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NodeKayles is an impartial game played on a simple graph. The SpragueGrundy theorem states that every impartial game is associated with a nonnegative integer value called a Nimber. This paper studies the Nimber sequences of various families of graphs, including 3paths, lattice graphs, prism graphs, chained cliques, linked cliques, linked cycles, linked diamonds, hypercubes, and generalized Petersen graphs. For most of these families, we determine an explicit formula or a recursion on their Nimber sequences.more » « less

Abstract The Cryogenic Underground Observatory for Rare Events (CUORE) is the most sensitive experiment searching for neutrinoless doublebeta decay (0 νββ ) in 130 Te. CUORE uses a cryogenic array of 988 TeO 2 calorimeters operated at ∼10 mK with a total mass of 741 kg. To further increase the sensitivity, the detector response must be well understood. Here, we present a nonlinear thermal model for the CUORE experiment on a detectorbydetector basis. We have examined both equilibrium and dynamic electrothermal models of detectors by numerically fitting nonlinear differential equations to the detector data of a subset of CUORE channels which are well characterized and representative of all channels. We demonstrate that the hotelectron effect and electricfield dependence of resistance in NTDGe thermistors alone are inadequate to describe our detectors' energydependent pulse shapes. We introduce an empirical secondorder correction factor in the exponential temperature dependence of the thermistor, which produces excellent agreement with energydependent pulse shape data up to 6 MeV. We also present a noise analysis using the fitted thermal parameters and show that the intrinsic thermal noise is negligible compared to the observed noise for our detectors.more » « less

Abstract The Cryogenic Underground Observatory for Rare Events (CUORE) is the first cryogenic experiment searching for
decay that has been able to reach the onetonne mass scale. The detector, located at the Laboratori Nazionali del Gran Sasso (LNGS) in Italy, consists of an array of 988$$0\nu \beta \beta $$ $0\nu \beta \beta $ crystals arranged in a compact cylindrical structure of 19 towers. CUORE began its first physics data run in 2017 at a base temperature of about 10 mK and in April 2021 released its$${\mathrm{TeO}}_{2}$$ ${\mathrm{TeO}}_{2}$ result of the search for$$3{\mathrm{rd}}$$ $3\mathrm{rd}$ , corresponding to a tonneyear of$$0\nu \beta \beta $$ $0\nu \beta \beta $ exposure. This is the largest amount of data ever acquired with a solid state detector and the most sensitive measurement of$$\mathrm{TeO}_{2}$$ ${\mathrm{TeO}}_{2}$ decay in$$0\nu \beta \beta $$ $0\nu \beta \beta $ ever conducted . We present the current status of CUORE search for$${}^{130}\mathrm{Te}$$ ${}^{130}\mathrm{Te}$ with the updated statistics of one tonneyr. We finally give an update of the CUORE background model and the measurement of the$$0\nu \beta \beta $$ $0\nu \beta \beta $$${}^{130}\mathrm{Te}$$ ${}^{130}\mathrm{Te}$ decay halflife and decay to excited states of$$2\nu \beta \beta $$ $2\nu \beta \beta $ , studies performed using an exposure of 300.7 kg yr.$${}^{130}\mathrm{Xe}$$ ${}^{130}\mathrm{Xe}$ 
Abstract The possibility that neutrinos may be their own antiparticles, unique among the known fundamental particles, arises from the symmetric theory of fermions proposed by Ettore Majorana in 1937 1 . Given the profound consequences of such Majorana neutrinos, among which is a potential explanation for the matter–antimatter asymmetry of the universe via leptogenesis 2 , the Majorana nature of neutrinos commands intense experimental scrutiny globally; one of the primary experimental probes is neutrinoless double beta (0 νββ ) decay. Here we show results from the search for 0 νββ decay of 130 Te, using the latest advanced cryogenic calorimeters with the CUORE experiment 3 . CUORE, operating just 10 millikelvin above absolute zero, has pushed the state of the art on three frontiers: the sheer mass held at such ultralow temperatures, operational longevity, and the low levels of ionizing radiation emanating from the cryogenic infrastructure. We find no evidence for 0 νββ decay and set a lower bound of the process halflife as 2.2 × 10 25 years at a 90 per cent credibility interval. We discuss potential applications of the advances made with CUORE to other fields such as direct dark matter, neutrino and nuclear physics searches and largescale quantum computing, which can benefit from sustained operation of large payloads in a lowradioactivity, ultralowtemperature cryogenic environment.more » « less

null (Ed.)Abstract The CUORE experiment is a large bolometric array searching for the lepton number violating neutrinoless double beta decay ( $$0\nu \beta \beta $$ 0 ν β β ) in the isotope $$\mathrm {^{130}Te}$$ 130 Te . In this work we present the latest results on two searches for the double beta decay (DBD) of $$\mathrm {^{130}Te}$$ 130 Te to the first $$0^{+}_2$$ 0 2 + excited state of $$\mathrm {^{130}Xe}$$ 130 Xe : the $$0\nu \beta \beta $$ 0 ν β β decay and the Standard Modelallowed twoneutrinos double beta decay ( $$2\nu \beta \beta $$ 2 ν β β ). Both searches are based on a 372.5 kg $$\times $$ × yr TeO $$_2$$ 2 exposure. The deexcitation gamma rays emitted by the excited Xe nucleus in the final state yield a unique signature, which can be searched for with low background by studying coincident events in two or more bolometers. The closely packed arrangement of the CUORE crystals constitutes a significant advantage in this regard. The median limit setting sensitivities at 90% Credible Interval (C.I.) of the given searches were estimated as $$\mathrm {S^{0\nu }_{1/2} = 5.6 \times 10^{24} \, \mathrm {yr}}$$ S 1 / 2 0 ν = 5.6 × 10 24 yr for the $${0\nu \beta \beta }$$ 0 ν β β decay and $$\mathrm {S^{2\nu }_{1/2} = 2.1 \times 10^{24} \, \mathrm {yr}}$$ S 1 / 2 2 ν = 2.1 × 10 24 yr for the $${2\nu \beta \beta }$$ 2 ν β β decay. No significant evidence for either of the decay modes was observed and a Bayesian lower bound at $$90\%$$ 90 % C.I. on the decay half lives is obtained as: $$\mathrm {(T_{1/2})^{0\nu }_{0^+_2} > 5.9 \times 10^{24} \, \mathrm {yr}}$$ ( T 1 / 2 ) 0 2 + 0 ν > 5.9 × 10 24 yr for the $$0\nu \beta \beta $$ 0 ν β β mode and $$\mathrm {(T_{1/2})^{2\nu }_{0^+_2} > 1.3 \times 10^{24} \, \mathrm {yr}}$$ ( T 1 / 2 ) 0 2 + 2 ν > 1.3 × 10 24 yr for the $$2\nu \beta \beta $$ 2 ν β β mode. These represent the most stringent limits on the DBD of $$^{130}$$ 130 Te to excited states and improve by a factor $$\sim 5$$ ∼ 5 the previous results on this process.more » « less