Let Ps(n) denote the n-th s-gonal number. Consider the Diophantine equation Ps(n)=tm for integers n,s,t and m>2. All solutions to this equation are known for m>2 and s∈{3,5,6,8,10,20}. Here we extend these results to the cases s=2k+4 (where k=4,6 or 5≤k≤97 is a prime number) and s=k+4 (where k=9,15 or 3≤k≤97 is a prime number). The proofs of our results use the modular and hypergeometric methods, linear forms in logarithms and extensive calculations. We were unable to completely solve the above Diophantine equations, but we expect (based on GRH and the weak effective abc conjecture) that there will be no additional solutions beyond those explicitly shown in Theorem 1, Theorem 2, Theorem 3.
Renewable-rich areas often experience curtailment because local demand and export capacity cannot absorb short-duration surpluses. Flexible loads and storage can help, but nameplate capability does not necessarily represent reliable market capacity when availability, internal state, rebound, telemetry quality, and baseline error vary between events. This paper develops a credible flexibility envelope by eroding a multi-period physical feasible set with a calibrated trajectory-error set. A class-conditioned recurrent state captures recent operation, a hierarchical Bayesian decoder updates uncertain physical parameters, and weighted block-conformal calibration adjusts residual margins under temporal dependence and drift. Offers, cleared capacity, dispatch instructions, and verified delivery are modeled separately, and the resulting envelopes are embedded in a loss-aware, multi-corridor network clearing model with voltage, thermal, feeder, settlement, and credit constraints. A controlled three-year synthetic benchmark uses a modified IEEE RTS-24 system, three radial feeder equivalents, seven resource classes, 20 random seeds, and 540 stress-test combinations. At a 95% coverage target, the proposed method achieves 95.0% empirical coverage, reduces non-delivery from 10.1% to 4.6%, increases curtailment reduction from 22.6% to 40.0%, and raises mean net surplus from 62.7 × 103 to 193.0 × 103 USD. Ablation and stress tests further examine drift, forecast error, telemetry loss, baseline bias, correlation, network changes, and extreme renewable conditions. The results are specific to the modified RTS-24 benchmark and motivate further evaluation on other networks and measured data.
The origin of obscuration in active galactic nuclei (AGN) is still a matter of contention. It is unclear whether obscured AGN are primarily due to line-of-sight effects (Orientation model), a transitory, dust-enshrouded phase in galaxy evolution (Evolution models), or a combination of both. The role of an inner torus around the central supermassive black hole also remains unclear in pure Evolution models. We use cosmological semi-analytic models and semi-empirical prescriptions to explore obscuration effects in AGN at cosmic noon, in the range 1 < z < 3. We consider a realistic object-by-object modelling of AGN evolution including different AGN light curves (LCs) composed of phases of varying levels of obscuration, usually (but not uniquely) with a larger degree of obscuration before the peak of AGN activity, mimicking the possible clearing effects of strong AGN feedback. Evolution models characterized by AGN LCs with relatively short pre-peak obscured phases followed by more extended optical/ultraviolet (UV) visible post-peak phases, struggle to reproduce the high fraction of obscured AGN at z similar to 2-3 inferred from X-ray surveys. Evolution models characterized by AGN LCs with sharp post-peak declines or persistent or multiple obscuration phases are more successful, although they still face challenges in reproducing the steady drop in the fractions of obscured AGN with increasing luminosity measured by some groups. Invoking a fine-tuning in the input LCs, with more luminous AGN defined by longer optical/UV visible windows, can improve the match to the decreasing fractions of obscured AGN with luminosity. Alternatively, a long-lived central torus-like component, with thickness decreasing with increasing AGN power, naturally boosts the luminosity-dependent fractions of obscured AGN, suggesting that small-scale orientation effects may still represent a key component even in Evolution models. We also find that in our models major mergers and starbursts, when considered in isolation, fall short in accounting for the large fractions of highly obscured faint AGN detected at cosmic noon.
Why do some public service organizations outperform others despite the same institutional constraints? Drawing on dynamic capabilities theory, we examine how dynamic managerial capabilities (DMCs) enable dynamic organizational capabilities (DOCs) influencing performance in institutionalized contexts. Using multi-respondent survey data from 127 English hospitals, combined with independent performance ratings, we test these relationships using PLS-SEM. Findings show DMCs - collective leadership (human capital), internal and external social capital, and bias for action (managerial cognition) - strongly predict DOCs. In turn, DOCs directly affect evolutionary fitness performance. The study makes three contributions. First, it advances the microfoundations agenda by showing how managerial attributes aggregate into organizational adaptation. Second, it clarifies when direct versus mediated capability-performance models are appropriate by linking this to the evolutionary-technical fitness distinction. Third, it demonstrates hierarchical PLS-SEM for modeling multidimensional capabilities. For practice, results suggest investing in relational infrastructure, distributed leadership, risk-tolerant cultures, and stakeholder engagement to enhance adaptive performance.
Water and air pollution are pervasive issues, impacting the majority of global citizens. In response, geopolymers functionalized with photocatalytic metal oxides have emerged as sustainable materials for environmental remediation. This study examines how the Si/Al molar ratio (1 and 2) and photocatalytic WO3 loading (WO3/Al molar ratios of 0.10, 0.15, and 0.20) impact the microstructural, optical, and photocatalytic properties of WO3-geopolymer composites. XRD analysis showed the complete conversion of WO3 into Na2WO4, impairing the intended photocatalytic function under visible light. Solid-state Al-27 NMR showed incomplete geopolymerization, owing to NaOH consumption due to the Na2WO4 conversion. With WO3 loading, the BET surface for the Si/Al = 1 series stabilized at similar to 11.7 m(2)/g after a significant initial decline; whereas the Si/Al = 2 series showed a substantial initial reduction and further reductions from 14.5 to 2.5 m(2)/g. The photocatalytic activity was evaluated by the decolorization of aqueous methylene blue (160 mg/L) and degradation of gaseous alpha-pinene (1 ppm) under UV irradiation. The Si/Al = 1 series showed increased decolorization with WO3 loading, whereas the Si/Al = 2 series showed a decrease. Surprisingly, the pristine samples with no WO3 addition outperformed the WO3-geopolymers in both series, as assessed by the decolorization of methylene blue, attributed to naturally occurring photocatalysts within the metakaolin feedstock and a larger surface area. This shows that the photocatalytic performance of geopolymers is not solely dependent on external photocatalytic WO3 loading but governed by the underlying geopolymer chemistry and network connectivity. These findings demonstrate the complexity in designing advanced ceramic photocatalytic systems while highlighting their potential for environmental remediation.
