Papers

We determine the $k$-colour size–Ramsey number of even cycles up to absolute constant factors. For every $k\ge2$ and every even $n\ge100\log k$, \[ \widehat R_k(C_n)=\Theta(k^2\log k)n. \] The lower bound follows from the corresponding result of Beke, Li and Sahasrabudhe for paths, while our upper bound improves the previous best estimate $O(k^{34}n)$ of Javadi, Kohayakawa and Miralaei.

The main aim of this note is to provide a characterisation of coextensive morphisms in the category of bounded hoops. This characterisation is then used to show that several categories of bounded hoops are coextensive as categories. Among these are the variety of bounded Wajsberg hoops, or more generally the variety of bounded $\vee$-hoops. Our characterisation also yields the coextensivity of the category $\mathbf{Heyt}$ of Heyting algebras and recovers the known coextensivity of the category $\mathbf{MV}$ of MV-algebras.

We study the fast-robber version of the Cops and Robbers game on finite strongly connected abelian Cayley digraphs, including undirected Cayley graphs as the symmetric case. For bounded out-degree $D$, we show that $c_{1,\infty}\left(\Gamma\right)=O_D\left(n^{1-\frac{1}{D}}\right)$; in the undirected case with $D\ge2$, this improves to the optimal exponent $1-\frac{1}{\left\lfloor \frac{D}{2}\right\rfloor}$. We also establish the degree-independent bound $c_{1,\infty}\left(\Gamma\right)=O\!\left(\frac{n\left(\log\log n\right)^2}{\log n}\right)$. These estimates follow from an optimized character-theoretic cyclic sweep over subgroup quotients.

Differential privacy has become the standard for generating privacy-protected data releases. However, differential privacy does not translate intuitively to disclosure risk. In particular, it remains unclear how much an adversary's belief about an individual's inclusion in a dataset can change after observing a protected release. To address this question, we derive upper and lower bounds on the posterior-to-prior ratios of inclusion beliefs under bounded probabilistic and approximate differential privacy. By assuming a worst-case adversary with all-but-one auxiliary information, i.e., knowledge of all except for one of the participants in a dataset, we obtain bounds that apply to any adversary. Because these bounds may fail with non-zero probability, we study the corresponding failure probability for the Gaussian mechanism. We derive a theoretical upper limit on this probability and compare it with Monte Carlo estimates across a wide range of parameter settings. The observed failure rate is several orders of magnitude smaller than its theoretical upper limit, indicating that the latter is highly conservative. These findings suggest that the inferential privacy guarantees provided by differentially private mechanisms may be substantially stronger in practice than what is implied by the theoretical upper limit.

We introduce and analyze a barrier-regularized symmetric Nitsche method for the scalar Signorini problem. Applying a logarithmic barrier to the nonnegative slack variable in an augmented Lagrangian and then eliminating that variable yields a smooth positive-part operator and the perturbed complementarity relation on a primal-dual central path, with barrier parameter $\mu=\gamma s$. For every $s>0$, the discrete problem is continuously differentiable, uniquely solvable, and has a symmetric positive definite Newton matrix when the Nitsche parameter is sufficiently large. Under a mild barrier-feasibility condition, the continuous logarithmic energy has a unique minimizer $u_\mu$ for every $\mu>0$, its gap is positive almost everywhere, and $\|u-u_\mu\|_{H^1(\Omega)}\lesssim\mu^{1/2}$. Under the Sobolev regularity used for finite element approximation, this minimizer satisfies the \(L^2\) central-path boundary law. If the obstacle is locally the trace of an $H^2$-function, we also prove a uniform local $H^2$ bound in the interior of each planar contact face, on neighborhoods that may contain a free-boundary point of the limiting Signorini solution. The method is exactly consistent and quasi-optimal relative to $u_\mu$, with constants independent of $s$. If $u_\mu$ is uniformly bounded in $H^r(\Omega)$, $3/2<r\leq k+1$, then $\|u-u_h\|_{H^1(\Omega)}\lesssim h^{r-1}\|u_\mu\|_{H^r(\Omega)}+\mu^{1/2}$; hence the sufficient balance $s\lesssim h^{2r-1}$ preserves the available energy-norm rate. We also derive rates for discrete penetration and the complementarity residual. Numerical experiments with $P_1$ and $P_2$ elements on structured and unstructured meshes support the predicted rates and the local Newton theory, while showing that a rate-preserving smoothing may still resolve the contact set poorly.

The read process of DNA-based data storage systems generates multiple noisy copies of the stored DNA sequences, affected by edit errors consisting of substitutions, deletions, and insertions. Motivated by the challenge of ensuring reliable data retrieval in the presence of edit errors, we present a concatenated coding scheme that accounts for practical design constraints in DNA storage. We introduce and apply the marker-delimited code (MDC) as the inner code, which enables fast and reliable computation of symbolwise a posteriori probabilities (APPs). We combine MDC with an outer LDPC code. The LDPC is decoded via belief propagation using the soft information generated by MDC. Our results show that, in comparison with prior work, this construction provides more efficient error correction over multi-read edit channels in the short-blocklength and high-rate regime.

We study the algebraic and differential structure of the Riemann curvature tensor of the higher-dimensional Kerr-NUT-(A)dS spaces, motivated by the eventual goal of finding an IDEAL (Intrinsic, Deductive, Explicit and ALgorithmic) characterization of this family of metrics. The special geometry of these spaces is governed by the principal tensor $h_{ab}$, a non-degenerate closed conformal Killing-Yano $2$-form, whose existence singles out the Kerr-NUT-(A)dS family; we therefore focus on the algebraic relationship between the Riemann curvature $R_{abcd}$ and $h_{ab}$. In our investigations we encountered an obstruction, which results in a no-go theorem: no non-trivial $2$-form, including $h_{ab}$ itself, can be constructed covariantly from the undifferentiated Riemann tensor alone. Despite that, we do characterize the family of Riemann-symmetric tensors that could be the curvature of a Kerr-NUT-(A)dS metric by an algebraic and a differential condition, stemming from the integrability of the defining equation of $h_{ab}$ and the second Bianchi identity. Our calculations apply in all higher dimensions, which becomes feasible by a judicious application of representation theoretic techniques related to a semi-direct product group $U(1)^n\rtimes S_n$ that stabilizes $h_{ab}$.

Starting from a three-dimensional model of linearized magnetoelasticity, we investigate the asymptotic behavior of a thin shell as its thickness approaches zero. Our analysis focuses on a family of linearly elastic elliptic membrane shells. Using $\Gamma$-convergence, we rigorously derive a two-dimensional linear magnetoelastic shell model and establish its relationship to the original three-dimensional model in terms of the convergence of minimizers.

We consider the barotropic Euler equations in dimension $d > 1$ with decaying density at spatial infinity. The phase portrait of the nonlinear ODE governing the equation for spherically symmetric self-similar solutions has been introduced in the pioneering work of Guderley. The existence of smooth decaying solutions through the sonic line has been obtained in the work (F. Merle et al., 2022) in a suitable range of parameters. We work in this paper in a different range of parameters and show the existence of holosphere like solutions which are smooth through the sonic line, but display vanishing density at a sphere.

We show that for any $g\geq2$, there exist uncountably many non-isomorphic smooth affine threefolds that are Stein deformation equivalent to $T^\ast\Sigma_g\times\mathbb{C}$, the product of the cotangent bundle of a genus $g$ oriented surface with the complex plane, answering a question of Ivan Smith. In fact, we prove that our family of smooth affine threefolds is parametrized by a $(6g-7)$-dimensional complex orbifold.

By focusing on algorithmic stability as a means of establishing out-of-sample bounds, we provide a system-theoretic interpretation of generalization in learning-enabled dynamical systems arising in data-driven optimization and feedback control approximation. Given two neighboring datasets, we specifically model sample replacement as an exogenous disturbance acting on a sensitivity system, while the incremental behavior of the data-dependent operator is encoded through an integral quadratic constraint. By relying on dissipativity arguments, we establish a matrix inequality-based certificate and a uniform stability bound that separates the one-sample sensitivity of the learned operator, and an algorithm-dependent dynamical gain. The latter can then be optimized, offering a tractable tool for certifying and comparing generalization capabilities of learning dynamics. We show that our results recover classical ones for gradient descent, apply naturally to momentum-based methods such as heavy-ball and Nesterov acceleration, and extend to data-driven control.

We investigate the iterative conception of set in its most general form, allowing urelements without assuming that they form a set. We formulate this conception in two ways, as stage theory and as level theory, and develop a general theory of levels with urelements. Unlike their pure-set counterparts, the resulting stage and level theories are not set-theoretically equivalent; moreover, second-order level theory with urelements is not weakly quasi-categorical. We then consider further principles governing stages and levels, motivated by directedness, unboundedness, and reflection. Some of these principles restore set-theoretic equivalence between the corresponding theories, while their level-theoretic versions yield forms of quasi-categoricity. These principles form strict implication hierarchies, thereby revealing distinct stronger conceptions of set beyond the basic iterative conception.

Healthcare workforce scheduling is an NP-hard optimization problem requiring simultaneous satisfaction of labor regulations, coverage requirements, employee preferences and cost objectives. Existing approaches (genetic algorithms, integer programming, constraint programming) model 6-12 constraints at shift-level granularity and cannot guarantee regulatory compliance. They also lack support for multi-role, multi-skill heterogeneity, mandatory break scheduling with midpoint control, acuity-weighted workload equity, sub-shift granularity, inter-week stability, and cross-midnight shifts. This paper presents CP-SAT: a Constraint Programming formulation for multi-role, multi-skill healthcare scheduling. CP-SAT enforces 14 hard constraints guaranteeing zero regulatory violations, while optimizing 15 soft objectives via a unified weighted penalty function. Contributions include a shift-window decomposition enabling break scheduling with centrality control, acuity-weighted workload equity, multi-granularity resolution from 15 minutes to 1 day, inter-week stability, and grid-offset preprocessing mapping cross-midnight shifts into a single scheduling day without solver changes. CP-SAT is evaluated on 18 instances: five synthetic hospital units (10-33 nurses), 10 INRC-II benchmarks (5-80 nurses, up to 8-week horizons) and 3 NRP-23 compatible instances (10-25 nurses) with cross-midnight Night shifts. Results: zero hard-constraint violations across all 18 instances by construction; proven optimality on INRC-II n005w4 (objective 118, gap 0.0%, 104s); feasible schedules scaling to 179,800 variables and 351,425 constraints (80 nurses); service quality improved 50-67% over MOGA; and model size scaling near-linearly at approximately 4,400 variables per employee. The formulation enforces 29 total constraints (14 hard, 15 soft), nearly three times the industry average.

We give a complete characterization of equivariant multi-head self-attention (MHSA): if an MHSA layer is equivariant to a symmetry group $G$, then $G$ can only act by permuting head-clusters, with QK and OV matrices satisfying an equivariance constraint tied to the group action. As a consequence, we prove that any fixed MHSA architecture that achieves exact equivariance by polynomially parameterizing unconstrained MHSA parameters inevitably leads to expressivity loss within the class of equivariant maps: the equivariance locus of unconstrained MHSA forms a union of extremely many Zariski-irreducible components in a reduced parameter space, and any single architecture covers at most one. For $G=D_4$ acting on $C$ copies of the regular representation as the token feature space, we show that there are $\Omega(C^{64})$ components for eight attention heads. We illustrate this with experiments on artificial tasks and show that the predicted head-cluster structures can arise naturally when the attention layer is pressured to become equivariant.

Free circle action on manifolds has been explored in several articles. Gonzalez and Velasco considered some free circle actions on the finite product of spheres. In this paper, we introduce generalized complex projective product spaces, extending their definition and the concept of Dold manifolds. This produces infinitely many different classes of new smooth manifolds. First, we study the integral cohomology rings and stable tangent bundles on certain generalized complex-projective product spaces. Then, we discuss the product inequality for strongly equivaraint TC. We exhibit the closeness of the lower and upper bounds for the LS-category and topological complexity of several classes of generalized complex projective product spaces. In many cases, we compute the exact value of the LS-category.

We study gradient estimates and volume growth on locally finite weighted graphs satisfying the $CD\psi(n,-K)$ condition with $K\geq0$. We establish a variational inequality for the heat semigroup and derive from it a family of Li-Yau type gradient estimates. Furthermore, under suitable assumptions on $\psi$, we establish a curvature dependent heat retention estimate for metric balls. Combined with a heat kernel Harnack inequality obtained from the established gradient estimate, this yields a curvature dependent exponential volume doubling estimate \begin{equation*} V(x,2r)\leq C e^{c\sqrt{K}r}V(x,r). \end{equation*} When $K=0$, the result reduces to a uniform volume doubling and further implies that the bottom of the spectrum of $-\Delta$ vanishes on infinite graphs.

Let $FG$ be the group algebra of a finite $p$-group $G$ over a finite field $F$ of characteristic $p$, where $p$ is an odd prime. Let $*$ be the classical involution of $FG$ and let $FG^-$ be the Lie subalgebra of the skew symmetric elements with respect to $*$. In this paper, we prove that the Lie nilpotency index of $FG$ is determined by its Lie subalgebra $FG^-$, in the case when $G$ is a $p$-group with a commutator subgroup of order $p$. The structure of the terms of the lower Lie central series of $FG^-$ has also been described.

Jaramillo-Velez recently introduced edge codes, a new class of toric evaluation codes constructed from the edges of a (hyper)graph $\mathcal{H}$. In the case that $\mathcal{H}$ is a tree, Jaramillo-Velez computed both the minimum distance and the weight distribution of the associated code. In this paper, we study edge codes associated to unicyclic graphs. Our most striking result is that computing the parameters of these codes is subtle in the case that the induced cycle has an even length because these values will depend on certain conditions regarding the length of the cycle and the size of the base field.

For every quadratic number field $K$, we prove a uniform $3$-unknown Diophantine definition of integer tuples in $\mathcal{O}_K$, allowing finitely many polynomial nonvanishing conditions. This yields an effective $+3$ transfer principle and a $13$-unknown representation of every recursively enumerable integer relation. Consequently, there exists an absolute degree bound $D_0\geq1$ such that for every quadratic number field $K$, there is no algorithm that, given \[ P(Y_1,\ldots,Y_{13})\in\mathbb{Z}[Y_1,\ldots,Y_{13}],\quad \mathrm{deg}\ P\leq D_0, \] decides whether $P=0$ has a solution in $\mathcal{O}_K^{13}$. The arithmetic input is a fourth-order Pell–Lucas congruence. It is a specialization of the norm-one Lucas multiplication formula, which yields exact valuations for the deviation of a Lucas quotient from its linear term, together with deviation criteria for Lucas–Wieferich and Wall–Sun–Sun primes. We also establish local surjectivity and $\ell$-adic density for second-order correction terms for norm-one Lucas sequences.

Popular adaptive stochastic gradient descent (SGD) methods to train artificial intelligence (AI) systems include the RMSprop, the Adam, and the AdamW optimizers, where the adaptivity parts in Adam and AdamW basically just coincide with RMSprop. Such adaptive methods involve several hyperparameters including the regularization parameter $\epsilon$ (which ensures that one does not divide by 0 and is often chosen to be very close to zero such as $10^{-8}$ in PyTorch by default) and the second moment decay parameter $\beta$ (which is often chosen to be very close to $1$ such as 0.99 (RMSprop) and 0.999 (Adam and AdamW) in PyTorch by default). Despite the high relevance of such methods, it remains an open research problem to provide error estimates for such methods with the error constants being not exploding but uniformly bounded with the respect to the hyperparameters, even in the situation of convex stochastic optimization problems. It is the key contribution of this work to essentially solve this problem for RMSprop. Specifically, we bound the expectation of the stopped evaluation of the objective function at the RMSprop process from above by the sum of an initialization term that decays exponentially in the training time, a stochastic approximation remainder of order $\gamma_n$, and a memory error of order $( 1 - \beta)^2$ with the error constants being uniformly controlled over all admissible choices of the step sizes, the second moment decay parameter $\beta$ and the regularization parameter $\epsilon\in[0,1]$ (also covering $\epsilon=0$). Our non-asymptotic error estimates hold not just for all sufficiently large n but hold for every gradient step $n=1,2,3,...$ with all error constants being explicitly specified. The key innovative new feature in the proof of our analysis are suitable inverse moment bounds for the second moment process in RMSprop.

Given Banach algebras with bounded approximate identities that are Morita equivalent in the sense of Paravicini, we give explicit formulas for the induced correspondence of closed ideals, characterize it via localized Morita submodules, determine when corresponding ideals are Morita equivalent via restricted bimodules, and show that corresponding quotients are always Morita equivalent. We prove preservation of geometric cores associated to compatible dense algebraic cores. For equivalent étale groupoids with paracompact unit spaces, Morita induction preserves dynamical ideals and inner supports. For the Morita equivalence between the $\ell^p$ uniform Roe algebra and the $\ell^p$ uniform algebra, we prove that the latter has a bounded approximate identity of projections, identify its geometric ideals explicitly, and identify corresponding complete-lattice block decompositions of the full ideal lattices with matching extremal ideals. Under property A, or for $p=1$, these blocks collapse; for $p=2$, collapse of the minimal block characterizes property A.

We study null-space estimation from a noisy matrix. For a simple left null space, we first derive an exact compact expression for the error of the smallest left singular vector. We then give an all-order series for the SVD vector and projector, followed by compact and consistently truncated series forms for the fixed-realization empirical risk and conditional population generalization risk. The recursion extends to a multiple-dimensional null space by following the complete invariant subspace. The convergence radius is not inferred from an error plot: it is computed independently from the nearest complex exceptional point that joins a retained eigenvalue branch to its complement. A reduced-nullity experiment shows that moving this spectral boundary can increase the radius, although the improvement is not monotone in the retained nullity. For individually ordered null directions under Gaussian training with \(\tau\geq m\), we prove that the Wishart splitting matrix \(W\) gives a strict second-order empirical ranking. Gaussian averaging equalizes the leading generalization risks at both small and very large noise, while a column-swap theorem proves strict expected generalization ranking for an isotropic signal subspace. For unequal spikes, an exact population-overlap criterion and a simultaneous \(99\%\) Monte Carlo confidence certificate explain the observed intermediate ranking. A sixth-order risk correction improves the lower-crossover estimate in the reported experiment. This equal–ranked–equal phenomenon is a finite-sample diagnostic related to spectral mixing, but its tolerance crossings, the exceptional-point radius, and the asymptotic BBP threshold are three distinct quantities.

Fix the spectrum of a bipartite density matrix and randomize its eigenbasis according to Haar measure. We study the probability distributions induced on the spectra of the two marginal states. For arbitrary subsystem dimensions $m$ and $n$, the joint characteristic function of the reduced density matrices is expressed as a Harish-Chandra-Itzykson-Zuber integral whose external eigenvalues are the pairwise sums $x_i + y_j$. Repeated external eigenvalues are handled by confluent determinant limits. In the two-qubit case, we derive an explicit alternating-spline formula for the joint density of the two marginal Bloch radii. Its support is the Bravyi-Klyachko compatibility region. We also obtain a compact truncated-power formula for the Bloch-radius density of either individual qubit marginal. In the qubit-qutrit case, we derive a truncated-power formula for the qubit Bloch-radius density and a bivariate spline formula for the joint density of the largest and smallest eigenvalues of the qutrit marginal. The latter two variables determine the full qutrit spectrum because the trace is fixed. The derivations combine confluent HCIZ integrals, distributional Fourier inversion, orbital measures, and the SU(2) and SU(3) derivative principles. The resulting densities are piece-wise polynomial on chambers determined by subset sums of the fixed global eigenvalues, in agreement with the Duistermaat-Heckman description of projected coadjoint-orbit measures.

For finite metric spaces, magnitude is an invariant known to be discontinuous everywhere with respect to the Gromov-Hausdorff distance. While much of the literature focuses on positive definite metric spaces, here we consider finite metric spaces admitting a nonnegative weighting. In this paper, we focus on the class of finite metric spaces that admit a nonnegative weighting. This class includes ultrametric spaces, for which specific bounds on magnitude are known. We generalize these bounds by proving that, for any two finite metric spaces with nonnegative weightings, the ratio of their magnitudes is exponentially bounded by their Gromov-Hausdorff distance. Consequently, we show that within this class, the logarithmic magnitude function is Lipschitz continuous, and the magnitude function itself is locally Lipschitz continuous, but not Lipschitz continuous.

This article studies mean convergence of Banach space-valued random elements indexed in a family of finite measure spaces. We derive $L^p$-convergence theorems under (compact) uniform integrability in two regimes: a decaying-index-mass regime and a bounded-index-mass regime, the latter requiring a new dependence structure which is called diagonal negative dependence for the random elements and expressed via the self-product of the index measure. We provide examples showing that the conditions to obtain the results are sharp and strictly weaker than related conditions in the literature. As a further illustration for the index measure space framework, a functional law of large numbers in $L^p$ on the space of continuous functions is derived, where the random elements are solutions of stochastic differential equations driven by Brownian motions extracted from a common Brownian sheet.

We study the forbidden poset analog of the maximal anti-Ramsey problem introduced for graphs by Burr, Erd\H os, Graham, and Sós. For integers $m\le 2^n$ and poset $P=(P,\preceq)$, we introduce $\mathrm{ar_m}(n,m,P)$ (and $\mathrm{ar^*_m}(n,m,P)$) to denote the minimum integer $k$ such that there exists a family $\mathcal{F}\subseteq 2^{[n]}$ with $|\mathcal{F}|=m$ and a coloring $\mathcal{F}\rightarrow [k]$ with all weak (strong) copies of $P$ being rainbow. As long as there exist $P$-free families of size $m$, these parameters equal 1. It is known that the largest size $La(n,P)$ ($La^*(n,P)$) of weak (strong) $P$-free families has order of magnitude $\Theta(\binom{n}{\lfloor n/2\rfloor})$ unless $P$ is the antichain $A_k$ on $k$ elements. In this paper we study $\mathrm{ar_m}(n,m,P)$ and $\mathrm{ar^*_m}(n,m,P)$ in two regimes of $m$. We determine the asymptotics of these parameters for all posets $P$ when $m=2^n$. We also consider the case $m=\Theta(\binom{n}{\lfloor n/2\rfloor})$. It is shown that for any connected poset $P$ and integer $k$, there exist integers $m_{P,k}$ and $m^*_{P,k}$ such that to color the middle $k$ layers of the Boolean lattice with all weak or strong copies of $P$ being rainbow, one needs $\Theta(n^{m_{P,k}})$ or $\Theta(n^{m^*_{P,k}})$ colors. For tree posets $T$, one has $m_{T,k}=m^*_{T,k}$. We conjecture that for any tree poset $T$, and positive real $\varepsilon$, $\mathrm{ar_m}(n,m,T),\mathrm{ar^*_m}(n,m,T)=\Omega(n^{m_{T,k}})$ holds provided $m\ge (k-1+\varepsilon)\binom{n}{\lfloor n/2\rfloor}$. We prove our conjecture on $\mathrm{ar_m}(n,m,T)$ for an infinite class of tree posets.

Smith's conjecture asserts that in every $k$-connected graph with $k\geq 2$, any two longest cycles intersect in at least $k$ vertices. In this work, we establish an $\Omega(k^{8/11})$ bound for this conjecture, improving upon the $\Omega(k^{2/3})$ bound of Ma and Zhao. Our proof combines a Ramsey theoretic refinement of the traditional Turán-type approach with computer search.

Using window theory, Bodzenta–Donovan showed that the derived category of a root stack $\sqrt[n]{X/D}$ has a $2n$-periodic $2$-term semiorthogonal decomposition. We define a categorical generalization of the root stack construction and interpret this as a pullback on the B-side of the 3d mirror symmetry equivalence of Gammage–Hilburn–Mazel-Gee. We analyze this pullback using categorical representation theoretic results of Ben-Zvi–Francis–Nadler and Ben-Zvi–Nadler–Preygel applied to SODs. We also show that the pullback is equivalent to a pushforward of perverse schobers on the A-side, allowing us to deduce periodicity from a simple decomposition of an A-side Lagrangian skeleton. Additionally, we adapt the construction of Bodzenta–Donovan to a new GIT problem, which yields an embedding of Coh($\sqrt[m]{X/D}$) into Coh($\sqrt[n]{X/D}$) for $m<n$ coprime, and prove $2n$-periodicity of the resulting $2$-term SOD.

Perfect state transfer on graphs has attracted extensive attention due to its application in quantum information and quantum computation. Explicit characterizations of connection sets admitting perfect state transfer in Cayley graphs are rare and, so far, are known only for a few abelian Cayley graphs. In this paper, we characterize the conjugation-closed connection sets of connected Cayley graphs over dihedral groups that admit perfect state transfer. By applying Ramanujan sums, Möbius inversion, and arguments based on the $p$-adic exponential valuation of rational numbers, we convert the eigenvalue constraints imposed by perfect state transfer into explicit structural conditions on the connection set. This yields a complete and practical characterization, which gives an effective criterion for recognizing and constructing such Cayley graphs and also determines the exact minimum perfect state transfer time.

In this paper, we study how the classification complexity of natural orbit equivalence relations changes when the full homeomorphism group of a compact metrizable space is replaced by a dense non-closed subgroup. For a compact space $X$ and a subgroup $G \leq \mathcal{H}(X)$, we consider three canonical actions: the left shift action on $\mathcal{H}(X)$, the induced hyperspace action on $\mathcal{F}(X)$, and the conjugation action on $G$ We first analyze subgroups of the group $\mathcal{H}^+([0,1])$ of increasing interval homeomorphisms, focusing on bi-Lipschitz homeomorphisms, diffeomorphisms, and bi-absolutely continuous homeomorphisms. We show that, in contrast to the behavior of closed subgroups, passing to these subgroups strictly increases the complexities of the associated classification problems or makes them incomparable with the corresponding full-group relations. In the second part, we investigate hyperspace actions of bi-absolutely continuous homeomorphisms on the Cantor space and the Hilbert cube with respect to some Borel probability measure and show that a similar behavior occurs on these spaces as well.