Papers

We study the Gorenstein homological dimension $Ghd_k G$ of groups $G$ which are of type $FP_{\infty}$ over a commutative ring $k$. Our main result shows that, over an arbitrary ring $R$, every finitely presented Gorenstein flat $R$-module is projectively coresolved Gorenstein flat. Consequently, for a group $G$ of type $FP_\infty$ over $k$ with sfli$k<\infty$, the three natural dimensions for $G$, namely Gorenstein homological, Gorenstein cohomological and projectively coresolved Gorenstein flat, all coincide. Building on this collapse, we establish a Gorenstein homological analogue of Fel$'$dman's theorem, a formula for iterated $m$-fold self-extensions of a group $N$ of type $FP_\infty$ over $\mathbb Z$, and a field-detection theorem, showing that the Gorenstein homological dimension of a group of type $FP_\infty$ over a principal ideal domain is realized after passing to a suitable field.

The low-power error-correcting cooling (LPECC) codes and constant-power error-correcting cooling (CPECC) codes, introduced in [IEEE Trans. Inf. Theory, 64 (2018), 3062–3085; 66 (2020), 4804–4818], respectively, are two coding schemes designed to simultaneously control the peak temperature and average power consumption of on-chip buses while providing error-correction capability for transmitted information. This paper establishes new upper bounds for both $(n,1,w,w-2)$-CPECC codes and $(n,t,w,w-2)$-CPECC codes using graph-theoretic techniques, and constructs several new families of optimal CPECC codes using combinatorial configurations. Moreover, it completely resolves the conjecture concerning CPECC codes posed in [IEEE Trans. Inf. Theory, DOI: 10.1109/TIT.2026.3721101]. Finally, we derive a new upper bound for $(n,t,w,w-2)$-LPECC codes by probabilistic method, along with new optimal families, and establish the relationship between optimal $(n,t,w,w-2)$-LPECC codes and optimal $(n+1,t,w,w-2)$-CPECC codes.

We introduce and study monoidal and symmetrized versions of parametrized topological complexity. First, we develop parametrized analogues of the monoidal topological complexity theories of Iwase–Sakai, Aguilar-Guzmán–González (based on the Fadell–Husseini approach), and Dranishnikov. We investigate the parametrized Iwase–Sakai conjecture and provide sufficient conditions under which it holds. We then introduce symmetrized parametrized topological complexity and combine it with the monoidal perspective to define monoidal symmetrized parametrized topological complexity, showing that the resulting notions agree. Finally, we compute these invariants for Fadell–Neuwirth fibrations.

Integrated sensing and communication is turning the cellular infrastructure into an active observer of the physical world. When such infrastructure interacts with embodied agents capable of deliberately changing their states and surroundings, the physical world itself can become a communication medium. This paper studies the fundamental communication limits of this sensing-induced embodied communication paradigm in the near field. We consider an agent that maps messages to the positions of a controllable scatterer within a bounded three-dimensional (3D) region, while a base station decodes the selected position from multi-snapshot monostatic sensing echoes. Near-field spherical wavefronts resolve both angle and range, expanding the embodied-symbol space from a 2D plane to a 3D volume. This gain, however, comes with a position-dependent and anisotropic reliability geometry. We characterize this geometry through the pairwise Bhattacharyya distance and derive a local ellipsoidal representation of the resulting 3D confusability regions, whose principal axes quantify directional sensing resolution. The ellipsoid further degenerates into the 2D transverse ellipse in the far-field limit, unifying the two regimes. We then formulate the finite-snapshot $\epsilon$-capacity and translate reliable codebook design into a 3D packing problem. A face-centered cubic construction provides an achievable rate, while a geometric converse yields a complementary upper bound. Numerical results validate the proposed geometry and demonstrate the capacity gain of near-field volumetric packing over far-field planar packing. These results establish a unified geometric and information-theoretic framework for communication through deliberately configured physical states.

Vector fields on graph structures naturally arise in diverse biological and engineered systems, where vector-valued states are defined on the nodes and evolve through the network interactions. Existing methods primarily characterize either the graph topology or individual signals, but generally do not quantify how local interactions among node-associated vectors are organized across the graph. To address this limitation, a sheaf-theoretic framework, termed SheafIQ, is proposed to represent neighboring vectors in a common edge-associated coordinate system, map local incompatibilities to a residual energy distribution, and quantify its global organization through entropy. Across proteins, functional brain networks, urban traffic systems, and power grids, SheafIQ consistently reveals complementary organizational information beyond conventional graph- and signal-based descriptors. More broadly, it establishes a unified information-theoretic framework for quantifying the organization of vector-valued states on geometric graphs, extending network analysis beyond graph topology alone.

The divided generalized Bernoulli values $b_{\chi,j} = fB_{1,\chi\omega^{-j}} \bmod p$, for $\chi$ an odd primitive Dirichlet character of conductor $f$ and order $d$ with $d \mid p-1$, control (for $p \nmid \varphi(f)$) the odd isotypic components of the $p$-class group of $Q(\zeta_{fp})$ through the characterwise abelian Main Conjecture; a zero is a twisted irregular pair, a branch with positive Iwasawa lambda-invariant, in the tradition studied by Ernvall, Holden, Delbourgo-Knospe and Knospe. We survey these zeros in the regime complementary to existing tabulations: fixed small conductor and large $p$ ($f = 3, 5$ to $p < 10^5$; all odd primitive characters of conductor at most 20 to $p < 2 \cdot 10^4$), computing each spectrum by a residue-class weight formula and one Bluestein convolution over $F_p$, a direct finite-field alternative of the same quasi-linear order as the standard power-series method. The survey records 27,508 zero lines over 55,121 character-prime pairs, each verified by two independent code paths with an exact order of vanishing; the counts and digits are consistent with the random model, and eight lines are non-simple, including one of depth three at $(f,p,j) = (19,37,16)$, giving class components of order exactly $p^2$ and $37^3$. The main contribution converts zeros into explicit certified generators: the conductor-three catalogue of arXiv:2607.23177 is extended from $p < 500$ to $p < 10^5$, all 2,441 zero lines simple, each projected circular unit proven to generate its complete order-$p$ Hilbert class field component by a fresh split-prime Artin certificate – the largest in the degree-199,980 field $Q(\zeta_{299973})$. Ancillary files contain all tables, certificates, and a verification program.

The stability programme initiated by Ulam asks when approximate solutions to algebraic identities must lie near exact ones. For lattices, this leads to the question of when a map that nearly preserves joins and meets can be approximated by a genuine lattice homomorphism. Badora–Kochanek–Przebieracz developed a neighbourhood-based framework for distributive lattices, centred on a separation (sandwich) lemma that constructs an exact join homomorphism between a join-subhomomorphism and a join-superhomomorphism via an order envelope. We revisit this mechanism and identify the single step at which distributivity is used: a decomposition identity for elements lying below a join. Without distributivity, separation can fail already in the modular lattice $M_3$ and in a small finite orthomodular lattice. On the positive side, we show that separation holds in arbitrary lattices whenever the lower bounding map is isotone. For orthomodular lattices—algebraic models of quantum logic—we develop a blockwise stability theory on Boolean blocks. Approximate identities on compatible pairs yield exact homomorphic selections on each block (for joins, for meets, and for both operations under bi-admissibility). We present several gluing criteria for assembling blockwise selections, and we give a concrete finite example showing that gluing can fail when block overlaps are non-trivial. Finally, in the spirit of Kalton–Roberts, we obtain blockwise approximation results for nearly additive functions on orthomodular lattices by finitely additive measures, with an illustration on finite-dimensional projection lattices.

This paper is concerned with the Cauchy problem for a semilinear wave equation with arbitrary quadratic nonlinearities, posed on a cosmological supersymmetric spacetime of the product form (Open Milne) $\times K$, where $K$ is a compact, Ricci-flat $n$-dimensional Riemannian manifold. We establish the global future existence of solutions for small initial data. Our proof, based on a novel modified energy, avoids spectral decompositions of the Laplace operator on the internal manifold and does not require any special nonlinear coupling structure. This robust framework is expected to be readily extendable to tensorial wave equations, including the Einstein equations.

Many operational decisions require solving a linear program whose cost vector is unknown at decision time and must be predicted from contextual information. Because prediction and decision are only weakly aligned, the emerging integrated learning and optimization (ILO) paradigm trains the predictor through the downstream problem, judging a prediction by the decision it induces. However, predictions are inevitably imprecise, so robustness often enters the decision stage. To address this issue, we propose an integrated learning and robust optimization (ILRO) framework, where a robust decision problem is used both to define the training problem (termed the RSPO loss problem), and to produce the deployed decision. Thus, this framework simultaneously achieves both robustness and learning-decision alignment. To tackle its computational challenges, we develop a convex surrogate, RSPO+, and characterize when it is Fisher consistent. Moreover, the RSPO loss possesses informative gradients, allowing us to develop first-order computational methods. We also derive finite-sample excess risk bounds for both RSPO and RSPO+ predictors. Numerical experiments on transportation and portfolio problems, in comparison with multiple benchmarks, show the advantage in decision quality of the proposed framework. The gain is more pronounced for scenarios with limited samples, high-dimensional decisions, and model misspecification.

A metric graph is a metric space obtained from a finite collection of intervals whose endpoints are identified in groups. It can also be seen as a finite, edge-weighted graph where the continuum of points along the interior of each edge is taken into consideration, and each edge is locally isometric to an interval whose length is the edge-weight. The diameter of a metric graph $G$ is the maximum distance between all pairs of points of $G$. We show that the total length of a metric graph $G$ with $\ell(G)$ leaves, cyclomatic number $cyc(G)$, and diameter $diam(G)$ is at most $(cyc(G) + max\{1, \ell(G)/2\}) \cdot diam(G)$. Furthermore, we show that his bound is tight, and we characterize the metric graphs where equality holds. As an application, we provide tight bounds in certain cases for the diameter of metric graphs obtained from a cycle or a star by the identification of a fixed number of points (pairwise or in groups).

We settle two endpoint problems for Beurling–Carleson support conditions. For $0<s<\frac12$ and $\theta=\frac{1-2s}{1-s}$, we construct a nonatomic singular probability measure $\mu$, supported on a single $\theta$-Beurling–Carleson set, such that $S_\nu'\notin H^s$ for every nonzero submeasure $0<\nu\leq\mu$. For $m>3$ and $\alpha=\frac{m-3}{m-1}$, we construct a nonatomic probability measure supported on a single $\alpha$-Beurling–Carleson set that is not the deficiency measure of any nearly maximal solution of $\Delta u=(u_+)^m$. Thus hereditary failure persists at the first endpoint, while the critical support condition in the second problem is necessary but not sufficient. The proofs combine endpoint Cantor–Moran constructions with kernel divergence and a nonlinear energy obstruction.

We study interlacing between real polynomial quotients obtained from centered binomial samples of a fixed even or odd real entire function at consecutive odd degrees. For sources of order at most one, we determine the optimal vertical-strip half-widths guaranteeing weak interlacing for every adjacent pair: $\sqrt{15/28}$ for even sources and the explicit algebraic constant $h_*\approx 1.0656$ for odd sources. These constants are forced by the first nontrivial pair and suffice at every degree. Under the same sharp bounds, nonpolynomial sources have simple zeros in $(-2,2)$ and strict interlacing at every positive index. We also prove an eventual result: if a nonpolynomial source has growth order less than two and only finitely many zeros off the imaginary axis, then all sufficiently high adjacent quotients have simple interior zeros and strictly interlace; the starting index depends only on the finite factor formed from those zeros. The proof turns the sampling problem into an interval real-rootedness problem for a Jacobi differential operator. For fixed degree, we also determine the maximal range of scalar shifts preserving real-rooted polynomials with at most one zero outside the interval. Applications give strict interlacing for quotients attached to completed Dedekind zeta derivatives and to successive centered blocks of critical values of self-dual newforms satisfying explicit spacing and parity conditions.

We present a new avenue to the usual Schwinger effect via the open string pair production for a system of two Dp branes with $1 \le p \le 6$ in Type II superstrings. The two Dp are placed parallel at a separation with one of them carrying the most general constant electromagnetic background allowed for the pair production. By taking the so-called field theory limit, we obtain systematically, from the open string pair production rate, the respective explicit Lorentz invariant QED pair production rate in diverse dimensions for a pair of charged/anti-charged massive particles, which can be scalars, spinors or vectors.

Let $M$ be a connected smooth $n$-manifold without boundary, where $n\geq3$, and let $\kappa\in\mathbb{R}$, with $\kappa\leq0$ if $M$ is open. We prove that every smooth Riemannian metric $g_0$ with $\mathrm{Scal}_{g_0}\geq\kappa$ is a locally uniform limit of smooth Riemannian metrics $g_i$ with $\mathrm{Scal}_{g_i}=\kappa$ that are locally uniformly bounded in $W^{1,\infty}$. As a corollary, combining this with Gromov's $C^0$-stability theorem, we obtain the perhaps surprising identity \[ \overline{\{g:\mathrm{Scal}_g=\kappa\}}^{\,C^{0,\alpha}_{\mathrm{loc}}}=\{g:\mathrm{Scal}_g\geq\kappa\}, \quad \forall \alpha \in(0,1). \] The restriction $\alpha<1$ is sharp. At $\kappa=0$, this proves and strengthens the Riemannian reverse-Burnett conjecture of Huneau and Luk.

Kernel spectral clustering with a single bandwidth can be inadequate for data exhibiting multiple characteristic pairwise-distance scales, a problem particularly prevalent in the high-dimensional regime. We address this issue through a multi-kernel formulation that aggregates kernels with different bandwidths. The bandwidths are selected as prescribed empirical quantiles of the pairwise squared distances, thereby capturing the relevant distance scales without requiring prior population-scale information. We develop a rigorous theoretical analysis of the resulting method under a general high-dimensional, multi-scale mixture model with heterogeneous cluster centers and covariance geometries. We construct a blockwise constant, low-rank informative approximation to the empirical multi-kernel matrix and establish row-wise $\ell_{2,\infty}$ perturbation bounds for its leading spectral components, as well as for the associated normalized Laplacian matrix. These bounds yield observation-level control of the spectral embedding, which is more informative than conventional global eigenspace perturbation estimates. Under suitable eigen-gap and cluster-separation conditions, we show that approximate $K$-means applied to the multi-kernel spectral embedding achieves exact recovery with high probability.

An $(\alpha,\beta)$ Somos-$4$ sequence $S_n$ is defined by the recurrence $S_nS_{n-4}=\alpha S_{n-1}S_{n-3}+\beta S_{n-2}^2$ ($n\geq 4$), with suitable initial values, where $\alpha$ and $\beta$ are constant parameters. A widely studied question is the following: When does the Hankel transform of a generating function become an $(\alpha,\beta)$ Somos-4 sequence? In particular, how can $\alpha$ and $\beta$ be derived for such a function? A sufficient condition for this problem has been established by Wang and Zhang. In this paper, we obtain the following three main results. (i): We extend the Wang–Zhang sufficient condition by working over the rational function field. Then we combine this result with the Sulanke–Xin quadratic transformation to resolve all of Barry's currently unsolved $(\alpha,\beta)$ Somos-4 conjectures, which arise in diverse contexts, including generalized Catalan recurrences, Riordan arrays, generalized Bernstein arrays, and elliptic curves. (ii): We show that the odd and even subsequences of an $(\alpha,\beta)$ Somos-4 sequence are again $(\alpha,\beta)$ Somos-4 sequences with transformed parameters. This is employed to establish Barry's Hurwitz transform conjecture. (iii): Using the theory of orthogonal polynomials, we prove a Hankel determinant formula and thereby prove a conjecture related to the $(\alpha,\beta)$ Somos-4 sequence. In addition, we prove some conjectures on formulas for periodic Hankel determinants.

Let $\lambda_{\mathbb K}(m)$ denote the maximal absolute projection constant among $m$-dimensional Banach spaces over $\mathbb K=\mathbb R$ or $\mathbb C$. Its exact value is known only in a few cases, and determining it remains a challenging problem. In this note, we investigate several structured vector configurations that naturally arise in this context. We first recall the connection between maximal projection constants and tight frames, then consider biangular tight frames, and show that their relative and quasimaximal projection constants coincide. A particularly interesting example is provided by the midpoints of the edges of a regular simplex, which yield natural lower bounds for $\lambda_{\mathbb R}(m)$. Numerical evidence suggests that these bounds may be sharp in dimensions $6$ and $8$. We also discuss weighted spherical $(2,2)$-designs with a small number of vectors and their connection with maximal projection constants. Examples in dimensions $4$ and $5$ indicate that the sign patterns of their Gram matrices may play an important role. These observations lead us to formulate several conjectures concerning maximal absolute projection constants and the vector configurations associated with them.

The classical bond lattice of a graph was used in a formula of Whitney to compute the chromatic polynomial. In this paper, we use weighted versions of the bond lattice, to introduce and study a new polynomial and a new symmetric function invariant of a graph. Examples of the new polynomial invariant include the classical Narayana polynomials (for the path graph), the tree-Eulerian polynomials (for the complete graph), and the binomial-Eulerian polynomials (for the star graph). These examples suggest an interesting connection to $h$-polynomials of general graph-associahedra. We prove that for any chordal graph, our polynomial graph invariant is $\gamma$-positive. Multiweighted bond posets yield a symmetric function graph invariant that is an analog of the chromatic polynomial. The highest degree homogeneous component of this symmetric function is of particular interest. The parking function symmetric function introduced by Haiman arises as an example, as do symmetric functions studied by the first author in connection with multibracketed Lie algebras and with colored exterior algebras. The $\gamma$-positivity result mentioned above is a specialization of an $e$-positivity result for the highest degree homogeneous component, which we prove for any chordal graph using the theory of lexicographic shellability. We conjecture that this symmetric function is Schur-log-concave, which specializes to Huh's log-concavity theorem for the chromatic polynomial.

This paper introduces quasi-triangular Jordan D-bialgebras, which are constructed from solutions of the Jordan Yang-Baxter equation (JYBE) with invariant symmetric parts. We first develop a systematic operator approach to the JYBE by introducing the notion of extended relative Rota-Baxter operators on Jordan algebras. Such operators with their extensions are shown to induce new Jordan algebra structures. Moreover, the symmetrizer-antisymmetrizer decomposition of linear maps reduces the study of extended relative Rota-Baxter operators to both pairs of homomorphisms of Jordan algebras and relative Rota-Baxter operators, respectively. This operator framework subsequently yields a characterization of solutions of the JYBE whose symmetric parts are invariant. These operator forms are further investigated in the context of quadratic Jordan algebras and semi-direct product Jordan algebras, respectively, leading to explicit constructions of solutions of the JYBE. A factorizable Jordan D-bialgebra is then introduced as a special class of quasi-triangular ones, which naturally factorizes the underlying Jordan algebra. We prove that the Drinfeld classical double of any Jordan D-bialgebra admits a factorizable Jordan D-bialgebra structure. Finally, the operator perspective gives rise to the notion of quadratic Rota-Baxter Jordan algebras: those of zero weight yield triangular Jordan D-bialgebras, while those of nonzero weight are shown to be in one-to-one correspondence with factorizable Jordan D-bialgebras.

We prove the existence of optimizers for a class of homogeneous Gagliardo-Nirenberg inequalities of the form $$ \|D^{s}\phi\|_{L^q} \leq C \|D^{s_1}\phi\|_{L^p}^{1-\theta} \|D^{s_2}\phi\|_{L^2}^{\theta}, \quad s_1<s_2, $$ under the scaling relation $$ \frac{1}{q}-\frac{s}{d} = (1-\theta)\left(\frac{1}{p}-\frac{s_1}{d}\right) + \theta\left(\frac{1}{2}-\frac{s_2}{d}\right), $$ and the strict interpolation condition $$ s<(1-\theta)s_1+\theta s_2. $$ When $s>s_1$, the problem reduces, after shifting the derivative orders, to the result of Bellazzini, Frank and Visciglia, whereas the case $s\leq s_1$ follows from the present paper. As an application, we study normalized solutions of the associated Euler-Lagrange equations under the constraint $$ \|D^{s_1}\psi\|_{L^p}^p=\lambda>0. $$ In particular, the case $p=2$ includes the stationary equation arising from the Majda-McLaughlin-Tabak (MMT) model.

We study oscillatory integral operators in $(2+1)$-dimensions with a homogeneous binomial phase \[ \Phi(x,y,t)=x^{k-k_P}t^{k_P}+y^{k-k_Q}t^{k_Q}, \qquad 1\le k_P<k_Q<k. \] For compactly supported smooth amplitudes, we establish sharp \(L^2(\R)\to L^2(\R^2)\) estimates with logarithmic losses occurring only in certain critical cases. The proof is based on scale-dependent Phong–Stein estimates.

Let \[ \xi\!\left(\frac12+z\right) =\sum_{n\geq 0}\frac{\gamma(n)}{n!}z^{2n}, \qquad J^{d,n}(X) =\sum_{j=0}^{d}\binom dj\gamma(n+j)X^j . \] The Riemann hypothesis is equivalent to the hyperbolicity of $J^{d,n}$ for every $d,n\geq0$. We prove that there is an absolute constant $K>0$ such that \[ n^3\log^2(n+2)\geq Kd^5 \quad\Longrightarrow\quad J^{d,n}\ \text{is hyperbolic}. \] Along every sequence with $n,d\to\infty$ in this region, the empirical measure of the naturally centered and scaled zeros also converges to Wigner's semicircle law. This gives a simultaneous degree–derivative version of the global semicircle consequence of the fixed-degree Hermite limit of Griffin, Ono, Rolen, and Zagier.

Every RB-domain is an FS-domain. Whether the converse holds was a long-standing open problem. We prove that the domain of closed axis-parallel squares in the plane whose centres lie in the unit square $[0,1]^2$, with the whole plane adjoined and ordered by reverse inclusion, is an FS-domain but not an RB-domain. The proof is quantitative. A finite-grid argument first shows that, on every finite slab $[0,1]^2\times[0,m]$, for every approximate identity and every $\varepsilon>0$, one member of the approximate identity has radius excess (i.e., the output radius minus the input radius) uniformly at most $\varepsilon$. By contrast, for every deflation and every $m>0$, the radius excess is at least $m/(4m+1)$ at some point of the slab $[0,1]^2\times[0,m]$. This solution was obtained independently of the recent work of Chen, Kou, and Lyu and uses a different method.

In this paper, we investigate the moduli space of algebraic asymptotic hyperkähler structures on hyperkähler quotients arising from a quaternionic vector space $\mathbb{H}^n$. We consider the period map on this moduli space, and prove a Torelli-type theorem (i.e., the bijectivity of the period map) under mild assumptions including the surjectivity of the Kirwan map. This work provides an algebro-geometric generalization of the Torelli-type theorem for ALE gravitational instantons by Kronheimer (1989). In particular, this theorem applies to toric hyperkähler varieties and Nakajima quiver varieties.

Let $F$ be a finite graph with at least one edge, and let $W$ be a graphon. We show that if the density of $F$ rooted at each edge is almost everywhere constant, then either $t(F,W)=0$ or $W$ is constant. For edge-transitive $F$, one rooted equation suffices. This recovers the edge-rooted triangle theorem of Reiher and Schacht. In their terminology, our result also shows that every clique is $2$-forcing, answering a question they posed. We give an explicit stability estimate when $W$ is bounded away from zero. Our proof has two steps: an entropy argument turns constant rooted densities into an additive identity for $\log W$, and a Hoeffding decomposition determines all solutions of that identity. The same method gives exact classifications and quantitative stability estimates for symmetric uniform hyperkernels, dissociated Aldous–Hoover hypergraphons, directed kernels, and tournamentons.

We study the conditionally resampled sliding-window count kernel associated with the empirical counts of length-$n$ windows from a stationary finite-state reversible Markov chain. Although the resulting count process is generally not Markov, its stationary one-step conditional law defines a genuine Markov kernel. For every fixed strictly positive reversible kernel \(P\) on a finite state space, we present a Poincaré inequality for the induced count kernel $\tP_n$ of length $n$. In other words, we derive the lower bound of the spectral gap $\Gap(\tP_n)$ of $\tP_n$ as \[ \Gap(\tP_n)\ge \frac{c(P)}{n}, \] where \(c(P)>0\) depends only on \(P\). The proof combines a martingale oscillation inequality for the stationary path law with a direct comparison of coordinate oscillations to the Dirichlet form of the count kernel. A linear statistic of the count vector gives the matching \(O(1/n)\) upper bound, so for every fixed strictly positive reversible \(P\) one has \(\Gap(\tP_n)=\Theta_P(1/n)\). The resulting count-space Poincaré inequality yields a local-to-global variance bound for finite-window count statistics and, together with a general matrix-concentration principle, operator-norm concentration for matrix-valued empirical averages.

Volumetric parameterization, the process of mapping a 3-manifold onto a simplified volumetric domain, is important for many tasks in computer graphics and imaging science. However, most prior volumetric parameterization approaches have only utilized standardized domains such as a solid ball regardless of the overall shape of the given 3-manifolds, which introduces significant geometric distortion and affects the subsequent shape processing and analysis tasks. To overcome this issue, in this work we propose a novel volumetric parameterization framework for simply connected 3-manifolds. Specifically, the proposed framework jointly controls local shape and mass distortions, while adapting the target domain during the optimization process. It enables three progressively more flexible target-domain settings for the parameterization: a prescribed solid ellipsoid, a volume-normalized adaptive ellipsoid with variable radii, and a sea-embedded free-boundary domain. For each setting, the parameterization algorithm consists of a 3D quasi-conformality shape update, a diffusion-based density-equalizing update, and a geometric correction procedure for removing element foldings, thereby allowing for volumetric parameterizations with different desired effects. Experimental results are presented to demonstrate the effectiveness of our proposed framework. Moreover, our framework can be easily applied to multiresolution and localized adaptive volumetric remeshing, volumetric registration, and volumetric morphing. Altogether, our work provides a new way for the representation, processing, and analysis of 3-manifolds.

We prove that a compact discrete quotient of a complex homogeneous space with compact isotropy is Kähler whenever it admits both a balanced metric and a pluriclosed metric. Moreover, if its real first Chern class vanishes, then the pluriclosed flow starting from any invariant pluriclosed metric exists for all time and converges smoothly to a flat Kähler metric.

The single-selection prophet inequality is a canonical Bayesian online selection problem in which independent nonnegative values arrive sequentially and the decision-maker must irrevocably select at most one. Classical single-threshold guarantees are tight in the worst case, but the hard instances that prove tightness are highly irregular: the prophet's advantage is driven by rare, very large realizations of the maximum. We refine this worst-case picture by imposing a bound on the relative variance of the prophet's value, $\mathrm{Var}(\max_{i\in[n]}X_i)/\mathbb E[\max_{i\in[n]}X_i]^2$. This yields a nonparametric complexity measure that interpolates between deterministic instances, where the full prophet value can be recovered, and the unrestricted worst-case regime. Our main technical contribution is a general kernel method for single-threshold prophet inequalities. The method represents an instance by the quantile function of the maximum and rewrites the payoff of a threshold as a linear kernel functional of this quantile. This turns the worst-case analysis into an infinite-dimensional convex program, restores strong minimax duality in quantile space, and reduces the bounded-variance adversary's problem to a one-parameter variational family. Applying this framework, we obtain an exact characterization of the IID bounded-variance curve and asymptotically optimal finite-horizon thresholds, a closed-form expression for the fixed-order non-identical model, and a prophet-secretary lower-bound program together with a strict separation from the IID benchmark at every positive finite variance constraint. As a further application of the same kernel viewpoint, we derive an exact formula for IID random horizons under a convexity condition on the horizon pgf, which includes monotone-hazard-rate horizons, highlighting the broad applicability of this new technique for single threshold settings.

Fractional elliptic stochastic partial differential equations (SPDEs) are widely used in statistics and machine learning for computationally efficient and flexible modeling of Gaussian random fields. The computational efficiency of the SPDE approach relies on finite element approximations combined with numerical quadrature and mass lumping, which enable sparse matrix methods during inference. Although many works have studied finite element approximations of fractional SPDEs, the effect of the mass lumping and quadrature approximations used in practice has not been fully analyzed. To fill this gap, we derive convergence rates for numerical approximations of fractional SPDEs based on finite element discretizations combined with numerical quadrature and mass lumping. Specifically, we obtain explicit convergence rates for the mean-squared error of the covariance function in a general framework that covers the main settings where mass lumping is used in the SPDE approach. We also analyze non-stationary variance-control factors of the form $L^\beta(\tau u)=\mathcal{W}$, where $\tau$ is spatially varying, and derive covariance error estimates showing how the regularity of $\tau$ affects the convergence rate. As specific examples, we provide results for random fields on bounded Euclidean domains, Riemannian manifolds, and metric graphs. Numerical experiments are presented that confirm the theoretical results.