Papers

We prove that every monic palindromic real-rooted polynomial with nonnegative integer coefficients is the $h$-polynomial of a simplicial convex polytope. This resolves in the affirmative a longstanding problem circulated by Brenti since 2004.

Reverse-complement (RC) and palindromic (PAL) duplications copy a length-$k$ block, reverse the copy, and insert it immediately after the original block; an RC duplication also complements the copied symbols. We study $q$-ary codes correcting $t$ such operations performed sequentially, so a later operation may copy symbols created by an earlier one. For fixed $q\geq2$ and $k,t\geq1$, every length-$n$ code $C$ for either channel satisfies $n-\log_q|C|\geq t\log_q n-O_{q,k,t}(1)$; for fixed $q,k$ and $1\leq t=t(n)=o(n)$ the lower bound is $t\log_q(n/t)-O_{q,k}(t)$. For a single RC error over an even alphabet with a fixed-point-free complement, the previously known RC-specific construction applies at odd $k$ and does not cover even $k$. For every even $k$ and any involutive complement, we give a coordinate-wise bijection that turns each RC duplication into a PAL duplication. Applying this bijection to every codeword therefore converts any $t$-error-correcting RC code into a PAL code of the same size, and conversely; encoders and decoders transfer by adding linear-time coordinate passes. For every even $k$, we also determine the maximum number of distinct descendants produced by exactly two errors from one source word. Words alternating between any two distinct alphabet symbols attain this maximum for PAL, and the bijection gives the RC maximizers. For both PAL and RC at fixed even $k$, we prove the existence of two-error-correcting codes with redundancy $4\log_q n+O_{q,k}(1)$. In the binary two-error problem, the converse gives $2\log_2 n-O_k(1)$, leaving a factor-two gap in the best existence bounds.

Conventional quantum-code constructions encode pure logical states as pure code states, but this restriction can sacrifice performance. We show that intrinsic encoding randomness can improve optimal entanglement fidelity. We bound the loss from imposing a rank-one encoder and prove it is at most quadratic near perfect recovery after joint optimization. The optimized advantage survives small noise perturbations. An explicit noise family requires higher-rank encoders arbitrarily close to perfect recovery, with every optimal encoder mapping pure inputs to mixed code states.

Prime coefficients of elliptic curves exhibit murmurations, statistical patterns that support the prediction of arithmetic labels. We conjecture that root number weighted averages of unitary normalized coefficients at primes and prime powers sample the same leading profile when placed at the effective position $p^k/X$, where $X$ is the conductor scale. For weight $2$ newforms of squarefree level in the level aspect, we prove this principle for every fixed $k$ under suitable short-window and growth conditions, extending Zubrilina's prime case and the square-case analysis of Kundu and Müller. Experiments with elliptic curve isogeny class representatives show that the resulting prime power features improve root number prediction and give a smaller gain in distinguishing ranks $0$ and $1$.

There exists an adjoint pair (D, C) of functors between the category of diffeological spaces and that of topological spaces. In this article, by using the functors, we introduce two pairs of functors between the categories of sheaves on a diffeological space and on a topological space. Then, the adjointness of the novel functors up to the functors induced by the unit and counit of C and D is clarified. There is no natural functor between sites obtained by a diffeological space and a topological space. Therefore, the adjunctions on the categories of sheaves are elaborated without applying the general theory of topoi.

We give sufficient conditions for an extension of valuation rings to be a filtered colimit of complete intersection algebras.

Let $X$ be a compact log-canonical Kähler $n$-fold, $n\ge3,$ with trivial canonical sheaf and with isolated singularities. We prove that the generic fibres of a semi-universal deformation of $X$ have Du Bois invariant $b^{1,n-2}=0$ at the singular points. Under a certain topological hypothesis on $X$ the generic fibres have also link invariant $l^{1,n-2}=0.$ If $X$ is a projective log-canonical $n$-fold, $n\ge3,$ with ample anti-canonical sheaf and with isolated singularities then the generic fibres have $b^{1,n-2}=l^{1,n-2}=0$ (without the topological hypothesis). These are generalizations of recent results of Tenie `Global smoothing of singular Fano and Calabi–Yau varieties' and an older result of Namikawa `Deformation theory of Calabi–Yau threefolds and certain invariants of singularities.'

We improve the best known result on the number of variables needed for the smooth Hasse principle for homogeneous forms of degree \(d\geq5\).

We construct a complete local Gorenstein ring $R$ of dimension $1$ with a prime ideal $\mathfrak{p}\in \operatorname{Spec}(R)$ such that $R$ is an AB ring, but the localization $R_{\mathfrak{p}}$ is not an AB ring. This settles the localization problem for AB rings posed by Huneke and Jorgensen in the negative.

The quantum Rabi model occupies a distinguished role in the study of quantum light and matter, describing the most fundamental interactions. Elucidation of its spectral properties, along with those of its generalizations, is necessary for quantum optics and applications in areas such as quantum information technologies. In this paper, we explore a precise description of energy degeneracy of the two-photon asymmetric quantum Rabi model and the nature of the system's symmetries; as a result, we demonstrate that these have significant relationships with the structure of certain algebraic curves (hyperelliptic curves) and number theory. Concretely, we first prove the existence of degenerate eigenvalues using the monodromy data of the Fuchsian ODE, which draws the eigenvalue problem of the system, and the existence of associated symmetry operators, previously discovered and studied heuristically in physics. Subsequently, we give a complete characterization of spectral degeneracies and present a series of conjectures showing that the degeneracy may determine the entire energy spectral structure. More precisely, for sufficiently strong interaction, we obtain an excellent approximation of the energy curves by the zero locus of a two-variable polynomial associated to degenerate eigenstates. In addition, we postulate an explicit description of the relation between the missing symmetry at the Hamiltonian level (hidden symmetry) and the degeneracy of the two-photon quantum Rabi model.

We investigate the eigenvalue of a generalized Monge-Ampère equation. We proved that the eigenvalue of the equation on the Lp Minkowksi addition of two convex bodies satisfies the Lp Brunn-Minkowski inequality.

We study asymptotic properties of critical points of the two-dimensional vectorial Allen-Cahn energy with finitely many non-degenerate wells, subject to a homogeneous Neumann boundary condition. For any sequence with uniformly bounded energy, we prove that the limiting full and potential measures are supported on a closed countably 1-rectifiable set up to the boundary and satisfy the discrepancy relations. The potential measure defines a free-boundary stationary rectifiable varifold. Boundary mass may occur: weights are constant on regular boundary arcs, finite-type junctions obey projected balance, and the sole non-boundary branch meets the boundary orthogonally. This provides a Neumann-boundary extension of Bethuel's planar interior theory. The key idea is to read the boundary geometry from the limiting stress-energy tensor, which identifies the potential-energy measure as the stationary interfacial measure even in the presence of boundary concentration.

A classical result in modal logic states that $\mathsf{S4}$ has $14$ modalities, that is, every sequence of negations, boxes, and diamonds is equivalent to one in a set of $14$ such sequences. We study analogous results for the non-classical analogues $\mathsf{CS4}$, $\mathsf{IS4}$, $\mathsf{GS4}$, and $\mathsf{GS4^c}$ of $\mathsf{S4}$. First, we show that, while all these logics have finitely many $\{\Box,\Diamond\}$- and $\{\neg,\Box\}$-modalities, the logic $\mathsf{CS4}$ has infinitely many $\{\neg,\Diamond\}$-modalities. Second, we show that $\mathsf{IS4}$ and $\mathsf{GS4}$ have finitely many $\{\neg,\Diamond\}$-modalities, but they have infinitely many $\{\neg,\Box,\Diamond\}$-modalities. At last, we show that $\mathsf{GS4^c}$ has finitely many $\{\neg,\Box,\Diamond\}$-modalities.

We develop a rigorous analytical framework for a class of physiologically structured population models with two internal state variables, nonlocal ecological feedbacks, dynamic resources, inter-zone transfer, and selective harvesting. The full model is a coupled nonlinear PDE–ODE transport–renewal system with endogenous inflow at the recruitment boundary, a setting in which transport, nonlocal dependence, and boundary renewal interact at the same level. For this full nonautonomous multi-zone system, we prove finite-horizon well-posedness in a positive $L^{1}$-based state space, including global existence on arbitrary bounded time intervals, uniqueness, nonnegativity, and continuous dependence on initial data, environmental forcing, and harvesting effort. We then isolate an autonomous single-zone reduction at extinction and construct a positive compact next-generation operator on the recruit space. In a further nonlinear stationary reduction, we prove that supercriticality of the basic reproduction number $\mathcal R_{0}>1$ yields existence of a nontrivial stationary state under a parametrized compact-operator hypothesis encoding density-dependent renewal feedback. Finally, for a finite-horizon harvest objective over a compact Lipschitz-regular admissible class, we establish existence of an optimal control. The results separate what can be proved for the full climate-explicit system from what can be justified only after autonomous reduction, thereby clarifying the mathematical scope of threshold and control theory for structured populations.

Motivated by the theory of Riemann surfaces and specifically the significance of Weierstrass points, we prove general structure results about finite groups that have a faithful transitive action with fixity 4. We also explain examples for many different possibilities of such actions.

Large Language Models (LLMs) have shown remarkable promise in translating and reformulating complex mathematical optimization problems across modeling languages. However, validating such transformations through empirical solver executions alone is unreliable, as solver outcomes may be affected by local minima, structural timeouts, numerical artifacts, and subtle semantic divergence between formulations. We introduce SOVER, an LLM-assisted SMT framework that separates semantic mapping from formal certification: Z3 checks domain cross-feasibility and global objective-order preservation for mixed-integer linear formulations, while dReal provides tolerance-aware feasibility/range and $\epsilon$-argmin checks for continuous nonlinear formulations. We also introduce NLEquiv-150, a public benchmark of 100 equivalent and 50 deliberately hard non-equivalent nonlinear reformulation pairs. With LLM-extracted mappings, SOVER classifies 149/150 pairs (99.33%) correctly, including all 50 hard negatives; the sole error is an incomplete mapping extraction.

We prove the shuffle realization for quantum affine superalgebras of types $B,C,D$ in the loop realization, in that we construct an isomorphism between each of these algebras and a suitably defined double shuffle algebra. We explicitly describe the latter shuffle algebra using certain vanishing conditions that generalize the Feigin-Odesskii wheel conditions; a novel feature is the appearance of a so-called scary wheel, which is a particular order $2$ vanishing condition involving $7$ variables. Along the way, we fully develop the theory of shrubs in super types $A$ (affine and toroidal), which are important combinatorial tools in the study of the shuffle algebras associated to toric Calabi-Yau threefolds. Our techniques also allow us to define quantum toroidal superalgebras of types $B,C,D$. More importantly, we provide a general framework to formulate and prove shuffle realizations in the wide generality of quivers with parameters.

Let $\mathscr{O}_K$ be a mixed characteristic complete DVR with perfect residue field $k$, and let $\mathfrak{X}$ be a proper formal scheme over $\mathscr{O}_K$ with semistable reduction. Answering a question of Fontaine and Jannsen, a classical theorem of Hyodo and Kato gives a rational identification between the log crystalline cohomology of the special fiber $\mathfrak{X}_0$ over $W(k)^0$ with the de Rham cohomology of the generic fiber $\mathfrak{X}_K$. In this note, we use a log variant of the saturated de Rham–Witt complex of Bhatt–Lurie–Mathew to prove that such a comparison holds integrally.

We study a one-dimensional fractional $U(1)$ Ginzburg–Landau model whose quadratic part has Fourier multiplier $|k|^\sigma$, focusing on the marginal case $\sigma=1$. This dispersion yields logarithmic spin-wave fluctuations, suggesting BKT-like behavior despite the one-dimensional setting. We sample the equilibrium Gibbs measure using stochastic Gross–Pitaevskii dynamics and analyze correlation functions, dimensionless ratios, effective exponents, and twist responses. The correlation function shows a low-temperature algebraic branch with a temperature-dependent exponent, while the high-temperature regime exhibits a nonlocal-kernel-induced tail consistent with $C(r)\sim r^{-2}$. The correlation and Binder ratios are nearly size independent at low temperature and collapse with the BKT-type variable $(T-T_{\rm BKT})(\log L)^2$; finite-size effects set in around $T\simeq0.35\text{--}0.4$, consistent with $T_{\rm BKT}\simeq0.35$. Unlike the two-dimensional XY model, twist responses do not yield a finite helicity modulus: the ordinary linear-response quantity grows with system size, whereas the cusp twist response scales as $L^{-\eta(T)}$, like the squared zero-mode order parameter. Thus, the transition is BKT-like in correlation scaling, but lacks a universal helicity-modulus jump.

In Lazy Cops and Robbers, at most one cop moves on each cop turn. We study the lazy cop number of the generalized hypercube $Q(n,m)$, whose vertex set is ${\{0,1,\ldots,m\}}^n$. For each fixed integer $m\geq2$, we prove the asymptotic upper bound $$c_{\mathrm{L}}(Q(n,m))=O\!\left(\frac{{(m+1)}^n}{n^{3/2}}\right).$$ This result improves the upper bound of Sim, Tan, and Wong by a factor of $\log n$. The proof combines a moving dominating-set argument with an explicit dominating-set construction inside the support classes of each level. As a separate domination result, we show that, for fixed integers $m\geq2$ and $d\geq1$, the Hamming graph $K_m^{\square k}$ has a distance-$d$ dominating set of asymptotic size $O(m^k/k^d)$. This order is optimal up to a constant factor.

We establish Aleksandrov–Bakelman–Pucci-type estimates for viscosity subsolutions of Poisson equations associated with the weighted $1$-Laplacian and, more generally, with fully nonlinear operators acting on the Hessian in directions orthogonal to the gradient. A key feature of the proof is that the quasiconcave envelope plays the role of the concave envelope in the classical ABP estimate.

Fixed-width sequential stopping rules terminate a stochastic simulation once an estimated confidence interval reaches a prescribed width. Classical fixed-width theory typically relies on a strongly consistent estimator of the asymptotic variance. This makes the normalized stopping time asymptotically deterministic, allowing fixed-sample-size limit theory to be transferred to the estimator at termination. This mechanism can fail when simulation output has infinite variance or long-range dependence. Although self-normalization and subsampling can yield asymptotically valid confidence intervals at a fixed sample size, the scaling process and the stopping time may retain nondegenerate randomness, so fixed-sample-size quantiles need not provide valid coverage at termination. In this paper, we develop a unified framework based on a joint functional limit theorem for the estimation process and a scaling process. We characterize the asymptotic behavior of both the stopping time and the self-normalized estimator evaluated at termination, thereby obtaining asymptotically valid sequential confidence intervals in classical finite-variance and infinite-variance settings. Moreover, we introduce a sequential subsampling procedure that consistently estimates the distribution relevant at the stopping time without directly estimating nuisance parameters in the limit distribution. The framework is verified for heavy-tailed moving-average processes, stochastic approximation, and an M/G/1 queue with heavy-tailed service times.

Feedback-based coding schemes have demonstrated substantial performance gains over today's open-loop coding schemes. Unfortunately, these gains are usually achieved in idealized settings with perfect feedback. Over the last few years, machine learning-based schemes have been shown to be promising solutions for implementing feedback-based codes, particularly when combined with short-block-length open-loop error correcting codes (ECCs) in a concatenated coding structure. However, existing ML-based feedback schemes remain agnostic to the outer code's structure, potentially misallocating feedback resources on error patterns already correctable by the outer ECC. To address this, we propose MaskCode, a Transformer-based inner feedback code for concatenated coding systems, which explicitly incorporates structural knowledge of the outer linear block code into the inner feedback encoder design via two synergistic mechanisms: 1) a soft syndrome-based input that informs the encoder about potential parity constraint violations, and 2) a code-aware attention mask derived from the Tanner graph. We further show that end-to-end training with a differentiable belief propagation (BP) decoder offers no additional gain, as MaskCode's structure-aware design already internalizes the structural knowledge of the outer code; in fact, backpropagation through the iterative BP decoder introduces gradient explosion, which degrades rather than improves performance. Extensive evaluations on BCH and LDPC outer codes demonstrate that MaskCode consistently outperforms all baselines, achieving up to 1.5 dB SNR gain.

We study the behavior of trace ideals under taking syzygies. In particular, when $R$ is numerical semigroup ring and $I$ is a homogeneous ideal in $R$, we obtain an upper estimate for $\operatorname{tr}_R(\Omega^1_R(I))$, and we show this estimate is sharp when $I$ is the conductor ideal $\mathfrak{c}_R$ of $R$. Using this result, we characterize the numerical semigroup rings for which $\operatorname{tr}_R(M) \subseteq \operatorname{tr}_R(\Omega^1_R(M))$ holds for every finitely generated $R$-module $M$. In a similar vein, we characterize numerical semigroup rings for which $\operatorname{tr}_R(\Omega^1_R(\mathfrak{c}_R))=\mathfrak{c}_R$.

Functional tissue units (FTUs), including tertiary lymphoid structures (TLSs), blood vessels, and glands, encode localized immune, vascular, and epithelial organization in histopathology. Accurate quantification of these structures is important for studying tissue architecture and disease-associated tissue organization. However, FTUs are frequently sparse, heterogeneous, and surrounded by large amounts of morphologically similar background tissue, making automated segmentation in whole-slide images (WSIs) challenging. We therefore developed FTU-Seek, a pathology foundation model-guided framework that treats morphology-aware negative-patch selection as a key component of sparse FTU segmentation. FTU-Seek uses frozen multi-depth features from the UNI pathology foundation model to train a patch-level classifier that distinguishes FTU-containing from FTU-absent tissue. Target-absent patches are subsequently ranked according to their predicted target-containing probabilities, and the highest-scoring hard negatives are selected through a static Top$K$ strategy to construct compact segmentation training sets. The framework was evaluated using five-fold cross-validation and internal test cohorts across TLS, blood-vessel, and gland segmentation tasks, with an additional independent 30-WSI held-out cohort for TLS. Positive-only, all-tissue, random-negative, and matched random Top$K$ sampling strategies served as comparators. Segmentation-derived phenotypes were further explored in external TCGA cohorts.

Let \(n\ge1\), \(0<s<1\), \(c>0\), and let \(A\) be a positive definite symmetric matrix. We prove that the unique decaying viscosity solution of \[ \min\{u,\,(-\Delta)^s u-(c-\langle Ax,x\rangle)\}=0 \qquad\text{in }\mathbb R^n \] has the form \[ u(x)=K\max\{1-\langle Bx,x\rangle,0\}^{1+s} \] for some \(K>0\) and some positive definite symmetric matrix \(B\). In particular, its positivity set is an ellipsoid. For \(s=1/2\) and \(n\ge2\), this result, combined with the classification of Fernández-Real and Yu, implies that cubic global thin obstacle solutions with nonempty bounded positivity set have ellipsoidal positivity sets. This proves the conjecture of Fernández-Real and Yu in the nonempty bounded-positivity case.

In this paper, we develop a theory of generalized fractional integration for set-valued mappings for the Katugampola fractional integral, which unifies the classical Riemann-Liouville and Hadamard fractional integrals. The Katugampola fractional integral of a set valued mapping is studied using integrable selections. We study its properties with respect to the Hausdorff metric on the space of nonempty compact subsets of R and several fundamental analytical characteristics including convexity, boundedness and continuity are preserved under Katugampola fractional integration. Furthermore, we establish that both bounded variation and Lipschitz regularity of a set-valued mapping are preserved under its Katugampola fractional integral. We investigate the existence of regular selections associated with the Katugampola fractional integral and show that whenever the original set valued mapping admits a selection with a specified regularity property, the corresponding Katugampola fractional integral does as well.

We study the diffusive limit of the steady neutron transport equation with velocity-dependent in-flow boundary data in polygonal domains. On a bounded convex polygon, we construct an approximation by the interior solution, flat side layers, and wedge layers, and prove an algebraic error estimate in $L^\infty$. We also give a complete formulation and well-posedness theory for the kinetic wedge layer, including its algebraic decay. The proof combines a characteristic stability estimate, a weighted Mellin mapping theorem, a two-depth construction and matching scheme, and a shifted superharmonic barrier for the wedge corrector. We also prove the $L^2$ diffusive limit at rate $O(\varepsilon^{1/2})$ on general bounded simple polygons, including those with reentrant vertices.

Let $L^x$ be the total local time of one-dimensional super-Brownian motion with $(1+\beta)$-stable branching mechanism, $0<\beta<1$. We prove that $\{x\in\mathbb R:L^x>0\}$ is almost surely a bounded open interval $(\mathsf L,\mathsf R)$ and that \[ h_L(\mathsf L)=h_L(\mathsf R)=1+\frac{2}{\beta} \] almost surely, where $h_L$ denotes the pointwise Hölder exponent. Thus the pointwise $\gamma$-Hölder condition holds at both endpoints for every $\gamma<1+2/\beta$ and fails for every $\gamma>1+2/\beta$. The same conclusions hold under the canonical excursion measure.

In this paper, we establish a summation formula associated with the generalized Apostol-Bernoulli functions. This formula unifies the Euler-Maclaurin summation formula, the Euler-Boole summation formula, and the character analogues of these two formulas. We also apply it to obtain the general power sum formula and the special values of Berndt's generalized $L$-function at non-positive integers.