We continue work started in Deane and Gentile [A diluted version of the problem of the existence of the Hofstadter sequence, J. Differ. Equ. Appl. 31 (2024), pp. 48-65] concerning integer sequences $ q(n) $ q(n), $ n\in \mathbb N $ n is an element of N, defined by $ q(n) = q(n-q(n-1)) + f(n) $ q(n)=q(n-q(n-1))+f(n), with $ q(1) = 1 $ q(1)=1. Here, $ f(n) $ f(n), with $ f(1) = 0 $ f(1)=0, is a given sequence. We define $ \mathcal {F} $ F as the set of semi-infinite sequences f such that the resulting sequence q exists. This requires that the term $ q(n-q(n-1)) $ q(n-q(n-1)) be defined for all n, that is, $ 1\leq q(n)\leq n $ 1 <= q(n)<= n applies for all $ n\in \mathbb N $ n is an element of N. We use a variety of approaches to probe the structure of $ \mathcal {F} $ F, including explicit construction, analysis and computer-assisted proof.

We investigate the conditions on an integer sequence f(n), n 2 N, with f(1) = 0, such that the sequence q(n), computed recursively via q(n) = q(n - q(n - 1)) + f(n), with q(1) = 1, exists. We prove that f(n + 1) - f(n) in {0,1}, n > 0, is a sufficient but not necessary condition for the existence of sequence q. Sequences q defined in this way typically display non-trivial dynamics: in particular, they are generally aperiodic with no obvious patterns. We discuss and illustrate this behaviour with some examples.

Triboelectric nanogenerators (TENGs) have demonstrated outstanding potential as energy harvesters and sensors for future wearable electronics. However, TENGs still require major improvements in their theory and optimization, especially for the sliding-mode designs. Addressing this gap, a novel theoretical model based on the distance-dependent electric field (DDEF) theory for sliding mode TENGs is presented here. The model is used to simulate the electrical outputs and impedance behaviour of a sliding mode TENG, and the results are verified experimentally. The outcomes indicate that compared to existing theoretical models, this new model provides higher accuracy in representing experimental TENG. Next, all the primary parameters (material, structural and motion parameters) which affect the sliding mode TENG are analyzed, uncovering new optimization strategies and more comprehensive parametric analysis compared to previous models. More importantly, the theoretical approach is equally applicable to sliding mode TENG as well as contact-separation mode TENG. This eliminates the need for bespoke capacitor models for each TENG type, leading to a universal theoretical platform for TENGs. The new model facilitates cross-comparison between different TENG working modes, uncovering a range of previously unreported output trends. Hence, this work significantly expands the understanding of TENGs, paving way to more efficient future device designs.

For each natural number n and any bounded, convex domain 2 C Rn we characterize the sharp constant C(n, 2) in the Poincare ' inequal-ity 11f - f over bar omega 11Loo(omega;R) < C(n, 2)11Vf11Loo(omega;R). Here, f over bar omega denotes the mean value of f over 2. In the case that 2 is a ball Br of radius r in Rn, we calculate C(n, Br) = C(n)r explicitly in terms of n and a ratio of the vol-umes of the unit balls in R2n-1 and Rn. More generally, we prove that C(n, Br(omega)) < C(n, 2) < nn+1 diam(2), where Br(omega) is a ball in Rn with the same n-dimensional Lebesgue measure as 2. Both bounds are sharp, and the lower bound can be interpreted as saying that, among convex domains of equal measure, balls have the best, i.e. smallest, Poincare ' constant.

For bounded, convex sets Ω⊂ℝ^d , the sharp Poincaré constant C(Ω ) , which appears in ||f-f̅__Ω||_L^∞(Ω )≤ C(Ω )||∇ f||_L^∞(Ω ) , is given by C(Ω )=max __∂Ωζ for a specific convex function ζ [Bevan et al. in Proc Am Math Soc 151:1071–1085, 2023 (Theorem 1.1)]. We study C(· ) as a function on convex sets, in particular on polyhedra, and find that while a geometric characterization of C(Ω ) for triangles is possible, for other polyhedra the problem of ordering ζ (V_i) , where V_i are the vertices of Ω , can be formidable. In these cases, we develop estimates of C(Ω ) from above and below in terms of more tractable quantities. We find, for example, that a good proxy for C(Q) when Q is a planar polygon with vertices V_i and centroid γ (Q) is the quantity D(Q)=max _i|V_i-γ (Q)| , with an error of up to ∼ 8% . A numerical study suggests that a similar statement holds for k-gons, this time with a maximal error across all k-gons of ∼ 13% . We explore the question of whether there is, for each Ω , at least one point M capable of ordering the ζ (V_i) according to the ordering of the |V_i-M| . For triangles, M always exists; for quadrilaterals, M seems always to exist; for 5-gons and beyond, they seem not to.

For each natural number n n and any bounded, convex domain Ω ⊂ R n \Omega \subset \mathbb {R}^n we characterize the sharp constant C ( n , Ω ) C(n,\Omega ) in the Poincaré inequality ‖ f − f ¯ Ω ‖ L ∞ ( Ω ; R ) ≤ C ( n , Ω ) ‖ ∇ f ‖ L ∞ ( Ω ; R ) \| f - \bar {f}_{\Omega }\|_{L^{\infty }(\Omega ;\mathbb {R})} \leq C(n,\Omega ) \|\nabla f\|_{L^{\infty }(\Omega ;\mathbb {R})} . Here, f ¯ Ω \bar {f}_{\Omega } denotes the mean value of f f over Ω \Omega . In the case that Ω \Omega is a ball B r B_r of radius r r in R n \mathbb {R}^n , we calculate C ( n , B r ) = C ( n ) r C(n,B_r)=C(n)r explicitly in terms of n n and a ratio of the volumes of the unit balls in R 2 n − 1 \mathbb {R}^{2n-1} and R n \mathbb {R}^n . More generally, we prove that C ( n , B r ( Ω ) ) ≤ C ( n , Ω ) ≤ n n + 1 d i a m ( Ω ) C(n,B_{r(\Omega )}) \leq C(n,\Omega ) \leq \frac {n}{n+1}\mathrm {diam}(\Omega ) , where B r ( Ω ) B_{r(\Omega )} is a ball in R n \mathbb {R}^n with the same n − n- dimensional Lebesgue measure as Ω \Omega . Both bounds are sharp, and the lower bound can be interpreted as saying that, among convex domains of equal measure, balls have the best, i.e. smallest, Poincaré constant.

For each natural number n n and any bounded, convex domain Ω ⊂ R n \Omega \subset \mathbb {R}^n we characterize the sharp constant C ( n , Ω ) C(n,\Omega ) in the Poincaré inequality ‖ f − f ¯ Ω ‖ L ∞ ( Ω ; R ) ≤ C ( n , Ω ) ‖ ∇ f ‖ L ∞ ( Ω ; R ) \| f - \bar {f}_{\Omega }\|_{L^{\infty }(\Omega ;\mathbb {R})} \leq C(n,\Omega ) \|\nabla f\|_{L^{\infty }(\Omega ;\mathbb {R})} . Here, f ¯ Ω \bar {f}_{\Omega } denotes the mean value of f f over Ω \Omega . In the case that Ω \Omega is a ball B r B_r of radius r r in R n \mathbb {R}^n , we calculate C ( n , B r ) = C ( n ) r C(n,B_r)=C(n)r explicitly in terms of n n and a ratio of the volumes of the unit balls in R 2 n − 1 \mathbb {R}^{2n-1} and R n \mathbb {R}^n . More generally, we prove that C ( n , B r ( Ω ) ) ≤ C ( n , Ω ) ≤ n n + 1 d i a m ( Ω ) C(n,B_{r(\Omega )}) \leq C(n,\Omega ) \leq \frac {n}{n+1}\mathrm {diam}(\Omega ) , where B r ( Ω ) B_{r(\Omega )} is a ball in R n \mathbb {R}^n with the same n − n- dimensional Lebesgue measure as Ω \Omega . Both bounds are sharp, and the lower bound can be interpreted as saying that, among convex domains of equal measure, balls have the best, i.e. smallest, Poincaré constant.

Herein, we discuss the use of a multiscale method based on lookup tables (LUTs) for extremely fast and mathematically accurate emission current calculations. The technique is applicable to metallic emitters of arbitrary shape, which could be of great use in particle-in-cell codes for investigating electron sources. The LUTs are prepopulated with precise calculations of local emission current density (LECD) that can be upgraded to include any higher order physics that is critical for current state-of-the-art emitters, including nanoscale emitter curvature. The method for considering the effect of curvature is discussed in detail. Early results show that the use of a LUT for LECD can speed up numerical simulations by a factor of about 1000 times while retaining high precision with a maximum error of less than one percent when compared to direct numerical solutions.

In the framework of KAM theory, the persistence of invariant tori in quasi-integrable systems is proved by assuming a non-resonance condition on the frequencies, such as the standard Diophantine condition or the milder Bryuno condition. In the presence of dissipation, most of the quasi-periodic solutions disappear and one expects, at most, only a few of them to survive together with the periodic attractors. However, to prove that a quasi-periodic solution really exists, usually one assumes that the frequencies still satisfy a Diophantine condition and, furthermore, that some external parameters of the system are suitably tuned with them. In this paper we consider a class of systems on the one-dimensional torus, subject to a periodic perturbation and in the presence of dissipation, and show that, however small the dissipation, if the perturbation is a trigonometric polynomial in the angles and the unperturbed frequencies satisfy a non-resonance condition of finite order, depending on the size of the dissipation, then a quasi-periodic solution exists with slightly perturbed frequencies provided the size of the perturbation is small enough. If on the one hand the maximal size of the perturbation is not uniform in the degree of the trigonometric polynomial, on the other hand all but finitely many frequencies are allowed and there is no restriction arising from the tuning of the external parameters. A physically relevant case, where the result applies, is the spin-orbit model, which describes the rotation of a satellite around its own axis, while revolving on a Keplerian orbit around a planet, in the case in which the dissipation is taken into account through the MacDonald torque.

We advocate the use of lookup tables in the development of extremely fast and accurate multiscale models based on the coupling of a quantum-mechanical wave impedance approach and finite-element simulations to determine the local emission current density (LECD) from a metallic emitter of arbitrary shape. The lookup tables are prepopulated with numerical solutions of LECD that can be adjusted to accommodate any form of higher order physics, which is critical for current state-of-the-art emitters. Results show that the use of lookup tables can speed up numerical simulations of the field emission current from metallic cathodes by a factor of about 1000× while retaining high precision, with a maximum error of less than 1% when compared to direct numerical solutions. Implementation of nanoscale emitter physics into lookup tables is discussed and used to assess the validity of the Kemble approximation for nanoscale metallic cathodes. The use of lookup tables is illustrated through a calculation of the LECDs of a metallic field emitter with a rugged surface and from an array of ellipsoid-on-a-post emitters. Section V contains our conclusions and suggestions for future work.

We exhibit a family of convex functionals with infinitely many equal-energy $$C^1$$ stationary points that (i) occur in pairs $$v_{\pm }$$ satisfying $$\det \nabla v_{\pm }=1$$ on the unit ball B in $${\mathbb {R}}^2$$ and (ii) obey the boundary condition $$v_{\pm }=\text {id}$$ on $$ \partial B$$ . When the parameter $$\epsilon $$ upon which the family of functionals depends exceeds $$\sqrt{2}$$ , the stationary points appear to ‘buckle’ near the centre of B and their energies increase monotonically with the amount of buckling to which B is subjected. We also find Lagrange multipliers associated with the maps $$v_{\pm }(x)$$ and prove that they are proportional to $$(\epsilon -1/\epsilon )\ln |x|$$ as $$x \rightarrow 0$$ in B. The lowest-energy pairs $$v_{\pm }$$ are energy minimizers within the class of twist maps (see Taheri in Topol Methods Nonlinear Anal 33(1):179–204, 2009 or Sivaloganathan and Spector in Arch Ration Mech Anal 196:363–394, 2010), which, for each $$0\le r\le 1$$ , take the circle $$\{x\in B: \ |x|=r\}$$ to itself; a fortiori, all $$v_{\pm }$$ are stationary in the class of $$W^{1,2}(B;{\mathbb {R}}^2)$$ maps w obeying $$w=\text {id}$$ on $$\partial B$$ and $$\det \nabla w=1$$ in B.

We consider the Modified Kuramoto–Sivashinky Equation (MKSE) in one and two space dimensions and we obtain explicit and accurate estimates of various Sobolev norms of the solutions. In particular, by using the sharp constants which appear in the functional interpolation inequalities used in the analysis of partial differential equations, we evaluate explicitly the sup-norm of the solutions of the MKSE. Furthermore we introduce and then compute the so-called crest factor associated with the above solutions. The crest factor provides information on the distortion of the solution away from its space average and therefore, if it is large, gives evidence of strong turbulence. Here we find that the time average of the crest factor scales like $$\lambda ^{(2d-1)/8}$$ for $$\lambda $$ large, where $$\lambda $$ is the bifurcation parameter of the source term and $$d=1,2$$ is the space dimension. This shows that strong turbulence cannot be attained unless the bifurcation parameter is large enough.

We show that the N-covering map, which in complex coordinates is given by $$u_{_{\scriptscriptstyle {N}}}(z):=z \mapsto z^{N}/\sqrt{N}|z|^{N-1}$$ and where N is a natural number, is a global minimizer of the Dirichlet energy $$\mathbb {D}(v)=\int _B |\nabla v(x)|^2 \, dx$$ with respect to so-called inner and outer variations. An inner variation of $$u_{_{\scriptscriptstyle {N}}}$$ is a map of the form $$u_{_{\scriptscriptstyle {N}}}\circ \varphi $$ , where $$\varphi $$ belongs to the class $$\mathcal {A}(B):=\{\varphi \in H^1(B;\mathbb {R}^2): \ \det \nabla \varphi = 1 \ \text {a.e.}, \ \varphi \arrowvert _{\partial B}(x) = x\}$$ and B denotes the unit ball in $$\mathbb {R}^2$$ , while an outer variation of $$u_{_{\scriptscriptstyle {N}}}$$ is a map of the form $$\phi \circ u_{_{\scriptscriptstyle {N}}}$$ , where $$\phi $$ belongs to the class $$\mathcal {A}(B(0,1/\sqrt{N}))$$ . The novelty of our approach to inner variations is to write the Dirichlet energy of $$u_{_{\scriptscriptstyle {N}}}\circ \varphi $$ in terms of the functional $$I(\psi ;N):= \int _{B} N |\psi _{_R}|^2 + \frac{1}{N} |\psi _\tau |^2 \, dy$$ , where $$\psi $$ is a suitably defined inverse of $$\varphi $$ , and $$\psi _{_R}$$ and $$\psi _{\tau }$$ are, respectively, the radial and angular weak derivatives of $$\psi $$ , and then to minimise $$I(\psi ;N)$$ by considering a series of auxiliary variational problems of isoperimetric type. This approach extends to include p-growth functionals ( $$p>1$$ ) provided the class $$\mathcal {A}(B)$$ is suitably adapted. When $$1<p<2$$ , this adaptation is delicate and relies on the deep results of Barchiesi et al. on the space they refer to in Barchiesi et al. (Arch Ration Mech Anal 224(2):743–816, 2017) as $$\mathcal {A}_p$$ . A technique due to Sivaloganthan and Spector (Arch Ration Mech Anal 196:363–394, 2010) can be applied to outer variations. We also show that there is a large class of variations of the form $$v=h\,\circ \,u_2\,\circ \,g$$ , where h and g are suitable measure-preserving maps, in which $$u_2$$ is a local minimizer of the Dirichlet energy . The proof of this fact requires a careful calculation of the second variation of $$\mathbb {D}(v(\cdot ,\delta ))$$ , which quantity turns out to be non-negative in general and zero only when $$\mathbb {D}(v(\cdot ,\delta ))=\mathbb {D}(u_2)$$ .

This is a theoretical and numerical study of a model of a rope fountain subject to a drag force that depends linearly on the rope velocity. A precise, analytical description of the long-term shape adopted by the rope is given, and various consequences are derived from it. Using parameters that naturally appear in the model, we distinguish between cases wherein energy is conserved by means of a constant tension far from the rope source (the ‘free’ case) and where energy conservation is a consequence of a non-constant tension (the ‘braked’ case). The model is used, among other things, to generate rope fountain shapes based on approximate experimental estimates of the parameter values and a careful numerical treatment.

In this paper we give an explicit sufficient condition for the affine map u(lambda)(x) := lambda x to be the global energy minimizer of a general class of elastic stored-energy functionals I(u) = integral(Omega) W(del(u)) dx in three space dimensions, where W is a polyconvex function of 3 x 3 matrices. The function space setting is such that cavitating (i.e., discontinuous) deformations are admissible. In the language of the calculus of variations, the condition ensures the quasiconvexity of I(.) at lambda 1, where 1 is the 3 x 3 identity matrix. Our approach relies on arguments involving null Lagrangians (in this case, affine combinations of the minors of 3 x 3 matrices), on the previous work [J. Bevan and C. Zeppieri, Calc. Var. Partial Differential Equations, 55 (2015), pp. 1-25], and on a careful numerical treatment to make the calculation of certain constants tractable. We also derive a new condition, which seems to depend heavily on the smallest singular value lambda(1)(del(u)) of a competing deformation u that is necessary for the inequality I(u) < I(u(lambda)), and which, in particular, does not exclude the possibility of cavitation.

We consider, from a mathematical perspective, the power generated by a contact-mode triboelectric nanogenerator, an energy harvesting device that has been thoroughly studied recently. We encapsulate the behaviour of the device in a differential equation, which although linear and of first order, has periodic coefficients, leading to some interesting mathematical problems. In studying these, we derive approximate forms for the mean power generated and the current waveforms, and describe a procedure for computing the Fourier coefficients for the current, enabling us to compute the power accurately and show how the power is distributed over the harmonics. Comparisons with numerics validate our analysis.

Triboelectric nanogenerators (TENGs) are in the forefront of next-generation energy harvesting technologies, having been demonstrated as a leading candidate for numerous applications in energy harvesting and self-powered sensing. However, critical parameters affecting TENG output behavior and their optimization are not well understood. Herein, for the first time, the power output characteristics of TENGs are fully unveiled by vigorously analyzing their impedance behavior as a function of excitation source and device parameters. In this paper, Norton's theorem, first presented in 1926 for two terminal linear electrical networks, is extended to represent TENGs, allowing accurate visualization of their dynamic power output behavior via small signal analysis. TENG impedance plots are introduced to accurately determine the peak power point of a given design, which holds paramount importance in understanding and improving TENGs. The knowledge with empirical understanding for these variations results in the design and construction of more efficient TENG devices for future applications.

The Wirtz pump is not only an excellent example of alternative technology, using as it does the kinetic energy of a stream to raise a proportion of its water, but its mathematical modelling also poses several intriguing problems. We give some history of the Wirtz pump and describe its operation. Taking a novel dynamical systems approach, we then derive a discrete mathematical model in the form of a mapping that describes its hydrostatic behaviour. Our model enables us to explain several aspects of the behaviour of the pump as well as to design one that gives approximately maximal, and maximally constant, output pressure.

As part of a longer-term project to compare different methods of extracting emission-area data from ideal Fowler-Nordheim plots, this Poster investigates refinements to the extraction-parameter approach. It is shown that varying the choice of the scaled fitting parameter f(t), depending on the range of values used for the independent variable, does make a noticeable difference. However, the (usually neglected) chord correction is much smaller than previously thought; it is justified to continue neglecting it, but better practice is to include it.

A generic electromagnetic model for the working principles of triboelectric nanogenerators derived using Maxwell's equations, to a universally applicable framework.