Searcharxiv⌕ Search

arXiv subjects

Lili Wang

Publications and source records attributed to Lili Wang.

At least 19 recordsLinked to original sources
arXiv2026

Microscopic Origin and Josephson Signatures of $d$-Wave Superconductivity in Monolayer FeSe

Monolayer FeSe on SrTiO$_3$ exhibits high-temperature superconductivity with an electron-only Fermi surface, raising fundamental questions about its unconventional pairing mechanism. Motivated by recent Josephson scanning tunneling microscopy evidence for nodeless $d$-wave superconductivity, we investigate the microscopic role of the Fe $d_{xy}$ orbital using an experimentally constrained multiorbital theory. A tight-binding model fitted to quasiparticle interference and angle-resolved photoemission spectroscopy reveals substantial $d_{xy}$ character near the Fermi level and dominant nearest-neighbor hopping, in contrast to the conventional kinetic-frustration picture of iron-based superconductors. Both itinerant spin-fluctuation calculations and strong-coupling exchange approaches favor a robust $d$-wave pairing state, whose interband pairing components allow a fully gapped quasiparticle spectrum. We further establish a symmetry-based selection rule for spatial Josephson tunneling, revealing characteristic bond-centered maxima and mirror-enforced interference zeros that distinguish $d$-wave pairing from competing $s$-wave states. These symmetry-enforced nodal lines survive near Fe-site impurities, while a pronounced contrast between the $d$- and $s$-wave Josephson patterns persists around vortices despite the loss of mirror protection. Our results connect the unconventional $d_{xy}$ electronic structure to nodeless $d$-wave superconductivity and demonstrate the power of spatial Josephson interference for resolving pairing symmetry in multiorbital superconductors.

cond-mat.supr-con↗
arXiv2026

The Finite Basis Problem for Multiplicatively Idempotent Semirings

The variety of semirings satisfying $x^2\approx x$ is hereditarily finitely based in the signature consisting of addition and multiplication, without designated constants. Multiplication may be noncommutative, and the semirings need not be finite. A subdirect decomposition reduces the proof to the subvarieties defined by $3x\approx x$ and $3x\approx2x$. In the latter case, we express each polynomial identity using a common doubled part and isolated word occurrences. Four fixed identities align the first- and last-occurrence words of these occurrences. Induction on their content extracts protected identities. The associated word identities define a band variety, to which the finite basis theorem for bands applies.

math.RA↗
arXiv2026

Global solvability with arbitrary mass for a chemotaxis--fluid model with sensitivity growth between linear and quadratic

We study a Patlak--Keller--Segel--Navier--Stokes system on $\mathbb T\times\mathbb R$ with density-dependent diffusion and chemotactic sensitivity growing like $n^{1+α}$, where $0<α<1$. The coefficients satisfy the constitutive relation of Painter and Hillen (Can. Appl. Math. Q. 2002; 10(4): 501--543). Near a sufficiently strong Couette flow, we prove global classical solvability and uniform density bounds for smooth initial data with arbitrary finite cell mass. A key ingredient is a joint energy estimate for the full density and the non-zero-mode vorticity, which preserves the positive nonlinear diffusion and avoids the mixed diffusion terms arising from separate estimates of the density modes. Bounded Fourier corrections yield enhanced dissipation within this energy, allowing the non-zero-mode density--fluid interactions to be controlled. For the density zero mode, which is not mixed by Couette transport, we observe that $α<1$ allows one-dimensional interpolation to absorb its chemotactic production into diffusion for every finite mass. Together, these estimates control the nonlinear diffusion and fluid coupling without imposing a smallness condition on the initial cell mass.

math.AP↗
arXiv2026

Suppression of blow-up in the 3D Patlak-Keller-Segel-Navier-Stokes system via non-parallel shear flows

In this paper, we consider the three-dimensional Patlak-Keller-Segel system coupled with the Navier-Stokes equations near the non-parallel shear flow $( Ay, 0, Ay )$ in $\mathbb{T}\times\mathbb{R}\times\mathbb{T}$. We show that if the shear flow is sufficiently strong (A is large enough), then the solutions to Patlak-Keller-Segel-Navier-Stokes system are global in time without any smallness restriction on the initial cell mass as long as the initial velocity satisfies $A^{\frac{2}{3}}\|u_{\rm in}\|_{H^2}\leq C_0$, which seems to be the first result of studying the suppression effect of shear flows for the 3D Patlak-Keller-Segel-Navier-Stokes system. Moreover, it implies that the solutions of the 3D Navier-Stokes equations are global in time if the initial velocity satisfies $A^{\frac{2}{3}}\|v_{\rm in}-(Ay,0,Ay)\|_{H^2}\leq C_0,$ which also shows the transition threshold for the shear flow $(Ay,0,Ay)$ in $\mathbb{T}\times\mathbb{R}\times\mathbb{T}$.

math.AP↗
arXiv2026

Mal'cev products of rings satisfying $x^n\approx x$ and idempotent semirings

Let $\Rn$ be the variety of rings satisfying $x^n\eqid x$, where $n\geq2$, and let $\W$ be a variety of idempotent semirings with commutative addition. The equality $\Rn\circ\W=\Rn\vee\W$ holds, and this variety has subvariety lattice $L(\Rn)\times L(\W)$. Its subdirectly irreducible members that are not additively idempotent have exactly one nontrivial ring component, which is a finite field. Such a member is determined by this field and an idempotent semiring with a multiplicative identity; a separation condition on unary polynomial functions of the latter characterizes subdirect irreducibility. Every subvariety has a finite identity basis. A four-element semiring generates a variety with subdirectly irreducible members of unbounded cardinality. The results extend the Mal'cev product theorem of Wang and Shao from the absorption subvariety to arbitrary $\W\leq\Slp$.

math.GR↗
arXiv2026

Finite bases for full power semirings of finite nilpotent semigroups

We investigate finite equational bases for full power semirings of finite semigroups, including the empty set, in the constant-free signature with addition and multiplication. For a nontrivial finite nilpotent semigroup, we associate a finite relational structure recording the ordered products that are nonzero. We prove that the full power semiring is finitely based if and only if this structure has finite duality, and relate this condition to first-order definability and dismantling of the square of its core. Quantitative bounds connect obstruction size with the number of variables required in an identity basis. In the commutative case of nilpotency index $d$, the criterion reduces to the existence of an element with nonzero $(d-1)$st power. We also establish a nonfinite-basis obstruction for semigroups with a two-element group ideal and prove a finite lifting theorem. These results yield a direct-product criterion and a five-element counterexample to sufficiency of the identity-fibre condition. The arguments use equational logic, finite relational duality, and explicit algebraic constructions.

math.GR↗
arXiv2026

The Finite Basis Problem for Semirings of Order Four

The finite basis problem for small semirings differs from its semigroup counterpart even in order three. Recent work classifies several additive types of four-element additively idempotent semirings, including all 386 semirings whose additive reduct is a chain. We consider all four-element semirings with commutative addition, in the binary signature without named constants. Combining the existing classifications with structural reductions and polynomial normal forms, we obtain 2284 finitely based and 57 nonfinitely based isomorphism types among the 2341 types. The nonfinitely based cases consist of 45 additively idempotent semirings and twelve with nonidempotent addition. The positive arguments give explicit bases or finite constructions with specified bounds. The negative arguments use cited results, term retractions and a hypergraph obstruction. A complete catalogue records the applicable result for each representative; separate correspondence tables identify precisely the cases supplied by the earlier classifications.

math.GR↗
arXiv2026

On the sharp critical mass threshold for the 3D Patlak--Keller--Segel--Navier--Stokes system via Couette flow

As is well-known, the solution of the Patlak--Keller--Segel system in 3D may blow up in finite time regardless of any initial cell mass. In this paper, we are interested in the suppression of blow-up and the critical mass threshold for the 3D Patlak--Keller--Segel--Navier--Stokes system via the Couette flow $(Ay, 0, 0)$. It is proved that if the Couette flow is sufficiently strong ($A$ is large enough), the initial cell mass is less than $16π^{2}$, and the zero modes of the initial velocity are sufficiently small, then the solutions for the system are global in time. The threshold $16π^2$ seems to be optimal: for every finite \(A>0\) and every \(M>16π^2\), we show that there exist smooth positive initial densities of mass \(M\), with zero initial velocity perturbation, whose corresponding solutions blow up in finite time. A key ingredient is a time-integrable decay estimate for the nonconstant part of the zero-mode velocity, which allows the logarithmic Hardy--Littlewood--Sobolev inequality to be used throughout the range $\frac{M}{2π}<8π$. Moreover, the global existence is obtained by combining quasi-linear space-time estimates proposed by Wei--Zhang (Comm. Pure Appl. Math., 2021) and a free-energy argument as in Bedrossian--He (SIAM J. Math. Anal., 2017) for the zero mode. The blow-up result follows from an exact invariant zero-mode reduction to the two-dimensional Patlak--Keller--Segel system and a localized virial argument.

math.AP↗
arXiv2026

A separating identity for matrix semirings over $S_7$

Jiao and Ren proved that the varieties generated by the matrix semirings $\M_n(S_7)$ form an ascending chain of nonfinitely based varieties, and asked whether consecutive members are equal for all $n\geq2$. We give a negative answer by exhibiting an identity in five variables that holds in $\M_2(S_7)$ and fails in $\M_3(S_7)$. Consequently, $\V(\M_2(S_7))$ is a proper subvariety of $\V(\M_n(S_7))$ for every $n\geq3$. The proof uses the cubic-power property established by Jiao and Ren and a description of products equal to the constant matrix $[a]_n$. The identity forces two distinct indices to exhaust the index set in dimension two; an additional index yields an explicit counterexample in dimension three.

math.RA↗
arXiv2026

The strong nonfinite basis property of $S_7$

We prove that the three-element additively idempotent semiring $S_7$ is strongly nonfinitely based: every finite additively idempotent semiring whose generated variety contains $S_7$ has no finite equational basis. This answers Problem~1.2 in Yue, Ren and Gao's paper Explicit equational bases for the power semirings of $S_7$ affirmatively. The proof combines a rigidity property of set systems with private points and a probabilistic construction derived from semilinear sets of letter counts in finite semigroups. The argument allows arbitrary polynomial substitutions and does not assume commutative multiplication. As consequences, both power semirings of the multiplicative reduct of $S_7$, and both power semirings of every finite group of order at least three, are strongly nonfinitely based. All algebras are considered in the signature $(+,\cdot)$ without nullary operations.

math.RA↗
arXiv2026

Compatible additions on a six-element commutative semigroup: equational bases and subvariety lattices

Let $M$ be the six-element commutative semigroup occurring as the common multiplicative reduct of the semirings $SR_6$ and $TR_6$. The closing paragraph of Shao, Ren, and Gao~\cite{ShaoRenGao2026} asks for the finite-basis and subvariety questions for the four remaining compatible additions on $M$. We answer these questions for the four isomorphism types $R_{01},R_{02},R_{11},R_{12}$. First, we classify all compatible additions on $M$: there are nine labelled additions and six isomorphism types, parametrized by $R_{ij}$ with $0\leq i\leq j\leq 2$. For each of the four new types we give a graph-theoretic criterion for every identity, an explicit infinite basis, and a proof of nonfinite basability. The generated varieties $\V(R_{01})$ and $\V(R_{02})$ have eleven subvarieties each, while $\V(R_{11})$ has sixty-six. The lattice $\Sub(\V(R_{12}))$ is countably infinite. Every identity in this variety reduces to a subset of twenty-five fixed identities together with two monotone path families $γ_n$ and $\gammaD_n$. This yields a canonical signature $(H,p,q)$, complete normal forms, explicit meet and join operations, and a formula for all covers. There are 153 fixed nodes, 43 one-parameter families, and 9 two-parameter families; exactly eighteen subvarieties are finitely based, and the unique limit subvariety is $\V(SR_6)$. The strong nonfinite-basis status of the four finite semirings remains open.

math.GR↗
arXiv2026

The finite basis problem for varieties generated by all semirings of a fixed order

Let $\Avar_n$ and $\VV_n$ be the varieties generated by all $n$-element additively idempotent semirings (ai-semirings) and all $n$-element semirings with commutative addition, respectively, in the signature $(+,\cdot)$ without constants. We prove that both varieties are finitely based if and only if $n\leq2$. The five-element case follows from cyclic identities and a structural analysis of products of squares. For $n\geq6$, each variety contains a finite inherently nonfinitely based ai-semiring. We also prove $\VV_n\cap\AI=\Avar_n$ for every $n$, where $\AI$ is the variety of all ai-semirings.

math.RA↗
arXiv2026

Should This Case Be Adapted? Prediction Fragmentation Controls Test-Time Adaptation

Episodic test-time adaptation resets a frozen segmenter to source weights $M_0$ on each case and adapts for a fixed step count. A fixed horizon conflates a cohort-level question, how far to adapt, with an irreducibly per-case one, whether this case should be adapted at all. Cohort means hide that decision: on cross-vendor cardiac MRI the mean $Δ$Dice from adaptation is statistically indistinguishable from zero while 58.7% of cases are individually made worse. We quantify this harm as harmful accepted area (HA), the harmful fraction of the edited area a controller deploys. Held-out tuning gives a stronger baseline than a fixed horizon, but the budget it selects transfers on neither of the two main medical benchmarks, and no global budget can condition on the case. We show that prediction fragmentation---the disagreement geometry between $M_0$ and the adapted mask $M_k$---predicts HA with no labels or extra backward passes at decision time, comparably on three benchmarks (Spearman $ρ$ 0.50--0.60), at a quarter of gradient-norm's latency. A case-level router built on it cuts HA from 0.228 to 0.139 on a benchmark that took no part in its design, with the design frozen and only cut-points recalibrated there. On the cardiac benchmark the design was selected on, the router cuts HA from 0.129 to 0.013 at matched Dice and 1.10 deployed updates, against the retrospective-best budget found post hoc on evaluation labels, and reduces that 58.7% to 20.0%, an upper bound we quantify. Where the retained cases are not net-helped (as on prostate), the router still cuts HA but concedes accuracy, a boundary we report. Thresholds are fit once on a labeled split disjoint from evaluation; decisions use no labels or gradients. The template ports across architecture and domain (nnU-Net$\to$SegFormer, Cityscapes$\to$ACDC) with coordinate, thresholds and per-bucket actions instantiated per domain.

cs.CV↗
arXiv2026

The variety generated by all semirings of order three is nonfinitely based

We prove that the variety generated by all semirings of order three is nonfinitely based, where addition is not required to be commutative and the signature has no constants. The same conclusion holds for the variety generated by all additively idempotent semirings of order three. We establish these conclusions by excluding a uniform bound on the number of variables in an identity basis. Our proof uses identities associated with anchored odd cycles. Three small commutative test semirings isolate a polynomial equivalence class consisting of exactly two polynomials. For a cycle of length $n$, every first nontrivial deduction between them requires an identity with at least $n+1$ variables, even under polynomial substitutions. A retraction followed by a band quotient transfers absorption identities to arbitrary addition and identifies the ai-subvariety of the full joint variety with the joint variety of the ai-generators. Validity of the cycle identities follows from a structural analysis of chain and flat addition. An elementary sixth-power lemma for semigroups of order at most three supplies the retraction.

math.GR↗
arXiv2026

Weinstock Inequality on Regular Trees

Let $T_n$ be the infinite $n$-regular tree, $n\ge3$. We prove that every finite connected vertex set $Ω\subset T_n$ satisfies the sharp inequality \[ σ_1(Ω)\le \frac{n}{(n-1)|Ω|+1}. \] Equality holds if and only if $Ω$ is a ball. Since \[ |δΩ|=(n-2)|Ω|+2, \] the result is equivalently a sharp upper bound at fixed external boundary cardinality, and hence a discrete Weinstock inequality on $T_n$.

math.SP↗
arXiv2026

Counterexamples to Escobar's conjecture

Escobar (J Funct Anal 165(1):101-116, 1999) conjectured that for every $n\ge 3$, an $n$-dimensional compact Riemannian manifold with nonnegative Ricci curvature and all boundary principal curvatures bounded below by $κ>0$ must satisfy $σ_1\geq κ$. We disprove this conjecture for every $n\geq 3$ by constructing conformal deformations of the Euclidean unit ball. We first establish a perturbative criterion, then construct explicit polynomial conformal factors satisfying this criterion. For every sufficiently small $t>0$, the resulting metrics $g_t=e^{2tΦ}g_{\mathbb{R}^n}$ have positive Ricci curvature, every boundary principal curvature is strictly larger than $1$, and $σ_1(\mathbb{B}^n,g_t)<1$. The proof requires several computations, some of which were carried out in Mathematica. The Mathematica code is attached to this submission.

math.SP↗
arXiv2026

Reinforcement Learning-Based Output Feedback LQR for Continuous-Time MIMO Systems

This article studies model-free output feedback linear quadratic regulation (LQR) for continuous-time linear systems with an $n$-dimensional state, an $m$-dimensional input, and a $p$-dimensional output, using filtered input--output data. Since the system state is unavailable, existing methods rely on dynamic filters to parameterize the hidden state using measurable input--output signals. However, the intrinsic dimension of the resulting filter-based parametrization can be smaller than the dimension of the complete filtered vector, and this deterministic redundancy can make the Bellman regressions rank deficient. We characterize this intrinsic dimension and show that the conventional filtered vector contains only $2n$ independent components for single-input multi-output (SIMO) systems and $n(m+1)$ independent components for general multi-input multi-output (MIMO) systems. Based on this characterization, a reduced filtered vector is extracted directly from data and used to develop reduced model-free output feedback policy iteration and value iteration equations, eliminating the redundant directions and decreasing the number of unknown parameters while retaining a fully input--output data-based implementation. A numerical example illustrates the rank reduction and the effectiveness of the learned controller.

eess.SY↗