On an asymptotically log-periodic solution to the graphical curve shortening flow equation

<abstract><p>With the help of heat equation, we first construct an example of a graphical solution to the curve shortening flow. This solution $ y\left(x, t\right) \ $has the interesting property that it converges to a log-periodic function of the form</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ A\sin \left( \log t\right) +B\cos \left( \log t\right) $\end{document} </tex-math></disp-formula></p> <p>as$ \ t\rightarrow \infty, \ $where $ A, \ B $ are constants. Moreover, for any two numbers $ \alpha &lt; \beta, \ $we are also able to construct a solution satisfying the oscillation limits</p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \liminf\limits_{t\rightarrow \infty}y\left( x,t\right) = \alpha,\ \ \ \limsup\limits _{t\rightarrow \infty}y\left( x,t\right) = \beta,\ \ \ x\in K $\end{document} </tex-math></disp-formula></p> <p>on any compact subset$ \ K\subset \mathbb{R}. $</p></abstract>

Off diagonal charged scalar couplings with the Z boson: Zee-type models as an example

Models with scalar doublets and charged scalar singlets have the interesting property that they have couplings between one Z boson and two charged scalars of different masses. This property is often ignored in phenomenological analysis, as it is absent from models with only extra scalar doublets. We explore this issue in detail, considering $$h \rightarrow Z \gamma $$ h → Z γ , $$B \rightarrow X_s \gamma $$ B → X s γ , and the decay of a heavy charged scalar into a lighter one and a Z boson. We propose that the latter be actively searched for at the LHC, using the scalar sector of the Zee-type models as a prototype and proposing benchmark points which obey all current experimental data, and could be within reach of the LHC.

Certificate complexity and symmetry of nested canalizing functions

Boolean nested canalizing functions (NCFs) have important applications in molecular regulatory networks, engineering and computer science. In this paper, we study their certificate complexity. For both Boolean values $b\in\{0,1\}$, we obtain a formula for $b$-certificate complexity and consequently, we develop a direct proof of the certificate complexity formula of an NCF. Symmetry is another interesting property of Boolean functions and we significantly simplify the proofs of some recent theorems about partial symmetry of NCFs. We also describe the algebraic normal form of $s$-symmetric NCFs. We obtain the general formula of the cardinality of the set of $n$-variable $s$-symmetric Boolean NCFs for $s=1,\dots,n$. In particular, we enumerate the strongly asymmetric Boolean NCFs.

A note on Variance-Sum Third Order Slope Rotatable Designs

Response Surface Methodology (RSM) is a collection of mathematical and statistical techniques useful for analyzing experiments where the yield is believed to be influenced by one or more controllable factors. Box and Hunter (1957) introduced rotatable designs in order to explore the response surfaces. The analogue of Box-Hunter rotatability criterion is a requirement that the variance of i yˆ(x)/ x be constant on circles (v=2), spheres (v=3) or hyperspheres (v 4) at the design origin. These estimates of the derivatives would then be equally reliable for all points (x , x ,...,x ) 1 2 v equidistant from the design origin. This property is called as slope rotatability (Hader and Park (1978)).Anjaneyulu et al (1995 &2000) introduced Third Order Slope Rotatable Designs. Anjaneyulu et al(2004) introduced and established that TOSRD(OAD) has the additional interesting property that the sum of the variance of estimates of slopes in all axial directions at any point is a function of the distance of the point from the design origin. In this paper we made an attempt to construct Variance-Sum Third Order Slope Rotatable in four levels. Keywords: Response Surface Methodology. Third Order Slope Rotatable Design; TOSRD (OAD), Variance-Sum Third Order Slope Rotatable Design.

Ultra-light dark matter

Ultra-light dark matter is a class of dark matter models (DM), where DM is composed by bosons with masses ranging from $$10^{-24}\, \mathrm {eV}< m < \mathrm {eV}$$ 10 - 24 eV < m < eV . These models have been receiving a lot of attention in the past few years given their interesting property of forming a Bose–Einstein condensate (BEC) or a superfluid on galactic scales. BEC and superfluidity are some of the most striking quantum mechanical phenomena that manifest on macroscopic scales, and upon condensation, the particles behave as a single coherent state, described by the wavefunction of the condensate. The idea is that condensation takes place inside galaxies while outside, on large scales, it recovers the successes of $$\varLambda $$ Λ CDM. This wave nature of DM on galactic scales that arise upon condensation can address some of the curiosities of the behaviour of DM on small-scales. There are many models in the literature that describe a DM component that condenses in galaxies. In this review, we are going to describe those models, and classify them into three classes, according to the different non-linear evolution and structures they form in galaxies: the fuzzy dark matter (FDM), the self-interacting fuzzy dark matter (SIFDM), and the DM superfluid. Each of these classes comprises many models, each presenting a similar phenomenology in galaxies. They also include some microscopic models like the axions and axion-like particles. To understand and describe this phenomenology in galaxies, we are going to review the phenomena of BEC and superfluidity that arise in condensed matter physics, and apply this knowledge to DM. We describe how ULDM can potentially reconcile the cold DM picture with the small-scale behaviour. These models present a rich phenomenology that is manifest in different astrophysical consequences. We review here the astrophysical and cosmological tests used to constrain those models, together with new and future observations that promise to test these models in different regimes. For the case of the FDM class, the mass where this model has an interesting phenomenology on small-scales $$ \sim 10^{-22}\, \mathrm {eV}$$ ∼ 10 - 22 eV , is strongly challenged by current observations. The parameter space for the other two classes remains weakly constrained. We finalize by showing some predictions that are a consequence of the wave nature of this component, like the creation of vortices and interference patterns, that could represent a smoking gun in the search of these rich and interesting alternative class of DM models.

A dual-phase alloy with ultrahigh strength-ductility synergy over a wide temperature range

High-entropy alloys (HEAs), as an emerging class of materials, have pointed a pathway in developing alloys with interesting property combinations. Although they are not exempted from the strength-ductility trade-off, they present a standing chance in overcoming this challenge. Here, we report results for a precipitation-strengthening strategy, by tuning composition to design a CoNiV-based face-centered cubic/B2 duplex HEA. This alloy sustains ultrahigh gigapascal-level tensile yield strengths and excellent ductility from cryogenic to elevated temperatures. The highest specific yield strength (~150.2 MPa·cm3/g) among reported ductile HEAs is obtained. The ability of the alloy presented here to sustain this excellent strength-ductility synergy over a wide temperature range is aided by multiple deformation mechanisms i.e., twins, stacking faults, dynamic strain aging, and dynamic recrystallization. Our results open the avenue for designing precipitation-strengthened lightweight HEAs with advanced strength-ductility combinations over a wide service temperature range.

Fermat’s principle in constant gravitational field

Recently (Int. J. Mod. Phys. D 27 (2018) 1847025) an interesting property of closed light rings in Kerr black holes has been noticed. We explain its origin and derive a slightly more general result.

Maintaining and escaping feedback control in hierarchically organised tissue: a case study of the intestinal epithelium

The intestinal epithelium is one of the fastest renewing tissues in mammals with an average turnover time of only a few days. It shows a remarkable degree of stability towards external perturbations such as physical injuries or radiation damage. Tissue renewal is driven by intestinal stem cells, and differentiated cells can de-differentiate if the stem cell niche is lost after tissue damage. However, self-renewal and regeneration require a tightly regulated balance to uphold tissue homoeostasis, and failure can lead to tissue extinction or to unbounded growth and cancerous lesions. Here, we present a mathematical model of intestinal epithelium population dynamics that is based on the current mechanistic understanding of the underlying biological processes. We derive conditions for stability and thereby identify mechanisms that may lead to loss of homoeostasis. A key results is the existence of specific thresholds in feedbacks after which unbounded growth occurs, and a subsequent convergence of the system to a stable ratio of stem to non-stem cells. A biologically interesting property of the model is that the number of differentiated cells at the steady-state can become invariant to changes in their apoptosis rate. Moreover, we compare alternative mechanisms for homeostasis with respect to their recovery dynamics after perturbation from steady-state. Finally, we show that de-differentiation enables the system to recover more gracefully after certain external perturbations, which however makes the system more prone to loosing homoeostasis.

Infinite line of equilibriums in a novel fractional map with coexisting infinitely many attractors and initial offset boosting

The study of the chaotic dynamics in fractional-order discrete-time systems has received great attention in the past years. In this paper, we propose a new 2D fractional map with the simplest algebraic structure reported to date and with an infinite line of equilibrium. The conceived map possesses an interesting property not explored in literature so far, i.e., it is characterized, for various fractional-order values, by the coexistence of various kinds of periodic, chaotic and hyper-chaotic attractors. Bifurcation diagrams, computation of the maximum Lyapunov exponents, phase plots and 0–1 test are reported, with the aim to analyse the dynamics of the 2D fractional map as well as to highlight the coexistence of initial-boosting chaotic and hyperchaotic attractors in commensurate and incommensurate order. Results show that the 2D fractional map has an infinite number of coexistence symmetrical chaotic and hyper-chaotic attractors. Finally, the complexity of the fractional-order map is investigated in detail via approximate entropy.

Designing of an algebraic signature analyzer for mixed-signal systems and testing

While the design of signature analyzers for digital circuits has been well researched in the past, the common design technique of a signature analyzer for mixed-signal systems is based on the rules of an arithmetic finite field. The analyzer does not contain carry propagating circuitry, which improves its performance as well as fault tolerance. The signatures possess the interesting property that if the input analog signal is imprecise within certain bounds (an inherent property of analog signals), then the generated signature is also imprecise within certain bounds. We offer a method to designing an algebraic signature analyzer that can be used for mixed-signal systems testing. The application of this technique to the systems with an arbitrary radix is a challenging task and the devices designed possess high hardware complexity. The proposed technique is simple and applicable to systems of any size and radix. The hardware complexity is low. The technique can also be used in algebraic coding and cryptography.