Papers

In this work, we establish a criterion for classifying elliptic curves that have no solutions in the set of positive integers. In particular, we prove that if $E: y^2 = Ax^3 + Bx^2 + Cx + D$ is an elliptic curve where $A$, $B$, $C$, and $D$ are coefficients of the curve, and $p$ is a prime number where $p \equiv 1 \pmod{4}$, and $(x,y)$ is a positive integer point, then if $$ -3A + C \equiv -1 \pmod{4}, $$ $$ A - B + C \equiv -1 \pmod{4}, $$ and $$ p^2 = A - B + C - B, $$ then there are no positive solutions for the curve $E$, where $E(\mathbb{Z}^{+}) = \varnothing$. Furthermore, we generalize this criterion by incorporating techniques from linear algebra. Specifically, we introduce a matrix construction to establish the nonexistence of positive integer solutions for a family of elliptic curves.

Let $f_{0,1,4}(n)$ denote the number of partitions of $n$ into parts congruent to 0, 1 or 4 modulo 5 with each part used at most twice, and let $f_{0,2,3}(n)$ be defined analogously for parts congruent to 0, 2 or 3 modulo 5. We prove that for every prime $p \equiv 3 \pmod 4$ there are explicit non-negative integers $a(p)$ and $b(p)$, determined by $20a(p) \equiv -9$ and $20b(p) \equiv -1 \pmod{p^2}$, such that $f_{0,1,4}(p^2m + a(p))$ and $f_{0,2,3}(p^2m + b(p))$ are congruent modulo 3 to $f_{0,1,4}(m)$ and $f_{0,2,3}(m)$ when $p \equiv \pm 1 \pmod 5$, and to $f_{0,2,3}(m)$ and $f_{0,1,4}(m)$ when $p \equiv \pm 2 \pmod 5$. By quadratic reciprocity the two functions are preserved exactly when 5 is a quadratic residue modulo $p$, and interchanged otherwise. The smallest cases are $f_{0,1,4}(9m) \equiv f_{0,2,3}(m)$ and $f_{0,2,3}(9m+4) \equiv f_{0,1,4}(m)$. The proof reduces each generating function modulo 3 to the square of a Rogers-Ramanujan-type theta function by means of $(1-x)^{\ell} \equiv 1-x^{\ell} \pmod{\ell}$ and the Jacobi triple product, and then dissects the resulting binary quadratic form using the fact that $-1$ is a quadratic non-residue modulo $p$. As corollaries we obtain, for each such $p$, a congruence with exceptions on a progression of modulus $p$, and an infinite family of self-similarity congruences on progressions of modulus $p^{dn}$ with $d = 2$ or $d = 4$ and constants $9(p^{dn}-1)/20$ and $(p^{dn}-1)/20$.

The preprint arXiv:2609.04176 (Z.-W. Sun, 3 September 2026) claims a proof that Catalan's constant $G$ is irrational: it constructs for each $B$ a number $\hat q_B$ from weighted tails of the series and asserts that, if $G=a/q$, the integer $N_B=q^S H_B^{\min}\hat q_B$ satisfies $0<|N_B|<1$ for large $B$ (its Theorem 9.1). We report four exact computations. First, the printed proof of the rank theorem (Theorem 2.1) carries $T_i$ where its own recurrence requires $T_{i+1}$; the slip is repairable, and the statement is verified exactly at 44 parameter pairs with $S\le 4$, $B\le 30$. Second, with the paper's definitions and the worst-case denominator, attained at an explicit rational $a/q$, a lower bound for the quantity the derivation of Theorem 9.1 controls equals $+1.49$ to $+1.79\,B^2$ at eighteen indices $20\le B\le 119$, where the derivation claims at most $-0.00966\,B^2+o(B^2)$. Third, the paper's own denominator bound (Corollary 5.2) holds in every tested instance and is nearly sharp, so the discrepancy lies in the asymptotic ledger of Sections 6-9. Fourth, an exact identity locates it: the prime 2 contributes $v_2(F_D)\log 2=(2\log 2+o(1))B^2$ to the quantity, because the positive part at 2 vanishes while $|\hat q_B|$ carries the full power of 2 in $F_D$; no display in the ledger carries a term of this order, and Remark 9.3 assigns it to the derivation of the odd-prime constant $c_{\rm odd}=0.006$ without showing how. At the tested parameters the odd-prime layers agree with the paper's constants to within about $0.1\,B^2$, and the remainder of the discrepancy is the size of the residual minor at those indices. These computations test the paper's bounding method, which is uniform in $(a,q)$; they show that the published derivation does not justify the claimed estimate, so the proof of the main theorem is incomplete. Nothing is concluded about the irrationality of $G$.

This tutorial provides a unified view of the emerging area of bilinear learning and control. Using linear systems as a benchmark, it explains what fundamentally changes in the bilinear settings, how recent theory addresses finite-sample learning and control, and how these ideas connect to broader themes in nonlinear control, representation learning, and data-driven decision making. For learning, we emphasize tools that are particularly useful in the bilinear settings, such as one-sided Bernstein's inequality for dependent and heavy-tailed covariates, blocking arguments, and martingale concentration for input-dependent noise. We then apply these tools to obtain finite-sample learning guarantees for fully observed bilinear systems, partially observed bilinear systems, and linear systems with bilinear observations. For control, we discuss quadratic control from bilinear observations, where the classical separation principle fails, and review tractable approaches based on belief-space receding horizon control. We also cover stabilization of bilinear dynamics under state feedback using semi-definite programming, LMI relaxations, sum-of-squares methods, and Koopman-based lifting. We conclude by discussing connections to reinforcement learning and machine learning, and some open problems in combined learning and control of bilinear systems.

We prove the irreducibility conjecture for Legendre polynomials proposed by Stieltjes: for every integer $j\geq 1$, both $P_{2j}(x)$ and $P_{2j+1}(x)/x$ are irreducible over the rational numbers. The proof proceeds by contradiction. Starting from a hypothetical nontrivial factorization, we remove the zero root from $P_n$ when present, multiply by a suitable constant, and express the resulting polynomial as a product of two integral polynomials $A(x^2)$ and $B(x^2)$. We then construct the resultant and its odd part, $R=\operatorname{Res}_t(A,B), \mathcal{R}=\frac{|R|}{2^{v_2(R)}}.$ Put $k=\operatorname{deg} A\leq\operatorname{deg} B$ and $m=\lfloor n/2\rfloor$. Using the classical auxiliary polynomial $U_n$ and the differential identity $(1-x^2)(P_n'U_n-P_nU_n')=1-P_n^2,$ together with orthogonality, divisibility properties of the coefficients, and a least-common-multiple estimate, we obtain an upper bound for $\mathcal{R}$. On the other hand, the Legendre differential equation gives a lower bound for the absolute value of the derivative of $AB$ at each root of $A$. We use Chebyshev polynomials and Hadamard's inequality to bound the product of the squared pairwise differences of these roots, and combine this with estimates for the leading coefficients and the power of $2$ in $R$ to obtain a lower bound for $\mathcal{R}$. These estimates yield $m\log4-\log\frac{4(m+1)^2}{\sqrt m} <\frac{\log\mathcal R}{k} <(m+\sqrt{2n-1})\log3 (n\ge64).$ For every $n\ge256$, the lower bound strictly exceeds the upper bound, giving a contradiction. Combining this with established irreducibility results and explicit integer comparisons for the remaining degrees proves irreducibility in every degree.

We define a rank-two extension $\mathcal{H}^{(2)}$ of the spherical $\mathcal{H}$-transform, together with its associated transform $\mathcal{R}^{(2)}$. For sums, both transforms are additive. The transform $\mathcal{R}^{(2)}$ yields a $4\times4$ formula for the average product of two hermitized resolvents, and hence for off-diagonal eigenvector overlaps, recovering the Ginibre, elliptic, and bi-invariant formulas. We also treat correlated pairs of matrices, and obtain overlaps between their eigenvectors or singular vectors.

Using ideas of Digne and recent work of Gaitsgory–Rozenblyum–Varshavsky, we show that Lusztig induction commutes with (a minor variant of) the Glauberman correspondence for large prime degree field extensions. This allows one to reduce the Mackey formula in Deligne–Lusztig theory to the case of “large” base fields, in which case the result is due to Bonnafé.

We prove Stieltjes' common root conjecture: Legendre polynomials of distinct degrees have no common nonzero root. We construct an auxiliary polynomial and show that, for the relevant power of two $q\ge2$, the edge of slope $1/q$ in its $2$-adic Newton polygon has horizontal length less than $3q$. If a common nonzero root existed, Newton polygon theory and the Legendre differential equation would force the same edge to have horizontal length at least $3q$, yielding a contradiction.

For an infinite product P(q) = prod_{m>=1} (1-q^m)^{eps(m)} and real delta, write P(q)^delta = sum_{n>=0} c_delta(n) q^n. Schlosser and Zhou conjectured precise sign patterns for these coefficients, for several products and ranges of delta. We resolve four of their conjectures completely, in each case extending partial results already in the literature: those for the Goellnitz-Gordon product Q_8 = (q,q^7;q^8)_inf/(q^3,q^5;q^8)_inf (Conjecture 21), for Q_12 = (q,q^11;q^12)_inf/(q^5,q^7;q^12)_inf (Conjecture 24), and for the Borwein products G_7 and G_11 (Conjectures 20 and 23). Conjecture 23 is true. Conjecture 21 fails exactly on [beta, 8/3), where beta is approximately 2.66448; the threshold 8/3 is sharp, and the first counterexamples occur near n = 7.6 x 10^5. Conjecture 24 fails exactly on (delta_1, 0), where delta_1 = -0.64411... is a root of the 22nd coefficient polynomial. Conjecture 20 fails exactly on (delta_c, 5), where delta_c = 4.8735075853867342634... is a root of the 897th coefficient polynomial. All other stated ranges are proved. Parts of Conjecture 21 were settled independently by He and Li; what is new here is the range near delta = -1, where their threshold diverges. Each failure has the same mechanism: a leading circle-method amplitude vanishes, and a secondary term with the wrong sign overtakes it. The proofs are self-contained apart from classical facts. The cusp analysis is exact, via finite orbits of Weil representations; the Hardy-Ramanujan-Rademacher expansion is made fully explicit; and certified computations in exact and ball arithmetic handle the finite ranges and the degenerate regimes.

Despite the widespread use and success of generative AI techniques today, theoretical guarantees on learning a distribution supported in $d$ dimensions from $n$ samples degrade as $O(n^{-1/\Theta(d)})$, though shown to be minimax optimal. We hypothesize that present bounds are too pessimistic because smoothness assumptions are not enough to capture the structure of distributions that often appear in real applications. Consequently, we introduce the class of sparse priors and define the "Sparse Dimension" as a measure of sparsity of a prior over the space of all distributions. We show that distribution learning under a $k$-sparse prior achieves a Bayesian risk lower bound of $\Omega(\sqrt{k/n})$ under common distance metrics, and show a matching (up to logarithmic terms asymptotically in $n,k$) upper bound for the TV distance under mild additional assumptions. We show the statistical equivalence of distribution learning and learning to sample in the Bayesian setting so that our results apply to learning to sample as well. While $k$ can still depend on the dimension $d$, or a notion of intrinsic dimension, our results show that learning under an appropriate prior overcomes the curse of dimensionality with respect to the dependence on $n$.

We distinguish two regimes in task-oriented quantization with a known deterministic oracle action. For an unconstrained interior oracle and a smooth strongly concave utility, quantizing the oracle action by vector Lloyd-Max minimizes a mean-squared-error surrogate and achieves a $\beta/\alpha$ approximation to the optimal $K$-level task quantizer. The reduction is exact for isotropic quadratic loss, and the corresponding task rate–distortion function is bracketed by two ordinary rate–distortion functions. Budget-constrained quadratic scheduling is different: the oracle satisfies a variational inequality, so quantizing water-filled actions is not generally optimal. We derive the exact Lloyd-type conditions for this case. The optimal action for a quantizer cell is water-filling evaluated at the cell's conditional-mean load, and the optimal encoder partitions load space into affine power-diagram cells. Thus the correct prescription is to quantize the load and water-fill its centroid. The distinction is material whenever a cell crosses water-filling active-set boundaries.

Word-representable graphs are a class of graphs that can be represented by words, where edges and non-edges are determined by the alternation of letters in those words. The tensor product $G \times H$ (also known as the direct product or Kronecker product) is one of the four standard graph products. Problem 7.2.5 in Kitaev and Lozin's book Words and Graphs (Springer, 2015) raised three open questions regarding the word-representability of tensor products. Despite extensive research on word-representable graphs, the word-representability of tensor products seems to have received no attention. This paper not only answers 2.5 of the questions posed by Kitaev and Lozin in Problem 7.2.5, but also initiates a systematic study, focusing on four fundamental families: wheel graphs $W_n$, complete graphs $K_n$, the Mycielskian of the cycle graph $\mu_n$, and the extended Mycielskian of the cycle graph $\mu'_n$. Among our main results, we prove that $W_{2n} \times G$, $\mu_{2n} \times G$, and $\mu'_{2n} \times G$ are always word-representable for any graph $G$; that $K_n \times K_m$ is word-representable if and only if $\min\{n,m\} \leq 3$; and that tensor products $G \times H$ containing $W_{2n+1} \times W_{2m+1}$ or $\mu_{2n+1} \times \mu_{2m+1}$ or $\mu'_{2n+1} \times \mu'_{2m+1}$ as induced subgraphs are non-word-representable. Our proofs exploit the hereditary nature of non-word-representability and the presence of non-comparability neighbourhoods.

We study the action of quantum channels on states on a von Neumann algebra $\mathcal{M}$ with conserved quantities encoded by a subalgebra $\mathcal{N}$. We define a phase as an equivalence class of normal states on $\mathcal{M}$ under the action of quantum channels preserving $\mathcal{N}$. First, we establish a relative quasi-entropic characterization of phase equivalence using the Petz recovery map. We further introduce the operator algebra (OA) symmetry of a state and define an equivalence relation between OA symmetries via bishifts of biprojections. When $\mathcal{N}\subseteq\mathcal{M}$ is an irreducible finite-index subfactor of type II$_1$, we prove that phase equivalence implies OA symmetry equivalence. Consequently, OA symmetry breaking provides a sufficient mechanism for detecting phase transitions. Furthermore, combining finite-index subfactor theory with quantum Fourier analysis, we prove that the cardinality of each phase is finite and obtain an explicit upper bound depending only on the Jones index.

For any $0\le\gamma<1/3$ we construct weak solutions $(v,B,p)$ of the ideal MHD equations with $v,B\in C^\gamma(\mathbb T^3\times\mathbb R)$, which conserve neither the total energy nor the cross-helicity and have nontrivial magnetic helicity. We also establish anisotropic Hölder bounds with distinct velocity and magnetic exponents and stronger regularity along every magnetic field line. The admissible exponents connect the Goldreich–Sridhar \cite{GoldreichSridhar1995} spatial and parallel pair $1/3, 1/2$ with the $1/4$ spatial scaling of the Iroshnikov–Kraichnan weak-turbulence picture \cite{Iroshnikov1963,Kraichnan1965}. In the spirit of Arnold's formulation of ideal hydrodynamics, a solution is regarded as a path of volume-preserving diffeomorphisms; the proof is then based on the interplay between classical convex integration techniques and geometric constructions at the level of the Lie algebra of this Lie group. Our work substantially extends the recent result of Enciso, Peñafiel-Tomás and Peralta-Salas \cite{EnPePe} and can be used to reprove \cite{GiRa} of Giri and Radu without a Newton iteration.

The Total Coloring Conjecture (TCC) is a challenging unsolved problem posed by Behzad and Vizing independently, which states that every simple graph $G$ admits a ($\Delta(G)$ +2)-total-coloring, where $\Delta(G)$ denotes the maximum degree of $G$. This conjecture has been confirmed for graphs with $\Delta(G)\leq 5$. However, for planar graphs, the only open case is $\Delta(G)=6$. It was known that planar graphs with maximum degree 6 and without 4-cycles are 7-totally-colorable. In this paper, we improve this result by showing that any planar graph $G$ of maximum degree 6, which does not contain some special 4-cycles, is 7-totally-colorable.

We present a simple argument providing a very general Bombieri–Taylor-type result to compute Bragg peaks in diffraction theory as limits. At its abstract core the argument relies on considering the Eberlein convolution as an inner product. This inner product gives an abstract version of a Cauchy–Schwarz type inequality which implies the Bombieri–Taylor-type result as well as generalizations.

We study a finite-domain event for the mirror model on the Manhattan lattice: every boundary-to-boundary trajectory crosses a marked connector an even number of times, while internal cycles are unrestricted. Our main theorem gives a finite-scale confinement criterion: if this event's probability reaches a universal threshold in one even two-square rectangle, then almost surely every trajectory in the plane is periodic. We construct an exactly normalized three-color tensor representation and prove deterministic bounds for approximate contractions using the signed partial-permutation structure of directed continuations. These tools turn numerical evaluations into rigorous finite-volume statements. At mirror density \(p=2/5\), we prove that the boundary-parity probability is nonmonotone in the rectangle width, with a decrease followed by a certified increase. These values remain below the sufficient confinement threshold and do not resolve planar localization at this density. Further experiments track how the measured finite-size minimum varies with density, compare matching-based and entrywise error bounds, and exhibit endpoint-sensitive and cooperative responses to local mirror changes.

This paper continues the study of the $K$-theory of monoid schemes, using it to give a useful definition of the higher $K$-theory of a matroid via its Bergman fan.

We reduce the leading redundancy coefficient for binary codes correcting any fixed number $t\ge2$ of deletions from $2t$ to $2t-1$. This is the first improvement to that coefficient since Levenshtein's 1965 bound. More generally, over every fixed alphabet of size $q\ge2$, codes of length $n$ correcting any combination of at most $t$ insertions, deletions and substitutions exist with redundancy $(2t-1)\log_q n+O_t(\log_q\log_q n)$. For two deletions, the upper-bound coefficient drops from $4$ to $3$, halving the gap to the lower-bound coefficient $2$. The leading coefficient $2t-1$ also holds for $t$ bounded substring edits, where both the removed and inserted strings have bounded length. The proof combines random binary linear hashing of substring counts with a two-colouring argument. A linear dependence among differences of substring-count vectors forces their local changes to overlap, and this overlap lowers the redundancy coefficient from $2t$ to $2t-1$. The result provides an edit-distance analogue of Varshamov's 1957 bound in the Hamming metric, both in the leading redundancy coefficient and in the role of linear independence in the proof.

We prove the existence of a McKean–Vlasov stochastic process with jumps associated to the nonlinear parabolic equation $\partial_t u = \Delta_p u + \Delta_p^s u$ in $\mathbb{R}^N\times(0,\infty)$, where $\Delta_p$ is the $p$-Laplacian and $\Delta_p^s$ is the fractional $p$-Laplacian. Our approach is the following: first, after proving the existence of a solution for the PDE presented earlier, we rewrite it as a nonlinear Fokker-Planck-Kolmogorov equation whose measure-valued solution is guaranteed when $p\ge4$. Then we solve the martingale problem associated to our FPKE via a new superposition principle for non-local Fokker-Planck-Kolmogorov equations, which only requires an integrability condition on the local coefficients. Finally, thanks to the martingale solution obtained, we derive the existence of a weak solution for the McKean-Vlasov's type SDE with jumps whose infinitesimal generator is a << hybrid version >> of the operator $\Delta_p+\Delta_p^s$.

This work focuses on range characterization of spherical mean transform (SMT) in odd-dimension Euclidean space. In a recent work, a simpler description characterizing the range of SMT in odd dimensions was derived. This involved symmetry relations involving certain linear ordinary differential operators acting on the coefficients of the spherical harmonics expansion of the function in the range. The sufficiency part of this characterization was shown using an existing range characterization result due to Agranovsky-Kuchment-Quinto. In the current work, we provide a direct and constructive proof showing the sufficiency of the aforementioned symmetry conditions and we obtain this without invoking any of the existing range characterization results. As an added advantage, given a function $g$ satisfying the range conditions, our approach provides an explicit construction of a function $f$ such that the SMT of $f$ is $g$.

We study the maximal multiplicative persistence of $k$-cycles in random Čech complexes. Let $f:\mathbb{R}^d\to\mathbb{R}$ be a probability density function, let ${P}_n$ be a Poisson process with intensity $nf$, and let $\Pi_{k,n}$ denote the largest death-to-birth ratio among all non-essential $k$-cycles ($1\le k \le d-1$). For the uniform distribution in the unit hypercube, it was proved in [9] that $\Pi_{k,n} = \Theta\left( \left(\frac{\log n}{\log\log n}\right)^{1/k}\right)$. In this paper we sharpen and extend this result to a law of large numbers, for a broad class of distributions. Most significantly, we show that the limiting constant depends only on $d$ and $k$, and not on the probability density $f$. Thus the extreme value of multiplicative persistence exhibits a universality phenomenon. We show that the limiting constant is determined by the asymptotic covering density of the $k$-dimensional sphere. Our proof identifies the geometric mechanism underlying maximal cycles, a persistent isoperimetric inequality, which gives sharp bounds on the number of points needed to generate a highly persistent cycle. By combining covering-density estimates with isoperimetric inequalities, we show that this minimum is asymptotically attained by efficient coverings of a $k$-sphere. A key ingredient is a geometric measure theory argument that uses compactness to relate discrete covering counts to the volume of a limiting cycle.

We establish product-profile anti-concentration, density, and Remez estimates for multi-affine polynomials on the cube. Let $X$ be uniformly distributed on $[0,1]^n$, and let $P:[0,1]^n\to\R$ be a nonconstant multi-affine polynomial of exact degree $d$, with all variables active. For a partition $\cB$ of the variables such that no monomial contains two variables from the same block, define $$ w_d(\cB) = \sum_{\cI\subseteq\cB, \abs{\cI} = d-1} \left(\prod_{B\in\cI}\sqrt{\abs B}\right) \sqrt{\sum_{C\in\cB\setminus\cI}\abs C}. $$ We prove the all-center small-ball estimate $$ \sup_{u\in\R} \Prob\{\abs{P(X)-u}\le\rho\} \le \Phi_d\left( C_dw_d(\cB)\frac{\rho}{\osc(P)} \right), $$ where $\osc(P)$ is the length of the range of $P$ and $\Phi_d(t) = t\sum_{j = 0}^{d-1}\log^j(1/t)/j!$ on $(0,1]$, capped at one. Every multi-affine polynomial admits the singleton partition, which yields the universal scale $n^{d-1/2}$; an admissible partition into at most $q$ blocks yields the improved scale $q^{(d-1)/2}n^{d/2}$. If $\cB$ has exactly $d$ blocks, then $$ w_d(\cB) = d\prod_{B\in\cB}\sqrt{\abs B}, $$ and products of centered block averages show that both the small-ball profile and the dependence on the full block-size vector are optimal, up to constants depending only on $d$. We also prove that $P(X)$ has a density $f_P$ satisfying $$ \norm{f_P}_{L^p(\R)} \le C_dp^{d-1} \left( \frac{w_d(\cB)}{\osc(P)} \right)^{1-1/p}, \quad 1<p<\infty, $$ with matching $p$- and block-scale growth for $p\ge2$ on the block-product models. As an application, we derive translation-invariant quotient Remez inequalities with the same structural scale. The proof combines weighted block selection, affine cube slicing, degree-lowering contractions, and the exact recursion underlying the product profile.

Let $\Bbbk$ be an algebraically closed field of characteristic zero. A recent theorem of I. Pan characterizes the nonzero locally nilpotent derivations of $\Bbbk[x,y]$ in terms of their isotropy groups: such a derivation is locally nilpotent if and only if its isotropy group contains automorphisms of arbitrarily large degree. We study analogues of this characterization for several noncommutative algebras related to the affine plane. Our main result shows that Pan's characterization extends to the differential Ore extensions $A_h=\Bbbk[x][t;h(x)\partial_x]$, where $h\in\Bbbk[x]\setminus\Bbbk$. For the first Weyl algebra $A_1$, we establish the same equivalence for every nonzero locally finite derivation. For the quantum plane and the first quantum Weyl algebra, known results imply that the corresponding equivalence holds vacuously for nonzero derivations. In contrast, the converse fails for the free associative algebra $\Bbbk\langle x,y\rangle$: we construct a non-locally-nilpotent inner derivation whose isotropy group has unbounded degree. We further analyze this failure through the behavior of automorphisms and derivations under abelianization.

Let \(X:M^n\to\mathbb R^{n+p}\) be an \(n\)-dimensional complete self-shrinker in arbitrary codimension \(p\ge 1\). Suppose that the scalar curvature \(R\) is a positive constant and that the squared norm \(S\) of the second fundamental form satisfies \(S\le 1\). We prove that \(S\equiv 1\) and that \(X\) is isometric to either the round sphere \(S^n(\sqrt n)\) or the standard generalized cylinder \(S^k(\sqrt k)\times\mathbb R^{n-k}\), \(2\le k\le n-1\). In particular, no new higher-codimension examples occur under these assumptions. The proof uses an algebraic lemma which yields a uniform positive lower bound for the Bakry–Émery Ricci curvature, together with the comparison theorem of Wei–Wylie, the equivalence of finite Gaussian volume and polynomial volume growth due to Cheng–Zhou, and the gap theorem of Cao–Li.

Communication-driven multi-view sensing relies on routine communication transmissions for sensing acquisition, while the resulting sensing data at distributed devices must be uploaded to an edge server under limited communication resources. This creates a unique coupling between sensing acquisition and edge inference: the communication interval determines the source information, whereas the uplink condition determines how much information can be delivered to the server for sensing inference. To account for this coupling, we propose a source entropy-guided adaptive transmission framework. Specifically, we characterize the entropy of packet-triggered channel state information (CSI) as a function of the communication interval using a multi-output Gaussian process. The resulting analytical bound is compared with the available bit budget, determined by the transmission rate and latency requirement, to select between original-data and task-oriented transmission. For task-oriented transmission, we formulate the communication-constrained inference problem based on the information bottleneck and decompose it into adaptive distributed encoding and multi-view inference (ADE-MI), which avoids alternating optimization between the devices and the edge server. Experiments on the Widar3.0 multi-view CSI gesture recognition dataset show that the analytical bound closely follows the normalizing-flow numerical estimate, while ADE-MI outperforms task-oriented benchmarks under the same bit budget and the proposed framework further improves recognition accuracy under time-varying channels.

For any prime number $p$, the classic Hopkins-Mahowald theorem asserts that $H\mathbb{Z}/p$ is an $\mathbb{E}_2$-Thom spectrum over the $p$-complete sphere spectrum. Kitchloo extended this result to show that $H\mathbb{Z}/p^k$ is an $\mathbb{E}_2$-Thom spectrum over the $p$-local sphere spectrum, except for the case when $(p, k) = (2, 2)$. We solve the remaining case by showing that $H\mathbb{Z}/4$ is not an $\mathbb{E}_2$-Thom spectrum over the $2$-complete sphere spectrum, and consequently the $2$-local sphere spectrum. This work was developed with substantial assistance from GPT-5.6 Sol and GPT-6 Astra, and an early draft was subsequently reviewed using Claude Fable 5.1.

While kernel-based learning methods for scalar-valued functions have been extensively studied, comparatively little work considers nonlinear vector-valued operators. Such operators with vector-valued outputs, e.g., state vectors and trajectories, commonly arise in system identification problems. In this paper, we study the problem of learning a vector-valued operator under the framework of vector-valued kernel ridge regression. We assume that the outputs take values in an arbitrary Hilbert space and are corrupted by additive noise with bounded norm. Under the assumption that the unknown operator belongs to a vector-valued reproducing kernel Hilbert space, we derive a pointwise deterministic bound on the discrepancy between the kernel ridge regression estimator and the ground truth in the corresponding Hilbert space norm. Numerical simulations under two different scenarios are presented to validate the theoretical results.

Accurate methane source localization and emission-rate estimation at oil and gas facilities require wind fields that resolve site-scale spatial and temporal variability, which cannot be characterized by a single anemometer or coarse operational weather products. We develop a multiscale Weather Research and Forecasting (WRF) framework for a Permian Basin facility that dynamically downscales hourly, 3 km High-Resolution Rapid Refresh (HRRR) fields through four nested domains with grid spacings of 3 km, 1 km, 200 m, and 40 m. The two outer domains use planetary-boundary-layer parameterization, whereas the two inner domains operate in large-eddy-simulation mode. One-minute wind observations from two on-site anemometers are assimilated through wind-only observational nudging in the innermost domain, and the effects of nesting feedback and turbulence-closure choices are examined. Simulations for winter and summer 2025 periods are evaluated using near-surface wind-speed time series, power spectra, spatial fields, and statistical metrics. The baseline simulation reproduces the broad evolution of observed wind events but drifts during weak-wind periods. Observational nudging with one- and two-way nesting reduces the absolute bias of wind-speed by %65 at one sensor and %87 at the other, while reducing the corresponding root-mean-square errors by %33 and %44. Nudging also increases high-frequency energy and resolved spatial gradients, although improvements in turbulence statistics are not uniform. These results demonstrate a practical physics-based pathway for generating high-resolution wind fields for methane-plume modeling while emphasizing the need for independent spatial observations to validate accuracy away from assimilated sensors.

In recent work, Hirschhorn and the second author defined $a_k(n)$ to be the number of partitions of $n$ wherein the even parts come in only one color, while the odd parts may be “colored” with one of $k$ colors for fixed $k\geq 1$. This function generalizes the classical partition function and has been of significant interest because it satisfies a number of congruences. Although prior work studying $a_k(n)$ has resulted in congruences in arithmetic progressions in a somewhat ad hoc manner, this work gives a uniform framework for studying congruences modulo 3. This allows us to prove an infinite family of infinite families of non-nested congruences modulo 3.