Erdős–Turán conjecture’, Int. J. Number Theory 20 (2024), 1515–1523] showed that $\limsup _{m\rightarrow \infty }\ell _m\le 192$. We prove

$$ \begin{align*} \limsup\limits_{m\rightarrow\infty}\ell_m\leq 144 \end{align*} $$

and investigate parallel results on subtractive bases of $ \mathbb {Z}_m$.

Recent critical scholarship on terrorism has centred on matters of race, class, and gender regarding how counterterrorism policies are connected to multiple systems of hierarchical power relations. This article builds upon this scholarship and looks towards the future. It engages with understandings of emancipatory futures in critical scholarship on terrorism while drawing upon abolitionist and anarchist political thought to expand understandings of such futures. Anarchist and abolitionist thinking are useful for considering futures beyond the ‘global war on terror’ (GWOT) because of their anti-state and anti-domination orientations and focus on building alternatives to prevent and manage violence apart from contemporary ‘counterterrorism’. After providing an outline of anarchist and abolitionist thought, the article connects these to contemporary examples drawn from the United States and Nepal. In doing so, it theorises and imagines futures for preventing violence and building public security that are linkedto anarchist and abolitionist understandings of violence and the state. In contrast to ‘power politics’ which centres on the state, an anarchist abolitionist approach explores how safety and security can be reimagined and remadein the absence of a state.

Microorganisms symbiotic with insects, whether permanently or temporarily, play a crucial role in the nutrition, development, reproduction, defence, and metamorphosis regulation. In some Lepidoptera, oviposition-deterrent pheromones (ODPs) on egg surface were used by pregnant females to modify the behaviour of conspecifics to avoid excessive competition for limited resources. In this study, we constructed four different Spodoptera litura groups, including, OH, OA, SH, and OA, which either feed on different hosts or grow in different environments. The 16S rDNA libraries of microbes from the egg surface of the four groups were constructed and sequenced. According to alpha and beta diversity indices, the microbes in environments and diets considerably influenced the richness, diversity, and community compositions of the microbiota on egg surfaces. The quantity of the main ODP components and the corresponding oviposition-deterrent activity among four groups were significantly differed among the four groups. The result of this study revealed that altering of microbes in environments or diets considerably changed the contents of ODP and oviposition-deterrent activity. As ODPs impart oviposition-deterrent activity towards closely related species, the findings of this study suggest that we should pay more attention to the role of symbiotic microorganisms in changing the ability of insects, especially sympatric species, to occupy the optimal niche when developing novel pest-control strategies.

The capacity of river mouths to reduce storm surge water levels upstream, referred to as along-estuary attenuation, has been assessed by several studies. The coastal protection function of semi-enclosed water bodies such as lagoons and channels with narrow inlets remains less explored and generalization is hampered by differences in morphology and hydrodynamic forcing. Here we use a hydrodynamic model to investigate surge attenuation along a microtidal channel with a narrow inlet at the Baltic Sea coast of Germany called The Schlei. We quantify the importance of wind and the contribution of the barrier spit system, which is constricting the inlet, to the reduction of water levels at the landward end of the channel. In addition, we explore the role of dikes in the region for the reduction of peak water levels and coastal flooding. We find effective along-channel attenuation inside The Schlei in its current state, which is mostly a result of the channel’s narrows. However, reduction rates decrease under simplified sea-level rise scenarios. Furthermore, along-channel attenuation is highly variable and can change to substantial amplification depending on hydrometeorological forcing. The barrier spit contributes to along-channel attenuation whereas the effect of existing dikes (or their removal) for along-channel attenuation is negligible.

In the present paper, we consider the following fractional double phase problem with nonlocal reaction:

\begin{align*}\qquad \quad \left\{\begin{array}{lll}(- \Delta)_p^s u + (- \Delta)_q ^s u + V(\varepsilon x) (|u|^{p - 2} u + |u|^{q - 2} u) = \left(\frac{1}{|x|^\mu} * F(u)\right) f(u) & \ \text{in}\ \mathbb{R}^N, \\u \in W^{s,p}(\mathbb{R}^N) \cap W^{s,q}(\mathbb{R}^N) ,\ u \gt 0 & \ \text{in}\ \mathbb{R}^N,\end{array}\right\}\end{align*}

where ɛ is apositive parameter, $0 \lt s \lt 1$, $2 \leqslant p \lt q \lt \min\{2p, N / s\}$, $0 \lt \mu \lt sp$, $(- \Delta)_t^s$ $(t \in \left\{p,q\right\})$ is the fractional t-Laplace operator, the reaction term $f : \mathbb{R} \mapsto \mathbb{R}$ is continuous, and the potential $V \in C (\mathbb{R}^N , \mathbb{R})$ satisfying a local condition. Using a variational approach and topological tools (the non-standard C1-Nehari manifold analysis and the abstract category theory), multiplicity of positive solutions and concentration properties for the above problem are established. Our results extend and complement some previous contributions related to double phase variational integrals.

A long standing conjecture states that the ropelength of any alternating knot is at least proportional to its crossing number. In this paper we prove that this conjecture is true. That is, there exists a constant $b_0 \gt 0$ such that $R(K)\ge b_0Cr(K)$ for any alternating knot K, where R(K) is the ropelength of K and Cr(K) is the crossing number of K. In this paper, we prove that this conjecture is true and establish that $b_0 \gt 1/56$.

The causal relevance of local flow conditions in open-channel turbulence is analysed using ensembles of interventional experiments in which the effect of perturbing the flow within a small cell is monitored at some future time. When this is done using the relative amplification of the perturbation energy, causality depends on the flow conditions within the cell before it is perturbed, and can be used as a probe of the flow dynamics. The key scaling parameter is the ambient shear, which is also the dominant diagnostic variable for wall-attached perturbations. Away from the wall, the relevant variables are the streamwise and wall-normal velocities. Causally significant cells are associated with sweeps that carry the perturbation towards the stronger shear near the wall, whereas irrelevant ones are associated with ejections that carry it towards the weaker shear in the outer layers. Causally significant and irrelevant cells are themselves organised into structures that share many characteristics with classical sweeps and ejections, such as forming spanwise pairs whose dimensions and geometry are similar to those of classical quadrants. At the wall, this is consistent with causally significant configurations in which a high-speed streak overtakes a low-speed one, and causally irrelevant ones in which the two streaks pull apart from each other. It is argued that this is probably associated with streak meandering.

Across Latin American countries, there is remarkable heterogeneity in abortion legislation, ranging from full prohibition to legal elective abortion.1 However, abortion policy does not seem to clearly map onto implementation on the ground. On the one hand, even in countries with very restrictive abortion laws, (clandestine) abortion rates are comparatively high, and legislation that criminalizes abortion is rarely enforced (Blofield 2006; Htun 2003). On the other hand, in countries that allow abortion under all or some circumstances, access to abortion is not guaranteed. In Argentina, for instance, different reports have stressed the difficulty in accessing abortion procedures both before and after legalization was enacted in early 2021.2

In vitro production of porcine embryos is a complicated process that includes in vitro maturation (IVM), in vitro fertilization (IVF) and in vitro culture (IVC). Insufficient cytoplasmic maturation, slow zona reaction and improper embryo culture conditions will compromise the efficiency of porcine embryo production in vitro. Previous studies have shown that insulin-transferrin-selenium (ITS) in IVM or IVC medium could improve porcine oocyte maturation, decrease polyspermy fertilization and promote subsequent embryonic development in vitro. However, the effect of ITS both in IVM and IVC media on porcine embryo production in vitro hasn’t been elucidated. In this study, we found that 1.0% ITS supplementation in IVM/IVC media promoted the expansion of cumulus cells, raised mitochondrial membrane potential, increased ATP content and reduced ROS level in matured oocytes, improved blastocyst rate and the cell number of blastocyst, simultaneously. In conclusion, the IVM/IVC media supplemented with 1.0% ITS can improve the efficiency of porcine embryo production in vitro.