Waring–Goldbach Problem of Even Powers in Short Intervals

In this paper, we study the average behaviour of the representations of n = p 1 2 + p 2 4 + p 3 4 + p 4 k over short intervals for k ≥ 4 , where p 1 , p 2 , p 3 , p 4 are prime numbers. This improves the previous results.

On Schneider’s Continued Fraction Map on a Complete Non-Archimedean Field

Let $${\mathcal {M}}$$ M denote the maximal ideal of the ring of integers of a non-Archimedean field K with residue class field k whose invertible elements, we denote $$k^{\times }$$ k × , and a uniformizer we denote $$\pi $$ π . In this paper, we consider the map $$T_{v}: {\mathcal {M}} \rightarrow {\mathcal {M}}$$ T v : M → M defined by $$\begin{aligned} T_v(x) = \frac{\pi ^{v(x)}}{x} - b(x), \end{aligned}$$ T v ( x ) = π v ( x ) x - b ( x ) , where b(x) denotes the equivalence class to which $$\frac{\pi ^{v(x)}}{x}$$ π v ( x ) x belongs in $$k^{\times }$$ k × . We show that $$T_v$$ T v preserves Haar measure $$\mu $$ μ on the compact abelian topological group $${\mathcal {M}}$$ M . Let $${\mathcal {B}}$$ B denote the Haar $$\sigma $$ σ -algebra on $${\mathcal {M}}$$ M . We show the natural extension of the dynamical system $$({\mathcal {M}}, {\mathcal {B}}, \mu , T_v)$$ ( M , B , μ , T v ) is Bernoulli and has entropy $$\frac{\#( k)}{\#( k^{\times })}\log (\#( k))$$ # ( k ) # ( k × ) log ( # ( k ) ) . The first of these two properties is used to study the average behaviour of the convergents arising from $$T_v$$ T v . Here for a finite set A its cardinality has been denoted by $$\# (A)$$ # ( A ) . In the case $$K = {\mathbb {Q}}_p$$ K = Q p , i.e. the field of p-adic numbers, the map $$T_v$$ T v reduces to the well-studied continued fraction map due to Schneider.

Value-Distribution of the Riemann Zeta-Function Along Its Julia Lines

For an arbitrary complex number $$a\ne 0$$ a ≠ 0 we consider the distribution of values of the Riemann zeta-function $$\zeta $$ ζ at the a-points of the function $$\Delta $$ Δ which appears in the functional equation $$\zeta (s)=\Delta (s)\zeta (1-s)$$ ζ ( s ) = Δ ( s ) ζ ( 1 - s ) . These a-points $$\delta _a$$ δ a are clustered around the critical line $$1/2+i\mathbb {R}$$ 1 / 2 + i R which happens to be a Julia line for the essential singularity of $$\zeta $$ ζ at infinity. We observe a remarkable average behaviour for the sequence of values $$\zeta (\delta _a)$$ ζ ( δ a ) .

On Average Behaviour of Regular Expressions in Strong Star Normal Form

For regular expressions in (strong) star normal form a large set of efficient algorithms is known, from conversions into finite automata to characterisations of unambiguity. In this paper we study the average complexity of this class of expressions using analytic combinatorics. As it is not always feasible to obtain explicit expressions for the generating functions involved, here we show how to get the required information for the asymptotic estimates with an indirect use of the existence of Puiseux expansions at singularities. We study, asymptotically and on average, the alphabetic size, the size of the [Formula: see text]-follow automaton and of the position automaton, as well as the ratio and the size of these expressions to standard regular expressions.

Turbulent Rayleigh–Bénard convection described by projected dynamics in phase space

Rayleigh–Bénard convection, i.e. the flow of a fluid between two parallel plates that is driven by a temperature gradient, is an idealised set-up to study thermal convection. Of special interest are the statistics of the turbulent temperature field, which we are investigating and comparing for three different geometries, namely convection with periodic horizontal boundary conditions in three and two dimensions as well as convection in a cylindrical vessel, in order to determine the similarities and differences. To this end, we derive an exact evolution equation for the temperature probability density function. Unclosed terms are expressed as conditional averages of velocities and heat diffusion, which are estimated from direct numerical simulations. This framework lets us identify the average behaviour of a fluid particle by revealing the mean evolution of a fluid with different temperatures in different parts of the convection cell. We connect the statistics to the dynamics of Rayleigh–Bénard convection, giving deeper insights into the temperature statistics and transport mechanisms. We find that the average behaviour is described by closed cycles in phase space that reconstruct the typical Rayleigh–Bénard cycle of fluid heating up at the bottom, rising up to the top plate, cooling down and falling again. The detailed behaviour shows subtle differences between the three cases.

3. Bridging matter

‘Bridging matter’ introduces statistical thermodynamics, which provides the link between the notional insides and outsides of atoms and molecules. It identifies the bulk properties of a sample with the average behaviour of all the molecules that constitute it. A key concept is the Boltzmann distribution, which shows the exponentially decaying function of the energy of molecules as temperature is increased. It captures two aspects of chemistry: stability and reactivity. Statistical thermodynamics is used by physical chemists to understand the composition of chemical reaction mixtures that have reached equilibrium. It also provides an explanation of Le Chatelier's principle, which states that a system at equilibrium responds to a disturbance by tending to oppose its effect.