Papers

We present a solution to some problems posed by the author and Kalenda. We show that the closed ball of nonseparable Hilbert space in its weak topology is homeomorphic to its positive part, as well as to its product with the Hilbert cube. In the separable setting we obtain that there is a weak homeomorphism of the closed unit ball of $\ell_2$ onto its positive part that preserves the norm, and via a result of Dijkstra and van Mill, the same is true for the ball of $\ell_\infty=\ell_1^*$ in the weak$^*$ topology. All spaces $B(\kappa,a,b)$ considered by the author and Kalenda are shown to be homeomorphic. The solution has been found by AI (Chatgpt 5.5), the role of the author has been to ask the right questions, check and understand the answers, and adapt the writing to his personal human taste.

Communication systems designed for reliable data reconstruction, rather than task-oriented communication, typically rely on separate source and channel coding and incur high latency under limited spectrum availability and fading channels. To address this, we propose a transmission framework with opportunistic spectrum access, in which the transmitter sends discrete latent representations learned via a vector-quantized variational autoencoder (VQ-VAE) over idle licensed channels using standard digital modulation. The AI-powered receiver is still able to reconstruct task-related information from the heavily compressed data. We develop a cross-layer latency model that accounts for compression, block errors, retransmissions, and stochastic channel access. Results on latency-accuracy trade-offs show that the proposed scheme achieves at least 79- and 3.3-fold latency reductions with only 5.7% and 2.4% drops in classification accuracy compared to benchmarks using conventional source and channel coding. The framework enables low-latency communication and reliable task execution even under limited spectrum availability and challenging channel conditions.

We study late-time identification of the Caputo order in a linear diffusion equation generated by a strictly positive self-adjoint operator with compact resolvent. For signed scalar observations \(M_\alpha(t)=\sum_n a_nE_{\alpha,1}(-\lambda_nt^\alpha)\) satisfying \(\sum_n|a_n|/\lambda_n<\infty\), we show that, after eigenspace grouping, every nontrivial observation has a finite first nonzero resolvent moment \(S_m=\sum_n a_n/\lambda_n^m\). A uniform differentiated large-argument expansion of the Mittag-Leffler factor yields eventual strict monotonicity of \(\alpha\mapsto M_\alpha(t)\) on admissible intervals avoiding the zeros of \(1/\Gamma(1-m\alpha)\), hence uniqueness from one sufficiently late scalar measurement. For two measurements, \(M_\alpha(\rho t)/M_\alpha(t)=\rho^{-m\alpha}(1+O(t^{-\alpha_0}))\), giving a log-ratio estimator with asymptotic-bias and relative-noise error bounds. For bounded observations, \(S_m=\langle\mathcal A^{-m}\varphi,h\rangle\); for a finite rod, the leading point-sensor condition is \((\mathcal A^{-1}\varphi)(x_*)\ne0\). Counterexamples show the sharpness of the exclusions and noise interpretation.

We show the existence of a topologically free minimal action of $\mathbb F_\infty$ on the Cantor space that does not have dynamical comparison. Moreover, we show that this phenomenon can happen both in the presence and in the absence of invariant measures. We also show that strict comparison of the reduced crossed product C*-algebra does not imply dynamical comparison for minimal actions. Our technique involves constructing a monoid which is not almost unperforated, embedding it into a countable refinement monoid and then realising it as the type semigroup associated to a dynamical system.

We study ridge-regularized log-density-ratio estimation in the Gaussian location model with a common covariance matrix. By affine invariance, the model is written as q $\sim$ N(0, I), p $\sim$ N($\Delta$, I), with linear features, where $\Delta$ is a mean vector. The variational estimator is the empirical Kullback-Leibler (KL) log-normalized fit with a squared L2-penalty on its nonconstant coefficient, and the spectral estimator recently introduced in [1] replaces a single variational problem by a continuum of ridge-regularized least-squares problems. We derive high-dimensional deterministic asymptotic equivalents when the numbers of observations and dimension tend to infinity with fixed ratios. The regularized variational limit is characterized by a scalar entropy minimization problem derived from the convex-Gaussian-min-max theorem (CGMT), while the regularized spectral limit follows from deterministic equivalents for resolvents of weighted sums of two independent Gaussian sample covariance matrices. We use these formulas to compare population risks, with experiments focused on fixed-signal aspect-ratio sweeps and optimized regularization. Our conclusion is that with many observations, under the criteria and asymptotic regimes analyzed here, the well-specified variational estimator has the smaller risk, while with fewer observations, the spectral estimator is favored because its covariance-based construction has lower variance. We also study how a nuclear penalty can be used and partially analyzed to perform feature learning.

A defending state facing sustained economic, media, and psychological pressure from an adversary that continuously re-optimises its campaign poses a problem that existing attacker-defender models in operations research do not directly resolve, because they treat the defender's state as a discrete allocation rather than a continuous, slow-moving institutional process. We formulate a coupled dynamical system in which grievance and institutional resilience evolve continuously while pressure intensity is chosen by an optimising Markov decision process, prove existence and local stability of the resulting equilibrium, and prove a formal result distinguishing it from standard feedback-stability analysis and from a stationary Markov chain treated in isolation. We validate the framework computationally using thirty randomised network instances, full value iteration, and a documented case study of cognitive warfare directed at Iran (2017–2026). The historically calibrated case sits approximately twenty-five times above the computed operational collapse boundary, and a greedy seeding policy reaches eighty-seven percent average network penetration across the randomised instances, significantly above a degree-centrality baseline. A practitioner can use the equilibrium and boundary computation to assess where a specific case sits relative to collapse, rather than relying on an unverified comparison between opposing pressure intensities.

Classical equivariant bordism theories require computing the full collection of equivariant characteristic numbers to detect whether an equivariant manifold bounds equivariantly or not. This paper establishes simplified equivariant bordism characterizations for two families of equivariant manifolds with isolated fixed points: unitary $T^k$-manifolds and closed smooth $(\mathbb{Z}_2)^k$-manifolds. For any unitary $T^k$-manifold $M$ with isolated fixed points, we establish an equivariant unitary bordism criterion built entirely from a single polynomial of equivariant Chern classes. We further introduce the minimal distinguishing degree and obtain two key inequalities that capture the interplay between $\dim M$ and the Euler characteristic $\chi(M)$ through this minimal distinguishing degree. These inequalities settle the existence problem of a linear lower bound for $\chi(M)$ within the framework of Kosniowski's conjecture and partially verify the conjecture under natural admissible assumptions. We also provide an alternative proof settling the toric generalization of Kosniowski's conjecture when $\dim M=2k$. By contrast, for a closed smooth $(\mathbb{Z}_2)^k$-manifold with isolated fixed points, we derive a more concise equivariant bordism criterion relying solely on the powers of the top equivariant Stiefel-Whitney class. Our new criteria substantially reduce computational demands.

We develop a variational nonlocal phase-field model for dynamic fracture in elastic solids. The proposed formulation is distinguished by three main features. First, the model is formulated through nonlocal kinematics and kernel-dependent function spaces, allowing weaker regularity requirements while recovering the classical local theory as the nonlocal interaction domain vanishes. Second, a nonlocal crack-surface functional is introduced as an integral counterpart of the Ambrosio–Tortorelli regularization, so that the characteristic length of the diffusive crack is implicitly determined by the nonlocal interaction domain rather than by a prescribed length scale. Third, the degraded nonlocal elastic energy and the nonlocal crack-surface functional are combined into a variationally consistent dynamic fracture system, consisting of a nonlocal momentum balance and an irreversible nonlocal gradient-flow evolution law for the phase field. The coupled system is solved using two temporal discretization strategies: a structure-preserving scalar auxiliary-variable scheme and a staggered alternating scheme, both combined with finite element discretization in space. Numerical examples involving Mode-I fracture, dynamic crack branching, Kalthoff–Winkler-type shear fracture, and fragmentation show that the proposed model captures complex crack initiation, propagation, branching, and interaction without explicit crack tracking. Quantitatively, the predicted crack-tip velocities remain below $0.6c_R$ in the dynamic branching and shear-loading tests, and the shear-loading benchmark gives an inclined crack path of approximately $48^\circ$, consistent with the characteristic Kalthoff–Winkler fracture pattern.

Let $W$ be an even Coxeter group. We prove that among all Coxeter systems generating $W$ the unique even Coxeter system realizes the minimal exponential growth. Our proof relies on comparing the exponential growth rates in the explicit algorithm of Mihalik which from any Coxeter system of an even Coxeter group eventually produces the unique even one. The main new ingredient is that blow downs along pseudo-transpositions do not increase the exponential growth rate.

Let $G$ be the orbifold fundamental group of the moduli space of smooth cubic surfaces $\mathcal{M}_{\mathsf{sm}}$ in $\mathbb{P}^3_{\mathbb{C}}$ with base point at the Clebsch surface $X_{\mathbf{1}}$. The image of the monodromy action $G \to \lbrace \text{Permutations of $27$ lines on $X_{\mathbf{1}}$} \rbrace$ is famously the Weyl group of type $E_6$. Here we give a description of this monodromy action in terms of the Petersen graph by working out the action of ten explicit generators of $G$ by elementary calculation. These ten generators were found in joint work with Allcock and Looijenga while studying the description of $\mathcal{M}_{\mathsf{sm}}$ as a discriminant complement in a complex $4$-ball quotient.

We consider locally compact Lévy trees conditioned to be large, with respect to different criterion: its height, its maximal ”size” vertex and its total ”mass”. In the critical case, we provide a coupling with a truncated Kesten tree which then allows to directly prove the local convergence in distribution of the conditioned Lévy tree to be large towards the Kesten tree. We also consider the sub-critical and super-critical cases. In the former case the results can be partial, due to a possible condensation phenomenon which is outside the mathematical framework used in this paper.

We partially answer two open questions concerning diameter two properties in absolute sums. First, we identify the conditions that a super $\Delta$-point in an absolute sum of Banach spaces imposes on the coordinates. Secondly, we show that the convex diametral local diameter two property (convex-DLD2P) passes from an absolute sum $X\oplus_N Y$ to its factors whenever $N$ is not the $\ell_\infty$-norm.

Typicality arguments for emergent spacetime rely on the Lubkin-Page bounds, which show that generic quantum states have vanishing correlations between subsystems. These bounds assume a tensor-product Hilbert space (a Type~I von~Neumann algebra), but the observable algebras in quantum field theory and quantum gravity are generically Type~II or Type~III, raising the question of whether the bounds survive. We prove that they do for all Type~II von~Neumann factors. For the hyperfinite Type~II$_1$ factor with a tripartite decomposition $R \cong A \otimes B \otimes E$, the mutual information between subsystems $A$ and $B$ vanishes as $O((d_A d_B / d_E)^2)$ in finite-dimensional approximations, provided $d_A d_B \leq d_E$ (Theorem~1). For Type~II$_\infty$ factors, including the gravitational algebras constructed via the crossed-product method by Witten and by Chandrasekaran, Longo, Penington, and Witten, the bound acquires an additional exponential suppression controlled by the Bekenstein-Hawking entropy (Theorem~2). We identify the obstructions to extending the result to Type~III factors and discuss the open question of whether the commutant of the observable algebra can serve as a natural thermal bath that tightens the bound further.

We show that the language of geodesic words representing elements of a stable subgroup $H$ of a group $G$ with finite generating set $A$ is regular, and that there is a sublanguage which bijects $H$. Consequently, the growth function of $H$ with respect to $A$ is rational, and in many cases, one can deduce a growth rate gap between $H$ and $G$. In particular, this applies to convex cocompact subgroups of $\mathrm{Out}(F_n)$, handlebody groups, and Torelli groups of surfaces of sufficient complexity. We also provide an example of a finitely presented, relatively hyperbolic, and Morse local-to-global group which contains a stable subgroup with unsolvable membership problem, answering a question of Cordes, Russell, Spriano, and Zalloum.

We compute the tensor of the multiplicative group scheme with the mod-$2$ motivic cohomology spectrum in normed motivic spectra over the complex numbers, and find that the resulting algebra is free on a generator in bidegree (2,1). This gives a motivic analog of Bökstedt periodicity. The proof proceeds by comparing the tau-inverted and tau-reduced forms of the tensor. After inverting tau, the calculation reduces to classical Bökstedt periodicity via Betti realization. The reduction modulo tau is governed by a comparison between normed algebra structures and derived algebra structures on cellular modules over motivic cohomology mod tau. This comparison produces divided power operations and leads to mixed Cartan and Adem relations intertwining normed and topological power operations. A key input is a detailed analysis of motivic extended powers of spheres and their tau-torsion structure. In contrast with the corresponding simplicial-circle calculation due to Dundas-Hill-Ormsby-Østvær, the large families of tau-torsion classes disappear for the Gm-tensor, leaving a considerably more rigid algebraic structure.

This article investigates the long-time behavior of (possibly infinite-dimensional) impedance passive systems under saturated output feedback and external disturbances. We assume that, in the absence of saturation and disturbances, the underlying linear output feedback exponentially stabilizes the system. Our main contribution is to show that fractional Sobolev regularity of the free output is preserved by the nonlinear feedback. More precisely, if the initial condition belongs to a suitable interpolation space associated with the linear closed-loop generator, then both the output and the state of the nonlinear closed-loop system inherit the corresponding fractional regularity. This regularity is sufficiently weak to be shared by the linear and nonlinear closed-loop systems, thereby avoiding the identification of the nonlinear generator domain. Combining this regularity result with observability estimates for the linear system yields a characterization of the asymptotic behavior of the nonlinear closed-loop system in the presence of disturbances. In particular, under the impedance passivity framework and exact observability and regularity assumptions, we establish a semi-uniform input-tostate stability property. The theory is illustrated by a multidimensional wave equation with nonlinear boundary damping.

We study the Cauchy problem associated with a class of triangular cross-diffusion systems of Shigesada-Kawasaki-Teramoto type. We develop a self-contained well-posedness theory in C 0 ([0, T ]; H s (T d )) based on regularity estimates for scalar Kolmogorov equations. The diffusion coefficient of each species depends only on species of lower index, yielding a hierarchical structure that allows for refined blow-up criteria. Finite-time singularities can occur only through the divergence of the L $\infty$ (T d ) norm of the solution. Assuming polynomial growth of the nonlinearities, this criterion is refined to an L p -based blow-up condition for some finite exponent p, yielding a substantially weaker obstruction to global existence than classical Sobolev blow-up criteria. The proof is achieved through refined tame estimates for composition in Sobolev spaces. As an application, we prove global existence of non-negative strong solutions for two-species systems with logistic-type reaction terms in dimensions d $\le$ 2.

We prove global existence for a degenerate Keller-Segel equation with small initial values in a critical Lorentz space.

We establish an analogue of the Brezis-Nirenberg local linking theorem for a class of Szulkin-type functionals arising from relativistic action principles. In this framework, compactness of Palais-Smale sequences is formulated with respect to a topology induced by the effective domain of the functional, replacing the classical strong Palais-Smale condition. The proof combines the original construction of the min-max geometry, based on a negative gradient flow, with the Ekeland-Lasry regularization. The main difficulty is that the regularized functional is naturally associated with the strong topology of the underlying functional space, whereas compactness for the original functional is formulated in the topology induced by the effective domain. We overcome this obstacle through a new perturbative construction that recovers the required min-max structure. We apply our abstract multiplicity result to two representative relativistic models: the Lorentz force equation, describing the dynamics of a charged particle in an electromagnetic field, and the Dirichlet problem for the prescribed mean curvature operator in Minkowski space. As a consequence, under natural assumptions, each problem admits at least two non-constant solutions.

We study the one-time weight on strict partitions obtained from the modified odd Greaves–Jing–Zhu operator. The shifted $t$-Schur functions generated by this operator are obtained from the classical Schur $Q$-functions by the plethystic substitution $X\mapsto X-tX$. Thus the corresponding weight \[ \lambda \longmapsto \mathcal Q_\lambda(X;t)P_\lambda(Y) \] is a shifted Schur weight with a virtual first alphabet. We give its normalization, its Pfaffian correlation kernel, its Fredholm Pfaffian for the largest part, and its size cumulants. For $t=-q$ with $q\geq 0$ the virtual alphabet becomes the positive alphabet $X+qX$, giving a genuine probability measure. This positive specialization is the one-time marginal of the two-color lift considered in a companion note.

We prove a general lower bound for the Lebesgue constants $\Lambda_n$ of Leja sequences for every nonpolar compact set \(K\subset\mathbb C\): $\textstyle \limsup_{n \to+\infty} \Lambda_n \, \sqrt{n^{-1} W_n(K)}\geq \tfrac{1}{\sqrt{2}}$, where \(W_n(K)\) denotes the Chebyshev–Widom factor. In particular, bounded Widom factors imply a universal square root obstruction. We also establish improved polynomial upper bounds on compact subsets $K$ of the real line, building on the approach of Andrievskii and Nazarov. Our first ingredient is a refinement of their Key Lemma: we lower its exponent from $9/8 = 1.125$ to a near-optimal $\gamma^\star \approx 0.6304$. Our second ingredient is a localisation of the separation argument: instead of using the global minimum spacing among the first $n$ Leja points, we retain the local Green-function scale. For finite unions of intervals, a packing argument and sharp estimates on the latter function near and away from the endpoints yield $\Lambda_n = O(n^{1 + \gamma^\star})$, uniformly over Leja sequences. This improves the previously known best exponent $13/4 = 3.25$ to around $1.6304$ for such sets (including the benchmark case $K = [-1,1]$), hence substantially narrowing the gap with the near-linear growth observed numerically. Corresponding polynomial bounds for $\tau$-Leja sequences are derived.

Let $\Gamma$ be a complex reflection group acting on the complex affine or hyperbolic space $X$ with the set of reflecting hyperplanes $\mathcal{H}$. We define an augmented rack $(G, \mathcal{K}, p)$ associated to the orbifold fundamental group $G := \pi_1^{\operatorname{orb}}( \Gamma \backslash (X - \mathcal{H}))$ which plays the role of the fundamental rack of a framed link complement as defined by Fenn and Rourke. This yields representations of the orbifold fundamental group $G$ on the cohomology of the associated rack space.

We express the log discrepancies of co-rank one foliations on normal varieties in terms of arc codimensions and differential orders. On smooth threefolds, if a foliation has simple singularities and its separatrices form an algebraic simple normal crossings divisor, we identify its foliated discrepancies with those of an ordinary pair for every divisor centred on the separatrices. This gives adjunction and log canonical threshold formulae. For algebraic images of the separatrices, we estimate changes in boundary orders caused by failure of ideal generation and by descent through the conductor. For locally complete intersection images, comparison on a common resolution yields criteria for equality of discrepancies and bounds on log canonical thresholds.

We prove the finite-time stability of two-soliton solutions of the three-dimensional gravitational Hartree equation whose centers remain close to an elliptic solution of the Kepler two-body problem.

In this article, we consider the $r$-logarithm for defining three-parameter family of Rényi relative entropies that are generalization of the $\alpha$-$z$-Rényi relative entropies. All the members of $r$-deformed $\alpha$-$z$-Rényi relative entropies satisfy the necessary axioms to be a divergence. We expose the range of parameters $\alpha$, $z$ and $r$ for which the data processing inequality holds. We also establish that $r$-deformed $\alpha$-$z$-Rényi relative entropy is an upper bound of the Tsallis relative entropy. Now, we have two upper bounds of the Tsallis relative entropy, which are $r$-deformed $\alpha$-$z$-Rényi relative entropy and the other one, which is discussed in literature. We investigate the order relationship between these two upper bounds of the Tsallis relative entropy. We observe that our new upper bound is more tighter when applicable to the density operators.

Sparse autoencoders (SAEs) decompose internal activations of neural networks into sparse linear combinations of learned features by fitting an overcomplete dictionary $\mathbf{W}\in\mathbb{R}^{m\times n}$ with $m<n$, and inferring a sparse code $\mathbf{x}\in\mathbb{R}^n$ from $\mathbf{h}\approx\mathbf{W}\mathbf{x}$. This inference problem closely resembles the canonical setup of compressed sensing, but dense decoders requires $O(mn)$ learned values, which becomes costly at large feature counts. We introduce Expander SAEs: TopK SAEs whose decoder and tied encoder are supported on a left-$d$-regular expander mask with $d\ll m$, learning only $dn$ decoder values while keeping the sparse-coding problem $(m,n,k)$ fixed. The same structure reduces storage and turns the matching-pursuit correlation step $\mathbf{W}^\top \mathbf{r}$ in OMP into an $O(dn)$ gather-and-reduce operation. Our experiments show that across Pythia-70M/160M, Qwen2.5-3B, and Llama-3.2-1B residual-stream activations, varying $d$ traces a consistent storage–fidelity frontier, and that at the most compressed modern-LM setting, Qwen2.5-3B with $d=7$ uses $293\times$ fewer learned decoder values than the full dense decoder while retaining $84$% of dense CE-loss recovered. Control experiments show that the improved storage–fidelity tradeoff is driven by sparse, diverse decoder support structure rather than by fewer learned decoder values, and that when sparse and dense decoders are compared at matched parameter count, part of the remaining gap comes from encoder amortisation. On the theoretical side, we show that expansion and column flatness are sufficient for identifiability of noiseless $k$-sparse codes, and we derive complementary sufficient conditions under which OMP recovers the support exactly.

M. Gorelik and Th. Heidersdors in the papers \cite{GH} investigated Euler characters for Lie superalgebra $\frak{gl}(m,n)$ and $\frak{osp}(m.2n)$. In the present paper we also investigate Euler characters for Lie superalgebra $\frak{gl}(m,n)$ but we use a different approach and our results are formulated in different terms.

A common Morse Hamiltonian bridges the planar Kepler–Coulomb and hyperbolic Landau systems, linking flat electric dynamics to curved magnetic dynamics. A radial Liouville transformation with coupling-constant metamorphosis maps the separated Kepler problem to the Morse system, while fixed-horocyclic-momentum reduction maps the hyperbolic Landau problem to the same system. In a common normalization, the Coulomb coupling is the signed product of the magnetic field and conserved horocyclic momentum, while the quantum Landau reduction has the universal half-density shift $1/4$. Beyond this shared reduction, we construct a direct classical orbit map on the regular nonradial sectors for all signs of the Kepler energy. It maps the Landau height to a scaled inverse Kepler radius, Kepler ellipses, parabolas and hyperbolas to magnetic circles, horocycles and hypercycles, and Hamilton's hodograph affinely to the Landau carrier circle. The reduced Kepler symplectic form and the Binet-selected complex structure determine a Poincaré Kähler metric. Its nondegenerate Binet orbits have constant geodesic curvature of magnitude equal to the inverse eccentricity, while zero coupling gives geodesics. The orbit map identifies this metric isometrically with the Poincaré configuration-space metric of the hyperbolic Landau problem. Composing the quantum Liouville maps eliminates the Morse wavefunction and yields an exact weighted intertwining identity between the radial Kepler and horocyclic Landau operator pencils. At distinguished half-integer magnetic fields, the angular-momentum grading of a fixed Kepler shell is reorganized into the finite reduced Landau tower and its threshold endpoint. These exact correspondences hold in the stated reduced sectors despite the unitary inequivalence of the complete parent systems.

In the present paper, we provide an explicit construction for generators of a $\lambda$-constacyclic code $\mathcal{C}$ of arbitrary length $\ell$ over a finite chain ring(FCR) $\mathcal{R}$ in terms of certain minimum degree polynomials of the ring $\mathcal{R}[x]/ \langle x^{\ell}-\lambda \rangle$. Moreover, the proposed construction achieves the minimum possible number of generators. We prove certain properties of this set of generators, using which we obtain a minimal spanning set of $\mathcal{C}$. We also obtain that the rank of $\mathcal{C}$ is $\ell-n_0$, where $n_0$ is the degree of the minimal degree polynomial in $\mathcal{C}$. Finally, we derive necessary and sufficient conditions under which an arbitrary length $\lambda$-constacyclic code $\mathcal{C}$ over $\mathcal{R}$ is Maximum Hamming Distance with respect to Rank(MHDR) as well as Maximum Distance Separable(MDS) in terms of a torsion code of $\mathcal{C}$ over the residue field $\mathbb{F}_q$ of $\mathcal{R}$. We further determine the exact values for $n_0$ for which $\mathcal{C}$ over $\mathcal{R}$ is MHDR.

In this paper, we study the existence problem for spherical \(T\)-designs on the \(d\)-dimensional sphere, where \(T\) is an infinite subset of \(\mathbb N\). We show that, if \(d\ge 2\), then a finite subset of \(S^d\) has infinite harmonic strength if and only if it is antipodal. For \(d=1\), we show that infinite strength spherical designs are exactly cyclotomic designs, and we characterize their existence in terms of certain \(0\)-\(1\) polynomials. We also prove that the harmonic strength of every infinite strength spherical design has the weak GCD property. Finally, for a given infinite subset \(T\subset \mathbb N\) with the weak GCD property, we give a finite procedure to decide whether there exists \(X\subset S^1\) such that \(\operatorname{Hst}(X)=T\), and apply this criterion to concrete existence and non-existence examples.