In this article we address the regularity of stable solutions to semilinear elliptic equations $-Δu = f(u)$ with MEMS type nonlinearities. More precisely, we will have $0\leq u \leq 1$ in a domain $Ω\subset \mathbb{R}^n$ and $f:[0,1)\to (0,+\infty)$ blowing up at $u=1$ and nonintegrable near 1. In this context, a solution $u$ is regular if $u<1$ in all $Ω$ or, equivalently, if $-Δu = f(u)<+\infty$ in $Ω$. This paper establishes for the first time interior regularity estimates that are independent of the boundary condition that $u$ may satisfy. Our results hold up to the optimal dimension $n=6$ (there are counterexamples for $n\geq 7$) but require a Crandall-Rabinowitz type assumption on the nonlinearity $f$. Our main estimate controls the $L^\infty$ norm of $F(u)$ in a ball, where $F$ is a primitive of $f$, by only the $L^1$ norm of $u$ in a larger ball. Under the same assumptions, we also give global estimates in dimensions $n\leq 6$ for the Dirichlet problem with vanishing boundary condition, improving previously known results. For $n\leq 2$, we do not need a Crandall-Rabinowitz type assumption and, thus, our global estimate holds for all nonnegative, nondecreasing, convex nonlinearities which blow up at 1 and are nonintegrable near 1.

This paper investigates Levi flat structures from the perspective of structure sheaves. We employ formal integrability to construct a class of differential complexes, thereby providing a resolution for the structure sheaf and a global realization of the Treves complex. Drawing inspiration from Morse theory and Grauert's convexity, we introduce notions of convexity and positivity that fully exploits Levi flatness, which ensures the global exactness of the differential complex and demonstrates Sobolev regularity in the compact case. As applications, we establish the global solvability of the Treves complex for Levi flat structures, together with results on singular cohomology and the extension problem for canonical forms in the elliptic case.

A Weyl structure on a Riemannian manifold $(M,g)$ is a torsion-free linear connection $\nabla$ such that there is a $1$-form $θ$ (called the Lee form) satisfying $\nabla g = 2\, θ\otimes g$. We examine the case in which there exists a $\nabla$-parallel distribution of codimension $1$ on which the Lee form vanishes identically. We prove that if $(M,g)$ is complete with $θ$ closed, then the Weyl structure must be flat or exact. We apply this to prove the conjecture of Lotta (Eur. J. Math., 2023), namely, every homogeneous Kenmotsu manifold is isometric to the real hyperbolic space.

Models which solve the strong CP problem by employing discrete spacetime symmetries generically suffer fine-tuning and quality problems. We demonstrate that these issues are greatly ameliorated when the only source of spontaneous CP breaking is from the chiral condensate of a strongly coupled hidden sector. This is shown explicitly in a model with the SM extended by a vector-like quark family and a complex scalar portal to QCD-like dark sector with $N_f$ families of dark fermions that confines at a high scale. The dark pions of the hidden sector are natural dark matter candidates, with the correct relic abundance obtained via freeze-in. These "confining" Nelson-Barr solutions connect phenomenological questions regarding the strong CP problem to recent developments in the understanding of confining gauge theories, and present ample room for further model building.

Dimension in physical systems determines universal properties at criticality. Yet, the impact of structural perturbations on dimensionality remains largely unexplored. Here, we characterize the attraction basins of structural fixed points in scale-invariant networks from a renormalization group perspective, demonstrating that basin stability connects to a structural phase transition. This topology-dependent effect, which we term geometric criticality, triggers a geometric breakdown hitherto unknown, which induces non-trivial fractal dimensions and unveils hidden LRG flows toward unstable structural fixed points. Our systematic study of how networks and lattices respond to disorder paves the way for future analysis of non-ergodic behavior induced by quenched disorder.

Hypergraphs extend traditional graphs by enabling the representation of N-ary relationships through higher-order edges. Akin to a common approach of deriving graph Laplacians, we define function spaces and corresponding symmetric products on the nodes and edges to derive hypergraph Laplacians. While this has been done before for Euclidean features, this work generalizes previous hypergraph Laplacian approaches to accommodate manifold-valued hypergraphs for many commonly encountered manifolds.

Designing satellite constellation systems involves complex multidisciplinary optimization in which coverage serves as a primary driver of overall system cost and performance. Among the various design considerations, constellation configuration, which dictates how satellites are placed and distributed in space relative to each other, predominantly determines the resulting coverage. In constellation configuration design, coverage may be treated either as an optimization objective or as a constraint, depending on mission goals. State-of-the-art literature addresses each mission scenario on a case-by-case basis, employing distinct assumptions, modeling techniques, and solution methods. While such problem-specific approaches yield valuable insights, users often face implementation challenges when performing trade-off studies across different mission scenarios, as each scenario must be handled distinctly. In this paper, we propose a collection of five mixed-integer linear programs that are of practical significance, extensible to more complex mission narratives through additional constraints, and capable of obtaining provably optimal constellation configurations. The framework can handle various metrics and mission scenarios, such as percent coverage, average or maximum revisit times, a fixed number of satellites, spatiotemporally varying coverage requirements, and static or dynamic targets. The paper presents several case studies and comparative analyses to demonstrate the versatility of the proposed framework.

We investigate the two-dimensional frustrated quantum Heisenberg model with bond disorder on nearest-neighbor couplings using the recently introduced Foundation Neural-Network Quantum States framework, which enables accurate and efficient computation of disorder-averaged observables with a single variational optimization. Simulations on large lattices reveal an extended region of the phase diagram where long-range magnetic order vanishes in the thermodynamic limit, while the overlap order parameter, which characterizes quantum spin glass states, remains finite. These findings, supported by a semiclassical analysis based on a large-spin expansion, provide compelling evidence that the spin glass phase is stable against quantum fluctuations, unlike the classical case where it disappears at any finite temperature.

The next generation of cosmology surveys will probe the matter distribution of the universe to unparalleled precision. To match this level of precision in cosmological parameter estimation, we need to use information at small scales of $\sim$ 1 Mpc, which requires an accurate model of baryonic feedback. In this paper, we employ the Dark Matter + Baryon (DMB) model, a flexible halo model that is well-fit to various hydrodynamical simulations, to extract information on baryonic feedback from galaxy cluster observables. Using a sample of thermal Sunyaev-Zeldovich (tSZ) selected galaxy clusters from the Atacama Cosmology Telescope (ACT) - with masses calibrated via weak lensing from the Dark Energy Survey (DES) - we develop a robust end-to-end pipeline that directly models the calibrated observables. Our analysis demonstrates that the tSZ Y-M relation can constrain several DMB model parameters, providing key insights into baryonic feedback effects on cosmic shear at the several percent level. We find a preference for intermediate to strong levels of feedback, which is both consistent with several hydrodynamic simulations and competitive with similar analyses performed on complementary probes. Finally, we discuss the implications of our results in the context of current and upcoming cosmic shear surveys.

The dispersion phenomenon of mass and heat transport in oscillatory flows has wide applications in environmental, physiological and microfluidic flows. The method of concentration moments is a powerful theoretical tool for analyzing transport characteristics and is well-developed for steady flows. However, the general solutions of moments derived by Barton (J. Fluid Mech., vol. 126, 1983, pp. 205-218) cannot be applied directly to unsteady flows. Prior studies needed to re-solve the governing equations of moments from scratch, encountering the complication induced by the time-periodic velocity, leaving higher-order statistics like skewness and kurtosis analytically intractable except for specific cases. This work proposes a novel approach based on a two-time-variable extension to tackle these challenges. By introducing an auxiliary time variable, referred to as oscillation time to characterize the inherent oscillation in the dispersion due to the oscillating flow, the transport problem is extended to a two-time-variable system with a "steady" flow term. This enables the direct use of Barton's expressions and thus avoids the prior complication. This approach not only offers an intuitive physical perspective for the influence of the velocity oscillation but also clarifies the solution structure of concentration moments. As a preliminary verification, we examine the transport problem in an oscillatory Couette flow. The analytical solution agrees well with the numerical result by Brownian dynamics simulations. The effects of the point-source release and the phase shift of velocity on the transport characteristics are investigated. By extending the classic steady-flow solution to the time-dependent flows, this work provides a versatile framework for transient dispersion analysis, enhancing predictions in oscillatory transport problems.

We apply the Tensor-Backflow method to investigate the Fermi-Hubbard model on two-dimensional lattices up to 256 sites, exploring various interaction strengths $U$, electron fillings $n$, next-nearest-neighbor hopping $t'$, and boundary conditions. By considering backflow terms from nearest- or next-nearest-neighbor sites, we achieve competitive results without enforcing geometric symmetries on the variational wave-function. The optimizations were stable from a prior unrestrictied Hartree-Fock state, followed by adding backflow corrections. Meanwhile, changing interaction strengths in the prior unrestrictied Hartree-Fock state is helpful to bypass the local minima. When $t'$=0, by considering nearest-neighbor backflow terms, linear stripe order emerges successfully for the case of $n$=0.875 and $U$=8 on a $16 \times 16$ lattice with periodic boundary conditions. In a similar case with open boundary conditions, the energy obtained is only $4.5 \times 10^{-4}$ higher than the state-of-the-art method fPEPS with bond dimension $D$=20. Compared to state-of-the-art neural network methods, the energies obtained using the Tensor-Backflow approach are competitive, with relative errors below $5 \times 10^{-3}$. For $n$=0.8 and $n$=0.9375, direct optimizations yield results consistent with the phase diagram from AFQMC. When $t'$=-0.2, considering next-nearest-neighbor backflow terms leads to energies that are either competitive with or even lower than those from state-of-the-art neural network approaches. For instance, for $n$=0.875 and $U$=8 on a $12 \times 12$ lattice with periodic boundary conditions, the energy obtained is $8.1 \times 10^{-4}$ lower than that from the neural network result. Thus, the Tensor-Backflow method demonstrates strong representational capabilities for solving the Fermi-Hubbard model.

We present the first cosmological constraints from a joint analysis of galaxy clustering and galaxy-galaxy lensing using extended SubHalo Abundance Matching (SHAMe). We analyse stellar mass-selected Galaxy And Mass Assembly (GAMA) galaxy clustering and Kilo-Degree Survey (KiDS-1000) galaxy-galaxy lensing and find constraints on $S_8\equivσ_8\sqrt{Ω_{\rm m}/0.3}=0.793^{+0.025}_{-0.024}$, in agreement with Planck at 1.7$σ$, with $σ_8$ the mass density fluctuation amplitude in 8 $h^{-1}{\rm Mpc}$ sphere at present and $Ω_{\rm m}$ the density parameter in total matter. These results are in agreement with the Cosmic Microwave Background results from Planck. We are able to constrain all 5 SHAMe parameters, which describe the galaxy-subhalo connection. We validate our methodology by first applying it to simulated catalogues, generated from the TNG300 simulation, which mimic the stellar mass selection of our real data. We show that we are able to recover the input cosmology for both our fiducial and all-scale analyses. Our all-scale analysis extends to scales of galaxy-galaxy lensing below $r_\mathrm{p}<1.4\,\mathrm{Mpc}/h$, which we exclude in our fiducial analysis to avoid baryonic effects. When including all scales, we find a value of $S_8$, which is 1.26$σ$ higher than our fiducial result (against naive expectations where baryonic feedback should lead to small-scale power suppression), and in agreement with Planck at 0.9$σ$. We also find a 21% tighter constraint on $S_8$ and a 29% tighter constraint on $Ω_\mathrm{m}$ compared to our fiducial analysis. This work shows the power and potential of joint small-scale galaxy clustering and galaxy-galaxy lensing analyses using SHAMe.

We describe four hyperbolic knot complements in $\mathbb{S}^3$, each of which covers a prism orbifold: the quotient of $\mathbb{H}^3$ by the action of a discrete group generated by reflections in the faces of a polyhedron that has the combinatorial type of a triangular prism. The prism orbifolds are rigid-cusped and contain compact, totally geodesic hyperbolic triangle sub-orbifolds; as a result, the knot complements covering them have hidden symmetries and contain closed, embedded, totally geodesic surfaces.

In the paper, we study a kind of Oscillatory singular integral operator with Calderón Type Commutators $T_{P,K,A} $ defined by \[T_{P,K,A} f(x)=\text { p.v.} \int_{\mathbb{R}^{n}} f(y) \frac{K(x-y)}{|x-y|}(A(x)-A(y)-\nabla A(y))(x-y) e^{i P(x-y)} d y, \] where $P(t)$ is a real polynomial on $\mathbb{R},$ and $K$ is a function on $\mathbb{R}^{n},$ satisfies the vanishing moment and $CZ(δ)$ conditions. Under these conditions, we show that $T_{P,K,A}$ is bounded on $L^p(\mathbb{R}^{n})$ with uniform boundedness, which improve and extend the previous result.

We study block-diagonal random matrices with i.i.d. subexponential entries and show that, despite their highly structured form, they already guarantee exact sparse recovery from a nearly optimal number of measurements. When the matrix reduces to a single block, our framework collapses to the classical i.i.d. subexponential ensemble, and our bounds recover the well-known optimal rates previously established for unstructured random matrices.

Recent work by Kocheril \textit{et al.}\cite{kocheril2025} claimed that phenylium--the cyclic structure of the \ce{C6H5+} species--is unreactive toward key interstellar molecules such as molecular hydrogen (\ce{H2}) and acetylene (\ce{C2H2}). This finding challenges the previously proposed role of phenylium as a cornerstone in the formation of polycyclic aromatic hydrocarbons (PAHs) \cite{cherchneff1992,byrne2024}. The study focused on the reactivity of \ce{C6H5+}, formed via the radiative association between \ce{C4H3+} and \ce{C2H2}, believed to be a major pathway for phenylium formation in astrochemical model, e.g. \cite{byrne2024}. Here, we present new experimental and theoretical evidence that challenges this assumption. Our results demonstrate that phenylium does indeed react with \ce{C2H2} under astrophysically relevant conditions. Quantum chemical calculations support this finding by revealing a barrierless mechanism, indicating that the reaction is feasible even in cold interstellar environments. We believe this clarification is critically important, and that further investigations into the formation of the first aromatic ring in space--a process that remains a key bottleneck in our understanding of PAHs formation and growth--is essential.

The full 245-letter symbol alphabet for all planar massless two-loop six-point Feynman integrals was recently determined in arXiv:2412.19884 and arXiv:2501.01847. In a parallel mathematical development, it was shown in arXiv:2408.14956 that there is an embedding of the cluster algebra associated to the partial flag variety $Fl_{2,n-2;n}$, which describes the kinematics of $n$ massless particles, into that of the Grassmannian Gr$(n{-}2,2n{-}4)$. In this paper we connect these developments by showing that most of the rational symbol letters can be expressed in terms of flag cluster variables, and that all of the algebraic symbol letters arise from infinite mutation sequences.

We make use of the perturbation theory for modified gravity models that we developed in previous works and apply it to construct the fullshape galaxy power spectrum for the Symmetron modified gravity model. First, we study the growth rate, that is a scale dependent quantity, and compare our results with those of the $n=1 $ Hu-Sawcki (HS) model, finding that the Symmetron has a growth quite similar to the HS F6 in the wavenumber interval $0.01 \leq k \leq 0.1 $ and for redshifts where Symmetron model is viable. We also propose a growth parametrization that turns to be a good approximation for the HS and Symmetron models, with a deviation less than $0.6 \%$. To compute the RSD multipoles we employ an expansion of the velocity moments generating function that is suitable for general modified gravity models. Later, we apply the fk-Perturbation Theory (fkPT) approximation to reduce the computation time of nonlinear kernels, to find the fullshape galaxy power spectrum for the Symmetron, and study the differences with HS model. The RSD multipoles of the Symmetron result similar to those of the HS F6 model. Next, we integrate this theory to an MCMC sampler and validate our results by fitting our parameters to EZMocks to recover the parameters that bring the model to GR. We found a similar agreement in the model validation between Symmetron and F6 model, recovering the simulation cosmological parameters, and concluding that our pipeline is ready to make cosmological parameters' inference with real data.

We report on three improvements in the context of Feynman integral reduction and $\varepsilon$-factorised differential equations: Firstly, we show that with a specific choice of prefactors, we trivialise the $\varepsilon$-dependence of the integration-by-parts identities. Secondly, we observe that with a specific choice of order relation in the Laporta algorithm, we directly obtain a basis of master integrals, whose differential equation on the maximal cut is in Laurent polynomial form with respect to $\varepsilon$ and compatible with a particular filtration. Thirdly, we prove that such a differential equation can always be transformed to an $\varepsilon$-factorised form. This provides a systematic algorithm to obtain an $\varepsilon$-factorised differential equation for any Feynman integral. Furthermore, the choices for the prefactors and the order relation significantly improve the efficiency of the reduction algorithm.

In the quest to unravel the dark sector, feebly interacting freeze-in dark matter presents an intriguing possibility, plausibly explaining the consistent null results from various dark matter experiments. We propose a unique imprint in the form of gravitational waves generated during the freeze-in production of dark matter from heavy particle decay in the early universe. This characteristic gravitational wave signature can serve as a powerful probe for freeze-in dark matter. Our study indicates that future high-frequency gravitational wave experiments can detect these waves, offering a novel avenue to critically test the underlying conditions and requirements of this dark matter paradigm, which typically lie beyond the reach of current and planned dark matter detection experiments.

The real-time correlators of quantum field theories can be directly probed through new approaches to simulation, such as quantum computing and tensor networks. This provides a new framework for computing scattering observables in lattice formulations of strongly interacting theories, such as lattice quantum chromodynamics. In this paper, we prove that the proposal of real-time estimators of scattering observables is universally applicable to all scattering observables of gapped quantum field theories. All finite-volume errors are exponentially suppressed, and the rate of this suppression is controlled by the regulator considered, namely, a displacement of the spectrum of the theory into the complex plane. A partial restoration of Lorentz symmetry by averaging over different boosts gives an additional suppression of finite volume errors. Our results also apply to the simulation of wavepacket scattering, where a similar averaging is performed to construct the wavepackets that regulate the finite volume effects. This result represents a necessary key step towards determining a broad class of scattering observables via quantum computing that are currently inaccessible via classical computing. Such observables are relevant for various applications, including hadron spectroscopy, hadron structure, and precision tests of the Standard Model. We also comment on potential applications of our results to traditional computational schemes.

Although bibliometrics has become an essential tool in the evaluation of research performance, bibliometric analyses are sensitive to a range of methodological choices. Subtle choices in data selection, indicator construction, and modeling decisions can substantially alter results. Ensuring robustness (meaning that findings hold up under different reasonable scenarios) is therefore critical for credible research and research evaluation. To address this issue, this study introduces multiverse analysis to bibliometrics. Multiverse analysis is a statistical tool that enables analysts to transparently discuss modeling assumptions and thoroughly assess model robustness. Whereas standard robustness checks usually cover only a small subset of all plausible models, multiverse analysis includes all plausible models. The benefits of multiverse analysis are illustrated by assessing the robustness of the findings reported by Wu et al. (2019), who observed that small teams tend to produce more disruptive research than large teams. While we found robust evidence of a negative effect of team size on disruption scores, the effect size depends substantially on the model specification. Our findings underscore the importance of assessing the multiverse robustness of bibliometric results to clarify their practical implications.

The Quantum Approximate Optimization Algorithm (QAOA) is a variational ansatz that resembles the Trotterized dynamics of a Quantum Annealing (QA) protocol. This work formalizes this connection formally and empirically, showing the angles of a multilayer QAOA circuit converge to universal QA trajectories. Furthermore, the errors in both QAOA circuits and QA paths act as thermal excitations in pseudo-Boltzmann probability distributions whose temperature decreases with the invested resource -- i.e. integrated angles or total time -- and which in QAOA also contain a higher temperature arising from the Trotterization. This also means QAOA and QA are cooling protocols and simulators of partition functions whose target temperature can be tuned by rescaling the universal trajectory. The average cooling power of both methods exhibits favorable algebraic scalings with respect to the target temperature and problem size, whereby in QAOA the coldest temperature is inversely proportional to the number of layers, $T\sim 1/p$, and to the integrated angles -- or integrated interactions in QA.

This letter introduces a pioneering, training-free and explainable framework for High-Resolution Range Profile (HRRP) automatic target recognition (ATR) utilizing large-scale pre-trained Large Language Models (LLMs). Diverging from conventional methods requiring extensive task-specific training or fine-tuning, our approach converts one-dimensional HRRP signals into textual scattering center representations. Prompts are designed to align LLMs' semantic space for ATR via few-shot in-context learning, effectively leveraging its vast pre-existing knowledge without any parameter update. We make our codes publicly available to foster research into LLMs for HRRP ATR.

Characteristic formulae give a complete logical description of the behaviour of processes modulo some chosen notion of behavioural semantics. They allow one to reduce equivalence or preorder checking to model checking, and are exactly the formulae in the modal logics characterizing classic behavioural equivalences and preorders for which model checking can be reduced to equivalence or preorder checking. This paper studies the complexity of determining whether a formula is characteristic for some process in each of the logics providing modal characterizations of the simulation-based semantics in van Glabbeek's branching-time spectrum. Since characteristic formulae in each of those logics are exactly the satisfiable and prime ones, this article presents complexity results for the satisfiability and primality problems, and investigates the boundary between modal logics for which those problems can be solved in polynomial time and those for which they become (co)NP- or PSPACE-complete.

Variational quantum algorithms (VQAs) offer a promising near-term approach to finding optimal quantum strategies for playing non-local games. These games test quantum correlations beyond classical limits and enable entanglement verification. In this work, we present a variational framework for the Magic Square Game (MSG), a two-player non-local game with perfect quantum advantage. We construct a value Hamiltonian that encodes the game's parity and consistency constraints, then optimize parameterized quantum circuits to minimize this cost. Our approach builds on the stabilizer formalism, leverages commutation structure for circuit design, and is hardware-efficient. Compared to existing work, our contribution emphasizes algebraic structure and interpretability. We validate our method through numerical experiments and outline generalizations to larger games.

This paper investigates code LLMs' capability of static analysis during code intelligence tasks such as code summarization and generation. Code LLMs are now household names for their abilities to do some programming tasks that have heretofore required people. The process that people follow to do programming tasks has long been understood to require static analysis. For example, human programmers navigate the call graph of large programs to comprehend the different parts of those programs. Education in programming includes static analysis under the assumption that better static analysis skills beget better programming. While popular culture is replete with anthropomorphic references such as LLM ``reasoning'', in fact code LLMs could exhibit a wholly alien thought process to humans. This paper studies the specific question of static analysis by code LLMs. We use three different static analysis tasks (callgraph generation, AST generation, and dataflow generation) and three different code intelligence tasks (code generation, summarization, and translation) with two different open-source models (Gemini and GPT-4o) and closed-source models (CodeLlaMA and Jam) as our experiments. We found that LLMs show poor performance on static analysis tasks and that pretraining on the static analysis tasks does not generalize to better performance on the code intelligence tasks and vice versa.

For any $m\geq 3$ we show that the Hamming graph $H(n,m)$ admits an imbalanced partition into $m$ sets, each inducing a subgraph of low maximum degree. This improves previous results by Tandya and by Potechin and Tsang, and disproves the Strong $m$-ary Sensitivity Conjecture of Asensio, García-Marco, and Knauer. On the other hand, we prove their weaker $m$-ary Sensitivity Conjecture by showing that the sensitivity of any $m$-ary function is bounded from below by a polynomial expression in its degree.

We consider an evolving random discrete tree model called Preferential Attachment with Vertex Death, as introduced by Deijfen. Initialised with an alive root labelled $1$, at each step $n\geq1$ either a new vertex with label $n+1$ is introduced that attaches to an existing alive vertex selected preferentially according to a function $b$, or an alive vertex is selected preferentially according to a function $d$ and killed. In this article we introduce a generalised concept of persistence for evolving random graph models. Let $O_n$ be the smallest label among all alive vertices (the oldest alive vertex), and let $I_n^m$ be the label of the alive vertex with the $m^{\mathrm{th}}$ largest degree. We say a persistent $m$-hub exists if $I_n^m$ converges almost surely, we say that persistence occurs when $I_n^1/O_n$ is tight, and that lack of persistence occurs when $I_n^1/O_n$ tends to infinity. We identify two regimes called the infinite lifetime and finite lifetime regimes. In the infinite lifetime regime, vertices are never killed with positive probability. Here, we provide conditions under which we prove the (non-)existence of persistent $m$-hubs for any $m\in\mathbb N$. This expands and generalises recent work of Iyer, which covers the case $d\equiv 0$ and $m=1$. In the finite lifetime regime, vertices are killed after a finite number of steps almost surely. Here we provide conditions under which we prove the occurrence of persistence, which complements recent work of Heydenreich and the author, where lack of persistence is studied for preferential attachment with vertex death.

We consider the addition to the Standard Model of a scalar $SU(2)$ multiplet $Δ_n$ with dimension $n$ going from $1$ to $6$. The multiplet $Δ_n$ is assumed to have null vacuum expectation value and an arbitrary (free) hypercharge. We determine the shape of the phase space for the new terms that appear in the scalar potential (SP); we observe in particular that, in the case of a 6-plet, the phase space is slightly concave along one of its boundaries. We determine the bounded-from-below and vacuum stability conditions on the SP for each value of $n$.