Papers

The constrained equilibrium problem for the logarithmic potential was introduced by Rakhmanov (1996) as he realized that the asymptotic distribution of the zeros of discrete orthogonal polynomials in a compact interval could be described in terms of the equilibrium measure of this interval in a class of measures subject to a certain constraint. Other authors such as Dragnev and Saff (1997), and Kuijlaars and Van Assche (1999), extended this approach to more general settings. These problems have proven to be useful for describing asymptotic distributions in different settings. In the current paper, we consider constrained equilibrium problems in the Riesz setting, that is, for $s$-Riesz potentials in the hyperplane $\mathbb{R}^d,$ with $d\geq 1$ and $\max (0,d-2) < s < d$. Along with some general results, an illustrative example consisting of the solution of a constrained equilibrium problem in the unit ball is studied in detail. This is the main part of the paper. Finally, a number of numerical experiments show how the constrained equilibrium measure, the solution of the problem, may be discretized using the so-called 'constrained Leja points' introduced by Coroian and Dragnev (2001).

In this paper, we establish variational formulas and a rearrangement inequality for principal eigenvalues of asymmetric second-order difference operators on periodic lattices. For a general irreducible nearest-neighbor operator, positive right and left eigenvectors give an explicit saddle point and hence equal minimum–maximum and maximum–minimum formulas over positive profiles and periodic logarithmic correctors. The corrector is unique and satisfies a nonlinear discrete flux-conservation law. For exponentially tilted symmetric diffusion, the formula separates the discrete Dirichlet and potential terms from an exact nonlinear periodic correction. In one dimension the flux is constant, and an elementary scalar-flux representation yields a bell-shaped cyclic rearrangement that maximizes the tilted principal eigenvalue for every tilt.

In this paper, we introduce a family of infinite-dimensional Lie algebras $B(q)$ of Takiff Block type, containing Heisenberg-Virasoro algebra, $W$-algebra $W(2,2)$, and BMS-Kac-Moody algebra as subalgebras. For $q\in\mathbb C^*$, we classify the $ B(q)$-modules that are free of rank one over $U(\mathfrak h)$, where $\mathfrak h=\mathbb C L_{0,0}\oplus\mathbb C W_{0,0}$, and determine their irreducibility and isomorphism classes. We also establish irreducibility and isomorphism criteria for their tensor products with irreducible restricted modules. Finally, restricting the above $B(-1)$-modules to several natural subalgebras of the central quotient $B(-1)/(\mathbb C C_1\oplus\mathbb C C_2)$ yields families of non-weight modules.

The weak-strong uniqueness of bounded solutions is established for a broad class of cross-diffusion systems, with or without volume-filling effects, and with or without full coercivity. Compared to existing results, the regularity and positivity assumptions on the strong solution are significantly relaxed. The analysis also extends to non-coercive systems, with the volume-filling constraint compensating for the lack of coercivity. The proof is based on two key ideas: The positivity condition is eliminated through the construction of a glued entropy, while the regularity requirement can be relaxed by using the gradient estimate and the Gagliardo-Nirenberg inequality.

Let $m\geq 1$, let $f:\mathbb N\to\mathbb Z/m\mathbb Z$ be a standard morphism and let $T(m)$ be the least integer $N$ such that every such $f$ admits a primitive monochromatic Pythagorean triple with hypotenuse at most $N$. The aim of this note is to prove that every standard morphism has infinitely many identity-valued Pythagorean triples and infinitely many primitive monochromatic Pythagorean triples. Thus the qualitative part of Problem~4.3 of Eliahou, Fromentin, Marion-Poty and Robilliard is solved for every $m$. Moreover, $T(m)$ is finite, the morphism $n\mapsto v_3(n)\pmod m$ has least possible hypotenuse $(9^m+1)/2$ and $T(d)\leq T(m)$ when $d\mid m$.

We use a variety of analytic and numerical techniques to approximate the geometry of individual dark matter halos when modeled as collections of ultralight scalar bosons (ULDM), followed by an analysis on the future directions and implications of this work. Ultimately, we aim to understand the morphology and creation of complex structures (like spirals) in galaxies from a geometric standpoint and to investigate the connection between dark matter and baryonic matter in galaxies. We find that superpositions of gravitational eigenmodes for ULDM in a fixed background potential can yield spiral density perturbations which rotate on the order of one hundred million years and appear within ten to twenty kpc of the halo center, both features comparable to realistic galaxies, and that this model provides a natural explanation for the density waves necessary to seed baryonic spirals.

This article studies a modular semistable elliptic curve $E$ over a totally real number field $F$ such that, upon base change to a totally imaginary quadratic extension $K$, it has analytic rank one. Assuming the Iwasawa main conjecture, along with a substantial number of assumptions, we prove a variant of the $p$-part of the Birch and Swinnerton-Dyer formula over $K$, where $p$ is an odd prime. More precisely, up to a $p$-adic unit, we have $$ \frac{L'(E/K,1)}{\Omega^{\mathrm{cong}}_{\mathbf{f}} \operatorname{Reg}(E/K)} = \# Sha(E/K)[p^\infty]\prod_{u} c_u(E/K), $$ where $\Omega^{\mathrm{cong}}_{\mathbf{f}}$ is the congruence period of the Hilbert modular form $\mathbf{f}$ associated to $E$ via the modularity conjecture.

We study the extended persistent homology of the radial filtration of a shape $M\subseteq \mathbb{R}^n$. A radial filtration is formed by choosing a center point $c$ and taking points of $M$ within distance $r$ of $c$. We show that, under mild assumptions, we can recover the radial extended persistence of a manifold with boundary from the radial persistence of its boundary. We also establish algorithms to compute radial extended persistence when $M$ is a set of pixels in a 2D digital grid. The methods are similar to those used to compute the extended persistent homology of a height filtration of a shape embedded in Euclidean space. We envisage these results will be useful in biomedical image analysis settings where it is natural to consider a radial filtration with respect to a fixed center, for example in the study of neuronal structures.

We prove a conifold gap theorem for the topological recursion of toric mirror curves: for every local analytic family acquiring a generic one-node degeneration and every fixed genus at least two, the conifold-polarized free energy has one universal polar term, and the remainder is jointly holomorphic in the transverse and spectator parameters. The result covers separating and nonseparating nodes and allows general family deformations beyond the pure filling-fraction case.

We study the one-dimensional strong-weak Lotka-Volterra competition-diffusion system \[ u_t=u_{xx}+u(1-u-av),\qquad v_t=dv_{xx}+rv(1-v-bu), \] with compactly supported initial data under the parameter condition $0<a<1<b$. Existing literature only gives leading-order spreading speed asymptotics without refined logarithmic corrections for wave fronts over the full parameter space. We convert the competitive system into an equivalent cooperative parabolic system and establish moving-domain comparison principles. Based on heat-kernel estimates, we derive sharp logarithmic asymptotic expansions for the rightmost level set of the stronger species $u$. According to the magnitudes of three characteristic speeds, five long-time dynamical regimes are classified, including pulled, nonlocally pulled, pushed and critical transition fronts, with explicit logarithmic and double-logarithmic phase corrections. Our results capture delicate long-time phase offsets of wave profiles neglected in previous leading-order theories.

The efficient randomized-adaptive design attains the minimal possible variance of the allocation proportion for any pre-specified target and has been extended from two to several treatment arms, but both versions target a single response. Confirmatory trials increasingly rely on several prioritized endpoints, typically efficacy followed by safety, that cannot be reduced to one summary without losing information. The efficient randomized-adaptive design is extended here to targets driven by the net treatment benefit of generalized pairwise comparisons, allowing any number of arms and endpoints of mixed type, including censored time-to-event outcomes. Strong consistency, a law of the iterated logarithm, and asymptotic normality of the allocation proportions are established by combining a sequential Hájek projection for the net benefit with the martingale technique used for the original design, and the design is shown to attain the semiparametric efficiency bound within the class of designs that track a smooth function of the net benefit. In simulations calibrated to a three-arm phase III melanoma trial, the proposed design concentrates allocation on the superior treatment more efficiently than designs driven by a single endpoint, while preserving type I error and power. A redesign of the trial illustrates the practical benefit, and the approach is discussed in relation to current regulatory thinking on adaptive designs for confirmatory trials.

Lewy's theorem says that a one-to-one harmonic mapping between plane domains has nonvanishing Jacobian. In dimensions at least three this statement is false for general harmonic homeomorphisms. We prove a four-dimensional Lewy theorem under the additional complex-analytic assumption of pluriharmonicity. More precisely, if \[ F:\Omega\subset \mathbb C^2\to \mathbb C^2 \] is a \(C^2\) pluriharmonic mapping which is one-to-one in a neighborhood of a point \(p\), then the real Jacobian of \(F\) is nonzero at \(p\). The proof is local. If the Jacobian vanished, a nontrivial real linear projection of \(F\) would be the real part of a holomorphic function \(f\) with \(df(p)=0\). The level hypersurface of this projection would therefore be a real analytic germ of the form \[ \{\operatorname{Re} f=0\}. \] We prove that such a germ cannot be locally flat at a critical point of \(f\). The obstruction is topological: local flatness forces the local homology of the hypersurface germ to agree with that of a real hyperplane, and hence forces every sufficiently small admissible link to have the integral homology, in particular the Euler characteristic, of \(S^2\). In the reduced case the Milnor open book of the plane curve singularity \(f^{-1}(0)\) gives a contradictory Euler-characteristic formula, while in the nonreduced case the local normal form produces more than two local complementary components. We then prove a separate bi-Lipschitz criterion for harmonic quasiconformal mappings: a harmonic quasiconformal homeomorphism from the unit ball onto a bounded \(C^1\)-Dini domain is bi-Lipschitz, provided it is already a local \(C^1\)-diffeomorphism.

This note removes the dimension one restriction of the main results in the paper with the same title by the second author.

For anisotropic diffusion with mixed derivatives, standard compact coordinate-aligned stencils and some local finite-volume or finite-element constructions can lose nonnegative coefficients unless suitable coefficient or mesh conditions are imposed. Here nonnegative coefficients are generated directly from conditional expectation rather than from an algebraic stencil decomposition. We construct a positivity-preserving scheme for possibly degenerate Hamilton–Jacobi–Bellman equations with controlled oblique Robin boundary conditions. At interior nodes the construction stems from a one-step reflected Feynman–Kac identity: an $m$-dimensional Rademacher vector generates $P=2^m$ equally probable weak-Euler branches, an exterior branch is mirrored about its oblique projection, and twice its overshoot serves as the discrete boundary local time $D$. The Robin coefficient enters only through the attenuation factors $e^{-\kappa D}$ and $De^{-\kappa D/2}g$, leaving the equal branch probabilities unchanged. Boundary nodes use a separate same-level closure over a spatial offset $\ell_h\asymp h$, which may be implicit; uniform obliqueness and $\mathbb P_1$ interpolation give a mesh-independent positive weight on interior nodes, so fixed linear Robin data yield a sparse nonsingular $M$-matrix system and the general control set yields a monotone contraction. Positivity is preserved for nonnegative data without a diagonal-dominance condition and without any CFL-type relation between $\Delta t$ and $h$.

Induction is the process by which empirical evidence is transformed to knowledge. Hume famously argued—and Popper and others agree—that there can be no logical justification for induction. A weaker form of induction, due to Bayes, expresses the aforementioned knowledge in terms of probabilities, and we review some well-known and not-so-well-known criticisms of the Bayesian solution. We then investigate the relatively new possibilistic inferential model (IM) framework, showing that, in addition to the IM's strong, statistical reliability guarantees that it uniquely enjoys, it is safe from those criticisms that damage the Bayesian foundations. For illustration, we reconsider the classical sunrise problem and compare our proposed solution with Laplace's famous rule of succession.

Inverse Optimal Control (IOC) aims to infer the underlying cost functional of an agent from observations of its expert behavior. This paper studies the finite-horizon continuous-time inverse LQR problem from closed-loop state–input trajectories, where both the system matrices and the quadratic cost are unknown. The finite horizon induces a time-varying optimal gain, and this endogenous excitation serves as the structural mechanism that makes joint recovery possible. We quantify this mechanism through three computable conditioning indices, which measure state richness, gain-variation richness, and injectivity of a structured cost operator. Using these indices, we establish joint identifiability conditions for the inverse problem considered here. Crucially, these conditions guarantee recovery of the ground-truth system matrices $(A,B)$ and the true cost weighting matrices, rather than merely a behaviorally equivalent surrogate. We also develop a conditioning-aware sampled-data reconstruction method that reconstructs the gain $K(\cdot)$ and the closed-loop dynamics matrix $A_c(\cdot)$ from noisy measurements, recovers $(A,B)$ in closed form, and identifies the quadratic weights through a convex semidefinite program. We further establish the non-asymptotic perturbation bounds and the consistency of the full reconstruction method under sub-Gaussian observation noise, with explicit dependence on the same conditioning indices. Numerical experiments support the theory and illustrate the diagnostic value of the conditioning indices.

We prove a nonlinear parabolic characterization of stochastic completeness at infinity for weighted graphs. For the filtration equation \[ (\partial_t + \Delta \Phi)u =0 \] where $\Delta$ is the non-negative formal graph Laplacian and $\Phi u =\phi \circ u$ with $\phi \colon \R\to\R$ nonconstant, continuous and increasing, stochastic completeness at infinity is equivalent to uniqueness of bounded pointwise solutions for every bounded initial datum. For \(\Phi=\id\), this recovers the classical heat equation characterization of stochastic completeness at infinity, and of stochastic completeness when the killing term is trivial, i.e., when \(\kappa=0\). If stochastic completeness at infinity fails, then every bounded initial datum admits infinitely many bounded pointwise solutions of the filtration equation. Admissible nonlinearities include the signed porous medium and fast diffusion powers $\phi(s)=s|s|^{m-1}$ for all $m>0$, as well as many others. Stochastic completeness at infinity is further characterized by a generalized mass balance: the total mass of a positive pointwise solution at time $t$, augmented by the mass $\int_0^t\sum_{x}\kappa(x)\phi(u(s,x)) \dd s$ dissipated by the killing term $\kappa$, equals the initial mass. This balance holds for every bounded positive solution on graphs of finite measure and for bounded finite-mass data on graphs of arbitrary measure under the sharp condition $\limsup_{r\to0^+}\phi(r)/r<\infty$. It also extends to positive pointwise solutions in $\ell^1$ that are bounded on every positive time interval. When the killing term is trivial, stochastic completeness at infinity reduces to stochastic completeness and generalized balance to conservation of mass.

For a graph $G$, its vertex deck is the multiset of graphs obtained by deleting one vertex. Bowler, Brown, and Fenner (BBF) proposed $2\lfloor(n-1)/3\rfloor$ as the maximum possible overlap between the decks of two nonisomorphic $n$-vertex graphs, for all sufficiently large $n$. We first give an explicit pair of connected nonisomorphic graphs on $78$ vertices with at least $51$ common cards, exceeding BBF's predicted value of $50$. We then construct, for every even $r\ge4$, families at arbitrarily large orders whose overlap fraction is asymptotically at least $1-1/r$. Consequently, for every $\alpha<1$, infinitely many pairs have more than $\alpha n$ common cards, so the attainable fraction is arbitrarily close to the full deck. For representative instances, the predicted overlaps were also checked by complete deck generation and isomorphism testing with Brendan McKay's nauty tools.

We introduce $C$-lifting functors, which axiomatize lifting properties of non-abelian tensor and exterior products with respect to prescribed classes of group extensions. For a homomorphism $f\colon\Gamma\to G$, we associate to every $C$-lifting functor $F$ a relative quotient $F_f(G)$. This quotient maps epimorphically onto the subgroup determined by $F$ in every $f$-extension belonging to $C$. We show that the construction is functorial in $f$ and that, for an $f$-extension $p\colon\widetilde G\to G$, it gives a morphism of natural exact sequences. In degree two this yields an epimorphism $ \frac{H_2(G;\mathbb Z)}{f_*H_2(\Gamma;\mathbb Z)} \longrightarrow \ker p\cap[\widetilde G,\widetilde G]. $ Applying the construction to iterated tensor and exterior powers, we obtain relative Schur-Baer theorems for the lower central and derived series. We also compare the relative tensor and exterior squares, relate the exterior construction to the relative Schur multiplier $H_2(G,\Gamma;\mathbb Z)$, and derive applications to orderability. As a consequence, we prove that a virtual knot group is left-orderable if and only if it is circularly orderable.

For integers $d\geq 3$, let $F_{2,d}(n)$ be the largest size of a subset of $[n]$ containing no two distinct elements whose product is a perfect $d$-th power, and let $f_{2,d}(n)$ denote the analogous quantity when the two elements need not be distinct. Fleiner, Juhász, Kövér, Pach, and Sándor proved that both complements have order $n^{2/3}$ when $d=3$, and asked for a leading constant. They also asked whether, more generally, $n-F_{k,d}(n)$ and $n-f_{k,d}(n)$ have order $n^{k/d}$ for $1<k<d$. We establish asymptotic formula in the case $k=2$ for every fixed $d\geq3$, \[ n-F_{2,d}(n)\sim n-f_{2,d}(n) \sim C_d\, n^{2/d}(\log n)^{d-3}, \] where $C_d>0$ is given explicitly by an Euler product and a polytope volume. In particular, the extra logarithmic factor gives a negative answer to the second question for every $d\geq4$. For $d=3$ we obtain \[ C_3=\frac{\pi^2}{4} \prod_p\left(1-\frac3{p^2}+\frac2{p^3}\right), \] which answers the first question. The proof uses an exact decomposition into complementary $d$-free kernel classes, a squarefree sieve in multiplicative boxes, and a two-height polytope calculation.

The geometric properties of quadrature points in one and multiple dimensions are a classical topic in numerical analysis. Recently, generalized quadrature methods, where discrete points and weights are replaced by integration along curves, have attracted growing interest. This paper focuses on positive generalized quadratures that are exact for multivariate trigonometric polynomials. We derive upper bounds on the covering radius of such quadratures using sign-localized test functions, a technique that also extends naturally to algebraic polynomials on intervals, spherical polynomials, and hyperbolic cross trigonometric polynomials. For lower bounds on the length of quadrature curves, we compare two recent approaches in the context of multivariate trigonometric polynomials. The second part of the paper studies a well-known, geometrically simple curve that underlies rank-1 Korobov lattice rules for periodic functions. We prove that these curves are quasi-optimal with respect to multiple optimality criteria and demonstrate its adaptability to the 2-sphere.

Let $\mathbb{Q}^{(d)}$ be the composite field of all number fields of degree at most $d$. In 2001 Bombieri and Zannier proved that $\mathbb{Q}^{(2)}$ has the Northcott property and asked what happens for $d\geq 3$. Here we study the absolute Weil height for elements in the composite ring of the rings of integers of such number fields. In particular, we consider $\mathbb{Q}^{(3)}$ as the composite field of $\mathbb{Q}^{(2)}$ and a minimal infinite family of cubic fields, and we show that the composite ring of the rings of integers of these fields does have the Northcott property. Our results follow from new height lower bounds, expressed in terms of the degree. Moreover, we introduce a notion of size for subfields of $\mathbb{Q}^{(3)}$. For instance, $\mathbb{Q}^{(3)}$ has size $1$ and the maximal abelian subfield of $\mathbb{Q}^{(3)}$ has size $1/2$. We show that there is a subfield of $\mathbb{Q}^{(3)}$ of size $1$ which has the Northcott property.

Rosenthal's classical theorem says that, for every infinite compact space $X$, the Banach space $C(X)$ admits a quotient isomorphic to either $c_0$ or $\ell_2$. The corresponding question for $C_p(X)$, the space $C(X)$ endowed with the topology of pointwise convergence, is much subtler and still open; the only compact spaces for which the existence of an infinite-dimensional metrizable quotient is not presently settled in ZFC are Efimov compacta. We prove that the Banach space case cannot be reproduced with the usual sequence spaces carrying their pointwise topologies: Let $X$ be a Tychonoff space and let $E\subseteq c_0$ be a non-trivial symmetric sequence ideal endowed with the subspace topology inherited from $\mathbb{R}^{\mathbb{N}}$. Then the existence of a continuous linear surjection $T:C_p(X)\rightarrow E_p$ implies $E=c_0$, where $E_p$ means $E$ with the topology inherited from $\mathbb{R}^{\mathbb{N}}$. Hence, no proper non-zero symmetric sequence ideal of $c_0$ can be realized as a continuous linear image of a $C_p$-space. Combining this result with the characterization of the Josefson–Nissenzweig property for $C_p(X)$ obtained by Banakh, Kąkol, and Śliwa, we derive a complete characterization of all pairs $(X,E)$ for which such a surjection exists. In particular, for every $0<q<\infty$, there is no continuous linear surjection $C_p(X)\rightarrow(\ell_q)_p$.

Glauberman's characteristic $p$-functor $K^{\infty}$ has a number of remarkable properties. For example it controls $p$-transfer in any finite group provided $p \geq 5$. These notes are the author's attempt to understand and simplify this work.

In this work, we construct fast eighth-order Pade schemes for the direct Zakharov-Shabat scattering problem. The schemes are based on an eighth-order exponential integrator obtained from the Magnus expansion. A direct extension of the conventional fast Pade representation to the eighth-order case leads to insufficiently accurate fast variants in the considered tests. To overcome this difficulty, we reformulate the spectral dependence on a finite real interval using the Joukowski mapping and represent the local numerators and denominators in the Chebyshev polynomial basis. This makes it possible to construct the global transition matrix by a fast product tree while retaining a compact polynomial representation. Numerical experiments for chirped hyperbolic secant potentials with both signs of dispersion show that the proposed Chebyshev-based fast schemes substantially improve the accuracy of the corresponding direct fast variants and can be used for efficient computation of the continuous nonlinear spectrum.

We present an algebraic and geometric condition for bivariate quaternionic polynomials of arbitrary bidegree to have a univariate linear left or right factor. We apply this quaternionic factorization theorem to the bidegree (1,1) case and recover a classical theorem of Clifford in elliptic geometry. By applying to the bidegree (2,2) case, we obtain decompositions into two circles of celestial surfaces, namely surfaces in the 3-dimensional sphere that contain two circles through a general point. This results in an alternative proof and refinement for a theorem by Skopenkov and Krasauskas from 2019, which states that a non-quartic celestial surface is Möbius equivalent to either the pointwise product of circles in the unit-quaternions, or an inverse stereographic projection of the pointwise sum of circles in Euclidean space. Our proposed method extends this decomposition result to the quartic case and we show that surfaces are, up to Möbius equivalence and stereographic projections, not both a sum and product of circles.

Let \( \Lambda=\{\lambda_j\}_{j\in\mathbb N}\subset[0,1) \) be a finite union of Carleson sequences, and let \( \Lambda^p=\{\lambda_j^p\}_{j\in\mathbb N} \) for \(p>0\). We prove that \(\Lambda\) and \(\Lambda^p\) admit compatible Hartmann-type cluster decompositions and that, under the natural coordinate identification, \( H^2(\Lambda)=H^2(\Lambda^p). \) After taking account of the change in the normalization weights, this trace-space identity gives \( \operatorname{Ran}(T_\Lambda) = \operatorname{Ran}(T_{\Lambda^p}) \) for the associated weighted evaluation operators. The characterization of finitely generated dynamical frames for normal operators then implies that, for every positive operator \(A\in\mathcal B(H)\), every finite index set \(I\), and every \(p>0\), \( \{A^nf_i\}_{n\in\mathbb N,\,i\in I} \text{ is a frame for }H\) if and only if \( \{A^{pn}f_i\}_{n\in\mathbb N,\,i\in I} \text{ is a frame for }H. \) In particular, for every \(q\in\mathbb N^+\), the subfamily \( \{A^{qn}f_i\}_{n\in\mathbb N,\,i\in I} \) is still a frame, and hence every such dynamical frame has infinite excess. An example shows that the positivity assumption cannot in general be replaced by normality.

Let $\Lambda$ be the Dirichlet-to-Neumann operator on the boundary of a compact three-dimensional Riemannian manifold. Using the Lee–Uhlmann full-symbol formula and Weyl's invariant theory, we compute the Guillemin–Wodzicki residues of $\Lambda$ and $\Lambda^2$ explicitly, and hence the first two logarithmic coefficients in the Steklov heat trace. As an application, we prove that geodesic balls in simply connected three-dimensional space forms are determined by their Steklov spectra among smooth domains in the same space form. More generally, we obtain a rigidity theorem in the class of compact constant-curvature manifolds with smooth, not necessarily connected, boundary. In the Euclidean case, we also prove spectral uniqueness for concentric spherical shell regions.

Coherences between different energy levels are strongly constrained by thermodynamics. Here we ask whether several coherence transfers that can be optimized separately can also be optimized simultaneously by the same thermodynamic process. Using an energy-resolved Stinespring representation of thermal operations, we express elementary transfer amplitudes as inner products of bath-weighted vectors. Joint saturation imposes collinearity and phase-closure conditions on these vectors, which must all arise from one energy-preserving system–bath unitary. This gives necessary compatibility conditions for arbitrary finite-dimensional Hamiltonians. We show that joint incompatibility persists in a system with both degenerate and nondegenerate energy levels: rigorous computer-assisted semidefinite-program bounds, combined with explicit thermal constructions, prove that three individual thermal optima cannot be approached simultaneously. We then determine how well different coherences can be protected when their optima are incompatible. For a four-level family at infinite temperature, we obtain the complete closed retention region and the cost of balancing the coherences relative to their individual optima. At positive temperature, covariance and Gibbs preservation permit joint retention that no thermal operation can approach. Finite-bath constructions and certified semidefinite relaxations bound the balanced thermal performance. When two coherences carry independent phases, these bounds describe simultaneous preservation of interference visibility and phase Fisher information.

We consider Delone point patterns with self-similarity. Under mild conditions, the similarity factor is a Pisot number if and only if the pattern is uniformly discrete. The classical case is a Meyer set $\Lambda$ with $\Lambda\supset \theta\Lambda$ for some $\theta>1,$ for which $\theta$ must be a Pisot number or a Salem number. When $\Lambda$ contains several similar copies of itself, the case of a Salem number drops out for $\theta<2.$ On the other hand, strictly self-similar patterns with a Pisot factor must be Meyer sets. Various examples are given.