Papers

In this paper, we consider energy identity for intrinsic stationary biharmonic maps into homogeneous spaces in supercritical dimensions, extending the corresponding result of Hornung-Moser [Anal. PDE. 2012] in critical dimension. The proof follows a similar strategy as that of Lin-Rivière [Duke Math. J. 2002]. A key ingredient is a conservation law for intrinsic biharmonic maps into homogeneous spaces, which allow us to derive higher regularity of the map.

We study the time-dependent Hartree–Fock equation in $\R^3$ with physical Coulomb interactions in both direct and exchange channels, with exchange coupling $\eta$, linearized about homogeneous equilibria $\gamma_f=g(-i\nabla)$ having compact momentum support. Nguyen and You showed that, without exchange, two purely imaginary plasmon poles $\pm i\tau_0(|k|)$ exist below a survival threshold $\kappa_0>0$. Existing nonlinear Hartree–Fock theory requires a short-range direct interaction and a smooth small exchange kernel, excluding the present Coulomb setting. On every compact band $0<k_-\le|k|\le k_+<\kappa_0$, we prove unconditionally for Coulomb exchange that, for small $|\eta|$, the poles persist, remain simple and purely imaginary, and depend real-analytically on $\eta$. Their first-order shift $\tau_X$ splits into explicit static self-energy and dynamic vertex corrections. Each pole yields an exact undamped mode with a closed-form unit-density eigenfunction, embedded in the fiber generator's essential spectrum. The causal density Green function splits into an explicit undamped sine wave and a remainder whose Laplace transform is holomorphic near the poles, while the plasmon channel satisfies uniform $t^{-3/2}$ dispersive decay and endpoint Strichartz estimates. A first-order Ward-type cancellation gives $|\tau_X(r)|\le Cr^2$ down to $r=0$; hence exchange leaves the plasma gap unchanged to first order, and we obtain a closed formula for the stiffness correction. Numerics for a $C^9$ smoothed Fermi ball support the analysis: self-energy and vertex terms cancel within $1.2\%$ on the computed range, and exchange softens the dispersion most strongly near $0.7\,\kappa_0$.

In this paper, we extend a large part of the uniformization theory of Bonk-Heinonen-Koskela [Asterisque 2001] to length spaces that are not necessarily proper or geodesic. Among other things, we show that there is a one-to-one correspondence between the quasiisometry classes of complete roughly starlike Gromov hyperbolic spaces and the quasisimilarity classes of bounded uniform spaces, which provides an affirmative solution to an open question of Bonk-Heinonen-Koskela. Our approach relies crucially on the work of Väisälä [Expo. Math. 2005], who investigated in depth Gromov hyperbolic spaces that are not necessarily proper or geodesic. One key new ingredient is to use the so-called (quasihyperbolic) $(c,\mu)$-quasigeodesic as a suitable substitute for quasihyperbolic geodesic.

We prove a critical Morrey rigidity theorem for the five-dimensional stationary Navier–Stokes equations. More precisely, every smooth solution on $\mathbb R^5\setminus\{0\}$ satisfying \[ \sup_{R>0}R^{-2}\int_{B_R}|u|^3\,dx<\infty \] is identically zero, up to an additive constant in the pressure. This replaces the pointwise Type-I control in the known higher-dimensional rigidity theory by a velocity-only, scale-invariant averaged condition that allows spatial concentration. The proof develops a weak head-pressure mechanism that does not rely on pointwise pressure estimates or classical normal traces. We reconstruct a canonical pressure from the velocity, derive a renormalized inequality for the positive head pressure, and introduce two monotone radial fluxes. Annular energy estimates, suitable-weak compactness, and blow-up and blow-down limits are then used to identify the endpoint fluxes and force rigidity. As an application, we obtain a removable-singularity criterion in dimension five: if a suitable weak solution is smooth away from one point and either its scale-invariant Dirichlet energy or its cubic velocity Morrey quantity remains bounded near that point, then the singularity is removable. Thus, within the isolated-singularity class, the smallness assumption in the classical stationary regularity criterion is replaced by boundedness. We also prove the corresponding velocity-only cubic Morrey rigidity theorem in dimension four by a different finite-energy argument.

Let $D\subset\Bbb Z$ with cardinality $q\ge 2$, and let $b\in \Bbb Z$ with $q<b$, and let $\mu:=\mu_{b,D}$ be the associated self-similar measure. It is well known that if there exists $L\subset \Bbb Z$ such that $(b,D,L)$ be a Hadamard triple, then the Beurling dimension of the spectrum of $\mu$ exhibits an intermediate structural property. In this paper, we establish a stronger result that both Beurling dimension and Beurling density of the spectra of $\mu$ can achieve full flexibility simultaneously. More precisely, for any $t\in(0, \frac{\log q}{\log b})$ and $s\in [0,\infty]$, there exists a spectrum $\Lambda:=\Lambda_{t,s}$ of $\mu$ such that $$\dim_{Be}(\Lambda)=t,\quad D_t^+ (\Lambda)=s.$$ Here, $\dim_{Be}$ and $D_t^+$ denote the Beurling dimension and the $t$-Beurling density, respectively. We further prove that the set of such spectrum whose Beurling dimension and Beurling density are equal to any fixed $t$ and $s$ has the cardinality of the continuum. \par This work generalizes a previous result of Lu \cite{Lu}, answers an open question raised by Dai, Fu and He \cite[Conjecture 5.3]{DaiFuHe}, and sheds new light on the fine structural properties of spectra for singularly continuous spectral measures.

We discuss the Babylonian division of trapezoids and the construction of such trapezoids with rational sides and transversals from Pythagorean triples.

We introduce the Tempered Bouncy Particle Sampler, a state-dependently tempered generalisation of the Bouncy Particle Sampler. Adapting the hypocoercivity framework of Dolbeault, Mouhot and Schmeiser (2010), we establish explicit, non-asymptotic exponential convergence bounds for a broad class of heavy-tailed targets that need not be log-concave or radially symmetric, including distributions with polynomial tails. The convergence rate is linked to the spectral gap of a tempered Langevin diffusion with the same invariant distribution, making a weighted Poincaré inequality the key assumption. We give admissible tempering choices for polynomial and stretched-exponential tails and derive mixing-time bounds with polynomial dependence on the dimension.

In the spirit of Conway and Coxeter, we classify all $\mathbf{Y}$-frieze patterns of type $A_n$. In particular, we settle a conjecture made by de Saint Germain that all such $\mathbf{Y}$-frieze patterns arise from Conway-Coxeter frieze patterns. Moreover, our approach naturally leads to the enumeration of these $\mathbf{Y}$-frieze patterns in terms of Fuss-Catalan numbers.

We develop a high-order physical-constraint-preserving (PCP) alternative finite difference weighted essentially non-oscillatory (AFD-WENO) scheme for the special relativistic hydrodynamics equations with general equations of state. The proposed scheme comprises two key limiters: a state limiter, which acts after the WENO state interpolation step, and a flux limiter, which acts on the final high-order fluxes. The state limiter ensures that the interpolated states are physically admissible, while the flux limiter ensures that the numerical fluxes are physically admissible. The resulting scheme is rigorously proved to satisfy the physical constraints. Incorporating multiple WENO interpolation techniques, including an improved adaptive-order formulation (WENO-AOI), the method is validated through extensive one- and two-dimensional numerical benchmarks with various equations of state. The numerical results demonstrate high-order accuracy, sharp resolution of discontinuities, and robust stability in extreme relativistic regimes.

Assuming the validity of the recently proposed “A compact complex threefold fibred by tori over the projective line, and the six-sphere”, we construct an exotic complex structure on $\mathbb P^{3}$, distinct from the point-blowup structures of Huckleberry, Kebekus, and Peternell. Then we perform an Atiyah flop to produce a complex structure on the standard smooth manifold $S^{2}\times S^{4}$.

We develop a geometric framework for implicit discrete Hamiltonian systems based on discrete Morse families, Lagrangian relations, and discrete analogues of Tulczyjew's triple. The main idea is to regard the Lagrangian submanifold defining the discrete dynamics, rather than an explicit symplectic evolution map, as the fundamental geometric object. This viewpoint naturally accommodates implicit, constrained, and degenerate discrete systems. Within this framework, we formulate a Type–II discrete Hamilton–Jacobi theory in terms of the propagation of Lagrangian submanifolds between consecutive discrete steps. When these submanifolds are locally represented by exact one-forms $dW_k$ and $dW_{k+1}$, the resulting equations provide a discrete Hamilton–Jacobi relation between consecutive generating functions. More generally, when the projection onto configuration space becomes singular and a single-valued generating function ceases to exist, we show that the evolution can be described by the composition of Type–II discrete dynamics with Morse families. This yields a generating family for the propagated Lagrangian submanifold without requiring the dynamics to be represented as a graph. As an application, we consider the propagation of optical wavefronts through fold caustics. A Type–II discrete Hamiltonian yields a symplectic ray integrator, while a Morse family represents the multivalued wavefront near the caustic. Their composition provides a discrete propagation rule for the complete Lagrangian manifold across the singularity. In this way, the same geometric construction simultaneously provides a discrete Hamiltonian integrator and a regular representation of multivalued Hamilton–Jacobi solutions.

We give a detailed account of the recent work of Hickman and Srivastava on counting integral points in a $ \delta $-neighbourhood of a dilate $ q \mathcal{M} $ of a non-degenerate curve $ \mathcal{M} \subset \mathbb{R}^{n} $, $ n \ge 3 $. Our aim is expository: we simplify the proof, provide additional details, refine the statement of the theorem and provide calculations from a more fundamental perspective. We also formulate a new conjecture based on recent work of Chen, Seeger, Srivastava and Technau.

Clark's syntactic concept lattice (SCL) records two-sided distributional structure, and Wurm extended it to tuples of arbitrary finite arity. We study \(\operatorname{cmp}_f(L)\), the minimum image size of a finite-monoid observation that preserves guarded tuple substitution through arity \(f\) on the principal layer. For regular languages, we characterize \(\operatorname{cmp}_f(L)\) exactly as the least cardinality of the codomain of an \(f\)-separating relational morphism from the pointed syntactic monoid. Let \(\operatorname{ch}(\mathbf V)\) denote the least arity at which these compression numbers stabilize uniformly over a pseudovariety \(\mathbf V\). Our main result is the following trichotomy of possible uniform heights: \(\operatorname{ch}(\mathbf V)\in\{1,2,\infty\}\), with \(\operatorname{ch}(\mathbf V)=\infty\) if and only if \(\operatorname{Synt}(\{ab\})\in\mathbf V\). Thus no finite uniform compression height \(3,4,\ldots\) occurs. The infinite case is sharp: inside \(\langle\operatorname{Synt}(\{ab\})\rangle\), every boundary \(d\to d+1\) admits unbounded compression gaps, and arbitrary finite strict prefixes of the arity hierarchy are realizable. On the finite side, commutative monoids and bands stabilize at arity one, while every completely regular syntactic monoid stabilizes by arity two; finite group kernels show that the binary bound is sharp. At unary arity, every nonempty finite simple graph is realized by an explicit length-three language, yielding an exact chromatic-number formula and NP-completeness of deciding \(\operatorname{cmp}_1(L)\le 3\) for explicitly listed length-three languages. The structural boundary between compression heights one and two remains open.

Tensor representations have emerged as a fundamental paradigm for modeling multidimensional data by preserving intrinsic correlations across multiple modes. This paper proposes a novel theoretical framework, termed Tensor Orthogonal Subspace Split (TOSS), which explicitly splits a tensor, along a prescribed mode, into two orthogonal components: a dominant component lying in a prescribed subspace and a residual component lying in the corresponding orthogonal complement. We first present the general formulation of TOSS and systematically investigate its fundamental properties. As an important and practically meaningful special case, we further introduce the rank-one TOSS, which imposes a separable rank-one structure along the splitting mode and admits a clear geometric interpretation. This formulation naturally captures dominant consistent patterns while effectively isolating orthogonal residual component. The proposed framework establishes a unified theoretical foundation for tensor-domain orthogonal split and opens new avenues for structured tensor modeling across diverse applications. Building upon the developed TOSS theory, hyperspectral image restoration and color video background modeling are considered as two representative tasks, for which corresponding optimization models are formulated. Efficient algorithms are developed to solve the resulting problems. Extensive experimental results validate the effectiveness and superiority of the proposed approaches.

In this paper, we study the generalized $L$-function considered by Berndt (1975). We introduce the generalized Apostol-Bernoulli polynomials and the generalized Apostol-Bernoulli functions, and establish some properties for them, including the Fourier series for these functions. We show that the values of Berndt's generalized $L$-function at integers are explicitly evaluated in terms of the generalized Apostol-Bernoulli functions.

Gélinas introduced the delooping level of an Artin algebra as a homological invariant that bounds the big finitistic dimension of the opposite algebra. A natural question is whether such invariants are preserved under derived equivalences. Chen recently showed that the finiteness of the classical delooping level and its sub-derived variant is not derived-invariant, but the case of the finer derived delooping level of Guo and Igusa remained open. In this paper, we prove that the finiteness of the derived delooping level is degree-shift invariant under derived equivalences: if two algebras are derived equivalent via a tilting complex of width $k_T$, then finiteness of the $(k+k_T)$-derived delooping level on one side implies finiteness of the $k$-derived delooping level on the other. Consequently, the finiteness of the derived $\infty$-delooping level is invariant under derived equivalences. Furthermore, we introduce the global derived delooping level and prove that, under a derived equivalence, its $\infty$-version changes by at most $k_T$. In particular, its finiteness is a derived invariant.

In this work, we propose an efficient decoding algorithm for codes arising from simplicial complexes, a family of binary linear codes for which no decoding method of this type was previously known. Although the algorithm does not always attain the maximum theoretical error-correcting capability, it provides an explicit bound that can be computed directly from the structure of the complex. Moreover, this bound is asymptotically optimal: the ratio between the guaranteed correcting capability and the theoretical maximum converges to $1$ as the code length increases, under natural assumptions on the dimension of the maximal faces. The correction capability is also presented in specific examples. Finally, we introduce specific families of simplicial complexes where the algorithm successfully reaches this theoretical bound.

We study nonzero-frequency phase-locked motions in asymmetrically coupled Kuramoto networks. Such motions are relative equilibria with fixed phase differences and a nonzero common angular velocity, and we call them dancing equilibria. Their existence requires all coupling sums to have the same nonzero value. We show that neither symmetric coupling nor an acyclic associated digraph can support a dancing equilibrium. We introduce structurally equitable and $q$-twisted state equitable partitions and prove a partition-based criterion for the resulting class-constant profiles, with standard labeled $q$-twisted profiles recovered from singleton partitions. For the forward $m$-neighbor model, we characterize existence by an exact indivisibility criterion. Stability is studied modulo the common phase-shift direction. For general directed networks, strong connectivity and edgewise phase differences in $\left(-\pi/2,\pi/2\right)$ imply local orbital exponential stability and yield an explicit positively invariant set contained in the local basin of attraction. For arbitrary twisted indices, this contraction argument gives a low-winding stability regime with explicit positively invariant neighborhoods. For each existing $q$-twisted branch of the forward model, a discrete Fourier transform criterion yields local orbital exponential stability when all nonzero Fourier-mode factors are positive and nonlinear instability when at least one is negative. In the unstable case, the proof constructs explicit escaping real Fourier perturbations. We further derive additional explicit stability and instability ranges for arbitrary twisted indices in terms of constants $N$, $m$, and $q$. For the first two twisted branches, sharper arguments yield a first-mode transition criterion for $q=1$ and a complete finite-size classification for $q=2$, with the degenerate case in each branch handled separately.

Safety-critical optimization applications, such as real-time power system operation, maintain feasibility at every intermediate step, not merely at convergence. Existing approaches either violate constraints mid-solve (interior-point methods) or enforce feasibility through per-instant quadratic programming subproblems with cubic computational cost and unbounded worst-case execution time. We propose a continuous-time optimization framework for smooth constrained nonlinear problems that preserves feasibility throughout the optimization process without requiring projection operators, quadratic programming subproblems, or other per-iteration optimization routines. The method is built around a reciprocal multiplier manifold, which establishes an explicit relationship between inequality constraints and their associated Lagrange multipliers. By designing a continuous multiplier update law, the manifold is shown to remain forward invariant, while the resulting dynamics are equivalent to continuous-time logarithmic barrier gradient descent. The proposed framework naturally extends to multiple inequality constraints, equality constraints, nonconvex feasible sets, and infeasible initial conditions. The method is further enhanced through an augmented Uzawa flow that eliminates oscillatory transients commonly observed in classical primal-dual saddle-point dynamics. The effectiveness of the proposed approach is applied to the AC Optimal Power Flow problem of IEEE 9-bus and IEEE 57-bus systems. Numerical results show convergence to solutions within 0.4% of the benchmark optimum while maintaining strict feasibility of all constraints. A computational complexity analysis shows that the proposed dynamics reduce the per-step computational cost from cubic to linear complexity. Finally, dynamic tracking studies under time-varying operating conditions demonstrate reliable feasibility preservation.

Multi-access speculative inference (Multi-SPIN) extends SPIN to multi-device edge networks to accelerate cooperative token generation. It allows on-device small language models (SLMs) to autoregressively draft multiple tokens for individual generation tasks, while an edge-server large language model (LLM) verifies them in parallel. The major communication overhead arises when a drafted token is rejected by the server, in which case sampling the correction token requires access to both the SLM-output draft distribution and the LLM-output target distribution over the full token vocabulary. Existing designs typically perform correction at the server by uploading the draft distribution, but transmitting a vocabulary-wide distribution creates a critical uplink (UL) bottleneck. Alternatively, the correction can be performed at the device by downloading the target distribution, leveraging the high transmission rates available on the downlink (DL). Motivated by this insight, we introduce communication-mode selection as a new design dimension for Multi-SPIN. Specifically, each device can adaptively switch between the UL and DL modes to balance the UL bottleneck against the shared DL resource constraint, thereby relieving the overall communication burden. We formulate a sum-token-goodput maximization problem that jointly accounts for mode selection, draft-length control, and power allocation. For mode selection, we reveal a simple optimal structure that enables efficient search over the number of UL devices, with the corresponding transmit powers optimized accordingly. For draft-length control, we develop a greedy-search algorithm that adapts device-specific draft lengths to heterogeneous computation and communication capabilities. Experimental results on Qwen2.5 and DeepSeek-R1 model pairs demonstrate that the proposed framework significantly improves token goodput.

The average performance of FirstFit online coloring on trees in the random order model is completely determined in recent works of Frei et al. and Bosek et al., showing $\Theta(\log n /\log\log n)$ number of colors, improving the $\Theta(\log n)$ colors in the adversarial model. We provide a few further results on slightly more general graph classes. Firstly, we extend their method to obtain a simple path-counting principle for sparse graph classes, which immediately yields for example that cactus graphs and uniform hypertrees exhibit a similar improvement. We then show that FirstFit uses only $O(1)$ colors on crown graphs, a standard example where adversarial arrival forces $\Theta(n)$ colors. We further show that density alone (even linear minimum degree) is insufficient to guarantee $O(1)$ colors even on bipartite graphs. Finally, we identify graph classes, including unit interval graphs and some graphs of high chromatic number, for which random arrival provides only limited improvement. We end with some open problems.

We prove that every Poisson action of positive intensity of a noncompact locally compact second countable sofic group has completely positive measure sofic entropy with respect to every locally compact sofic approximation. The proof constructs random microstates for arbitrary measurable factors by completing local Poisson configurations with an independent exterior Poisson process. Probability estimates and a packing argument then show that every nontrivial factor has positive entropy. We also give a compact-group example showing that the noncompactness assumption is essential.

This document reports over 15 counterexamples found using GPT Pro over the course of 12 months, for problems in combinatorics, number theory, convexity, analysis, and other areas. The counterexamples concern both (reasonably) well-known problems as well as lesser known instances. The counterexamples are accompanied by the repo https://github.com/suvrit/count-ex-machina, and we warmly welcome new contributions toward helping building a rich “open-source countex library.”

Let $p$ be an odd prime and let $K/\mathbb{Q}_p$ be a finite unramified extension. From a finite presentation by roots of unity of order prime to $p$, we construct a localized Iwasawa pseudomeasure on $\mathbb{Z}_p^\times$. Although the pseudomeasure depends on the chosen presentation, its image modulo bounded measures depends only on the associated Bloch class: it is the Frobenius-depleted Coleman regulator of that class multiplied by a universal half-shifted zeta principal part. Consequently, every nonexceptional weight component is bounded, while the exceptional component has at most a simple pole with explicitly determined residue. For $p>3$ and $\mathbb{Z}_p$-valued coefficients, vanishing of the principal part is equivalent to vanishing of the corresponding class in $K_3(K;\mathbb{Z}_p)$. Cyclotomic refinements preserve the Bloch class and act on the associated pseudomeasures by explicit Iwasawa multipliers. Normalized finite linear combinations of refinements interpolate arbitrary finite jets of the bounded weight-space data, subject only to the normalization at the exceptional point. We also establish a half-shifted complex Mellin factorization, compare the construction with the GSWZ germ family, and derive, at simple degree-one places above primes $p>3$, a finite-polylogarithm criterion for the local $K_3$-class of the knot $5_2$.

Let $J_r^{SU(n)}(K;q)$ denote the reduced $SU(n)$ quantum invariant of a zero-framed knot $K$, colored by the $r$th symmetric power of the defining representation and normalized to be $1$ for the unknot. For every fixed $n\ge2$ we prove the Chen–Liu–Zhu cyclotomic expansion conjecture: there are unique coefficients $H_k^{(n)}(K;q)\in\mathbb{Z}[q^{\pm1}]$ such that \[ J_r^{SU(n)}(K;q)=\sum_{k=0}^{r} \left(\prod_{i=0}^{k-1}\{r-i\}\{r+n+i\}\right) H_k^{(n)}(K;q), \] where $\{m\}=q^m-q^{-m}$. The finite dual interpolation formula of Beliakova–Gorsky gives an integral one-sided factorial expansion. After identifying their reduced scalar with the Habiro–Lê convention, we restrict the completed center to one-row colors. Completed Harish–Chandra reflection then yields inversion symmetry in the variable $z$, and integral descent through $X=z+z^{-1}$ converts the one-sided expansion into the two-sided Newton basis. Cyclotomic-local interpolation and a UFD denominator-removal argument prove Laurent integrality of the Newton coefficients. We also determine a natural coefficient ring for a rank-uniform expansion. For every zero-framed knot there are unique Laurent differential coefficients $G_k(K;A,q)\in\mathbb{Z}[A^{\pm1},q^{\pm1}]$. The associated Newton coefficients are Laurent polynomials in $A$ over $\mathbb{Q}(q)$ whose values at every geometric node $A=q^n$, $n\ge2$, lie in $\mathbb{Z}[q^{\pm1}]$. They define a two-variable Newton inverse-limit element whose positive-rank specializations recover all symmetric-color HOMFLY–PT polynomials. The completion is taken in the Newton kernels rather than coefficientwise at roots of unity. After a positive rank and a color have been fixed, the series is finite and may be evaluated at a root of unity.

Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a right Auslander-Reiten-Serre (ARS for short) duality $(\tau, \eta)$ in the sense of Iyama, Nakaoka and Palu. We show $(\tau, \eta)$ induces right ARS dualities on the relative theories of $(\mathcal{C},\mathbb{E},\mathfrak{s})$. Under relative structures, we show that the functor $\tau$ is indeed an exact functor between extriangulated categories and preserves almost split exangles. Finally, we give some applications and examples on these results. For instance, we generalize a recent result by A. Hubery to a categorical framework.

Let $A, T \in M_n(\mathbb{C})$ and let $\Delta_A(T) = AT - TA$ denote the inner derivation induced by $A$. We determine when $T$ and $\Delta_A(T)$ are nilpotent under the second-order relation $$ \Delta_A^2(T) + \alpha\, \Delta_A(T) + \beta\, T = 0, \qquad \alpha, \beta \in \mathbb{R}, $$ according to the location of the roots of $z^2 + \alpha z + \beta$. If the roots have nonzero real parts of the same sign, then both $T$ and $\Delta_A(T)$ are nilpotent. If the roots are purely imaginary and nonzero, no nilpotency conclusion holds in general, whereas if one root is zero and the other is nonzero, $\Delta_A(T)$ is nilpotent but $T$ need not be. The double zero root gives the Kleinecke–Shirokov theorem. For real roots of opposite signs, the answer is governed by an arithmetic threshold: writing $z_1/z_2=-p/q$ in lowest terms, every solution is nilpotent when $p+q>n$, while for $p+q\le n$ an explicit cyclic construction yields solutions for which both $T$ and $\Delta_A(T)$ are invertible. Finally, for a relation $$ \sum_{k=0}^{m} c_k\,\Delta_A^k(T)=0 $$ of arbitrary order, every iterated commutator $\Delta_A^j(T)$ is nilpotent whenever the roots of the associated polynomial lie in an open half-plane whose boundary passes through the origin.

Function correcting codes (FCCs) are designed to protect a specified function evaluation of messages at a higher level than the level of protection for messages, against errors while reducing the redundancy required for reliable communication. FCCs have thus far been studied for channels matched to various distances, including the Hamming and Lee distances. Every function partitions the message space into preimage sets corresponding to its distinct function values. Existing Plotkin-type bounds on the optimal redundancy of FCCs under the studied distances, applicable to arbitrary functions on the message space, depend on the pairwise distances among all the message vectors. This makes these bounds difficult to compute. We derive a general Plotkin-type bound on the optimal redundancy of FCCs under Wyner Graham distances, which include the Hamming and Lee distances as special cases. Our bound depends only on the cardinalities of the preimage sets and the sum of pairwise distances only among vectors within each preimage set. This approach significantly reduces the computations required to evaluate the existing Plotkin-type bounds and yields simplified bounds that are easier to compute for specific functions. We obtain simplified Plotkin-type bound for linear functions under the Wyner-Graham distance. Furthermore, the existing simplified bounds for linear functions under the Hamming and Lee distances are recovered as special cases of the proposed bound. We also obtain simplified bounds for several important classes of functions, including the Hamming weight function, the Hamming weight distribution function, the monomial functions under the Hamming distance, and the modular sum function, the Lee weight function, and the Lee weight distribution function under the Lee distance.

Fix an integer $p\geq 2$ and $0<s<1$. We construct an ergodic $\times p$-invariant measure $\mu$ on $\mathbb T^2$ of dimension $s$, such that every line projection preserves dimension, including when the corresponding projected IFS has exact overlaps. In fact, our measure assigns mass $O(w^s)$ to every planar tube of width $w$. The result is sharp at the endpoint $s=1$ as shown by Pyörälä, Shmerkin, Suomala and Wu (2025). In contrast, we show that every quasi-Bernoulli $T_p$-invariant measure with positive dimension admits a dimension-dropping projection.

The main focuses of the article are limit theorems of Wasserstein projection in the convex order which are useful for inference tasks. The main results rely on the establishment of dual attainment, stability and several useful observations. The first is a clean criterion to invoke the Wasserstein projection dualities on the classical cost $c=h(x-y)$. A new strategy was introduced to establish the dual attainment. Backward and forward dual transport are proved to enjoy a conjugate relationship. Sufficient conditions are given that ensure uniqueness of the optimal dual potential. These ingredients laid the groundwork for the proof of optimal dual potential stability, thereby permitting us to prove the central limit theorems under mild moment assumptions. Additional discussions show that the moment assumptions are sharp. These results also answered several open questions raised by Professor Benjamin Jourdain.