Papers

We introduce polynomial permutation stability, extending classical permutation stability from homomorphisms to polynomial maps of groups. The universal polynomial group $\operatorname{Pol}_s(G)$ provides a natural framework for this theory, and we prove that polynomial stability of degree $s$ is equivalent to permutation stability of $\operatorname{Pol}_s(G)$. This yields, in particular, polynomial stability of all finite groups in degree two. We develop a corresponding theory of polynomial sofic approximations and show that, for countable groups, it does not give a new notion of soficity. Nevertheless, the associated universal polynomial groups contain new structural information: in degree two they embed into a wreath product, which allows us to characterize their soficity in terms of the original group. We also formulate and study weak polynomial stability through the weak stability theory of universal polynomial groups. For finite groups, we investigate the structure of higher-degree universal polynomial groups. We obtain general obstructions to amenability and prove largeness results for a family of finite groups with perfect derived subgroup and abelianization of order two.

The return probability in the simple symmetric random walk on the 4-dimensional lattice $\mathbb{Z}^4$ is evaluated in closed form in terms of the generalized hypergeometric function.

Hepatitis B (HBV) is a potentially fatal liver infection that remains a primary widespread disease around the globe, despite the availability of vaccinations, owing to ongoing obstacles such as vertical transmission and delayed treatment. In the present study, a compartmental mathematical model that accounts for the effects of vertical transmission and harmonic mean type incidence is put forward for the analysis of the transmission dynamics of HBV. The model's threshold behavior is investigated through the basic reproduction number $R_0$, and the qualitative dynamics around the disease-free and endemic equilibria are examined to establish conditions for their asymptotic local and global stability. To identify key epidemiological drivers, the global sensitivity analysis of the system is performed through the incorporation of Latin Hypercube Sampling (LHS) and Partial Rank Correlation Coefficient (PRCC) methodologies. We formulate the optimal control problem using Pontryagin's maximum principle. Effective strategies for diminishing the prevalence of HBV are emphasized through numerical simulations. Numerical simulations demonstrated that optimized control measures play a significant role in reducing the number of infected individuals and increasing the recovered population. In addition, we have characterized three distinct strategies for reducing infection and observed that each has limitations in eliminating infections. However, the simultaneous administration of treatment and vaccination is useful and effective for the reduction of HBV infection.

For a stochastic partial differential equation posed on a Gelfand triple and satisfying the fully local monotone conditions of Röckner, Shang and Zhang, we compute the logarithmic derivative of the law of the solution at a fixed time along a prescribed direction of the state space. The formula is intrinsic, being expressed through the Hilbert-Schmidt Malliavin derivative $\Phi_r = \mathcal{D}_r X(t)$ and the covariance $\gamma_t = \int_0^t \Phi_r \Phi_r^{*} \,\mathrm{d}r$ alone, so that neither the inversion of the first variation used in finite dimensions nor uniform Malliavin-Sobolev bounds on the Tikhonov family are called upon. It is obtained from an integration-by-parts identity on an abstract Hilbert space, a Moore-Penrose construction of a covering field on Wiener space, and a trace formula for the Tikhonov limit, specialised to the equation through the representation $\Phi_r = Y(t,r)\mathcal{B}(r,X(r))$ of the Malliavin derivative by the first variation; the stochastic $p$-Laplacian and the two-dimensional Navier-Stokes equation are treated in detail. Beyond the variational class, a scalar reduction gives an integration-by-parts identity for the law of a pairing $\langle u(t),\varphi\rangle$ which passes to the renormalised limit for the singular equations of Bruned, Chandra, Chevyrev and Hairer, and which is represented by a logarithmic derivative under second-order Malliavin smoothness and negative-moment hypotheses.

We classify the differential operators \(T=z^j\partial_z^j+z^m\partial_z^\ell\), where \(0\le m<\ell\) and \(1\le j<\ell\), whose monic eigenpolynomials satisfy a finite-term recurrence relation. Writing \(k=\ell-m\), such a recurrence exists if and only if \(j=1\) and \(k\mid\ell\). In that case we determine all recurrence coefficients in closed form and prove that the associated difference operator has order \(\ell\). We also give an explicit factorization showing that every admissible operator is of Type~(2) in Conjecture~1.10 of Horozov–Shapiro–Tater; the equality of the differential and difference orders is the conclusion predicted by their Conjecture~1.9.

Kostana [5] asked for natural conditions on a class $\mathcal{K}$ to ensure that forcing with finite members of $\mathcal{K}$ on an underlying set $X$ is forcing equivalent to Cohen forcing. We show that if $\mathcal{K}_{\triangle}$ is the class of finite triangle-free graphs and $|X|\geq\aleph_2$, then $ \mathbb{C}_X(\mathcal{K}_{\triangle}) \not\simeq \mathbb{C}[X]. $ We then use a characterization of Cohen algebras due to Milovich to obtain a positive result. We show that if $\aleph_n\leq|X|$ and $\mathcal{K}$ has $(n+1)$-disjoint amalgamation, then $\mathbb{C}_X(\mathcal{K})$ is Cohen.

Let $X\subset \mathbb C^n$ be a closed pure $d$-dimensional complex analytic set. We associate to $X$ its set of projective asymptotic directions $$ \Sigma_\infty(X) :=\{\ell \in \mathbb P^{n-1}(\mathbb C):\ell\cap C_\infty(X)\ne\{0\}\}, $$ where $ C_\infty(X)$ is the total tangent cone at infinity. We prove the metric Chow-type characterization $$X\text{ is algebraic} \quad \Longleftrightarrow \quad \mathcal{H}^{2d} \bigl( \Sigma_\infty(X)\bigr)=0. $$ As an application, if $C_\infty(X)$ is definable in an o-minimal expansion of $(\mathbb R,+,\cdot)$ and $\dim_{\mathbb R} C_\infty(X)\le 2d$, then $X$ is algebraic.

We study the linear recurrence order of the Potts model partition function on thermal 2D grid graphs. By restricting the transfer matrix (TM) to the real coupling axis, the physical operator maintains diagonalizability under the Spectral Theorem. We show that this thermal regularization allows the Krylov subspace to saturate the unconstrained planar state capacity, locking the recurrence order to the Dyck path up to reversal (OEIS A007123) for $q \ge 4$. Furthermore, the recurrence order collapses to height-restricted Dyck paths up to reversal (OEIS A001998) for $q=3$ due to finite-index Jones-Wenzl projections, and to the zero-magnetization conservation sector (OEIS A001405) for $q=2$. This framework bridges graph-theoretic combinatorics with the representation theory of physical loop gas models.

We prove stability estimates for the volume of central hyperplane projections of unit balls in $\ell_p^n(\mathbb{R})$, establishing the dual counterpart of the result of Chasapis, Nayar and Tkocz on stability of sections.

We introduce the notion of a Penner-type class of an exact symplectomorphism on a Liouville domain with an $A_k$-configuration of Lagrangian spheres for $k\geqslant 2$, and prove that such a class has positive Floer-theoretic entropy. As a corollary, we construct infinitely many smoothly trivial symplectic isotopy classes of exact symplectomorphisms with positive Floer-theoretic entropy on any $4n$-dimensional Liouville domain that admits an $A_2$-configuration of Lagrangian spheres. We also prove that the Floer-theoretic entropy of an exact symplectomorphism on a Liouville domain provides a lower bound for its topological entropy.

We study semi-orthogonal decompositions of perfect derived categories of gentle algebras via marked ribbon surfaces. We characterize such decompositions in terms of suitable disjoint union decompositions of full formal arc systems, and relate this description to good cuts of the corresponding surfaces. For gentle algebras, rotations of curves induce fully faithful functors from extension-closed subcategories of module categories to the components of the associated semi-orthogonal decompositions. Under additional Abelian and extension-comparison conditions, these constructions give derived decompositions of the module categories.

A B-coloring of a graph $G$ is a proper edge-coloring in which every $4$-cycle is rainbow. Let $q_B(G)$ be the minimum number of colors in such a coloring. Gyárfás and Sárközy (2023) determine $q_B(G)$ when $G=P_m\square P_n$ is a rectangular grid. In this paper, we completely determine $q_B(G)$ for cylindrical and torus grid graphs $G$. For a torus grid $G=C_m\square C_n$, where $m,n\ge3$ are integers, we prove that $q_B(G)=4=\Delta(G)$ if both $m$ and $n$ are even and $G\not\cong C_4\square C_{4k+2}$ for any integer $k\ge1$, and that $q_B(G)=5=\Delta(G)+1$ if at least one of $m,n$ is odd or $G\cong C_4\square C_{4k+2}$ for some integer $k\ge1$. For a cylindrical grid $G=C_s\square P_m$, $q_B(G)$ also depends on the parity of $s$ and the length of $P_m$. For integers $m\ge2$ and $n\ge2$, we have $q_B(C_{2n}\square P_m)=4$. For integers $m\ge2$ and $n\ge1$, we have $q_B(C_{2n+1}\square P_m)=4$ if $2\le m\le n$, whereas $q_B(C_{2n+1}\square P_m)=5$ if $m\ge n+1$. In higher dimensions, we discuss the B-coloring of discrete torus and $\ell$-cylindrical grid, obtaining some exact results and certain bounds.

For a flat $\mathbb{Z}_{(p)}$-scheme $X$, we introduce an obstruction class to the existence of a Frobenius lift on its reduction $X_1$ modulo $p^2$. This class is defined by replacing the cotangent complex in the classical construction of the Kodaira–Spencer class with its arithmetic analogue, the Frobenius–Witt cotangent complex. We further prove that this class coincides with the obstruction to Frobenius lifting defined by classical deformation theory. We also define its relative version and prove that this is an obstruction class to the existence of a Frobenius lift compatible with the given one on the base scheme.

We provide an explicit algebraic construction showing that, uniformly for integers $3 \leq s \leq t$, as $t \to \infty$, \[ R( s,t ) \geq t^{(1-o(1)) \log s / \log(\log s + 1) }. \] For large fixed $s$, this improves the dependence on $s$ in the general off-diagonal construction of Alon and Pudlák. In particular, $R(33, t) \geq t^{2.1-o(1)}$, to our knowledge, the first explicit construction showing $R(s, t) \geq t^c$ for some fixed $s$ and some $c > 2$. In the diagonal case, it improves the leading constant in the exponent of the classical Frankl–Wilson bound from $1/4$ to $1$, while being almost as simple to describe.

We study quantum shadow inversion under Markovian locality constraints for four-partite systems arranged along the chain $A$–$B$–$C$–$D$. The goal is to reproduce the expectation value of a fixed endpoint observable $O_{AD}$ after an unknown global unitary, without requiring full unitary inversion. We formulate the task using Markov-admissible supermaps and introduce the Markov-implementable centralizer to describe the remaining endpoint gauge freedom. We show that unrestricted endpoint post-processing is too broad, and impose an endpoint-local refinement. Under this condition, every implementable endpoint unitary must factorize across $A|D$, so the Markov constraint strictly reduces the centralizer-induced shadow ambiguity whenever the full centralizer contains non-product unitaries. This provides a structural mechanism by which Markov locality enhances identifiability in quantum shadow inversion.

Data-driven methods for earthquake and tsunami early warning rely on large ensembles of rupture scenarios and their resulting waveforms, but generating such datasets with repeated high-fidelity seismic and tsunami simulations is prohibitively expensive. We exploit the linear time-invariant structure of both dynamics to precompute elastic Green's functions and acoustic-gravity adjoint responses, reducing the source-to-waveform map to two consecutive convolution operators. We evaluate these convolutions with a distributed, FFT-accelerated GPU pipeline that partitions the large seafloor grid across GPUs and directly generates the final observation waveforms. We demonstrate the scalability of this pipeline for the Cascadia Subduction Zone with 963 subfaults, 2,416,530 seafloor grid points, 64 observation locations, and 256 timesteps, requiring 9.45 TiB of aggregate GPU memory. On 64 GB200 GPUs within one NVL72 domain, the pipeline generates waveforms in 24 ms per rupture once the response operators are resident, enabling large rupture ensembles to be evaluated within minutes.

In this paper, we study Ricci curvature bounds for statistical submersions from a direct Ricci-curvature perspective. Although Chen–Ricci inequalities for statistical submersions have already been established, a complementary lower estimate is needed to obtain a two-sided description of the Ricci curvature. Motivated by this observation, we combine the Ricci curvature relations of a statistical submersion with Hineva's algebraic inequality. We first obtain a Chen–Ricci upper estimate and a Hineva-type lower estimate along the vertical distribution, expressed in terms of the intrinsic Ricci curvature of the fibres and the fundamental tensors $T$ and $T^{*}$. The two estimates together provide upper and lower bounds for the vertical Ricci curvature. We then derive a Chen–Ricci inequality along the horizontal distribution involving the fundamental tensors $A$ and $A^{*}$. By combining the vertical and horizontal curvature relations, we further obtain corresponding Chen–Ricci and Hineva-type estimates for the mixed distribution. The equality conditions are characterized in terms of the components of the fundamental tensors and their duals. Explicit examples are given to illustrate both equality and strict inequality cases. Thus, the paper provides a unified Ricci-curvature approach to statistical submersions and, in particular, gives the first Hineva-type lower Ricci curvature estimates for statistical submersions, leading to two-sided Ricci curvature bounds.

This paper proposes modified polarization-adjusted convolutional (MPAC) codes and their hybrid decoding that achieves an improved performance-complexity tradeoff. For MPAC codes, only a subset of the information bits undergo the convolutional transform. The output is then combined with the remaining information bits for the inner polar transform. Correspondingly, the convolutionally transformed bits are recovered by Fano decoding, while the remaining information bits are recovered by the successive cancellation (SC) decoding, constituting the hybrid Fano-successive cancellation (HFSC) decoding. The MPAC codes are further designed by the coset-wise analysis that characterizes the number of minimum weight codewords (MWCs). It is discovered that a partially convolutional transform can improve the codeword through utilizing the row combinations of the frozen set efficiently. This property enables the MPAC codes to outperform their prototype polarization-adjusted convolutional (PAC) codes and cyclic redundancy check (CRC)-polar codes. Furthermore, MPAC codes can be optimized by reducing the number of MWCs. Our numerical results demonstrate that, with a similar decoding complexity budget, the MPAC codes offer competent decoding performance when compared with PAC codes using Fano decoding and CRC-polar codes using SC list (SCL) decoding.

It has long been known that well-trained neural networks can be compressed very strongly without affecting their performance, an important phenomenon that remains poorly understood. We prove a uniform compressibility theorem for deep multilayer perceptrons with analytic activations. For a deep, wide fixed teacher network, there exists a narrow (same depth) network that approximately represents the same function as the original. The reachable compressed width is strikingly independent of the original width, but is $O((\log(1/\varepsilon))^{d_{in}})$, where $\varepsilon$ is the error budget and $d_{in}$ is the effective input dimension. Our construction involves a novel derivative-matching technique which is aware of the low-dimensional input, and a layer-wise reweighting that preserves the input-output mapping.

The general subgroup permanental-dominance conjecture was previously known only through matrix order three. This paper proves its complete order-four case: for every subgroup $H\leq S_4$, every irreducible complex character $\chi$ of $H$, and every $4\times 4$ Hermitian positive-semidefinite matrix $A$, it establishes $d_\chi^H(A)/\chi(1)\leq \operatorname{per} A$. Unlike the usual immanant specialization, the result covers all thirty-seven irreducible-character cases arising from the eleven conjugacy classes of subgroups of $S_4$. Thirty-five cases follow from general principal-minor, moment, and block-contraction inequalities. The two non-real $A_4$ characters are reduced to polynomial nonnegativity on the cone of $3\times 3$ positive-semidefinite Gram matrices and are resolved by a rank-one sum-of-squares identity, an exact positive-definite interior certificate, rational Gram certificates, and closure of the Gram cone.

It follows by Puuska that, over a commutative Noetherian ring $R$, a morphism $i : M \rightarrow I$ of $R$-modules is an injective envelope if and only if its Matlis dual $Hom_R(i, E)$ is a flat cover for some injective cogenerator $E$, and equivalently for every injective $R$-module $E$. In this paper, we will generalize this result to non-commutative Noetherian algebras.

We prove a weighted Bossel-Daners transfer principle for the first Robin eigenvalue of the $p$-Laplacian under $w=m^{1/p'}$, where $p'=p/(p-1)$. The argument combines double-density isoperimetry with spectral admissibility for singular weights and normalized-flux monotonicity. It uses an exact $BV$ zero-extension formula, an $L^{p'}(m\,dx)$ selection lemma, and a nonatomic rank map that remains well defined on positive-measure level sets. For the singular pair \[ m_b(x)=|x|^b, \qquad w_b(x)=|x|^{b/p'}, \] known power-weight isoperimetry provides the geometric input. We establish spectral admissibility up to the critical exponent $p=N$ and, for the positive radial first eigenfunction $z=z(r)$ on the centered ball $B_R$, derive the integrated singular radial equation and center asymptotics and prove directly that \[ \theta_R'(r)>0 \quad(0<r<R), \qquad \theta_R(r)=\frac{|z'(r)|^{p-1}}{r^{b/p'}z(r)^{p-1}}, \] with $\theta_R(0)=0$ and $\theta_R(R)=\beta$. Let $\Omega\subset\mathbb{R}^N$ be a finite union of bounded connected Lipschitz domains whose closures are pairwise disjoint, let $\Omega_b^\sharp$ be the centered ball of equal $|x|^b$-weighted volume, and let $\lambda_{1,\beta}^b$ denote the first eigenvalue for the displayed pair. Consequently, \[ \lambda_{1,\beta}^{b}(\Omega_b^\sharp) \le \lambda_{1,\beta}^{b}(\Omega) \] whenever $N\ge 2$, $1<p\le N$, $-p<b<0$, and $\beta>0$. The range $1<p<N$ is complementary to the previously known $p\ge N$ weighted Talenti theory; at the shared endpoint $p=N$, the present proof also permits the singular contact $0\in\partial\Omega$.

Learned joint source-channel coding (JSCC) enables robust wireless image transmission by jointly optimizing the transmitter and receiver over differentiable channel models. For bandwidth-limited and time-varying visual links, a single model should support user-adjustable transmission rate and adapt to changing wireless channel conditions, while also dynamically allocating resources according to spatial content. Existing content-adaptive or dynamic allocation schemes often rely on entropy coding, context/probability prediction, explicit rate maps or masks, or auxiliary allocation networks, complicating the encoder-decoder pipeline and increasing side-information overhead. We propose TS-JSCC, a single-model adaptive JSCC framework with tail-structured sparsification. First, an L1-based tail-structured sparsification objective encourages each token to retain an active feature-channel prefix while suppressing trailing ones. This enables content-adaptive feature-channel allocation with compact side information through active-prefix transmission. Second, lightweight stage-wise neural regulating modules use a normalized sparsity-control coefficient and the channel signal-to-noise ratio (SNR) to rescale intermediate features for single-model transmission rate and SNR adaptation. Experiments on CIFAR-10, Kodak, and CLIC2021 under additive white Gaussian noise (AWGN) and Rayleigh fading show that TS-JSCC achieves strong rate-distortion performance against the latest learned-JSCC baselines and remains competitive with the considered idealized separation baselines, while retaining a simple one-shot encoder-decoder without extra structures or computations.

We investigate the existence and non-existence of large-amplitude boundary layer solutions to the outflow problem for the one-dimensional full compressible Navier-Stokes equations on the half-line $\mathbb{R}_+$. Through a delicate global phase-plane analysis, we substantially extend the small-amplitude boundary layer solutions obtained via the center-manifold approach to the large-amplitude regime. Based on the sign of $M_+-1$ (with $M_+$ denoting the Mach number at the right end state) and the Prandtl-number-related parameter $\zeta$, we provide a complete characterization of the existence and non-existence of large-amplitude boundary layer solutions for this half-space outflow problem. In sharp contrast to the corresponding inflow problem, boundary layer solutions to the outflow problem are permitted to be non-monotone, and their existence region in the phase plane can be an open set.

The aim of this paper is to classify some special Riemannian manifolds with cyclic parallel Ricci tensor, i.e. \begin{equation} D_{ijk}=\nabla_{i}R_{jk}+\nabla_{j}R_{ki}+\nabla_{k}R_{ij}=0\nonumber \end{equation} These structures include non-compact gradient shrinking Ricci soliton, compact $(m > 1)$-quasi-Einstein manifolds with boundary and critical spaces. We will construct some integral identities and make use of the curvature conditions reasonably to prove that the Ricci tensor is parallel.

Let $G$ be a finite group. In this note, we construct an element ${\sigma}^G$ in the Burnside ring of $G$ over $\mathbb{Q}$, which gives a rational number $m(G)$ that is the sum of the inverses of the element orders in $G$, by using the deflation map ${\mathrm{Def}}^G_{G/N}$ induced by the $(G/N,G)$-biset $G/N$ for a normal subgroup $N$ of $G$. As a corollary to Theorem, we obtain another expression for the rank of the crossed Burnside ring $B^{\rm c}(G)$ of $G$ and the determinant of the Cartan matrix of $p$-local Mackey algebra of $G$ over a field $k$ of characteristic $p>0$ with big enough.

We introduce a continuous data assimilation method for particle-in-cell simulations of the Vlasov-Poisson equation when only hydrodynamic moments are observed. The forecast state is an empirical measure on phase space, whereas the observed fields (density, bulk velocity, and temperature) constrain only a few velocity moments and leave the velocity-space shape of the distribution undetermined. We construct the moment feedback as a Wasserstein gradient flow of a moment-mismatch functional over phase-space measures. The resulting drift acts directly on particle positions and velocities, couples the density, momentum, and energy residuals through a single variational structure, and vanishes on the entire moment-compatible set. Under the standard Wasserstein metric, the energy residual produces a position correction that grows quadratically with the particle speed, and the particle system falls outside standard well-posedness theory. Our primary formulation pairs the quadratic moment mismatch with a velocity-weighted Wasserstein metric that penalizes spatial transport at large peculiar velocity relative to the observed bulk flow, which removes this growth. A direction-split variant retains the plain metric instead. Under the same weighted metric, an alternative moment-relative-entropy functional yields an affine, shape-preserving velocity correction and explicit global moment balances for the space-inhomogeneous system. We prove that the finite-particle scheme with linear Lenard-Bernstein collisions is globally well posed. In 1D1V and 2D2V experiments with several collision models, the nudged formulations reduce bulk-velocity and temperature errors by up to two orders of magnitude relative to an unassimilated run.

We study positive solutions of the partially confined stationary nonlinear Schrödinger equation $$-\Delta u+|y|^2u+\lambda u=g(u),\quad (y,z)\in\mathbb{R}^d\times\mathbb{R}^{m},\quad 1\leq d<N,\quad m:=N-d.$$ The spectrum of $-\Delta + |y|^2$ is given by $[d,\infty)$. We prove that $\lambda\geq-d$ is necessary for positive solutions in the classes considered here, and the threshold $\lambda=-d$ yields several Liouville type results. In particular, for the pure-power equation $$(-\Delta + |y|^2-d)u=u^p,$$ we prove that any nonnegative $H^1(\mathbb{R}^N)$ weak solution is trivial whenever $1\leq p\leq \max\{(N+2)/(N-2), m/(m-2)\}$ for $m\geq 3$ or $p\geq 1$ for $m=1,2$. The proof combines a half-space oscillator gap, moving planes at the spectral threshold, a Picone inequality, a strict Gaussian second-moment inequality, and anisotropic Pohozaev identities. For $\lambda>-d$, we establish the existence of positive solutions under some standard assumptions. Furthermore, every positive solution decaying at infinity is radially symmetric and strictly decreasing in the confined variables and, up to one common translation, radially symmetric and strictly decreasing in the free variables. \vskip 0.2in Dedicated to our supervisor Prof. Wenming Zou on the occasion of his 60th birthday.

In this paper, we introduce a new class of operators, called $q$-Berezin sectorial operators, as an extension of the class of $q$-sectorial operators. By presenting examples on the Hardy-Hilbert space, we show that there exist operators which are $q$-Berezin sectorial but not $q$-sectorial. Several new inequalities for the $q$-Berezin number associated with this class of operators are also derived. In addition, we investigate the geometric structure of the $q$-Berezin range for various classes of operators on the Bergman space, including weighted shift and certain composition operators.

Localized forcing in the fluid inside or outside a viscous drop can drive drop translation. Using the Lorentz reciprocal theorem, we derive an integral expression for the translational velocity of a spherical Newtonian drop subject to localized force and source distributions in either fluid and to interfacial traction. For a clean drop, we obtain explicit responses to Stokeslets, force dipoles, rotlets, general second force moments, and source dipoles as functions of position, orientation, and viscosity ratio. Interior forcing obeys a finite selection rule: only force moments through second order and the first source moment contribute directly to translation. Exterior forcing can couple to multipoles of all orders and produces distance-dependent responses. Although different enclosed singularities can produce the same drop velocity, resolving their exterior flows in drop-centered spherical Stokes modes provides additional constraints on the underlying forcing. We also distinguish regularized force distributions, governed by prescribed kernel moments, from resolved rigid particles, governed by low-order surface-traction moments and prescribed slip. The framework unifies these representations and shows how exterior-flow measurements provide information beyond drop translation, laying the foundation for constructing squirmer-like viscous drop solutions with controllable far-field behaviors.