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1. q,t$q,t$‐Catalan measures

   https://doi.org/10.1112/blms.12989  

   Cavey, Ian (March 2024, Bulletin of the London Mathematical Society)

   Abstract We introduce the ‐Catalan measures, a sequence of piece‐wise polynomial measures on . These measures are defined in terms of suitable area, dinv, and bounce statistics on continuous families of paths in the plane, and have many combinatorial similarities to the ‐Catalan numbers. Our main result realizes the ‐Catalan measures as a limit of higher ‐Catalan numbers as . We also give a geometric interpretation of the ‐Catalan measures. They are the Duistermaat–Heckman measures of the punctual Hilbert schemes parametrizing subschemes of supported at the origin. 

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2. Proca Q-balls and Q-shells

   https://doi.org/10.1007/JHEP10(2021)103  

   Heeck, Julian; Rajaraman, Arvind; Riley, Rebecca; Verhaaren, Christopher B. (October 2021, Journal of High Energy Physics)

   A bstract Non-topological solitons such as Q-balls and Q-shells have been studied for scalar fields invariant under global and gauged U(1) symmetries. We generalize this frame-work to include a Proca mass for the gauge boson, which can arise either from spontaneous symmetry breaking or via the Stückelberg mechanism. A heavy (light) gauge boson leads to solitons reminiscent of the global (gauged) case, but for intermediate values these Proca solitons exhibit completely novel features such as disconnected regions of viable parameter space and Q-shells with unbounded radius. We provide numerical solutions and excellent analytic approximations for both Proca Q-balls and Q-shells. These allow us to not only demonstrate the novel features numerically, but also understand and predict their origin analytically. 

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3. q-Rational and q-real binomial coefficients

   https://doi.org/10.1016/j.aam.2023.102618  

   Machacek, John; Ovenhouse, Nicholas (February 2024, Advances in Applied Mathematics)
4. Self-organization of amorphous Q-carbon and Q-BN nanoballs

   https://doi.org/10.1016/j.carbon.2022.03.003  

   Narayan, J.; Khosla, N. (June 2022, Carbon)
5. Excited Q-balls

   https://doi.org/10.1140/epjc/s10052-022-10772-5  

   Almumin, Yahya; Heeck, Julian; Rajaraman, Arvind; Verhaaren, Christopher B. (September 2022, The European Physical Journal C)

   Abstract Complex scalars in U (1)-symmetric potentials can form stable Q-balls, non-topological solitons that correspond to spherical bound-state solutions. If the U (1) charge of the Q-ball is large enough, it can support a tower of unstable radial excitations with increasing energy. Previous analyses of these radial excitations were confined to fixed parameters, leading to excited states with different charges Q . In this work, we provide the first characterization of the radial excitations of solitons for fixed charge, providing the physical spectrum for such objects. We also show how to approximately describe these excited states analytically and predict their global properties such as radius, energy, and charge. This enables a complete characterization of the radial spectrum. We also comment on the decay channels of these excited states. 

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