Papers

We construct an equivalent norm on $\mathbb{R}\oplus\ell_2$ and a nonempty closed convex unbounded subset having the fixed point property for nonexpansive self-mappings. The ambient renormed space has the fixed point property on bounded closed convex subsets. The main point is a geometric localization estimate at a distinguished point of the unbounded set; the fixed point conclusion then follows from the Kirk–Ray localization theorem.

In this work, long-time Sobolev stability and error estimates with optimal $\varepsilon$-dependence are established for the Strang splitting method applied to the Allen–Cahn equation with periodic boundary conditions. By combining the modified energy dissipation law, and Hölder-Gagliardo-Nirenberg product estimates, the numerical solution is proven to admit uniform-in-time $H^r$-seminorm bounds $\mathcal{O}(\varepsilon^{-r}+1)$, consistent with the corresponding Sobolev bounds of the exact solution. A second-order error estimate $\mathcal{O}(\varepsilon^2\tau^2)$ is further established on any prescribed finite time interval, exhibiting quadratic dependence on $\varepsilon$. Numerical experiments are presented to verify the optimal $\varepsilon$-dependence of both the stability and error estimates.

We give an explicit countably infinite orthomodular lattice generated by three specified elements and a direct argument that it admits no injective homomorphism into any $\sigma$-complete orthomodular lattice. In particular, it does not embed into any set-sized complete orthomodular lattice. The source is a generated subalgebra of a cofinite reduced product of finite-dimensional real subspace lattices. A nonzero finite term $a$ is orthogonal to the orbit of a terminal line under every finite word in three Sasaki projections, while another finite term $X$ satisfies $\pi_a\pi_X(t)=a$. If the image of this countable orbit has a supremum in a target orthomodular lattice, Sasaki adjunction forces that supremum to commute with the generator images. This forces the image of $a$ to vanish. The proof imposes no atomicity, modularity, Hilbert representation, cardinality bound on the target, or preservation of source infinitary operations. All target lemmas are derived from the orthomodular axioms.

Gauging fermion parity in a $(2+1)$-dimensional fermionic topological order requires lifting its symmetries to the resulting bosonic theory. We give a direct tensor-categorical proof that, for a slightly degenerate braided fusion category $\mathcal{C}$ and a fixed minimal nondegenerate extension ${\cal E}$, the restriction homomorphism $r\colon\operatorname{Aut}_{\mathrm{LR}}(\mathcal{E})\to \operatorname{Aut}_{\mathrm{LR}}(\mathcal{C})$ on classes of complex-linear locality-respecting (LR) braided autoequivalences is surjective with central kernel $\mathbb{Z}/2$, generated by fermion attachment. Thus each symmetry class has exactly two LR lift classes, related by attaching the fermion to every vortex. We apply graded Morita theory to reduce the kernel to two module types on the vortex sector, corresponding to tensoring with the vacuum or the fermion. We then prove relative rigidity to show that each type determines a unique LR class, which are distinguished by fermionic self-braiding even when their actions on simple-object labels coincide. We prove surjectivity by extending graph algebras in $\mathcal{C}^{\mathrm{rev}}\boxtimes\mathcal{C}$ to Lagrangian algebras in $\mathcal{E}^{\mathrm{rev}}\boxtimes\mathcal{E}$. Our results require neither unitarity nor a ribbon structure. We illustrate the lifting ambiguity in the Laughlin state, the $SO(3)_3$ theory and the Ising topological order.

We proved recently that there are no unconditional Schauder frames of exponentials in the space $L^p(\Omega)$, $1 \le p < 2$, for any set $\Omega \subset \mathbb{R}^d$ of finite measure. For $p > 2$, we proved that the same holds provided that $\Omega$ has nonempty interior, but the general case remained open. In this note we close this gap, i.e., we establish that exponentials cannot form an unconditional Schauder frame in $L^p(\Omega)$, $p \ne 2$, without any additional assumptions on the set $\Omega$ of finite measure other than measurability.

We prove decay estimates for oscillatory integral operators along $(y_1, y_2, y_1^2+y_2^2)$ in the finite field and the continuous settings. In the continuous setting, our result allows a bi-parameter maximal modulation, which extends the one-parameter version previously established by Pierce and Yung. Our approach is different from theirs. Then we discuss three applications of our estimates: a power saving bound for certain pattern finding problem in finite fields, a short approach to $L^p$ boundedness of an one-parameter Pierce–Yung operator, and $L^p$ boundedness of a bi-parameter Pierce–Yung operator.

Variational inequalities (VIs) provide a unified framework for optimization, saddle-point, equilibrium, and learning problems, where operators need not be globally Lipschitz. The $\alpha$-symmetric generalized-smoothness condition allows local operator variation to grow with the operator norm, but existing extragradient methods often require unknown problem-specific constants. To address this issue, we propose PF-AEG, an explicit, line-search-free extragradient method whose stepsizes use quantities generated during the iterations, without prior knowledge of these constants. Under $p$-quasi-sharpness, we prove full-sequence convergence and last-iterate squared-distance rates: linear convergence for $0<p\le2$ and $\mathcal{O}(k^{-2/(p-2)})$ for $p>2$. The latter matches existing parameter-dependent rates under comparable assumptions, while the linear guarantee extends the previous $p=2$ result to $0<p\le2$. Building on the idea of PF-AEG, we develop AEG-VR, a variance-reduced adaptive extragradient method for finite-sum VIs. We prove almost-sure convergence and, under mild conditions, obtain analogous expected last-iterate rates, while retaining independence from the unknown problem-specific constants in stepsize selection. Experiments on synthetic generalized-smooth VIs and several learning problems show that the proposed methods are competitive with, and in several tested regimes outperform, existing baselines.

We introduce Rota-Baxter operators on Lie affgebras by an identity independent of the choice of origin and prove that they induce descendant Lie affgebra structures. We establish a bijection between ordered affine Lie factorizations and idempotent operators split at a chosen idempotent fixed point. A cokernel obstruction determines the existence of this point, while invariance under the structural endomorphisms characterizes splitting. We also determine when weighted descendants satisfy the affine Jacobi identity and when operators of nonzero weight give affine doubles. For canonical Lie affgebras, we describe all affine Rota-Baxter operators by Lie subalgebra decompositions and translation vectors, and classify them up to automorphism in dimensions at most two.

We study weighted Bergman spaces of quasiregular mappings on the unit ball in $\mathbb{R}^n$, with weight $(1-|x|)^\alpha$, $\alpha>-1$. For every fixed distortion bound $K\ge1$ and every $p>0$, we characterize Bergman integrability by averaged derivatives. We also give geometric conditions under which integration over a subset, or against another measure, controls the full Bergman integral. When $K$ is sufficiently large, the dominating sets are precisely the relatively dense sets. For comparison, we determine the dominating sets for the Möbius class $K=1$ in dimensions $n\ge3$. At the critical exponent $p=n+\alpha$, its domination criterion differs from that for every fixed $K>1$. Our results extend aspects of the Bergman theory developed by Nolder~\cite{Nolder90} and by Oleinik and Vuorinen~\cite{OV}, and complement the recent work of Adamowicz and González~\cite{AG21,AG25} and Adamowicz and Caamaño~\cite{AC} on quasiregular Hardy spaces.

The $(d+1)$-type $\Lambda$-Wright–Fisher process with polynomial drift is a Lévy-type process on the $d$-dimensional unit simplex whose generator entails a particular jump and diffusive component, as well as a polynomial drift component. In this work, the process is connected to a (Bernstein) dual process, which admits an interpretation as an urn model with coloured balls. Exploiting this duality, we identify various parameter regimes: one in which the $\Lambda$-Wright–Fisher process is absorbed almost surely at a vertex of the simplex, and others in which it admits a unique nontrivial ergodic probability measure. The results include verifiable criteria for absorption versus persistence, as well as representations of absorption and stationary measures. Moreover, the duality relation is exploited to devise and implement a simulation-based approach for estimating moments of the $\Lambda$-Wright–Fisher process.

We study a conjecture of Conlon, Fox, Gunby, He, Mubayi, Suk, Verstraëte, and Yu, which characterizes the $3$-graphs $H$ for which the off-diagonal Ramsey number $r(H,n)$ is polynomial. We prove the conjecture for $3$-graphs with at most five tight components. Additionally, we show that the full conjecture is equivalent to its specialization to linear $3$-graphs. We prove quasipolynomial upper bounds for a class of `forward-doubleable' $3$-graphs, which includes the Fano plane. Finally, we give an exponential lower bound for `non-forward-laminar' $3$-graphs and use this to completely classify the Ramsey growth of a natural family of 3-graphs called chains.

Portfolio decisions are often made under uncertainty about expected returns, and investors revise their beliefs as new information arrives. Although richer information generally cannot reduce optimized value ex ante, the trading strategy it induces need not improve on the strategy induced by a less informative information structure in every fixed-drift scenario. We introduce scenario-conditional expected utility and value of information to evaluate strategies selected under the investor's current belief. In a finite-horizon Gaussian portfolio choice model, we compare a fixed-belief investor, a price observer, and an attentive investor who also processes an auxiliary signal under an attention capacity constraint. Ex ante, price learning and auxiliary attention are valuable, and the total value of attention increases with capacity. Scenario-conditionally, learning creates a correction benefit by reducing the initial trading-position gap relative to the full-information strategy, but also a fluctuation cost through belief revisions, so the ex ante ranking can reverse. We derive the current-belief distribution of scenario-conditional value and the associated underperformance probability, defined as the probability of negative scenario-conditional value relative to fixed-belief choice. Over sufficiently short horizons, the scenario-conditional value of learning is negative when belief error is small relative to posterior uncertainty. For a given horizon, sufficiently high attention capacity yields a higher scenario-conditional value than any fixed finite-capacity benchmark across all drift scenarios. We also derive sharp convergence rates to full-information performance under Shannon and Tsallis technologies.

We compute cohomology spaces of the ortosymplectic Lie superalgebra $\mathfrak{osp}(1|2)$ on the (1,1)-dimensional real superspace with coefficients in the superspace $\frak{D}_{{\lambda},\mu}$ of $n$-ary differential operators acting on weighted densities (${\lambda}\in\mathbb{R}^n$ and $\mu\in\mathbb{R}$).

The TF-cone conjecture of Asai and Iyama predicts that the TF equivalence class of an integral $g$-vector is the open cone generated by the canonical summands of all its positive multiples. We give a counterexample to this conjecture. For the tensor product of the three-arrow Kronecker algebra with the path algebra of $s\to t$, a vector $w$ has a two-dimensional TF equivalence class, whereas every positive integral multiple $mw$ is generically indecomposable. The cone generated by these canonical summands is therefore a single ray. The same example disproves the relative-interior version proposed by Haerizadeh and Yassemi.

We determine the best constants in estimates of the $L^\infty$ norm of functions on the sphere with bounded Laplacian whose first spherical harmonic components vanish. In every dimension $d\geq2$, these constants are expressed through weighted $L^1$ extremal polynomials. On $S^2$ the constant is an explicit positive series, and the best coefficient of $n^{-2}$ is $7\zeta(3)/4$. On $S^3$ the best coefficient of $n^{-2}$ is a positive integral involving the Hörmander–Bernhardsson extremal function. In higher dimensions the rescaled extremal polynomials converge to weighted entire extremals, which give asymptotic formulas for the Bohr constants on $S^4$ and $S^6$. The most unexpected and, in our view, interesting observation is that, for every fixed $n\geq2$ on $S^2$ and $S^3$, the constants in best approximation estimates are strictly smaller than the corresponding Bohr–Favard constants.

We characterize the generic consistency of vertical systems in terms of the iterated circuits of the column matroid of the coefficient matrix. More precisely, we prove that such a system is generically consistent if and only if there exists an iterated circuit such that the dimension of the Minkowski sum of the associated Newton polytopes is equal to the rank of the coefficient matrix. We also give an equivalent characterization in terms of $G$-spanning flags of the dual matroid. With this, we derive a combinatorial formula for the codimension of the locus of parameter values for which the system has a solution, extending known results on dual defects of toric varieties. When this locus has codimension one, its defining equation yields a generalization of the sparse resultant to vertical systems, which applies also to $\mathcal{A}$-discriminants. Finally, we characterize when the solution set is contained, for generic parameter values, in a finite union of hyperplanes parallel to a coordinate hyperplane.

We establish a priori estimates and compactness results for positive solutions of the CR Nirenberg problem under natural weighted flatness conditions. The proofs combine refined blow-up analysis, Pohozaev identities, and the complete classification of positive entire solutions to the critical CR Yamabe equation recently obtained by Li–Liu–Ma \cite{LLM2026} and Li \cite{L2026}. Existence and degree results are established in our subsequent paper \cite{QTWPartII}.

This paper studies non-autonomous iterations of $\lambda e^z$, where at each step the parameter $\lambda$ is chosen from a bounded interval. Urbański and Zdunik introduced and developed a thermodynamic formalism for random dynamics of exponential maps, extending the autonomous result established independently by Rees and Lyubich to typical parameter sequences in $[A, B]^{\mathbb{N}}$ with $1/e<A<B<\infty$. In this paper, we show that the same conclusion indeed holds for all parameters in $[A, B]^{\mathbb{N}}$, Our proof is based on carefully crafted uniform versions of pullback arguments, inspired by the original methods of Rees and Lyubich. In a related direction, Urbański and Zdunik established a non-autonomous analogue of Misiurewicz's theorem for the exponential map, later refined by Lech. We further extend their work by providing a refined condition for the Julia set to be the entire complex plane.

We study a class of periodic NLS-Dysthe (NLS-Dy) equations, also referred to as higher-order nonlinear Schrödinger equations (HNLS), of the form $i\partial_t u+a\partial_x^2u+b|u|^2u+i\varepsilon(c\partial_x^3u+d|u|^2\partial_xu+eu^2\partial_x\overline{u})=0$ with $ x\in\mathbb T$, $a,b,c,d,e\in\mathbb R$, $abc\neq0$, and $\varepsilon>0$. For $0<\varepsilon\ll1$, equations of this type arise as higher-order modulation models for some nonlinear dispersive PDEs, including the Klein-Gordon equation and the Serre-Green-Naghdi equation modeling the evolution of nonlinear shallow water waves. We prove local well-posedness in $H^s(\mathbb T)$ for $s>3/2$ for arbitrary admissible coefficients, and establish an $H^1$ continuation criterion for $s\ge2$. In the parameter regime $e\neq0$, a lower-order conservation law provides uniform $H^1$ control and, together with the continuation criterion, yields global well-posedness for $s\ge2$. This result does not require the non-resonance condition appearing in Takaoka's periodic theory, while the condition $e\neq0$ naturally arises in the modulation setting considered here. Finally, motivated by the long-time regime relevant for modulation, we establish uniform long-time control without imposing a smallness assumption on the initial datum. Using the conservation laws of cubic NLS and their invariance under the commuting complex mKdV flow, we obtain uniform $H^s$ bounds on time intervals of order $\varepsilon^{-1}$ for integer $s\ge2$. For non-integer real $s\ge2$, the maximal $H^s$ solution exists on the same time scale, while uniform control is obtained at a lower regularity, with a loss of less than one derivative.

Free classical communication can yield an unbounded single-shot saving in quantum compression, even though it leaves the asymptotic rate of an i.i.d. source unchanged. At every fixed error in $(0,1)$, we exhibit sources whose unassisted compression cost is arbitrarily large, yet a single qubit suffices with classical communication and no preshared entanglement. We establish this separation by determining the exact optimal source fidelity for compression through an $m$-dimensional noiseless quantum channel, with and without free classical communication, while preserving correlations with an inaccessible reference. Without classical assistance, the optimal fidelity is exactly the Ky Fan norm $\|\rho\|_{(m)}$, the sum of the $m$ largest eigenvalues of the source state. With classical assistance, we obtain an explicit spectral threshold formula and a protocol attaining it. Classical communication strictly improves fidelity whenever $m$ is smaller than the source rank. The corresponding compression costs are two smoothed versions of the log-rank entropy, also called max-entropy: trace-distance smoothing under local operations and quadratic smoothing under LOCC. For i.i.d. copies of a fixed source with positive entropy variance, however, the two costs have identical first-, second-, and third-order asymptotic terms. For nonlattice spectra, we determine the constant-order advantage: classical communication saves $\log_2(e)-1$ bits per block asymptotically, and the ratio of assisted to unassisted minimum communication dimensions converges to $2/e$.

Graphs defined on groups, such as power graphs, enhanced power graphs and commuting graphs, form a hierarchy under inclusion of edge sets. We develop an analogous theory for topological spaces. To a space X we attach five graphs on the vertex set X, all invariant under homeomorphisms: the specialization, inseparability, closure-meeting, Urysohn and functional graphs. They form a diamond-shaped hierarchy whose members are null exactly when X is T_1, Hausdorff, Urysohn or functionally Hausdorff. The Urysohn graph always contains the square of the inseparability graph, with equality for Alexandrov spaces, and the hierarchy is compatible with products via the strong product of graphs. Every countable graph, together with isolated vertices, is the inseparability graph of a second countable T_1 space. For finite spaces we characterize the graphs that occur as inseparability graphs (every edge lies in the closed neighbourhood of a simplicial vertex), determine all coincidences between members of the hierarchy, and show that the specialization graph determines the weak homotopy type but not the space itself. Using Dowker duality we prove that the complexes of inseparable and of closure-meeting families are always homotopy equivalent and, under an acyclicity hypothesis, recover the weak homotopy type. We show that if the closure-meeting graph is a cograph then it is chordal, an analogue of a recent result for enhanced power graphs; the same holds for the inseparability graph of an Alexandrov space but not in general. Finally, following Cameron's work on topologies preserved by permutation groups, we show that for a topology invariant under a primitive group each graph is null or connected, and that the neighbourhood filters of the five graphs reverse the hierarchy, giving spaces whose specialization graph contains the Rado graph as a spanning subgraph.

We prove that, on a non-compact Ahlfors-regular PI space, every function of bounded variation $f$ can be decomposed as a countable sum $\sum_{n=1}^\infty f_n$ of monotone functions of bounded variation $f_n$, in such a way that the total variation of $f$ is the sum of the total variations of the functions $f_n$. Our result extends a Euclidean result of Bianchini and Tonon to the metric setting.

Reconfigurable intelligent surfaces (RIS) combined with rate-splitting multiple access (RSMA) provide a promising solution for improving the spectral efficiency (SE) of sixth-generation (6G) communication systems. However, most existing studies neglect the practical coupling between the RIS phase shift and amplitude, which may cause phase-shift mismatch and significant performance loss. In this paper, we investigate the multiuser sum-rate maximization problem for practical RIS-enhanced RSMA systems considering amplitude-phase coupling. To address the resulting non-convex problem, an effective variable separation algorithm is proposed. The proposed algorithm is compatible with RIS reflecting optimization under the continuous phase shift or any amplitude-phase coupling relationship and can also be directly extended to practical RIS-enhanced NOMA and OMA systems. Simulation results demonstrate performance gain of over 190% compared with the ideal-RIS-based benchmark, while the proposed RSMA scheme significantly outperforms the corresponding NOMA and OMA schemes.

We study randomized row-action methods for problems with linear observations, convex constraints, and non-Euclidean geometry. Our starting point is a unified formulation in which the data-fitting term acts in observation space, while the feasible set and the regularizer encode structure in variable space. It covers constrained least squares, nonnegative and simplex-constrained models, regularized data fitting, and equality-constrained Bregman selection problems. The structure of these problems leads naturally to two algorithmic branches. The first uses primal Bregman-proximal row steps for soft data-fitting problems. The second uses dual Bregman–Kaczmarz steps for problems with hard equality constraints. We first derive a common one-step inequality. We then give deterministic and stochastic convergence results for the soft branch under relative smoothness. For constrained least squares, the normalized row oracle has exact first- and second-moment identities. They yield an expected ergodic bound in the inconsistent case, an $O(1/K)$ bound in the interpolating case, and linear convergence of the actual iterates under a residual error bound. For equality-constrained selectors, we prove linear convergence in expected Bregman distance under a residual-to-Bregman error bound. We recall global sufficient conditions for this bound and derive local constants for the nonnegative quadratic geometry and the entropy geometry on the simplex. Numerical experiments illustrate the noisy stepsize trade-off and the local entropy selector theory, and show that entropic row-action updates can recover interpretable factors in a simplex-structured nonnegative matrix factorization problem.

In 1997 Guo Wuwen proved that if $(\lambda_1,\lambda_2,\dots,\lambda_n)$ is the spectrum of a nonnegative matrix with Perron eigenvalue $\lambda_1$ and $\lambda_2$ real, then $(\lambda_1+t,\lambda_2\pm t,\lambda_3,\dots,\lambda_n)$ is the spectrum of a nonnegative matrix for every $t\ge 0$. Whether the same holds for symmetric nonnegative matrices has remained an open question. We give an affirmative answer, in a slightly stronger form: if $A$ is a symmetric nonnegative matrix with eigenvalues $\lambda_1\ge\dots\ge\lambda_n$ and $d_1\ge|d_2|+\dots+|d_n|$, then there is a symmetric matrix $B\ge A$ with eigenvalues $\lambda_1+d_1,\dots,\lambda_n+d_n$, and $B$ has the same diagonal as $A$ when $d_1+\dots+d_n=0$. This leads us to call a symmetrically realizable list Guo-extreme if none of Guo's moves can be reversed within the set of symmetrically realizable lists. Every symmetrically realizable list is a Guo-extreme list plus a vector from the cone of Guo's moves. Our second main result is a certificate, modelled on a theorem of Laffey on extreme symmetric matrices: if $A$ is irreducible and every nonnegative symmetric matrix that commutes with $A$ and vanishes wherever $A$ is positive is a multiple of the identity, then the spectrum of $A$ can be moved by a small amount in every trace-preserving direction (in every direction, if $A$ has a positive diagonal entry) and remain symmetrically realizable; in particular it is not Guo-extreme.

We give an explicit treatment of the third tensor power of the Clifford representation on $n$ qutrits, for $n\geq2$. This representation has five pairwise inequivalent irreducible constituents, with multiplicities $1,1,1,1,2$. The decomposition reduces the Clifford twirl of a tensor cube to four real parameters. Two of them are spectral moments of the input. The other two are cubic invariants expressed as sums of cubes of the Weyl characteristic function and the discrete Wigner function. On trace-one Hermitian matrices, the invariant polynomials of degree at most three are spanned by the constant function and four algebraically independent functions. These functions do not separate Clifford orbits. For pure inputs a single parameter suffices. We recover its range and the characterization of its maximum by stabilizer states. We bound the stabilizer fidelity, determine full separability across the three copies in the symmetric family, and recover the criterion for a complex projective $3$-design. We classify the jointly invariant quantum Boolean functions. For stabilizer testing and for testing unknown observables we give explicit bounds on copies and queries. We also give character formulas for the finite projective Clifford groups and record the complete character table for two qutrits.

Given a smooth domain $D\subset\mathbb S^{N-1}$, $N\geq 3$, and a positive constant $H$, we study radial graphs over $D$ with constant mean curvature $H$ which meet orthogonally the boundary of the cone spanned by $D$. The corresponding problem is given by a quasilinear elliptic equation on $D$ with homogeneous Neumann boundary condition. We investigate the structure of the solution set under smooth perturbations of the domain which are parametrized by an open set of $C^{3,\alpha}$-diffeomorphisms of the sphere. After pulling the equation back to a fixed reference domain, we apply a classical transversality theorem and prove that, in the class of domains for which $N-1$ does not belong to the Neumann spectrum of the Laplace-Beltrami operator, there is a residual set for which every solution is nondegenerate. A consequence of our result is that, for every such domain and every $R>0$, the set of solutions whose $C^{2,\alpha}$-norm is bounded by $R$ is finite.

Let $F$ be a $p$-adic field, let $k,c\ge1$, and let $\rho$ be a $(k,c)$ representation of $\mathrm{GL}_{kc}(F)$: a complex representation of finite length whose wave front set has $(k^c)$ as its unique maximal orbit, and which admits a $(k,c)$ functional that is unique up to scaling. We prove that the functions in the $(k,c)$ model of $\rho$ are of moderate growth, by relating this model to the Zelevinsky model of $\rho$, and obtain as a consequence a bound for the restriction of these functions to the standard Levi subgroup associated with the composition $(c^k)$ of $kc$. As applications, we study convergence problems of several integrals involving $(k,c)$ models: integrals from the generalized doubling method, and a new family of integrals which extends the Rankin–Selberg integrals for Speh representations of Lapid and Mao.

In this paper, we first apply the implicit function theorem to find nontrivial solutions to the following Bernoulli's overdetermined problem \begin{equation} -\Delta u=\gamma\,\, \text{in}\,\,\Omega \setminus\overline{D}, \,\, u=1,\,\,\partial_\mathbf{n} u=Q\,\,\text{on}\,\,\partial D, \,\,u=0\,\,\text{on}\,\,\partial\Omega \nonumber \end{equation} for a fixed boundary $\partial\Omega$ and a free boundary $\partial D$ in $\mathbb{R}^{N}$ with $N\geq2$, where $\gamma$ is a sufficiently small constant, the Bernoulli value $Q$ is determined by the construction, and $\Omega \setminus\overline{D}$ is a bounded ring-shaped domain. In dimension $N=n+1$ with $n\geq2$, we further study the existence of a nontrivial unbounded domain $\Omega \setminus\overline{D}$ in $\mathbb{R}^N$ such that the Bernoulli's overdetermined problem admits a solution. This unbounded domain we construct is periodic in one variable and radial in the other variables by bifurcation analysis.

We prove that the full Dirichlet-to-Neumann map for static isotropic elasticity uniquely determines both smooth Lamé parameters in a bounded smooth domain of \(\mathbb R^3\) under strong convexity. The proof uses an exact four-component Laplace-type reduction, but its decisive identities are obtained from physical elastic solutions. The characteristic quotient of the reduced system has rank three, whereas the leading physical adjoint lift supplies only two directions. We construct the missing polarization from the first nonvanishing correction of a reduced amplitude in the kernel of that lift. Paired with fixed primal and adjoint quotient frames, the resulting complex geometrical optics solutions yield a complete family of holomorphically weighted \(3\times3\) matrix moments through the physical Betti identity. Cauchy–Plemelj factorization converts these moments into an invertible relative transport normalized in the common exterior. Holomorphic dependence on the projective null direction and transport rigidity then determine a material ratio and the shear modulus, hence both Lamé parameters. No smallness, symmetry, separability, or analyticity hypothesis is imposed.