Papers

We study the Gray-Hervella classes of invariant almost Hermitian structures on the maximal real flag manifold of type $\mathcal{A}_3$. We first obtain an explicit parametrization of all invariant almost Hermitian structures and establish necessary and sufficient conditions for an invariant almost Hermitian structure to belong to each Gray-Hervella class. For each nonempty class, we give an explicit parametrization of the corresponding set of structures, describe it as a smooth manifold and determine its dimension.

We present a new approach to lower bounds on Borel rank via a connection between Borel sets and infinite circuits. We prove an infinite analog to Håstad's switching lemma called the restriction lemma, which shows that by fixing the values of some inputs, we can simultaneously reduce the complexity of one Borel function while largely maintaining the complexity of another. This result provides a purely combinatorial proof of the Borel hierarchy theorem, as well as simple proofs of various Ramsey-like properties of Borel sets and functions. We prove that the restriction lemma is sharp and also discuss counterexamples demonstrating the limitations of the restriction technique, in the context of both Borel functions and $\mathsf{AC}^0$ circuit families.

This paper analyzes the ergodic rate of a wideband reconfigurable intelligent surface (RIS)-assisted terahertz link under beam squint and finite-resolution phase control. Independent Rayleigh fading on the two RIS-assisted hops produces double-Rayleigh cascaded amplitudes. Exact second- and fourth-order channel moments are derived by incorporating the frequency-dependent array response and phase-quantization errors. Moment matching then provides a Gamma approximation for the received power on each subcarrier and a tractable closed-form approximation for the wideband ergodic rate. A first-null bandwidth approximation characterizes the aperture–bandwidth limitation of frequency-flat RIS control. Monte Carlo simulations validate the analytical framework.

We study finite polyhedral models inside the projectivized space of equivalent norms $\mathcal{N}'(X)$ on a finite-dimensional Banach space $X$, endowed with the logarithmic distortion metric. Given a finite symmetric direction set $E$, we introduce the class of $\alpha E$-norms, defined as Minkowski functionals of symmetric polytopes whose vertices lie on the rays prescribed by $E$. We identify the admissible weights $\alpha$ for which this parametrization is non-redundant and prove that they give a one-to-one parametrization of the corresponding finite direction model $\mathcal{N}(E)$. We then refine this description by introducing complete, symmetric, simplicial $E$-fans, which encode the conical regions on which the associated polyhedral norms are linear. For a fixed fan $\mathcal{F}$, we define the corresponding geometric and coordinate fan models $\mathcal{N}(\mathcal{F})$ and $\mathcal{R}(\mathcal{F})$, and show that they are naturally isomorphic as cones. After quotienting by positive scalar multiplication, the coordinate models become isometric to the corresponding norm models: the logarithmic distortion metric on $\mathcal{N}'(E)$ and $\mathcal{N}'(\mathcal{F})$ is represented exactly by the Hilbert projective metric on the admissible weight spaces. Finally, we prove completeness results for these finite models and show how they provide finite-dimensional polyhedral approximations to the metric geometry of $\mathcal{N}'(X)$.

Squashed entanglement is an additive bipartite entanglement measure. For a state $\rho_{AB}$, it is defined as the infimum of all conditional mutual information $I(A:B\mid E)$ evaluated on extensions $\rho_{ABE}$. It was unresolved whether there is a bound on the dimension of the conditioning system, $E$, needed for this optimization. We show that no such cardinality bound is possible. Explicitly, we find that the squashed entanglement for a partially dephased Bell pair on two qubits cannot be achieved for any finite dimensional extension $E$. A crucial ingredient of this proof is a direct sum step, which in a sense, symmetrizes two purifying systems while keeping the conditional mutual information unchanged and the coherence no worse.

Coven and Meyerowitz gave two cyclotomic conditions, (T1) and (T2), which characterize integer tiles whose cardinalities have at most two distinct prime factors. We prove that the same characterization holds without this restriction. The proof uses a reduction in Chinese remainder coordinates: slicing by the lowest base-p digit produces sets with a common tiling complement in a group of order smaller by a factor of p. This reduction preserves the cyclotomic data needed for an induction on the exponents in (T2).

In this paper, we show that for every p>2, there is an orthonormal basis of spherical harmonics on Sn with n>=2 which achieves the maximal Lp norm growth. More precisely, one basis achieves this growth for all p<=2(n+1)/(n-1), and another for all p>=2(n+1)/(n-1). This extends the results in Han (arXiv:1404.5016) to all dimensions and all p>2, and improves the positive density subsequence there to a whole basis.

This paper is concerned with the study on Legendrian submanifolds with conformal Maslov form in the unit sphere $\mathbb{S}^{2n+1}$, which admits a Sasakian structure $(\varphi,\xi,\eta,g)$ for $n\ge2$. As the main result, we classify such submanifolds with constant sectional curvature, motivated by the classification result of the minimal Legendrian submanifolds with constant sectional curvature. Moreover, we prove that, for a closed Legendrian submanifold $M^n$ in $\mathbb{S}^{2n+1}$ with conformal Maslov form, if its sectional curvature satisfies the pinching $0\leq\sec_g\leq1$, then either $M^n$ is the totally geodesic Legendrian sphere with $\sec_g=1$, or $M^n$ is a closed embedded weighted Clifford torus with $\sec_g=0$. This extends the corresponding pinching theorem of Dillen–Vrancken (J Math Pures Appl 69:85–93 1990) from minimal Legendrian submanifolds to the conformal Maslov class under the same curvature bounds.

Simultaneous wireless information and power transfer (SWIPT) is a critical technology for the future of the Internet of Things (IoT). However, ensuring a stable power supply in such networks remains a significant challenge. This work introduces dynamic polarization control as an additional degree of freedom (DoF) in SWIPT systems. We propose a system where both the base station (BS) and the users can adjust their antenna polarization, a technique known as polarforming. In addition, each user device is capable of splitting the incident signal to perform simultaneous information decoding (ID) and energy harvesting (EH). The resulting non-convex optimization, with many coupled variables, is solved using a graph neural network (GNN) that learns the sub-optimal beamforming, polarization, and power-splitting variables. Simulation results demonstrate that the proposed GNN-based dynamic polarforming optimization significantly outperforms fixed-polarization schemes, particularly under imperfect channel state information (CSI). Moreover, joint polarforming and GNN-based optimization maintain robust SWIPT performance under both polarization mismatch and imperfect CSI.

We present a general study of extending structures in the linear framework of Universal Algebra, which unifies many existing results and contains applications to various Poisson type algebras including transposed Poisson algebras.

This paper studies the simultaneous recovery of point-source locations and temporal strengths in both the drift and diffusion terms of a stochastic parabolic equation with Neumann boundary conditions, using only boundary observations on a nonempty portion of the boundary after the sources have become inactive. By separating the stochastic equation into its expectation and centered parts, the drift channel reduces to a deterministic source problem with scalar Laplace moments, while the diffusion channel is controlled by Itô's isometry, which preserves the full $L^2$-energy of the integrand. Under suitable nondegeneracy and separation assumptions, we establish Lipschitz stability for the location of a single source in either channel, logarithmic stability for the drift strength, and Lipschitz stability for the complete diffusion strength including its sign. For multiple sources, we obtain Lipschitz stability up to permutation for drift locations and weighted temporal moments in dimensions two and three, and for diffusion locations and strengths in dimensions one through three. The proof employs explicit adjoint probes, including complex-isotropic polynomial annihilators in dimensions two and three and spectral synthesis in one dimension, combined with boundary-to-source estimates derived from Green's identities. These results extend and improve upon existing deterministic stability theories by revealing the fundamentally different temporal information carried by stochastic integrals, a distinction that is further illustrated by numerical experiments.

We show that the strong maximal operator on ${\mathbb R}^2$ does not satisfy a linear weighted $L^2$ estimate in terms of the rectangular $A_2$ characteristic. More precisely, we construct weights with arbitrarily large characteristic for which the operator norm is bounded below by $[w]_{A_2^{\mathrm{str}}}\sqrt{\log [w]_{A_2^{\mathrm{str}}}}$. The proof is elementary. It uses an explicit two-parameter mass recurrence on a product of geometric interval partitions, together with a direct comparison of arbitrary rectangular averages with anchored averages.

We analyze how an arbitrary observation mask on a finite periodic ring of equally spaced sensors affects the posterior covariance, Fourier-mode coupling, and normalized posterior trace in Gaussian-process reconstruction. Under a rotationally stationary prior and homogeneous independent measurement noise, complete observation gives independent scalar posterior formulas for the Fourier modes. For an arbitrary mask, however, the matrix $Q_M=F D_MF^\ast$ is generally non-diagonal; its off-diagonal entries are finite Fourier components of the realized mask and couple modes in the posterior precision. Consequently, masks with the same unavailable-channel fraction can have different normalized posterior traces because their geometries differ. A dimensionless 64-channel synthetic benchmark illustrates this finite-matrix effect. A circumferential array of equally spaced wall-mounted microphones at a fixed axial station of a circular fan or compressor duct provides one concrete mechanical-engineering interpretation: failed, saturated, corrupted, or dropped-out channels form the observation mask. The analysis is a finite-dimensional reference calculation under the stated rotational-stationarity and common-noise assumptions, not a performance claim for a nonuniform or unequally instrumented operating duct.

This paper investigates polarization-aware transmission for multi-user multiple-input single-output (MU-MISO) short-packet ultra-reliable low-latency communications (URLLC) under imperfect channel state information (CSI). We consider a system in which the base station (BS) and users are equipped with polarization-reconfigurable antennas that enable adaptive polarization states through controllable polarization coefficients. A multi-objective optimization problem is formulated to jointly maximize the finite-blocklength (FBL) achievable sum rate and minimize the maximum decoding error probability (DEP), subject to transmit-power, latency, reliability, and discrete polarization-control constraints. The resulting multi-objective problem is scalarized using a normalized weighted-sum utility. To enable low-complexity online decision-making, a heterogeneous graph neural network (GNN) is developed to learn the joint mapping from estimated polarized CSI to digital beamforming and transmit/receive polarforming vectors (PFVs) while accounting for the system constraints. Numerical results demonstrate that the proposed GNN-based polarforming (PF) framework substantially improves the FBL sum rate while reducing the maximum DEP compared with conventional fixed-polarization schemes, particularly under channel depolarization and imperfect CSI. This highlights the potential of adaptive polarization control for reliable low-latency transmission.

Existing operation formulations for AC/DC distribution grid often oversimplify the interplay of AC/DC interconnections and VSC characteristics. Moreover, current distributed algorithms depend on gradient-based methods, which achieve only linear convergence rates and struggle to scale efficiently for large systems. This paper proposes a convex ALADIN variant that exploits second-order derivatives of convex subproblems, achieving locally quadratic convergence for distributed nonlinear but convex optimization problems. Then, this convex ALADIN is applied to a separable AC/DC OPF model for AC/DC hybrid distribution grid, explicitly incorporating complicated AC/DC interconnections and multiple droop-controlled VSCs while retaining computational traceability. To bridge theory with practice, we implement the distributed convex second-order optimization framework using the PROcess Orchestration Framework, a co-simulation platform that automates workflow design, model integration, and distributed execution via a user-friendly graphical interface, eliminating manual coding. Numerical results illustrate effectiveness of the proposed model.

In dimension at least two, a class of locally Lipschitz vector fields realizes every finite correspondence between distinct inputs and distinct targets at any prescribed positive time, provided that it is linear, its members generate global flows, and it approximates every smooth vector field of bounded support uniformly on bounded sets. The input and target sets may overlap. The proof constructs a smooth autonomous reference flow and finitely many localized correction fields, then uses uniform stability and Brouwer degree to obtain exact endpoints within the approximating class. The argument uses only uniform approximation of the vector fields; it does not require control of their derivatives. Applied to shallow ReLU vector fields, the result gives exact finite interpolation without time-dependent coefficients or additional state variables, together with bounds on width and normalized coefficient strength. In contrast, no continuous autonomous semiflow can exchange and compress two disjoint balls. Its time maps therefore fail to approximate all continuous maps uniformly on compact sets. The results distinguish exact interpolation at finitely many points from uniform control of neighborhoods.

In this paper we study local uniform, with respect to time, asymptotic expressions for the asymptotic behavior of the entrance probability of discounted aggregate claims to some rare sets, in a multivariate risk model with arbitrarily dependent insurance and financial risks. Our model is based on a multivariate version of a model, introduced by Guo (2022), and we consider that the logarithmic return process of the insurers investment portfolio is described by a jump-diffusion process. The dependence between the insurance and financial risks is implied by the dependence of the claim-vectors with the jumps of returns, and is arbitrary under some distributional conditions on the claim-vectors and the discounted claim-vectors. In opposite to previous papers on this topic, except the multidimensional extension, we consider that the model is driven by a common counting process, that is not necessarily renewal, the claim vectors are interdependent, and their distribution is not restricted to the class of (multivariate) regularly varying distributions. Under the condition that the claim vectors, and the discounted claim-vectors follows distributions from the class of multivariate consistently varying and positively decreasing distributions, and under some moment conditions on the jumps and the counting process, our main result shows the presence of multivariate linear single big jump principle of the discounted aggregate claims in this not necessarily Levy-Renewal environment. After restriction of the distributions to multivariate regular variation, we obtain more explicit expressions, under a slightly weaker moment condition on the jumps. We provide a corollary, in which the conditions of the main result are satisfied under a weak dependence structure and we find a more explicit asymptotic expression, using the technique of solution of the dependence.

We study the diffusion and reaction effects of Bartnik's quasi-spherical equation to deform boundary mean curvature in fill-in problems with scalar curvature bounded below. The diffusion effect yields an explicit $L^p$-to-$L^\infty$ estimate for $\partial_tu=u^2\Delta u$, thereby extending the known upper bound for the minimum boundary mean curvature of fill-ins to a quantitative upper bound for its harmonic mean. For spin fill-ins, this bound is explicit and involves only coarse intrinsic boundary data. By introducing an absorbing reaction term, we also construct a deformation that transforms any nonnegative initial mean curvature into a terminal mean curvature with a uniform positive lower bound. For Gromov's conjecture on total mean curvature, this reduces the $H\geq 0$ case to Theorem A of Frenck, Hanke, and Hirsch \cite{FHH}, which assumes $H\geq\kappa>0$. This covers the case of non-spin boundaries, complementing their result for spin boundaries (Theorem B) for this conjecture.

We introduce and study a Yamabe-type variational problem associated with the curl operator on middle-degree forms on a closed oriented Riemannian manifold $(M^n,g)$ with $n\equiv3\pmod4$. The corresponding curl–Yamabe constant is defined by minimizing a conformally invariant, gauge-invariant quotient built from $\|{\rm curl}\,\alpha\|_{\frac{2n}{n+1}}$ and $\int_M\langle{\rm curl}\,\alpha,\alpha\rangle$, and its Euler–Lagrange equation is the critical curl–Yamabe equation ${\rm curl}\,\alpha=\frac{n+1}{2}|\alpha|^{\frac{2}{n-1}}\alpha$. Using sharp curl–Sobolev inequalities on the sphere and an Aubin-type cut-off construction, we prove that $0<Y_{\rm curl}(M,[g])\le Y_{\rm curl}(\mathbb{S}^n)$ for every such manifold, and that the infimum is attained whenever $Y_{\rm curl}(M,[g])<Y_{\rm curl}(\mathbb{S}^n)$. Moreover, for $n>3$ we establish the strict inequality for manifolds that are not locally conformally flat, via a test-form expansion detecting the Weyl tensor, and deduce existence of a minimizer solving the curl–Yamabe equation.

We study overdetermined equations on the hyperbolic and de Sitter planes by means of the support-function representation of zero mean curvature surfaces in Lorentz–Minkowski three-space. On the hyperbolic plane, solutions of $\Delta u-2u=0$ give rise to branched spacelike maximal surfaces. We prove that, on a bounded simply connected domain, nontrivial constant Dirichlet and Neumann data force the domain to be a geodesic disk, provided that the associated support quadric is nonlightlike; moreover, the solution is a multiple of the hyperbolic cosine of the distance from the center. On the de Sitter plane, the corresponding equation is the Klein–Gordon equation $\Box u+2u=0$, and its solutions generate timelike minimal surfaces wherever the support tensor is nondegenerate. We identify the support quadric and the constant-angle condition determined by constant Cauchy data, prove that local rigidity fails along every analytic noncharacteristic curve, and recover rotational symmetry from constant data on a global Cauchy circle. Thus the same support-function formalism reveals a sharp transition from elliptic rigidity to hyperbolic flexibility when the signature changes.

We establish the global well-posedness of weak solutions to the Cauchy problem for a broad class of multi-dimensional $n\times n$ Keyfitz-Kranzer type systems with homogeneous flux. Our proof relies on a novel viscous approximation that yields the necessary compactness estimates. By exploiting the specific structure of this proposed approximation, we derive a priori uniform $L^{\infty}$ bounds and prove the $L^1_{loc}$ precompactness of the sequence of approximate solutions. This framework allows us to rigorously justify the vanishing viscosity limit. Furthermore, under suitable assumptions on the initial data, we employ the relative entropy method to analyze the $L^2$-time decay of large perturbations of the viscous shock, up to a dynamical shift.Evan

We prove that every separable infinite-dimensional Fréchet algebra $X$ admits a continuous linear operator $T$ supporting a dense invariant hypercyclic algebra, giving an affirmative answer to a question of Bayart, Costa Jr. and Papathanasiou. In fact, every dense countable-dimensional subalgebra $\mathcal A$ of $X$ can be prescribed as an invariant hypercyclic algebra. When $X$ admits a continuous norm, the operator can additionally be chosen in the form $T=I+K$, where $K$ is nuclear, so that $\mathcal A = \operatorname{span} \operatorname{Orb}(a,T)$ for any prescribed $a\in\mathcal A\backslash\{0\}$.

A matrix cascade is an iterative refinement process in which vector-valued data are repeatedly filtered by matrix-valued masks across successively finer scales. Such cascades arise in multichannel subdivision, Hermite refinement, multiresolution synthesis, and exponential-spline constructions, and in nonstationary schemes the masks may vary with scale. We develop a chronological stability framework for this setting, allowing both the transition operators and the associated matching spaces to change from level to level. If the chronological radius satisfies $\rho_{\rm ch}<1$ and the matching defects are summable, then the cascade converges uniformly and, for every $q\in(\rho_{\rm ch},1)$, \[ \|F_n-\Phi\|_\infty \lesssim q^n+\sum_{j=1}^n\delta_j q^{\,n-j} +\sum_{j>n}\delta_j . \] The time-weighted defect term records the level at which each perturbation is inserted, and the matching spaces need not converge; they may even alternate indefinitely. For compact piecewise-linear seeds, every finite ordered matrix cascade admits an exact fixed-width ReLU realization with $O(n)$ depth and parameters, and this linear depth order is worst-case optimal at fixed width. Consequently, exponentially decaying defects yield certified approximants of size $O(\log\varepsilon^{-1})$. We further construct stable two-channel classes in which the chronological radius and the leading metric-entropy coefficient are independent invariants, showing that stability does not determine finite-accuracy description complexity. The framework therefore provides tolerance-controlled finite representations for nonstationary multiresolution and exponential-spline refinement.

We prove that for every dimension $n \geq 2$ and parameter $p \in ]1,n[$, the hedgehog $u_0(x) = x/|x|$ is the unique minimiser of the $p$-energy $\mathcal{E}_p [u]:=\int_{\mathbf{B}^n}|\nabla u|^p\,{\rm d}x$ in the class of $W^{1,p}$-mappings from the $n$-dimensional unit ball $\mathbf{B}^n$ to the $(n-1)$-dimensional unit sphere $\mathbf{S}^{n-1}=\partial\mathbf{B}^n$ whose trace equal to the identity on $\mathbf{S}^{n-1}$. This answers in the affirmative an open question on p.29 in [Robert M. Hardt, Singularities of harmonic maps, Bull. Amer. Math. Soc. (N.S.) 34 (1997), 15–34].

We study iHopf algebras associated with quantum supergroups of basic type. For a quantum supersymmetric pair $(\mathbf{U}, \mathbf{U}^\imath)$, we realize $\mathbf{U}^\imath$ and $\mathbf{U}$ as the iHopf algebras of the Borel quantum group ${\hat{\mathbf{U}}}_q^{\geq0}$ and the tensor product algebra ${\hat{\mathbf{U}}}_q^{\geq0}\otimes {\hat{\mathbf{U}}}_q^{\geq0}$, respectively, while the coideal subalgebra structure is encoded by an embedding between the iHopf algebras. Moreover, we derive an explicit conversion formula between the iHopf multiplication and the original multiplication, which provides a general mechanism transforming the defining relations of quantum (super)groups into relations of iquantum (super)groups. In particular, all the Serre relations that occur in quasi-split iquantum supergroups of basic type are characterized.

Consider the linear form $L(x,y,z,w) = 3x+y-2z-2w$. For a positive integer $N$, denote by $r_L(N)$ the largest size of a subset of $\{1,2,\dots,N\}$ that avoids nontrivial solutions to $L = 0$. We show that $r_L(N) = \Omega(N^{0.5608687})$, improving the lower bound for Problem 16 in Green's list of open problems. Our proof uses the method of combinatorial degenerations to turn a finite certificate into large solution-free sets. In fact, we can improve Ruzsa's lower bound of $N^{1/2-o(1)}$ for many four-variable equations. Consider $L(x,y,z,w) = ax+by-cz-dw$ with $a,b,c,d\in \mathbb Z_{>0}$, $a+b=c+d$, $\{a,b\} \neq \{c,d\}$ and $abcd$ not a square. We show that there is $\varepsilon_L > 0$ such that $r_L(N) = \Omega_L(N^{1/2 + \varepsilon_L})$. Furthermore, we show that for primitive translation-invariant linear forms $L$ in $s$ variables, the lower bound $\Omega_s(N^{1/(s-1)})$ coming from a greedy construction is never optimal.

We construct a product spectral decomposition from power-normalised parabolic-cylinder functions. At noninteger real orders, these functions are not square integrable with respect to the corresponding full-line Gaussian measure. Bringing the admissible half-line branches into the same Gaussian space and matching them at the origin gives a corrected orthonormal eigenbasis. A unitary transformation to a weighted probability space makes the eigenfunction corresponding to the lowest eigenvalue constant and permits consistent countable products. The resulting law is non-Gaussian under deformation, and the decomposition recovers classical Wiener chaos when the deformation is removed. We identify the associated closed gradient, adjoint divergence and deformed number operator. The Rodrigues formula covers all orders occurring in the spectral branches, including negative orders. Under deformation, however, changing both branch orders by the same nonzero amount breaks the required relation between them. Fractional evolution defined through spectral calculus preserves the basis. After unitary identification, the one-coordinate fractional evolution operators converge in operator norm to Gaussian and three-dimensional radial Ornstein–Uhlenbeck fractional evolutions at the respective parameter endpoints. Convergence is uniform on compact time intervals away from zero. An application example with inverse stable clock is provided in the accompanying github repository.

In this paper, we settle a conjecture of Polujan and Pott by constructing an explicit, infinite family of Maiorana–McFarland bent functions $f_t$ in $2(2^t-1)$ variables with algebraic degree $\deg(f_t) = t + 1$ for any integer $t \ge 2$. Our construction builds upon a minimal commutative algebra $I_t$, which naturally induces a triangular set-partition polynomial permutation $P_t$. By identifying an elementary abelian subgroup within the direct sum $ I_t \oplus I_t^*$, we establish an explicit nonlinear coordinate transformation that pulls $f_t$ back to a canonical quadratic form. This linearizes the translation development $\operatorname{Dev}(D_{f_t})$ under an exotic group structure and proves that it is isomorphic to the classical symplectic design $S^\pm(2(2^t-1))$, thereby fully resolving the conjecture.

We classify the modular curves $X_0(N)$ that admit a smooth embedding in $\mathbb P^1 \times \mathbb P^1$ and obtain a partial classification for the Shimura curves $X_0^D(N)$. We also prove that, for $D>1$, a Shimura curve $X_0^D(N)$ admits a smooth plane model if and only if its genus is at most one.

Orbit-span and dimension problems for designs have been studied in several classical association schemes using a variety of methods. Recently, through a detailed analysis of total trades, Ghorbani et al. showed that the orbit of a fixed combinatorial design asymptotically attains the full dimension permitted by the design equations. Analogous dimension-saturation results for the global spans of index-one designs in the bilinear forms and Grassmann schemes were obtained via laborious eigenvalue computations. In this paper, we work with the top fiber $X$ of a graded poset carrying a compatible transitive action of a finite group $G$, and assume that the induced Schurian association scheme on $X$ is commutative. For the multiplicity-free decomposition $\mathbb{C}^{X}=U_{0}\oplus U_{1}\oplus\cdots\oplus U_{m}$, we prove that, under explicit spectral and quantitative conditions, the $G$-orbit of the characteristic vector of any $t$-design spans the maximal submodule $U_{0}\oplus U_{t+1}\oplus U_{t+2}\oplus\cdots\oplus U_{m}$ allowed by the $t$-design definition. We verify these conditions asymptotically for the Hamming, bilinear forms, Johnson, and Grassmann schemes. This recovers the fixed-orbit saturation theorem of Ghorbani et al. for combinatorial designs, gives a new result for orthogonal arrays, and asymptotically extends the previous bilinear forms and Grassmann results to orbit spans of individual designs of arbitrary fixed index.