Abstract. We introduce several new methods to obtain upper bounds on the number of solutions of the congruenceswith a prime p and a polynomial f , where (x, y) belongs to an arbitrary square with side length M . We use these results and methods to derive non-trivial upper bounds for the number of hyperelliptic curvesover the finite field F p of p elements, with coefficients in a 2g-dimensional cubethat are isomorphic to a given curve and give an almost sharp lower bound on the number of non-isomorphic hyperelliptic curves with coefficients in that cube. Furthermore, we study the size of the smallest box that contain a partial trajectory of a polynomial dynamical system over F p .

We prove that for any infinite-type orientable surface S there exists a collection of essential curves Γ in S such that any homeomorphism that preserves the isotopy classes of the elements of Γ is isotopic to the identity. The collection Γ is countable and has infinite complement in C(S), the curve complex of S. As a consequence we obtain that the natural action of the extended mapping class group of S on C(S) is faithful.

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Let p be a prime, ε > 0 and 0 < L + 1 < L + N < p. We prove that if p 1/2+ε < N < p 1−ε , thenWe use this bound to show that any λ ≡ 0 (mod p) can be represented in the form λ ≡ n 1 ! . . . n 7 ! (mod p), where n i = o(p 11/12 ). This slightly refines the previously known range for n i .

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The effect of refrigerant blends 410A and 507 on the operation of an ejector cooling system -ECS-is theoretically studied with the aid of a validated multi-geometry ejector mathematical model. For a system cooling capacity of 1 kW, a set of possible design conditions were obtained by means of a parametric study varying the generator, condenser and evaporator temperatures from 50ºC to 70ºC; 30ºC to 40ºC and 5ºC to 15ºC. The ejector's lower U and values are obtained when R410A is employed, meaning higher primary fluid mass flow rates and smaller ejectors, respectively. ECS's upper COP s values are achieved at higher T GE and T EV with lower T CO . An ECS operating with these blends has a higher COP s using R410A, around 0.53 for T GE , T CO and T EV of 70ºC, 30ºC and 10ºC, respectively. The system using R507 has a similar but slightly lower performance. Therefore, the systems employing R410A and R507 are good options when a maximum generation temperature of 70ºC is available from a thermal source, either solar or industrial. As well, an ECS working with R410A and R507 experiences higher pressures and has a robust construction. NomenclatureCOP coefficient of performance (dimensionless) d diameter (m) ECS ejector cooling system F f friction factor (dimensionless) h specific enthalpy (kJ/kg) l distance, length (m) m mass flow rate (kg/s) p pressure (MPa) Q heat flow rate (kW) T temperature (ºC) U entrainment ratio, 2 m / 1 m , (dimensionless) W mechanical power (kW) ejector area ratio, (d m /d t ) 2 , (dimensionless) efficiency (dimensionless) Subscripts CO condenser d diffuser e main nozzle exit EV evaporator GE generator m mixing chamber n main nozzle, between main nozzle exit and mixing chamber inlet p r reversible pump s system t main nozzle throat 1…6 thermodynamic cycle states Jorge I. Hernandez et al. / Energy Procedia 57 ( 2014 ) 3021 -3030 3023