Papers

Existence of canonical metric on a canonical model of projective singular variety was a long standing conjecture and the major part of this conjecture is about varieties which do not have definite first Chern class (most of the varieties do not have definite first Chern class). There is a program which is known as Song-Tian program for finding canonical metric on canonical model of a projective variety by using Minimal Model Program. In this paper, we apply Song-Tian program for mildly singular pair $(X;D)$ via Log Minimal Model Program where $D$ is a simple normal crossing divisor on $X$ with conic singularities. We show that there is a unique $C^\infty$-fiberwise conical Kähler-Einstein metric on $(X;D)$ with vanishing Lelong number which is twisted by logarithmic Weil-Petersson metric and an additional term of Fujino-Mori [72] as soon as we have fiberwise KE-stability or Kawamata's condition of Theorems 2.28, or 2.30(in $C^0$ case). In final we highlight that how the complete answer of this question can be reduced to CMA equation corresponding to fiberwise Calabi-Yau foliation(due to H.Tsuji) which is still open.

We study resolution-independent relative logarithmic source complexes for AKSZ-BV-BFV theory on face-oriented Joyce manifolds with generalized corners. Under the geometric hypotheses (G1)-(G3), the Dupont-Panzer-Pym total relative logarithmic complexes of admitted smooth monoidal resolutions represent $j_!\Omega_{X^\circ}^\bullet$ and carry the same compactly supported derived trace. The comparison is deliberately unfiltered: it identifies total relative cocycle classes through the common interior, but not individual resolved faces, separate BFV descendants, nonlinear mapping spaces, or absolute regularized integrals. For the intrinsic $b$-source ${}^{b}T[1]X$ we formulate the additional Stokes and transfer data needed to recover strict facewise structures. A multiplicative regularized Stokes trace system on Joyce's face category gives the presymplectic intrinsic BV-BFV identity and incidence descent. The finite datum (G4) is a linear subdivision/aggregation transfer criterion; (G5) is a separate cyclic field-level criterion for classical abelian BF theory. Neither is a general existence theorem, and no general strict multiplicative intrinsic trace is constructed here. We prove the interval contractions needed for codimension-two collars and analyze the positive real conifold through the common star refinement of its two diagonal resolutions. On an explicit finite product-Whitney logarithmic class the exceptional square satisfies (G4) after normal-face totalization, and the transferred differential is exactly the signed intrinsic incidence differential. Its finite algebraic BF dual gives a cyclic logarithmic-cellular shadow, but not the full continuum intrinsic $b$-de Rham datum (G5). General nonlinear continuum pushforward, existence of the intrinsic multiplicative trace, and loop-level logarithmic graph integrals remain open.

Let $B_1^n$ be the standard cross-polytope in $\mathbb R^n$, let $g_1,\ldots,g_m$ be independent standard Gaussian vectors in $\mathbb R^n$, and set $G_m=\operatorname{conv}{\pm g_1,\ldots,\pm g_m}$. For $m=n^3$ it is proved that $$ \mathbb P\left\{d_{\mathrm{BM}}(G_m,B_1^n)\geqslant c n^{5/8}(\ln n)^{-5/8}\right\}\geqslant 1-\frac2n $$ for a suitable absolute constant $c>0$. This independently improves the polynomial exponent $4/7$ in Friedland's preceding work. Independent concurrent work of Friedland, which appeared after completion of the present manuscript, obtains the same polynomial exponent with the stronger logarithmic factor $(\ln n)^{-1/4}$ by a different argument. The proof uses Friedland's discretization and conditioning argument together with the $K/U$ decomposition. A selected family of $K$ vectors is suppressed and the remaining $K$ vectors are quotiented out. In the resulting quotient simultaneous bounds are proved for every top-dimensional exterior product formed from the suppressed $K$ vectors and the $U$ vectors. A Dvoretzky-Rogers selection after L"owner normalization converts these determinant estimates into a bound for the minimum volume ellipsoid of the whole projected polytope and Maurey's empirical method then gives the required Gaussian measure estimate.

The bosonic thermal-loss channel is a fundamental model for quantum communication. This Gaussian channel models the transmission of bosonic systems subject to both loss and thermal noise. It describes many practical systems, including optical fibers, waveguides and free-space links, where background thermal noise is significant. Determining the channel's two-way-assisted quantum capacity, the maximum rate at which quantum information can be transmitted reliably through the channel with two-way classical assistance, remains an open problem. Here, we improve upon the best known lower bound for this capacity for a wide range of channel parameters. We achieve this by combining an improved qubit-bosonic distribution technique with a recently introduced technique for discovering entanglement distillation protocols.

We resolve two questions posed by Huang and Ye concerning full-scattering systems. First, we construct a four-symbol subshift which is $\Delta^*$-mixing but not full-scattering, thereby showing that $\Delta^*$-mixing does not imply full-scattering. Second, we prove that every finite product of full-scattering systems is full-scattering.

We study centered quadratic variations of fractionally filtered generalized Hermite processes. For every chaos order $q\ge2$, every admissible anisotropic monomial kernel $g_{\boldsymbol\gamma}$, and every fixed $0<h\le1/2$, we prove that $N^{2h+\theta-1}V_N$ converges to a Rosenblatt process with self-similarity exponent $1-\theta$, without subtracting any chaos projection. Thus the visible roughness $h$ determines the normalization, whereas the latent singularity exponent $\theta$ determines the memory of the limit. The proof uses a sharp paired-filter threshold and charged diagram power counting. At $h=0$, the bare kernel has logarithmically divergent energy. Variance-normalized $h$-regularization and a hard endpoint cutoff have the same residue and converge in finite-dimensional distributions to the universal Gaussian field $G_t=(\zeta_t-\zeta_0)/\sqrt2$. Although $G$ has no stochastically continuous modification, the iterated boundary-first quadratic energies converge to $\sqrt3B$. In the isotropic case, simultaneous limits $h_N\downarrow0$ are governed by $\lambda_N=h_NN^{1/2-\theta}$: $\lambda_N\to0$ yields a Brownian limit, $\lambda_N\to\lambda\in(0,\infty)$ yields an independent Brownian-Rosenblatt sum, and $\lambda_N\to\infty$ yields a Rosenblatt limit after division by $\lambda_N$. We also obtain an explicit polynomial rate in fixed Malliavin-Sobolev norms and functional convergence in little-Hölder spaces.

For a graph $G$ with $3\mid e(G)$, let $R(G,\mathbb Z_3)$ denote the least integer $N$ such that every edge weighting $w:E(K_N)\to\mathbb Z_3$ admits a copy of $G$ whose edge weights sum to zero. We prove that every $n$-vertex graph $G$ with $n\ge6$ and $3\mid e(G)$ satisfies $R(G,\mathbb Z_3)\le n+5$. Moreover, the stronger bound $R(G,\mathbb Z_3)\le n+4$ holds whenever $G$ contains an induced $P_4$ or an induced $2K_2$. Consequently, we resolve a problem posed by Caro and Mifsud: every graph $G$ with $3\mid e(G)$ satisfies $R(G,\mathbb Z_3)\le |V(G)|+8$, with equality if and only if $G\cong K_3$.

Adaptive coders are commonly primed with training data, turned into context tables or shipped dictionaries. The training data generally do not match the source being encoded, and the coder must decide how much to trust them. We make that trust a design variable: a Krichevsky–Trofimov (KT) estimator over an $m$-ary alphabet whose training counts are scaled by $\xi\in[0,1]$. For a training sequence of length $\ell$ at Kullback–Leibler divergence $D$ nats per symbol from the message source, the redundancy-minimizing weight is $\xi^*=d/(2\ell D+d)$, $d=m-1$. The effective training length $\xi^*\ell$ follows a harmonic law: any mismatch caps the usable training information at $d/(2D)$ symbols, however much was collected. Three implementable selectors supply the unknown $D$: offline, from the spread of the training set; by a plug-in loop, from the decoded prefix; and by a twice-universal mixture, with no estimation at all. On the ten English texts of the Calgary and Canterbury corpora, weighting removes up to $22.7\%$ of the redundancy of the classical KT code, the $\xi=0$ endpoint, and beats both endpoints on every file. An open-source implementation is provided.

We develop a Hamilton–Jacobi theory for finite-horizon optimal control problems governed by state-dependent delay equations. In this setting, the delayed-time map depends on the controlled trajectory itself, so that both the state and the point at which the past state is evaluated vary simultaneously. The natural state variable is therefore the entire history, and the value functional is defined on a space of Lipschitz histories. Under suitable growth, Lipschitz, and monotonicity assumptions, we establish the dynamic programming principle and characterize the value functional as the unique viscosity solution of the associated Hamilton–Jacobi equation with co-invariant derivatives. To accommodate Lipschitz histories, we introduce a notion of viscosity solution based on finite-dimensional projections generated by polygonal extensions, and prove a comparison principle in the resulting class of functionals. Under additional regularity assumptions, we derive a Pontryagin minimum principle whose adjoint equation contains advanced terms induced by the state dependence of the delay. We also obtain a generalized transversality relation formulated through the delay superdifferential of the value functional. Finally, we establish semiconcavity in the history variable and a joint semiconcavity estimate on an appropriate solution manifold.

Recently, in the study of topological properties in Domain theory, the first countable spaces and Frechet-Urysohn spaces receive close attention and are frequently employed. In this paper, we focus on sequential spaces weaker than Frechet-Urysohn spaces. The main results are: (1) For a well-filtered space X, the Scott topology and the upper vietoris topology on K(X) coincide if PS(X) is a sequential space; (2) every {\omega}-well-filtered coherent d- space X is sober when the procuct of X is a sequential space.

In a remarkable breakthrough, Faulhuber, Petersen, van Velthoven, and Voigtlaender disproved the Heil–Ramanathan–Topiwala conjecture by constructing twelve linearly dependent time–frequency shifts of a Schwartz function. The following day, the author compressed their vector-Zak and cohomological architecture to an explicit intrinsically subcritical four-point configuration, thereby attaining the minimum possible cardinality. Soon thereafter, Jasper and Mixon gave a transparent analytic refinement of the breakthrough mechanism and proved, for every Diophantine pair, that HRT counterexamples may be realized with windows in the Roumieu Gelfand–Shilov class $\mathcal S_1^1(\mathbb R)$; their theorem establishes the endpoint regularity phenomenon in broad arithmetic generality while leaving support cardinality unrestricted. The purpose of the present paper is to show that these two sharp advances can be attained simultaneously. We prove that the same explicit four points, coefficients, and matrix cocycle admit a nonzero window in $\mathcal S_1^1(\mathbb R)$. Moreover, if $0\ne f\in\mathcal S_s^\sigma(\mathbb R)$ and $s<1$ or $\sigma<1$, then every finite system of distinct time–frequency shifts of $f$ is linearly independent. Thus four is the least possible cardinality and $(1,1)$ is the sharp coordinatewise threshold within the Roumieu Gelfand–Shilov scale.

We give an exact computer-assisted counterexample to a direct common-degree quartic extension of the affine active-network shape theorem for demand-dependent Price of Anarchy. The instance is a directed network with five vertices, six edges, and three origin–destination paths, all with positive rational costs $c_e(x)=a_e+b_ex^4$. Exact rational interval certificates show that all three paths carry positive Wardrop flow throughout the demand interval $[17,24]$, while \[ \operatorname{PoA}(21)>\operatorname{PoA}(17),\qquad \operatorname{PoA}(21)>\operatorname{PoA}(24). \] Continuity therefore forces an interior local maximum despite a constant equilibrium active network. Krawczyk inclusions isolate the Wardrop and social-optimum KKT solutions, elementary boundary inequalities certify constant support, and interval social costs certify both strict comparisons. At differentiability points, we also derive a serial-edge identity explaining how a common quartic edge can alter the derivative of PoA without changing either route split. The proof and certificate use exact rational arithmetic for every interval and sign decision.

Modern supply chain networks increasingly rely on real-time data to navigate structural uncertainties, market volatility, and operational disruptions. This manuscript bridges the gap between statistical data-driven learning and robust decision-making frameworks in logistics and operations management. We present a comprehensive, mathematically rigorous treatment of supply chain analytics, moving from empirical demand forecasting to optimal inventory and network control under uncertainty. Key topics explored include sample minimization, dynamic programming recursions for time-varying inventory replenishment, network fulfillment frameworks, and advanced distributionally robust optimization (DRO) via transport theory to hedge against rare events. By integrating predictive statistical models with prescriptive control algorithms, such as column generation for vehicle routing and non-homogeneous queueing regimes, this text provides the foundational tools necessary for designing resilient, data-driven automated systems. It serves as both a theoretical blueprint and an algorithmic guide for researchers and practitioners operating at the intersection of machine learning, mathematical optimization, and applied probability.

This work develops tensor-train (TT) formulations for solving large-scale three-dimensional linear elasticity problems discretized by isogeometric analysis. By exploiting the tensor-product structure of the basis functions and the low-rank structure of geometry-dependent coefficient fields, the stiffness operator, mass operator, force vector, and displacement solution are represented in TT format. Two solution strategies are investigated: a block-operator formulation, in which the coupled elasticity operator is stored as separated TT blocks, and a single-operator formulation, in which the full coupled system is stored as one monolithic TT operator. A matrix-free three-field TT conjugate-gradient solver is introduced for the block formulation, while AMEn is used for the single-operator formulation. Numerical examples demonstrate substantial compression of both operators and solutions compared with conventional sparse full-grid representations, showing that TT-based formulations provide an efficient and scalable approach for large-scale three-dimensional elasticity simulations.

How a quantum state responds to small parameter changes underlies state distinguishability, many-body response, and metrological sensitivity. The Pauli probability spectrum provides a natural description of many-body quantum states, but it is not evident how much of this response survives in the corresponding classical distribution. Here we show that, for pure states, the complete labeled Pauli distribution retains the full local metric of the pure-state manifold, reproducing the quantum Fisher metric up to a fixed normalization. This correspondence has a direct measurement realization: preparing a state together with its complex conjugate and performing pairwise Bell measurements between corresponding degrees of freedom converts the quantum response into an ordinary classical distribution without losing any of the sensitivity available in the two inputs. The readout is fixed, factorizes into local pairwise measurements even for strongly entangled many-body states, and preserves the response along all parameter directions simultaneously. We trace this correspondence to the reality of the underlying operator amplitudes and show that the same mechanism extends beyond qubits. When the correspondence fails, we identify explicitly where the missing information resides: part of the response can remain hidden in phases for identical-copy and complex-coordinate readouts, while mixed states develop transverse operator-space motion that is invisible to the Bell probabilities. These results provide a direct link between Pauli/Bell statistics, many-body response, nonstabilizerness, and quantum geometry.

The quantum tunneling process is a fundamental concept that presents the unconventional nature of quantum dynamics. The main anomaly of this effect revolves around the choice of imposing the classical 4-momentum to become imaginary during the barrier traversal. This paper addresses this paradox by introducing a complex-time parametrization, which uses a complex temporal coordinate, $\tau = t + i\kappa$. Here, the real axis $(t)$ represents standard thermodynamic time, while the imaginary axis $(i\kappa)$ is correlated to spatial evanescence. We demonstrate that evaluating this extended complex-time framework yields a hyperbolic section of the causal lightcone, proving that the traversal is strictly bounded by the speed of light and preserves causal disconnection. Applied to quantum tunneling, we resolve the anomaly of imaginary momentum through a continuous geometric signature flip of the effective complex-time momentum norm from a timelike to a spacelike state. This theoretical model finds a physically phenomenological analogy in microwave scattering experiments using open, non-Hermitian systems, which link complex transmission time delay to spatial wave decay.

In this paper, we investigate the algebraic counterpart of the Fundamental Theorem of Algebra. We explore the concept of real closed fields and quadratic forms. We show, by means of Galois theory, that $F(\sqrt{-1})$ is algebraically closed if $F$ is real-closed. Lastly, we explain the algebraic closure of $\mathbb R(\sqrt{-1})=\mathbb C$ by demonstrating the real-closeness of $\mathbb R$.

For a formally real field $F$, we study totally positive field extensions $K$ over $F$. We show that, if $K /F$ is Galois and totally positive then so is the corresponding extension of their pythagorean closures $K_{\rm py}$ over $F_{\rm py}$. We also study the behaviour of weak isotropy and weak hyperbolicity of central simple algebras with an orthogonal involution over totally positive field extensions. We use some of these results to prove new cases for which a conjecture due to Becher holds.

Motivated by the geometric properties of curved tubes $T\left(P,a\right)$ defined by closed curves $P$, we compute the Cheeger constant $h\left(T\left(P,a\right)\right)$ in the real space forms of arbitrary dimensions with constant sectional curvature. The structure and properties of the system of Fermi coordinates allows us to parametrize the curved tube and straightforwardly compute the upper bound of $h\left(T\left(P,a\right)\right)$ using the exact formulas for the area and volume of $T\left(P,a\right)$. Next, we derive the lower bound by combining the geometry of tubes with a calibration-type argument. The key idea is to describe the tube using geodesic spheres moving along the underlying curve $P$, which provides a natural outward direction and makes the estimate geometrically transparent. This allows us to compute the lower bound via the divergence theorem. Finally, for the class of unbounded curved tubes in noncompact real space forms, we also compute the Cheeger constant and prove that there is no finite-volume Cheeger set.

We exhibit two explicit circulant graphs of prime order $251$ that are $K_4$-free and have independence number $19$. Consequently \[R(4,20)\ge 252.\] These improve the bound $R(4,20)\ge 237$ given by Nagda, Raghavan, Thakurta and the long standing bound $R(4,21)\ge 242$ recorded in Radziszowski's dynamic survey. The graphs are $32$-subsets of a pair of undirected quintic cyclotomic classes modulo $251$, in analogy with the quartic-residue circulant of order $313$ used for $R(4,22)$. Clique-freeness is elementary; the independence-number claims are certified by a bitset branch-and-bound on the $186$-vertex residual of a vertex.

We study integrable variable-mass sine-Gordon model (vmSG) with point defects. Using the Lax and Bäcklund-gauge formulations, we construct type-I and type-II defect matrices, derive the corresponding sewing conditions, and generate the bulk and defect contributions to an infinite hierarchy of conserved charges. The lowest members reduce to the standard sine-Gordon energy and momentum in the homogeneous limit, while for inhomogeneous backgrounds they define integrability-generated energy- and momentum-type quantities. Defect compatibility imposes matching conditions on the variable-mass functions across the defect. An analytical transmission factor $z$ characterizes soliton transmission, topological conversion, and absorption/emission processes. We then deform the type-I sewing conditions by parameters $\alpha$ and $\beta$ and derive the associated defect anomalies. Consistent one-soliton transmission is recovered for $\alpha \beta =1$, whereas for $\alpha \beta \neq 1$ parity-centered kink-kink and kink-antikink transmissions display vanishing lowest order integrated anomalies but no generic vanishing at higher order. Numerical simulations reproduce the integrable transmission/conversion regimes and show that non-integrable defects generate small residual disturbances and lowest order quasi-conserved charges, while quasi-conservation of the higher order charges remains unestablished. These results elucidate how exact defect integrability deforms into charge-dependent quasi-conservation and establish a framework for defect-controlled soliton transport in inhomogeneous media, with potential applications to nonuniform Josephson junctions, magnetic and nonlinear-optical systems, and effective molecular and DNA models.

Let $\mathcal O$ be a nonzero nilpotent orbit in a complex simple Lie algebra $\mathfrak g$. For $\mathfrak g$ of classical type, we classify the orbit closures $\overline{\mathcal O}$ that are isomorphic to $(T^*X)^{\mathrm{aff}}$ for a smooth quasi-affine variety $X$. In types $G_2$, $F_4$, and $E_8$, we show that no nonzero nilpotent orbit closure admits such a cotangent model.

Contrastive learning has become a cornerstone of modern representation learning, powering CLIP-style models that underpin text-to-image generation, vision-language models, and retrieval across a rapidly growing range of modalities. Despite this empirical success, the expressive power of these architectures remains poorly understood. To gain insight, we study expressivity by adopting a population-level, density-estimation viewpoint: each architecture comprises a parameterized set of densities whose parameters may be chosen to approximate the joint distribution of the modalities. This isolates a question of pure representational capacity: which joint distributions can a given contrastive family of parameterizations approximate to arbitrary accuracy? We show that expressivity is sharply architecture-dependent. For two modalities, the simple two-tower CLIP architecture is a universal approximator. A natural generalization of CLIP, widely used in practice when three or more modalities are present, is based on a loss found by summing over all pairwise similarities. This provably cannot represent arbitrary joint distributions, although we prove that it remains expressive enough to match all pairwise conditionals. Motivated by this gap, we propose Hadamard-CLIP, which adds a single learned weight vector on top of the existing encoders and restores universal approximation of the joint for any number of modalities while preserving CLIP's fast, precomputable-embedding retrieval.

Multipartite entanglement remains a challenging and not fully understood aspect of quantum information. Monogamy-of-Entanglement (MoE) games have been highly effective for studying limitations on the usefulness of entanglement imposed by monogamy constraints. To better reveal the extent to which multipartite entanglement can be useful, we introduce a class of quantum guessing games, termed One-at-a-Time Guessing (OTG) games. In these games, quantum players individually guess the outcomes of random measurements performed by a referee on a pre-shared entangled state. Unlike MoE games, OTG games select players individually at random according to a specified probability distribution, thereby probing each player's correlation with the referee. We show that, despite monogamy constraints, players sharing certain entangled states can moderately outperform those relying only on classical uncertainty. This advantage arises even in simple OTG games involving only Pauli measurements on qubits, where optimal entanglement increases the winning probability by at least 4%. This contrasts with MoE games, where shared entanglement has been shown in several settings to provide only limited (if any) advantage over classical strategies. We further establish a majorization property: the value of an OTG game respects the majorization ordering of the player-selection probability distribution. We also analyze in detail a two-player OTG game in which the referee measures one of the three Pauli observables on a qubit, and show that it is optimally played using a specific parameterized family of three-qubit $W$-like states. These results suggest that OTG games provide a useful framework for investigating the usefulness of multipartite entanglement in multiparty quantum correlations.

We establish a uniform asymptotic formula for the smoothly weighted shifted second moment of the Gaussian angular Hecke $L$-functions $L(s,\lambda^k)$. The completed moment is expressed as the sum of four explicit main terms, corresponding to the four functional-equation swaps, with an error of size $O_{\Phi,\epsilon}(K^{1/2+\epsilon})$.After the Archimedean factors are removed, the resulting formula agrees with the numerator-only specialization of the four-swap prediction of the $L$-functions Ratios Conjecture. The proof transforms the off-diagonal contribution into Weyl sums over the roots of $r^2\equiv-1\pmod C$, realizes these sums spectrally through incomplete Poincaré series evaluated at $i$, and separates the Eisenstein and Maaß spectra. The two $v$-type main terms arise respectively from the zero frequency and from the combined residues of two moving Eisenstein poles, while the cuspidal spectrum is absorbed into the square-root error term.

Contact Hamiltonian dynamics gives dissipative mechanics an intrinsic action variable, but explicit contact splittings reach only kinetic energies whose terms are exactly integrable: frozen-coordinate diagonal metrics (the spherical pendulum, a torus particle) are included, while dense metrics with momentum cross terms, with the double pendulum as flagship, are not. We introduce a projected Pihajoki-contact integrator for this non-separable setting, combining phase-space duplication, symmetric projection onto the physical diagonal, and constant-friction damping half-steps, with the action factor carried by an exact Herglotz update. As in the projected extended-phase-space framework it builds on, the construction needs no binding parameter, returns the copies to the diagonal at every step, and confines the nonlinear solve to the $2n$ projection variables. For constant friction the step rescales $\omega=d\eta$ by the exact factor $e^{-\gamma\tau}$ when the projection is solved exactly (a classical conformally symplectic identity, realized here for this class), while time-symmetry, consistency, and smoothness yield an $O(\tau^3)$ one-step contact-form residual, a bound not specific to the contact form. On the damped double pendulum, spherical pendulum, and torus particle the method is second-order accurate, reproduces the contact decay law, and controls long-time energy and contact drift in coarse or stiff regimes where the Tao baseline and the unprojected average lose the solution. A head-to-head with exact-contactomorphism splittings delimits the niche: where a frozen-coordinate splitting exists it preserves the contact form exactly and wins at matched cost; for the dense double-pendulum metric the realizable alternative is first-order with a prohibitive constant and the projected method prevails. The contact-form estimate is local, one-step, and constant-friction.

The notion of a regular point is extended to that of a weak regular point, and so the notion of a strong natural boundary may be described locally as the absence of weak regular points, just as a natural boundary is locally the absence of regular points. The main result is a sufficient condition for a strong natural boundary, which is the conjunction of the unboundedness of the coefficients of the underlying power series with either of two specific gap density conditions imposed on the exponents that yields a lacunary sequence.

For integers $n\ge 1$, consider assertions: $\mathbf{(P_n)}$: Let $\Omega \subset S^n$ be a spherical domain enclosed by totally geodesic $S^{n-1}$'s with non-obtuse dihedral angles. Then in $[0,2(n+1)]$, its Neumann spectrum can only take values among $\{0,n,2(n+1)\}$. Moreover, $n$ is a Neumann eigenvalue if and only if the corresponding eigenfunction is the restriction of a linear function in $\mathbb{R}^{n+1}$, while $2(n+2)$ is a Neumann eigenvalue if and only if the corresponding eigenfunction is restriction of a quadratic polynomial in $\mathbb{R}^{n+1}$. $\mathbf{(Q_n)}$: A weak solution $u$ to $\Delta u = f$ with the Neumann boundary condition, with $f$ Hölder continuous, in a conical polyhedral domain $\Omega$ in $\mathbb{R}^n$ with non-obtuse dihedral angles, is in $C^{2,\alpha}_{loc}(\overline{\Omega})$. We prove the implications \[\mathbf{(Q_n)} \Rightarrow \mathbf{(P_n)},\qquad \mathbf{(P_n)}\Rightarrow \mathbf{(Q_{n+1})}.\] Consequently, both assertions hold in all dimensions. These give the optimal Neumann eigenvalue lower bound and $C^2$ elliptic regularity in non-obtuse Riemannian polyhedral domains.

Thompson's group $F$ is the group of all piecewise-linear homeomorphisms of the unit interval whose breakpoints are dyadic and whose slopes are integer powers of $2$. If $H$ is a finitely generated subgroup of a finitely generated group $G$, its distortion measures the difference between the intrinsic word metric of $H$ and the metric induced from $G$. The subgroup is undistorted when these metrics are equivalent. Guba and Sapir asked whether $F$ contains a distorted copy of itself. The same question was also suggested by Brin. We answer this question negatively: every subgroup of $F$ isomorphic to $F$ is undistorted in $F$.

We prove finite-time singularity formation for the forced generalized surface quasi-geostrophic equation in the singular velocity regime $\gamma\in(0,1)$, where $\gamma=0$ corresponds to SQG. For every such $\gamma$, we construct a smooth, compactly supported initial datum and a time-dependent force $F$ for which the corresponding solution $\theta$ is classical on $[0,1)$ with finite energy for all times, but loses Sobolev regularity at $t=1$. More precisely, there exists \[ \kappa_0>2+\gamma+\frac{\gamma^2(1-\gamma)}{25(4+\gamma)}, \] such that the force satisfies $F\in L^1([0,1];H^\kappa(\mathbb{R}^2))$ for every $\kappa\in[2+\gamma,\kappa_0]$, whereas \[ \lim_{T\nearrow1}\int_0^T\|\theta(\cdot,t)\|_{H^\kappa}\,dt=\infty \] for every exponent in the same interval. At the same time, the solution remains uniformly bounded in $H^{\kappa_1}$ throughout its lifespan for any \[\kappa_1\in\left[0,2+\gamma-\frac{\gamma(1-\gamma)}{2(4+\gamma)}\right].\] Then, the singularity occurs within the Sobolev well-posedness regime and cannot be attributed to insufficient regularity of the force or the initial conditions. To the best of our knowledge, this is the first finite-time blow-up result for classical finite-energy solutions of the generalized SQG equations in a well-posedness regime.