Papers

Let $1<m<n$ and $0<q<m-1$. We classify positive, locally bounded weak solutions of \[-\Delta_m u=u^p|Du|^q\quad\text{in }\mathbb R^n,\qquad p=\frac{m-q}{n-m}\left(n+\frac{q}{m-1-q}\right)-1.\] Every solution is constant or belongs to an explicit family of radial solutions, up to translation and scaling. No global bound, decay, or finite-energy assumption is required. The proof is based on a scalar differential identity and a maximum principle that remains valid at critical points. These lead to a sharp gradient bound, classification of normalized solutions, and a first-contact argument for arbitrary entire solutions. Local regularity and the behavior at critical points are treated throughout in the weak solution class; global $C^2$ regularity is not assumed.

For the 2D incompressible Euler equations on the half-plane, we establish Lyapunov stability of the Lamb dipole without sign condition on the vorticity. The main challenge arises from growth of the $x_{2}$ weighted norm for mixed sign vorticities, whereas this norm is conserved and responsible for dipole stability in the single signed vorticity case. This difficulty is handled by a Lagrangian bootstrap argument, based on carefully dividing the solution into several pieces and analyzing interactions among them, with the most delicate interaction involving vorticity drifting to infinity in the vertical direction. We expect that the strategy developed here will provide a starting point for the stability analysis of more general vortex configurations, including interactions among multiple vortex dipoles and among multiple vortex rings in settings involving mixed-sign vorticity.

This paper investigates the boundary pointwise regularity of solutions to the p-Laplace equation. By treating curved boundaries as perturbations of flat ones, we establish the boundary pointwise Lipschitz regularity under the exterior $C^{1,\mathrm{Dini}}$ condition and the Hopf lemma under the interior $C^{1,\mathrm{Dini}}$ condition. We also establish the boundary pointwise $C^1$ regularity under the $C^{1,\mathrm{Dini}}$ condition by a compactness argument.

We study three particles on the square lattice with contact interactions of infinite strength, for all total quasimomenta, for three statistics: three identical bosons, two identical fermions plus a third particle, and two identical bosons plus a third particle. Previous work treated only the trimer and zero quasimomentum. Here the atom-dimer sector is analyzed on the whole Brillouin zone. A Feshbach-Schur reduction onto the infinitely degenerate eigenvalue 1 of the pair-contact operator gives explicit effective Hamiltonians with proved first-order error. The threshold problem reduces to a 4x4 Hermitian matrix criterion built from the two-point resistances of the square lattice, and hence to a quartic polynomial. This yields the complete atom-dimer zone and the critical mass ratio on the Brillouin zone, with bound between 1 and pi/(pi-2), and closed forms at all symmetry points. We prove: the zones in closed form; the critical mass ratio with exact value pi/(pi-2) at zero quasimomentum, reproducing a numerical value; at most two atom-dimer states for every quasimomentum and mass ratio; the onset laws, an essential singularity on the zone boundary and a linear-over-logarithm onset in the pure p-wave; two exact dualities showing that a change of statistics is equivalent to a shift of quasimomentum by pi; and the global range of the second fermionic threshold. The onset laws admit a renormalization-group reading: the inverse lattice Green function is a marginal s-wave coupling, binding energies arise by dimensional transmutation with lattice scale 16, and the zone boundary is of marginal (BCS/Kondo) type rather than of Berezinskii-Kosterlitz-Thouless type. All constants are verified independently by lattice quadrature, computer algebra, exact diagonalisation, and infinite-lattice solvers.

Dujella, Jak\v setić and Pe\v carić recently established in \cite{DJP}, a chain of power-sum inequalities, built from Hölder's and Cauchy's inequalities and their converse forms, and applied it to the Fibonacci sequence using the identities $\sum_{i=1}^nF_i^2=F_nF_{n+1}$ and $\sum_{i=1}^nF_iF_{i+1}=F_{n+1}^2-\frac{1+(-1)^n}{2}$. We show that this machinery applies uniformly across the one-parameter family of $k$-Fibonacci and $k$-Lucas numbers of Falcón and Plaza, via the generalized identities \begin{align*} \sum_{i=1}^nL_{k,i}^2=\frac{L_{k,n}L_{k,n+1}-2k}{k},\qquad \sum_{i=1}^nF_{k,i}^2=\frac{F_{k,n}F_{k,n+1}}{k}, \end{align*} and \begin{align*} \sum_{i=1}^nL_{k,i}L_{k,i+1}=\frac{L_{k,n+1}^2}{k}-k+\bigl((-1)^n-1\bigr)\left(\frac{2}{k}+\frac{k}{2}\right), \end{align*} which recover the classical Fibonacci and Lucas identities at $k=1$. We further exploit the cross-identity $F_{k,i}L_{k,i}=F_{k,2i}$, valid for every $k\ge1$, to obtain a Cauchy-Schwarz-type inequality linking the two families that has no counterpart in the Fibonacci-only setting. At $k=1$, this specializes to a fully explicit elementary inequality between ordinary Fibonacci and Lucas numbers, alongside the corresponding Hölder and Cauchy-conversion refinements we obtain for ordinary Lucas numbers.

Finite Fibonacci and Lucas sums involving zeta values at negative integers can have individual terms far larger than their total. In this paper, we study the distribution of these term magnitudes as the even degree increases. For non-negative weights of at most exponential growth, we prove a finite exponential expansion for the normalized absolute-term sum with an explicit remainder at every fixed order. The normalized magnitudes converge in total variation to a discrete distribution determined by the even exponential generating function of the weights. For the Fibonacci and Lucas sums, this distribution has its unique maximum at index eight and we prove that the same index gives the largest finite-sum term at every even degree at least twenty-six. We also derive the factorial growth of the summation condition number. The signed identities are placed within the established Bernoulli-polynomial reflection framework and extended to a trace-one recurrence family, including repeated roots and a vanishing scale. Exact rational calculations and precision-refined evaluations illustrate the proved bounds and distinguish a fixed cancellation profile from the growing length of the sum.

Let $T$ be a convolution Calderón–Zygmund operator satisfying Hytönen's non-degeneracy condition. We prove that, for every $p\in(1,\infty)$ and $m\ge3$, there exists a positive constant $C$ such that, for any $t\in[1,\infty)$, \begin{equation*} \sup_{[W]_{\mathcal A_p}\le t} \|T\|_{L^p(W)\to L^p(W)} \ge C t^{1+\frac{1}{p(p-1)}}, \end{equation*} where the supremum is taken over all matrix weights $W:\ \mathbb R^n\to\mathbb C^{m\times m}$ with their weight constants $[W]_{\mathcal A_p}\le t$. If, in addition, the odd part of the kernel is non-degenerate (such as the Hilbert transform and the Riesz transforms), the same conclusion holds for every $m\ge2$.

The Favard length of a planar set is the average length of its orthogonal projections. We provide sharp estimates for the Favard length decay of a random variant of the non-homogeneous four corner Cantor sets introduced by Garnett in the 1970s. As a corollary, we obtain examples which show that Mattila's classical lower bound on the Favard length in terms of the Riesz capacity is sharp for all possible decay rates. We also obtain a new family of examples of sets with positive analytic capacity and zero Favard length.

For $3\le n\le6$, we determine the sharp isoperimetric bound for the mass-normalized first weighted Neumann eigenvalue on bounded Euclidean domains. For every finite nonnegative nonatomic measure $\mu$ on $\overline\Omega$, \begin{equation*} \overline\lambda_1^N(\Omega,\mu)\le\frac{n(n-1)}{n-2}\omega_n\sin^2\vartheta_n\left(\frac{|\Omega|}{\omega_n}\right)^{(n-2)/n}, \end{equation*} where $\omega_n$ is the volume of the unit ball and $\vartheta_n\in(\pi/2,\pi)$ is the angle at the first stationary radius of the regular rotational harmonic-map profile. The bound is attained on balls by a smooth positive radial density. In fact, we affirmatively resolve Question~1.12 posed by Vinokurov \cite{Vinokurov2026}. Furthermore, for admissible domains with finite boundary measure, we also prove the sharp strict Steklov bound \begin{equation*} \sigma_1(\Omega)|\partial\Omega|\,|\Omega|^{(2-n)/n}<\frac{n(n-1)}{n-2}\omega_n^{2/n}\sin^2\vartheta_n, \end{equation*} whose constant is approached by $C^1$ perforated domains.

Let $M^n\to\mathbb{S}^{n+q}(1)$ be a closed connected minimal immersion, where $n\ge3$, and set $Q=S+\lambda_2$, with $S=|h|^2$ and $\lambda_2$ the second largest eigenvalue of Lu's fundamental matrix. We determine the sharp codimension range for Lu's second-gap conjecture. For every $2\le q\le n$, there exists $\gamma_{n,q}>0$ such that, if $Q$ is constant and $Q>n$, then $Q\ge n+\gamma_{n,q}$. Conversely, for every $q\ge n+1$, we construct closed connected homogeneous minimal embeddings, followed when necessary by totally geodesic inclusions, with constant scalar curvature and constant $Q$-values dense in $(n,2n)$. Thus, in every dimension $n\ge3$, Lu's conjecture holds precisely for $q\le n$. Combined with the theorem of Peng-Terng for hypersurfaces and the recent resolution of the two-dimensional case, this gives a complete resolution of Lu's second-gap conjecture: for every $n\ge2$, the conjecture holds exactly when $q\le n$ and fails when $q\ge n+1$. In codimension two we further obtain the explicit admissible gap $\gamma_{n,2}=\exp(-10^{16}n^2)$.

Automatic relevance determination (ARD), the default tool for variable selection in Gaussian-process (GP) regression, ranks inputs by inverse lengthscales – which measure how fast a function varies, not how much an input contributes to prediction – and offers no calibrated rule for deciding which inputs to keep. The prediction-centred alternative, the derivative sensitivity $\nu_j = \mathbb{E}[(\partial f/\partial x_j)^2]$, is available in closed form from a fitted GP, but turning it into a selection rule is harder than it looks: at a null input the estimator is a degenerate quadratic form, so Wald and Bernstein-von Mises cutoffs are anti-conservative, and the natural residual bootstrap is mis-scaled. We show that a studentized multiplier bootstrap of the GP derivative process repairs both, prove its validity through an invariance principle for quadratic forms, and obtain asymptotic family-wise and false-discovery-rate control across inputs. Over 100 replications the rule controls FDR wherever inputs are truly null, while uncalibrated derivative rankings breach the target by up to 2x and a Bernstein-von Mises cutoff by 2.2x; at matched FDR it loses no power; it holds under a Matérn kernel and input correlation up to 0.99; on real data with planted and authentic null inputs it admits 5-12x fewer spurious inputs; it costs 5-18% of the GP fit; and a block-averaged variant retains validity at cost linear in $n$.

We develop an analytical framework for studying a family of interacting particle systems that we refer to as the multi-particle Oja flow. The common interaction structure underlying this family arises in a variety of applications, including higher-order synchronization models, opinion dynamics, and self-attention dynamics derived from Transformer architectures, motivating a unified mathematical framework for their analysis. We first establish well-posedness for both the finite-particle and the mean-field systems. We also prove convergence of the finite-particle dynamics to the mean-field equation as the number of particles tends to infinity. We then investigate low-dimensional structures underlying the dynamics. In particular, we show that the mean-field solution admits a representation through a time-dependent normalized linear transformation acting on the initial distribution, with the transformation governed by a matrix ODE. Building on this representation, we identify a broad class of parametric families that are invariant under the mean-field dynamics. When the initial distribution belongs to one of these families, the evolution remains within the same family for all time and can therefore be characterized through the dynamics of the corresponding family parameters. As a concrete example, we develop this reduction within the angular central Gaussian family and illustrate how the resulting parameter dynamics provide reduced descriptions of the mean-field evolution for several representative systems.

As a non-abelian analogue of the Hodge locus, Simpson introduced the non-abelian Noether–Lefschetz locus and conjectured its algebraicity for $\mathbb Z$PVHS. In this paper, we study this question on the moduli space of curves $\mathcal M_g$. For a non-unitary representation \[ \rho:\pi_1(\Sigma_g)\longrightarrow \mathrm{SL}_2(\mathbb R) \] which admits a $\mathbb R$PVHS of weight one, we prove that algebraicity of a positive-dimensional non-abelian Noether–Lefschetz component is equivalent to discreteness of $\operatorname{im}\rho$, and in this case the component is precisely a marked fixed-target orbifold Hurwitz component. As applications, we construct two explicit rational families by slit surgery: one with non-discrete monodromy and non-algebraic Noether–Lefschetz image, and another with discrete monodromy whose period map is nevertheless non-uniformizing. The latter gives an affirmative answer to a question of Baldi–Lam concerning $\mathbb Q$PVHS.

Let $(R,\mathfrak m,k)$ be a Noetherian local ring, let $M$ be a nonzero finitely generated $R$-module, and let $J_1,\ldots,J_s$ be arbitrary proper ideals of positive height on $M$. We introduce the notion of directional mixed multiplicities, obtained from the multigraded quotients $J^{\mathbf u}M/J^{\mathbf u+\mathbf e_j}M$ by retaining one distinguished direction $j$. These invariants refine the mixed multiplicity sequences of arbitrary ideals while remaining compatible with the two classical extremes: for one ideal they recover the positive components of the Achilles–Manaresi multiplicity sequence, and for $\mathfrak m$-primary ideals their top-degree terms recover the usual mixed multiplicities individually. We establish additivity, an associativity formula, superficial reduction, and an exact decomposition of the diagonal coefficients into directional components. Over an infinite residue field, every positive directional coefficient is realized as an eventual length by a sufficiently general directional $(FC)$-system. We then identify an additional synchronization condition under which the directional coefficient becomes the Hilbert–Samuel multiplicity of a parameter ideal, yielding a Rees-type theorem and recovering the classical theorem in the $\mathfrak m$-primary case. In addition, we show a multihomogeneous product formula for $J_1^{n_1}\cdots J_s^{n_s}$, which specializes to the classical Bhattacharya–Rees–Teissier formula.

We study the elementary continued fractions of Ramanujan. The entries considered are precursors of his hypergeometric and $q$-hypergeometric continued fractions. They appear in the first third of Chapter 12 of Ramanujan's second notebook.

The inducibility problem asks for the maximum number of induced copies of a fixed graph among all graphs with a prescribed number of vertices. Inducibility has been an active area of research in extremal combinatorics, but determining all extremal graphs for explicitly defined patterns remains challenging, particularly when the description is required to hold at every host order. A graph $H$ is called a fractalizer if, for every positive integer $n$, every $n$-vertex graph maximizing the number of induced copies of $H$ is a balanced iterated blow-up of $H$, obtained by recursively repeating the same pattern in parts whose sizes differ by at most one. Previous probabilistic results show that large random graphs and random abelian Cayley graphs are fractalizers with probability tending to one, establishing their abundance without directly providing explicit families. We prove that every sufficiently large prime-order Paley graph is a fractalizer. Thus these classical arithmetic patterns determine the exact recursive structure of every extremal host, at every host order and without any algebraic assumptions on the host. We also construct a second explicit infinite family of nontrivial Cayley fractalizers, for which the fractalizer property admits a simpler proof than in the Paley case. Together, these results resolve the explicit-construction question discussed at the 2025 American Institute of Mathematics workshop "Flag Algebras and Extremal Combinatorics."

Metric graphs are generalisations of linear networks and provide a natural framework for the definition of continuously-indexed Gaussian processes. We define a new class of distances on these topologies, termed polyharmonic distances, which unify and extend the spectral construction underlying the effective resistance distance and the biharmonic one. We give both a spectral and a variational characterisation. Furthermore, we show an explicit class of stochastic processes whose variograms coincide with the squared polyharmonic distances. Finally, we show how these metrics can be composed with suitable completely monotonic functions to define isotropic processes having any prescribed finite-order mean-square differentiability along the edges and satisfying the Kirchhoff conditions up to order one at the vertices.

We introduce and study Reed–Solomon sunflowers, namely families of Reed–Solomon codes whose pairwise intersections are all equal to the same fixed subspace. This notion lies at the intersection of extremal subspace combinatorics and coding theory: it can be viewed as a structured version of the sunflower problem in the Grassmannian, and it naturally produces constant-dimension subspace codes with prescribed minimum distance. We focus mainly on the case in which the center is the one-dimensional space generated by the all-one vector. We give an algebraic criterion, expressed in terms of generalized $V$-matrices, ensuring that a family of Reed–Solomon codes forms such a sunflower. We then study the size of these families through counting and constructions. In dimension two, we show that all distinct Reed–Solomon codes form a sunflower and determine its size by counting Reed–Solomon codes up to affine equivalence of their evaluation vectors. For fixed dimension $k\geq3$ and length $\ell\geq2k-1$, we give an explicit recursive construction with $\Omega_{k,\ell}(q^{\lfloor\ell/(2k-1)\rfloor})$ petals and a greedy existence argument with $\Omega_{k,\ell}(q^{\ell-2k+2})$ petals as $q\to\infty$. We also apply the greedy argument to obtain families of $[\ell,k]_q$ MDS codes of size $\Omega_{k,\ell}(q^{2(\ell-2k+2)})$, whose pairwise intersections have dimension at most one but need not be equal.

We reduce the Alperin-McKay-Navarro conjecture to an inductive condition on the universal covering groups of non-abelian finite simple groups. The reduction yields height-zero character bijections compatible with Brauer correspondence and block relations between H-triples for arbitrary finite ambient groups. We record the resulting arithmetic and structural consequences, including preservation of $p$-rationality levels and character-theoretic criteria for Sylow subgroups. The proof combines semilinear centralization and transfer through quasisimple components with induction on the central index. It uses the Clifford and gluing theorems for block relations and a central-defect Dade-Glauberman-Nagao correspondence established in two related papers. We also verify the inductive condition for several sporadic groups at primes for which the Sylow subgroups have prime order, and in defining characteristic for the Suzuki groups ${}^{2}B_{2}(2^{2m+1})$ with $m \geq 2$ and the small Ree groups ${}^{2}G_{2}(3^{2m+1})$ with $m \geq 1$.

The loop group description of surfaces whose Gauss map is Lorentz harmonic is known for timelike surfaces of constant mean curvature in the flat Lorentzian 3-space, and for surfaces of constant Gaussian curvature in the 3-sphere. We carry out the corresponding construction for timelike surfaces in anti-de Sitter 3-space $\mathbb{H}^3_1\cong SL(2,\mathbb{R})$: those of constant mean curvature, whose Gauss map is harmonic for the conformal structure of the first fundamental form, and those of constant Gaussian curvature $K>-1$, $K\neq0$, whose Gauss map is then an immersion and is harmonic for the second fundamental form. For both classes we prove the harmonicity characterization, recover the surfaces from extended frames by Sym-Bobenko-type formulas, and solve the geometric Cauchy problem by the generalized DPW method, with potentials written explicitly in terms of the data and no normalization of the Gauss map. We then show that the two constructions differ only by a normalization: after the substitution $\lambda=\zeta^2$ and a constant gauge, a constant mean curvature extended frame $\hat F$ is an extended frame of its Gauss map, and the maps $\hat F|_{\lambda=e^{4\theta}}\exp(\theta e_3)\hat F|_{\lambda=1}^{-1}$ and $e_3\hat F|_{\lambda=e^{4\theta}}e_3\exp(\theta e_3)\hat F|_{\lambda=1}^{-1}$, with $e_3=\mathrm{diag}(-1,1)$, are timelike surfaces of constant Gaussian curvature $1/\sinh^2\theta$ and $-1/\cosh^2\theta$ wherever the constant mean curvature surface is not flat. The first is the classical parallel surface, the second its polar surface. Explicit families, with extended frames and surfaces in closed form, illustrate the constructions.

We study block relations between H-triples of finite groups. We prove a Clifford correspondence theorem in which a single pair of associated projective representations satisfies both the mixed comparison-function identities and the block conditions on every intermediate group. The resulting correspondence preserves relative character heights and yields a common defect group for each pair of corresponding blocks. We also establish induction and gluing results, including an induction criterion that allows unequal Clifford indices. Direct products, wreath products and changes of ambient group are treated within the same framework. These results provide character-triple tools for Galois-equivariant local-global correspondences.

What information is needed to turn a natural-gradient direction into a useful finite update? Under a population Kullback-Leibler (KL) budget, we call a step useful if it is feasible and loses at most a fraction $\varepsilon$ of the best feasible gain along the direction. We construct a four-state exponential family whose laws share their initial gradient, scalar Fisher information and natural gradient, yet two laws have disjoint useful-step sets. With these quantities supplied exactly and the law otherwise known only through draws, the family's worst-case sample complexity is $\Theta(\log(1/\delta)/(p\varepsilon^2))$ for small $\varepsilon$, where $p$ scales rare-state probabilities and $\delta$ is the failure probability. The budget is fixed and the optimal gain stays bounded away from zero, so the step length, not the direction, carries this cost. For succinctly described event-tilt models, returning a useful step is NP-hard even with the exact natural gradient and efficient exact sampling. Recovering the unit natural gradient to constant error is also NP-hard even in a two-parameter logistic family with Fisher condition number at most 3. We also give matching sample bounds for event tilts, sample bounds for damped Fisher solves and a population-KL certificate for affine classifiers. In frozen-feature classifier heads, stopping at a sampled KL boundary succeeds in about half of the trials, and a 10% KL margin raises joint success above 93% at a KL budget of 0.01. Thus, knowing where to move is not enough: how far to move can carry an update's entire cost.

Two irreducible representations of $U(d)$ whose Young diagrams differ by a single box are connected by four covariant quantum channels: the box can be removed or added, and one keeps either the new diagram or the box ($\mathbb{C}^d$ or its dual). We prove that for each of these channels the output of a coherent state majorizes every other output, so that coherent states minimize the output entropy, and we compute the optimal output explicitly in terms of hook lengths. For the two channels that keep the box we also determine all minimizers; these need not be coherent, even when they are pure. As an application we consider $N$ particles with a given permutation symmetry and find sharp bounds on the spectrum of the reduced density matrix of a single particle, as well as the exact set of spectra that mixed states can reach (for fermions these are the Pauli principle and Coleman's theorem).

We study the dimension-indexed marked-root polynomial maps introduced by Harish. For every $n\geq 4$, their lower coefficient block extends to a polynomial coordinate system, giving three-dimensional Keller maps of geometric degree $n(n-2)$ for every choice of the fixed coefficients. We determine the generic monodromy of the original maps and all successive coordinate descents: it is the cyclic wreath product for odd $n$, and an explicit subgroup of index two for even $n$. Its order is $(n-2)^n n!/\gcd(2,n-2)$. We construct the smooth finite normalization and identify its complement of the affine source as $n-2$ disjoint affine divisors. One is ramified; the others record unramified loss of inverse sheets. The normalization has Picard group $\mathbb{Z}^{n-2}$ and the homotopy type of a wedge of $n-2$ two-spheres. Its rational deck group is cyclic, while the polynomial deck group of the source map is trivial. By base change, the normalization describes the preimage of any subvariety of the target; when the target polynomial factors, a component with only constant units is an open subset of a Kummer cover of the simple roots of one factor. Finally, the monodromy separates these maps, for $n\geq 4$, from single weighted lifts under stable polynomial left-right equivalence.

Let $\mathscr{X}(\boldsymbol{\Sigma})$ be the toric Deligne–Mumford stack with stacky torus $\mathscr{T}$ associated to the stacky fan $\boldsymbol{\Sigma} = (N,\Sigma, \beta)$. A toric principal $H$-bundle over $\mathscr{X}(\boldsymbol{\Sigma})$ is a principal $H$-bundle $\mathscr{P}$ equipped with a $\mathscr{T}$-action lifting the $\mathscr{T}$-action on $\mathscr{X}(\boldsymbol{\Sigma})$, such that the $\mathscr{T}$-action and the $H$-action on $\mathscr{P}$ commute. For a connected reductive algebraic group $H$ over $\mathbb{C}$, we classify the isomorphism classes of framed toric principal $H$-bundles over $\mathscr{X}(\boldsymbol{\Sigma})$ in terms of piecewise linear maps from $|\Sigma|$ to the cone over the Tits building of $H$. Using this classification, we describe the equivariant automorphism group of a toric principal $H$-bundle, and give a necessary and sufficient condition for an equivariant reduction of the structure group. We also show that every toric principal $\mathrm{GL}(r)$-bundle over the weighted projective stack $\mathbb{P}(w_0,\dots,w_n)$ splits equivariantly whenever $r < n$, and that every toric principal $H$-bundle over the weighted stacky projective line $\mathbb{P}(a,b)$ splits equivariantly when $H$ is a connected reductive algebraic group.

We prove the conjecture of Laubie on dimensions of components of two operads completing Loday's operadic butterfly diagram. For one of these operads, the dimension of the $n$-th component is equal to $\binom{2n}{n-1}$ for all $n\ne 3$, and for the other it is equal to $\binom{2n}{n-1}+1$ for all $n\ge 5$.

We study vector-valued local minimizers of double phase functionals with two modulating coefficients and logarithmic-type growth. The energy density is $a(x)|Du|^p+b(x)|Du|^p\log(e+|Du|)$, where the nonnegative coefficients $a(\cdot)$ and $b(\cdot)$ may vanish but their sum is bounded away from zero. Assuming that $a(\cdot)$ is uniformly continuous and that $b(\cdot)$ satisfies a vanishing log-Hölder continuity condition, we prove that minimizers are locally Hölder continuous for every exponent in $(0,1)$. If both coefficients are Hölder continuous, then the gradient of a minimizer is locally Hölder continuous. The proof combines Gehring-type higher integrability with a comparison argument.

Examples of one-sided balanced shifts which do not have variable specification are provided. On the other hand, under synchronization condition, bi-balancedness is proven to be equivalent to variable specification.

Decision-focused surrogates can recover downstream decisions without identifying the quotient report. We characterize the equality set of the convex Smart Predict-then-Optimize surrogate (SPO+) population risk. Under central symmetry, the centered mean class is the unique Bayes minimizer exactly when every nonzero effective displacement makes the old optimizer leave the shifted optimal face with positive probability. This condition separates face crossing from selected-oracle disagreement and gives quantitative local coercivity. Without symmetry, strict crossing alone need not identify the mean; selection balance with reflected crossing restores quotient-report identification, and conditional versions extend the result to measurable predictors. These are population statements, without finite-sample report-recovery or generic transfer-regret guarantees. Closed-form mechanisms reproduce the analytic identities and rates. Portfolio, complete-matrix KuaiRec, and Energy/Storage studies measure predictive fidelity, shifted regret, and fitted-report geometry. A known data-generating process (DGP) companion retains their application geometries while isolating conditional-mean recovery and crossing, without testing the original observational assumptions.

A noetherian form over a category enables one to formulate and prove homomorphism theorems in that category, such as the isomorphism theorems and the diagram lemmas of homological algebra. In this paper we develop a strategy for establishing existence of a noetherian form over a given category, which enables one to find noetherian forms for a broad range of concrete categories. While it was already known that all semi-abelian categories, Grandis exact categories and algebraic categories have noetherian forms, we now establish that so do wide classes of essentially algebraic and topological categories, which include the categories of small categories, groupoids, topological spaces, extended pseudometric spaces, approach spaces, graphs, measurable spaces and axiom-free relational structures.