We are interested in studying sets of the form \[ \mathcal{U}(\alpha) := \left\{ x\in X: \ \exists M=M(x) \geq 1 \text{ such that } \forall N\geq M, \ \exists n\leq N \text{ such that } d(T^nx, x) \leq |\lambda|^{-\alpha N} \right\} \] where $(X,T,d)$ is our metric dynamical system and $|\lambda|>1$. Although a lot of results exist for the one dimensional case, not as many are known for systems in higher dimensions and especially in the hyperbolic case. We consider $X=\mathbb{T}^2$, $T(x) = Ax \pmod{1}$, where $A$ is a hyperbolic, area preserving, $2\times 2$ matrix with integer entries and $\lambda$ is the eigenvalue of $A$ of modulus larger than $1$ and we explicitly calculate the Hausdorff dimension of this set.

Let be an invertible matrix with integer elements. Then determines a self-map of the -dimensional torus . Given a real number , and a sequence of points in , let be the set of points such that for infinitely many . The Hausdorff dimension of has previously been studied by Hill-Velani and Li-Liao-Velani-Zorin. We provide a lower bound on the Hausdorff dimension of for any expanding matrix. For hyperbolic matrices, we compute the dimension of only when is a matrix. We give counterexamples to a natural candidate for a dimension formula for general dimension .

The life of cutting tool inserts is critically important for efficient machining, reducing manufacturing cost, embedded energy, and enabling more complex parts to be machined. For these applications, cemented carbide (WC-Co) materials are a prime candidate. The performance of these materials can be limited by early fracture, typically via an intergranular fracture path with respect to carbide grains. This motivates further studies to understand the character of the grain boundary network so that grain boundary engineering (GBE) of WC-Co tools can be used to improve tool life and performance. In this work, we have used Rohrer et al.'s five-parameter grain boundary character distribution (GBCD) analysis to examine the grain boundary network of WC-10wt%Co and WC-10wt%Co-1wt%Cr samples (Rohrer et al., 2004a [1]). It was found that the measured area fraction of the Sigma 2 boundaries was comparable to the values reported in the literature despite the relatively larger grain sizes (similar to 14 mu m) and higher cobalt contents. The result suggests that chromium doping increases the area fraction of Sigma 2 boundaries from 12.8 % to 14.8 %. It is proposed that this is a consequence of altering the Sigma 2 boundary energy, as associated with adding chromium.

Chunkwood fuels have a particle size larger than normal chips which enables good drying and storage properties and are therefore appreciated by small-scale users. However, small-scale boilers optimized for chunkwood are not commercially available and the research question is if modern wood chip stokers, selected for having a robust fuel feeding system could feed and combust the fuel. Chunkwood fuel feeding, and combustion tests are performed in a 27-kW and a 240-kW wood chip stoker. Both boilers fulfill Ecodesign emission requirements for carbon monoxide (CO) at nominal load, but further optimization is required to fulfil requirements for dust. Partial load combustion needs to be further studied. There were problems with high stress on the fuel feeding system in both stokers, traced to when excessively large fuel pieces passed the outlet of the fuel bin and when fuel discs became trapped between the auger screw and the lid of the conveyor. Suggestions to solve the fuel feeding problems includes redesign of the fuel bin auger screw to cut oversized pieces, alternatively use of previously developed prototype conveyors that worked. Further studies are required to optimize the fuel feeding system and the combustion performance including a solution for partial load operation.

We consider linear mappings on the $d$-dimensional torus, defined by $T(x) = Ax \pmod 1$, where $A$ is an invertible $d \times d$ integer matrix, with no eigenvalues on the unit circle. In the case $d = 2$ and $\det A = \pm 1$, we give a formula for the Hausdorff dimension of the set \[ \{ \, x \in \mathbb{T}^d : d (T^n (x), x) < e^{- \alpha n} \text{ for infinitely many } n \, \}. \]

Consider the quadratic family T-a(x)=ax(1-x), for x is an element of[0,1] and mixing Collet-Eckmann (CE) parameters a is an element of(2,4). For bounded phi, set (phi) over tilde (a):= phi - integral phi d mu(a), with mu(a) the unique acim of T-a, and put (sigma(a)(phi))(2) : = integral(phi) over tilde (2)(a) d mu(a) + 2 Sigma(i>0) integral(phi) over tilde (a) ((phi) over tilde (a) circle T-a(i))d mu(a). For any transversal mixing Misiurewicz parameter a(*), we find a positive measure set Omega(*) of mixing CE parameters, containing a(*) as a Lebesgue density point, such that for any Hoolder phi with sigma(a)*(phi) not equal 0, there exists epsilon(phi) > 0 such that, for normalised Lebesgue measure on Omega(*) boolean AND [a(*) - epsilon(phi), a(*) + epsilon(phi)], the functions xi(i)(a) = (phi) over tilde (a)(T-a(i+1)(1/2))/sigma(a)(phi) satisfy an almost sure invariance principle (ASIP) for any error exponent gamma > 2/5. (In particular, the Birkhoff sums satisfy this ASIP.) Our argument goes along the lines of Schnellmann's proof for piecewise expanding maps. We need to introduce a variant of Benedicks-Carleson parameter exclusion and to exploit fractional response and uniform exponential decay of correlations from Baladi et al [Whitney-Holder continuity of the SRB measure for transversal families of smooth unimodal maps. Invent. Math. 201 (2015), 773-844].

Suppose $(f,\mathcal{X},\mu)$ is a measure preserving dynamical system and $\phi \colon \mathcal{X} \to \mathbb{R}$ a measurable function. Consider the maximum process $M_n:=\max\{X_1 \ldots,X_n\}$, where $X_i=\phi\circ f^{i-1}$ is a time series of observations on the system. Suppose that $(u_n)$ is a non-decreasing sequence of real numbers, such that $\mu(X_1>u_n)\to 0$. For certain dynamical systems, we obtain a zero--one measure dichotomy for $\mu(M_n\leq u_n\,\textrm{i.o.})$ depending on the sequence $u_n$. Specific examples are piecewise expanding interval maps including the Gauss map. For the broader class of non-uniformly hyperbolic dynamical systems, we make significant improvements on existing literature for characterising the sequences $u_n$. Our results on the permitted sequences $u_n$ are commensurate with the optimal sequences (and series criteria) obtained by Klass (1985) for i.i.d. processes. Moreover, we also develop new series criteria on the permitted sequences in the case where the i.i.d. theory breaks down. Our analysis has strong connections to specific problems in eventual always hitting time statistics and extreme value theory.

We consider the set $\mathcal{R}_\mathrm{io}$ of points returning infinitely many times to a sequence of shrinking targets around themselves. Under additional assumptions we improve Boshernitzan's pioneering result on the speed of recurrence. In the case of the doubling map as well as some linear maps on the $d$ dimensional torus, we even obtain a dichotomy condition for $\mathcal{R}_\mathrm{io}$ to have measure zero or one. Moreover, we study the set of points eventually always returning and prove an analogue of Boshernitzan's result in similar generality.

Motivated by the random covering problem and the study of Dirichlet uniform approximable numbers, we investigate the uniform random covering problem. Precisely, consider an i.i.d. sequence $\omega =(\omega _n)_{n\geq 1}$ uniformly distributed on the unit circle $\mathbb{T}$ and a sequence $(r_n)_{n\geq 1}$ of positive real numbers with limit $0$. We investigate the size of the random set $$\begin{align*} & {\operatorname{{{\mathcal{U}}}}} (\omega):=\{y\in \mathbb{T}: \ \forall N\gg 1, \ \exists n \leq N, \ \text{s.t.} \ | \omega_n -y | < r_N \}. \end{align*}$$Some sufficient conditions for ${\operatorname{{{\mathcal{U}}}}}(\omega )$ to be almost surely the whole space, of full Lebesgue measure, or countable, are given. In the case that ${\operatorname{{{\mathcal{U}}}}}(\omega )$ is a Lebesgue null measure set, we provide some estimations for the upper and lower bounds of Hausdorff dimension.

Consider a mixing dynamical systems $([0,1], T, \mu)$, for instance a piecewise expanding interval map with a Gibbs measure $\mu$. Given a non-summable sequence $(m_k)$ of non-negative numbers, one may define $r_k (x)$ such that $\mu (B(x, r_k(x)) = m_k$. It is proved that for almost all $x$, the number of $k \leq n$ such that $T^k (x) \in B_k (x)$ is approximately equal to $m_1 + \ldots + m_n$. This is a sort of strong Borel--Cantelli lemma for recurrence. A consequence is that \[ \lim_{r \to 0} \frac{\log \tau_{B(x,r)} (x)}{- \log \mu (B (x,r))} = 1 \] for almost every $x$, where $\tau$ is the return time.

Chunkwood is a wood fuel with a particle size larger than normal chips. It was thoroughly investigated during the late 1970s and 1980s, but commercial small-scale use was not established, despite advantages compared to wood chips. Technologies for small scale automatic feeding and combustion were developed in the 1970s and 1980s but automatic feeding systems failed, and the boiler efficiency and emissions fall short of today’s standards. This paper describes combustion tests with automatic feeding of chunkwood in a 30 kW and a 200 kW wood chip boiler. The chunkwood combustion conditions was satisfactory for both boilers at nominal load though measurements performed for the 30 kW boiler showed relatively poor efficiency due to high flue gas temperature and high air factor. There were problems with fuel feeding in both boilers, traced to when excessively large fuel pieces passed the outlet of the fuel storage. Problems also occurred when larger fuel discs became trapped between the auger screw and the lid of the conveyor, but the later problem could be reduced by releasing the upper part of the lid so that it could move upwards. Design guidelines for chunkwood friendly boiler construction are provided, based upon the problems exposed.

A convertible, zoned ventilation system was field-tested in a modern, airtight Swedish home when occupied either by an experimental team or by a family. Indoor air quality in the master bedroom was monitored under four ventilation strategies. Relative to constant air volume strategies (CAV), demand-controlled ventilation (DCV) that was responding to CO2 concentration extracted more air when people were present, but less in total over 24 h. This elevated the indoor air humidity, beneficial in climates with dry winter air. Multiple monitors within the bedroom indicated that vertical CO2 stratification occurred routinely, presumably due to low mixing of supply air from a wall-mounted diffuse vent, spreading the air radially over the wall. This seemingly improved air quality in the breathing zone under local (ceiling) extract ventilation but worsened it during more typical, centralised extract ventilation, where air escapes the room via an inner doorway. The local extract arrangement thus seemed to yield both improved ventilation efficiency and reduced contaminant spread to other rooms. The noted air quality variations within the room highlight the importance of sensor placement in demand-control ventilated spaces, even in small rooms such as bedrooms.

A greenhouse water curtain heating system allows heating of the greenhouse by low-temperature water, which can be obtained from residual waste heat sources. The water curtain can be applied on the outside of the greenhouse roof or enclosed between two foils. But also enclosed water curtains suffer from high heat losses, which limits the integration of low temperature waste heat sources. An effective way of reducing the heat losses is to combine the water curtain system with retractable liquid foam enclosed between two foils. But until now, a systematic evaluation of the thermal performance of such system combination is still lacking. Thus, this study aims to fill in this research gap by evaluating the heat transfer characteristics of a double-foil greenhouse roof section installed with a combined water curtain and liquid foam system. Experimental tests have been conducted under a wide range of temperature scenarios in a climate chamber where the heat loss and heat gains from the water curtain is measured. The results have been compared with the data in the existing studies. This study revealed that combining the water curtain system with liquid foam, reduces the heat losses by half compared to using just the water curtain: the heat loss coefficient was reduced from 4.4 W.m(-2).K-1 down to 2.0 W.m(-2).K-1. The heat losses through the roof using the combined system are also lower than the heat losses from a double foil greenhouse with other heating systems. An average water curtain temperature of 5.1 degrees C above the inside greenhouse temperature can compensate for the heat losses through the roof at an outdoor temperature of -19 degrees C. Based on the study results, recommendations for market implantation of this technology are provided. This study confirms energetic benefit of combining water curtains and the liquid foam technology.

I prove a mass transference principle for general shapes, similar to a recent result by H. Koivusalo and M. Rams. The proof relies on Vitali's covering lemma and manipulations with Riesz energies. The main novelty is that it is proved that the obtained limsup-set belongs to the classes of sets with large intersections, as defined by K. Falconer. This has previously not been proved for as general shapes as in this paper.

We associate to a perturbation \begin{document}$ (f_t) $\end{document} of a (stably mixing) piecewise expanding unimodal map \begin{document}$ f_0 $\end{document} a two-variable fractional susceptibility function \begin{document}$ \Psi_\phi(\eta, z) $\end{document}, depending also on a bounded observable \begin{document}$ \phi $\end{document}. For fixed \begin{document}$ \eta \in (0,1) $\end{document}, we show that the function \begin{document}$ \Psi_\phi(\eta, z) $\end{document} is holomorphic in a disc \begin{document}$ D_\eta\subset \mathbb{C} $\end{document} centered at zero of radius \begin{document}$ >1 $\end{document}, and that \begin{document}$ \Psi_\phi(\eta, 1) $\end{document} is the Marchaud fractional derivative of order \begin{document}$ \eta $\end{document} of the function \begin{document}$ t\mapsto \mathcal{R}_\phi(t): = \int \phi(x)\, d\mu_t $\end{document}, at \begin{document}$ t = 0 $\end{document}, where \begin{document}$ \mu_t $\end{document} is the unique absolutely continuous invariant probability measure of \begin{document}$ f_t $\end{document}. In addition, we show that \begin{document}$ \Psi_\phi(\eta, z) $\end{document} admits a holomorphic extension to the domain \begin{document}$ \{\, (\eta, z) \in \mathbb{C}^2\mid 0<\Re \eta <1, \, z \in D_\eta \,\} $\end{document}. Finally, if the perturbation \begin{document}$ (f_t) $\end{document} is horizontal, we prove that \begin{document}$ \lim_{\eta \in (0,1), \eta \to 1}\Psi_\phi(\eta, 1) = \partial_t \mathcal{R}_\phi(t)|_{t = 0} $\end{document}.

The grain boundary network of WC in WC-Co is important, as cracks often travel intergranularly. This motivates the present work, where we experimentally measure the fracture energy of Σ2 twist grain boundaries between WC crystals using a double cantilever beam opened with a wedge under displacement control in a WC-10wt%Co sample. The fracture energy of this boundary type was compared with cleaving { $$10\overline{1}0$$ } prismatic planes in a WC single crystal. Fracture energies of 7.04 ± 0.36 Jm−2 and 3.57 ± 0.28 Jm−2 were measured for { $$10\overline{1}0$$ } plane and Σ2 twist boundaries, respectively.

We consider dynamical systems $$(X,T,\mu )$$ which have exponential decay of correlations for either Hölder continuous functions or functions of bounded variation. Given a sequence of balls $$(B_n)_{n=1}^\infty $$ , we give sufficient conditions for the set of eventually always hitting points to be of full measure. This is the set of points x such that for all large enough m, there is a $$k < m$$ with $$T^k (x) \in B_m$$ . We also give an asymptotic estimate as $$m \rightarrow \infty $$ on the number of $$k < m$$ with $$T^k (x) \in B_m$$ . As an application, we prove for almost every point x an asymptotic estimate on the number of $$k \le m$$ such that $$a_k \ge m^t$$ , where $$t \in (0,1)$$ and $$a_k$$ are the continued fraction coefficients of x.

An abstract is not available for this content. As you have access to this content, full HTML content is provided on this page. A PDF of this content is also available in through the ‘Save PDF’ action button.

We establish quantitative results for the statistical be\-ha\-vi\-our of \emph{infinite systems}. We consider two kinds of infinite system: i) a conservative dynamical system $(f,X,\mu)$ preserving a $\sigma$-finite measure $\mu$ such that $\mu(X)=\infty$; ii) the case where $\mu$ is a probability measure but we consider the statistical behaviour of an observable $\phi\colon X\to[0,\infty)$ which is non-integrable: $\int \phi \, d\mu=\infty$. In the first part of this work we study the behaviour of Birkhoff sums of systems of the kind ii). For certain weakly chaotic systems, we show that these sums can be strongly oscillating. However, if the system has superpolynomial decay of correlations or has a Markov structure, then we show this oscillation cannot happen. In this case we prove asymptotic relations between the behaviour of $\phi $, the local dimension of $\mu$, and on the growth of Birkhoff sums (as time tends to infinity). We then establish several important consequences which apply to infinite systems of the kind i). This includes showing anomalous scalings in extreme event limit laws, or entrance time statistics. We apply our findings to non-uniformly hyperbolic systems preserving an infinite measure, establishing anomalous scalings in the case of logarithm laws of entrance times, dynamical Borel--Cantelli lemmas, almost sure growth rates of extremes, and dynamical run length functions.

We introduce a parameter space containing all algebraic integers β∈(1,2] that are not Pisot or Salem numbers, and a sequence of increasing piecewise continuous function on this parameter space which gives a lower bound for the Garsia entropy of the Bernoulli convolution ν_β. This allows us to show that dim_H (ν_β)=1 for all β with representations in certain open regions of the parameter space.