Papers

We study Dyck paths refined simultaneously by proper area and water capacity, where water capacity is measured above the path and below its lattice-path upper hull. The finite-height ingredients used in the enumeration are classical bounded-height area-polynomial and continued-fraction objects. The upper-hull decomposition produces a coupled area–capacity substitution, which gives an exact four-variable height expansion with denominator branches indexed by the height levels. The full generating function is asymmetric in the two weights, while the height summands admit a symmetric unreduced denominator representation under interchange of the area and capacity weights. In the open square $0<p,q<1$, we prove that the length radius of $G(x,1,p,q)$ is the minimum of the positive real denominator branches. The proof combines uniform normal convergence of the height expansion below this first branch with a Perron-root representation of the branch locations and an interval log-submodularity theorem for spectral radii of weighted paths. On the diagonal $p=q=s$, the classical Chebyshev specialisation gives explicit branch crossings and a $(1-s)^{2/3}$ branch-envelope accumulation law at the Dyck critical point.

This paper proposes a novel flexible coupler antenna (FCA) that translates passive coupling elements around a fixed-position active antenna to reshape the induced currents on the passive elements for radiation. A new form of mechanical beamforming is achieved by moving only the passive coupling elements while keeping the active antenna stationary. The proposed design significantly reduces the antenna and radio-frequency (RF) chain costs of conventional active array beamforming with low mechanical control complexity and energy consumption. For the purpose of exposition, we consider a point-to-point communication system with one FCA at the transmitter and one fixed antenna at the receiver. Specifically, based on multi-port circuit theory, we establish both the line-of-sight (LoS) and multipath channel models and derive the mechanical beamforming weights of the passive couplers as functions of their positions. Then, we formulate a new problem to maximize the received signal-to-noise ratio (SNR) by optimizing the positions of passive couplers at the transmitter, subject to coupler movement and transmit power constraints. Solving the resulting problem is inherently difficult because coupled channel and mechanical beamforming create non-linearity in the objective function.To tackle this problem, we propose an efficient block-coordinate conditional gradient method to search for the best positions of all passive couplers by sequentially optimizing the position of each coupler with those of the other couplers fixed in an iterative manner.Simulation results demonstrate that the proposed system significantly outperforms benchmark schemes in terms of achievable rate, but with significantly reduced active antennas and RF chains.

Product decompositions of metric spaces are built from coordinate maps, but these maps are not part of the resulting metric space. We recover this missing coordinate structure through equivalence relations whose classes are candidate coordinate fibers, and the resulting quotient metrics reconstruct the coordinate factors. This framework characterizes exactly when a metric space admits a finite product or power presentation. We prove an equivalence of categories showing that these equivalence-relation data preserve exactly the ordered coordinate information of power presentations. For spaces with suitable $\ell^\infty$-prime factorizations, we use prime multiplicities to determine the existence and classification of roots. We also study metric spaces satisfying $X\cong X\times_\infty X$, where repeated coordinate splitting gives a family of metric quotients indexed by infinite binary sequences. We prove that these binary tree structures exactly characterize metric spaces satisfying $X\cong X\times_\infty X$. As an application to persistent homology, we show how to recover filtration parameters whose products or powers form a given space of intervals.

Clapeyron's Theorem of classical linear elasticity provides a way to explicitly express the energy stored in an equilibrium configuration in terms of the work of the forces applied on the boundary. We derive several new integral relations which can be viewed as nonlinear analogs of this classical result, reinterpreting them as rather general statements within Calculus of Variations. These relations reflect specific properties of Lagrangians, that we call “partial variational symmetries”, since they are more general than classical variational symmetries. In the framework of nonlinear elasticity, partial variational symmetries, such as scale invariance, or scaling homogeneity, lead, via Noether's analysis to different nonlinear generalizations of Clapeyron's Theorem that combine naturally the work of physical and configurational forces. We present a series of illuminating examples showing the effectiveness of the obtained general results in different problems of nonlinear elasticity.

We present a low-variance randomised numerical linear algebra approach for multi-query finite element systems arising from parametric elliptic partial differential equations with applications to digital twins and online model calibration. The method relies on Galerkin subspace projection for reducing the dimensionality, and then combines parameter-oblivious leverage-score Bernoulli sampling with a control variates scheme to yield a reduced-variance `forward' sketch and an invertible `inverse' sketch that are then fused to a single efficient regularised estimator. Effectively, this reduces the computational cost in computing the projected system of equations while preserving the structure, stability, and accuracy of the underlying FEM formulation. We derive probabilistic bounds for the sketching error, invertibility, and estimator variance, and then validate the method on large-scale example problems. The results show that when the parameter fields do not vary too sharply, the synergy of control variates together with the sketch fusion can largely offset the loss incurred by the sub-optimal parameter-oblivious sampling. In this regime, our method achieves substantial savings in time, memory, and communication while maintaining accuracy levels that are acceptable for scientific simulation.

Let $p>3$ be a prime and let $$f_{\lambda_1,\lambda_2}(x)=x^{p^2-p+1}+\lambda_1x^{p^2}+\lambda_2x^{2p-1}\in\mathbb F_{p^2}[x].$$ We determine all pairs $(\lambda_1,\lambda_2)\in(\mathbb F_{p^2})^2$ for which $f_{\lambda_1,\lambda_2}$ is a permutation polynomial of $\mathbb F_{p^2}$. The final classification consists of three explicit families. The first one is the binomial case $\lambda_1=0$. The other two are obtained from the condition $\lambda_2=c\lambda_1^3$, with $c\in \mathbb F_{p}^{*}$, and are defined by two simple equations involving the norm $\lambda_1^{p+1}$. The proof is based on the AGW criterion and on the study of a quartic curve naturally associated with the rational function induced on the unit circle $\mu_{p+1}$.

This work develops several constructive foundations for systems and control within Bishop-style constructive mathematics. For an engineer, the guiding principle is that an object claimed to exist, such as a trajectory, an optimal control law, a selector, or a viable solution, should come with finite data and an operation computing approximations to any prescribed precision. The style remains close to classical analysis, but existential statements are organized so that their computational content is visible. The paper begins with elementary geometric data in finite-dimensional Euclidean spaces: blocks, multiblocks, representable sets, regular functions, and certified integrals. This set-first integration route is meant to complement, rather than replace, abstract constructive integration theories such as Daniell-type or integration-space approaches. The developed apparatus is then applied to a constructive functional extremum-value theorem, selector extraction for multifunctions, Filippov-type and viable solutions of differential inclusions, regular probability densities, controlled Markov chains, and empirical density certificates. A short account of resolvent projectors and linear stability is included for completeness.

Gene regulatory networks (GRNs) are high-value targets for therapeutic and synthetic-biology control. Classical Hill models carry a structural defect: the production term vanishes when the activator is absent, causing loss of controllability under multiplicative actuation and a collapse of network coupling under additive actuation – precisely where biological operation is most common. Building on logistic functions as robust Hill alternatives, we develop an additive state-feedback framework for logistic-based GRNs, with two companion scalar results. A feedforward-plus-proportional law turns any positive setpoint into a closed-loop equilibrium, regardless of the uncontrolled dynamics. We prove local exponential stability under a Gershgorin gain bound and, via a common quadratic Lyapunov function built on the logistic sector bound, global exponential stability under the explicit condition $(\gamma_1+K_1)(\gamma_2+K_2)>\kappa_1\kappa_2\lambda^2/64$, with a closed-form rate. A diagonal Lyapunov certificate $P=\mathrm{diag}(B,A)$ yields an explicit settling-time bound (within $1\%$ of simulation) and an ISS ultimate-bound estimate. We further establish a parameter-uniform monostabilization budget $K^{*}=\kappa\lambda/4-\gamma$ for bistable self-activation switches, and a Halanay-type delay-uniform global exponential stability theorem under $\gamma+K>\kappa\lambda/4$, with two-sided closed-form rate bounds. Numerical comparison with the Hill counterpart confirms robust tracking in the nominal range while exposing the structural divergence near the boundary of the positive orthant: the Hill coupling $|[J_f]_{21}|=\Theta(x_{d,1}^{n-1})\to0$ as $x_{d,1}\to0$, whereas the logistic coupling stays strictly positive.

As it is well known, solutions of equations of nonlinear elastodynamics, representing extremals of the action functional, can form shocks. We adapt the classical approach of Emmy Noether to such singular extremals and derive the appropriately generalized integral relations within Calculus of Variations. We apply them to elastodynamical extremals with shocks, obtaining new integral relations involving kinetic and elastic energies. For the extremals representing thermodynamically admissible (entropy) solutions of the corresponding hyperbolic Euler-Lagrange equations, the classical equalities, characterizing the variational approach of Noether, expectedly transform into ineqalities. We show that, rather remarkably, despite the crucial role of material velocity in the fully inertial energy redistribution processes, the corresponding kinetic energy can be completely eliminated from the expression for the dynamically stored elastic energy, even in the presence of shocks.

Orthogonalized momentum updates, as used in Muon-style optimizers, have recently shown strong empirical stability in large-scale deep learning. However, most current orthogonalized methods are still paired with fixed, externally scheduled, or otherwise open-loop magnitude rules, so their scale is not directly calibrated from the realized optimization trajectory. Motivated by the closed-loop perspective behind Lipschitz-free and noise-adaptive methods, we propose OptMuon, a family of adaptive momentum orthogonalization methods for stochastic nonconvex optimization. OptMuon combines Muon-style polar-factor directions with a trajectory-dependent AdaGrad-Norm-type coefficient schedule, so that the update magnitude is determined by the observed gradient and momentum history rather than by a prescribed Lipschitz-dependent rule. The schedule does not use the smoothness constant, the variance level, or the bounded-gradient constant in parameter selection, and its running-maximum correction prevents isolated gradient spikes from causing excessive coefficient collapse. Under lower-boundedness, unbiased stochastic gradients with bounded variance, smoothness, and an almost-sure bounded stochastic-gradient condition, we prove two complementary expected-stationarity guarantees. OptMuon-A achieves the noise-adaptive rate \(\tilde{\mathcal O}(T^{-1/2}+\sigma^{1/2}T^{-1/4})\) under average smoothness, while OptMuon-I achieves \(\tilde{\mathcal O}(T^{-1/2}+\sigma^{1/3}T^{-1/3})\) under individual smoothness. In the zero-noise regime, both bounds automatically reduce to a nearly optimal deterministic first-order rate \(\tilde{\mathcal O}(T^{-1/2})\) without manual hyperparameter retuning. These results show that closed-loop scalar adaptation can be combined with Muon-style momentum orthogonalization while retaining noise adaptivity and zero-noise optimality up to logarithmic factors.

We study the quantitative approximation of certain stochastic integrals, where we use discrete time approximations under initial enlargement of filtration. It turns out that the approximation rate is in general the same as in the case of no additional information, however, the asymptotic constant improves.

Optical intelligent reflecting surfaces (OIRS) can improve the coverage of indoor visible light communication (VLC) systems, however, practical deployment requires a finite offline codebook to avoid repeated real-time optimisation of mirror orientations. A uniform codebook with fixed angular steps does not provide uniform coverage on the user plane, because the mapping from steering angles to reflection locations on the user plane is nonlinear. To address this problem, this paper proposes a geometric-optics-based non-uniform codebook design for OIRS-assisted VLC systems. The proposed method constructs an individual codebook for each IRS element according to its geometric position, so that the reflected beams are distributed more uniformly over the user plane. The codebook accuracy is evaluated using the Frobenius norm of the channel error matrix. Simulation results show that the proposed design provides more uniform spatial mapping with fewer codewords than the uniform codebook, and that the sweep-angle resolution has a stronger effect on the codebook accuracy than the tilt-angle resolution.

The meet of two points in $\mathbb{Z}^d$ is their coordinatewise minimum. We show that every finite antichain $A$ in $\mathbb{Z}^d$ has at least $c_d |A|^{d/(d-1)}$ distinct pairwise meets, where $c_d > 0$ depends only on $d$, and that the exponent $d/(d-1)$ is best possible. As a corollary we obtain an isoperimetric inequality for downsets: every finite downset $D$ in $\mathbb{Z}^d_{\geq 0}$ satisfies $|D| \geq c_d |\max(D)|^{d/(d-1)}$, where $\max(D)$ is its set of maximal elements. By prime factorization the meet bound also yields a lower bound on greatest common divisors: a primitive set of $N$ integers supported on at most $d$ primes has at least $c_d N^{d/(d-1)}$ distinct pairwise gcds.

We relate the properties of bivariate-bicycle-surface (BBS) codes, constructed from a pair of bivariate polynomials over a finite field, to the number and location of their common roots in the extension field. The number of roots $(x,y)$ with finite, non-zero coordinates – counted with algebraic multiplicity – determines the dimension of the codes. This dimension is invariant under monomial automorphisms of the Laurent polynomial ring. Conversely, roots with zero or infinite $x$- or $y$-coordinates indicate that specialized generators are required near the corresponding boundary (e.g., the left or right boundary for a root where $x$ is zero or infinite, respectively). These roots can appear or disappear under monomial transformations, which reveals the structure of tilted boundaries. Based on these results, we formulate a prescription for constructing BBS codes that works for regions with rectangular, diagonal, and arbitrarily tilted boundaries. A key advantage of this approach is that no corner corrections are needed, provided the polynomials satisfy orientation-specific edge conditions.

We study the maximal number $m_{k,\ell,n}$ of empty convex $k$-gons ($k$-holes) determined by an $n$-point set in the plane in general position that contains no empty convex $\ell~$-gon, focusing on the first nontrivial case $\ell=k+1$. Our main result determines the exact value in the small-excess regime: for $n=k+a$ with $a\le k/2-1$, we prove $m_{k,k+1,k+a}=2^a.$ We also describe the extremal configurations attaining equality. Beyond this exact range, we provide upper and lower bounds in the proportional regime $n=\alpha k$ and in the regime where $k$ is fixed and $n$ goes to infinity. In the last mentioned regime we prove that $m_{k,k+1,n}=\Omega_k(n^{\lfloor\frac{k}{3}\rfloor})$ and $m_{k,k+1,n}=O_k(n^{\lfloor k/2\rfloor+1})$.

Let $f$ be a Hénon-Sibony map of $\mathbb{C}^k$ of algebraic degree $d_+\geq 2$, whose inverse $f^{-1}$ has algebraic degree $d_-$. The topological entropy of $f$ is equal to $\log d_+^{p} = \log d_-^{k-p}$. We show that every ergodic $f$-invariant measure $\nu$ satisfying $h_\nu(f)>\log \max\{ d_+^{p-1},d_-^{k-p-1}\}$ is supported on the Julia set $\mathcal{J}$ of $f$.

We investigate the spectral properties of tight-binding quantum Hamiltonians defined on the bipartite Levi graphs of finite geometric configurations. In Part I, we analyze the $10_3$ Desargues and Kantor configurations, demonstrating how deterministic spatial deformations induced by real planar embeddings ($\mathbb{R}\mathbb{P}^2$) destroy translational symmetry, leading to structural wave localization. We show how embedding in the complex projective plane ($\mathbb{C}\mathbb{P}^2$) restores Bloch wave propagation via synthetic gauge phases. In Part II, we evaluate the Schläfli double six ($12_5, 30_2$) and the Cremona-Richmond ($15_3$) configurations in $\mathbb{R}^3$. We analytically derive their exact spectral decompositions, confirming the existence of macroscopic zero-energy flat bands ($E=0$) driven entirely by geometric frustration. In Part III, we establish a formal structural isomorphism between these discrete tight-binding networks and the flavor symmetry sector of the Standard Model. We map the Schläfli graph to the $SU(6)$ flavor multiplets, where the flat band corresponds to the kinematic freezing of ultra-heavy baryons. Finally, we discuss the complementary Cremona-Richmond $15_3$ topology, demonstrating how its distinct geometric nature (based on tritangent planes rather than point intersections) provides a purely algebraic, topological completion to the $W(E_6)$ symmetry of the 27 lines.

This paper uses theoretical results previously established in the literature to design search algorithms to find new linear codes over $\mathbb{Z}_4$ from cyclic and negacyclic codes of even length. As a result of these searches, we have found 2500 new cyclic codes and 730 negacyclic codes. These new codes exhibit improved parameters compared to previously known codes. Additionally, we have obtained binary quantum codes with good parameters from such $\mathbb{Z}_4$ codes.

We present a unified theoretical framework for the mapping between diffusive and wave-like dynamics, formulated as a spectral integral operator acting on temporal fields. By introducing an analytic continuation in the complex frequency plane, we establish an explicit correspondence between thermal diffusion and a virtual wave field governed by a hyperbolic equation. This mapping is shown to define a causal, compact Fredholm operator that acts as a nonstationary low-pass filter, thereby revealing the intrinsic information loss of diffusive processes and the fundamental ill-posedness of the inverse reconstruction. Within this operator framework, we demonstrate that commonly used excitation schemes-including pulse, lock-in, chirped, and coded excitations-emerge as distinct projections onto subspaces of a single underlying transformation, corresponding to different sampling strategies of its spectral structure. This unifies previously disparate virtual wave formulations and provides a systematic interpretation of excitation design in terms of operator sampling and information encoding. The framework further generalizes to matrix-valued systems and suggests a spectral-geometric interpretation of temporal evolution across diffusive and propagative regimes.

Let $f$ be a holomorphic automorphism of a compact Kähler manifold $X$ with simple action on cohomology. We show that every ergodic measure with sufficiently large entropy is supported on the Julia set of $f$. In particular, when $X$ is a surface, any ergodic measure with positive entropy is supported on the Julia set. The proof relies on quantitative estimates for the speed of convergence towards the Green currents of $f$, with respect to a suitable norm on an adapted functional space of non-necessarily closed currents.

We consider some extremal combinatorial questions for bipartite graphs definable in stable one-based (and related) structures. We show that they satisfy both strong Erdős-Hajnal property and linear Zarankiewicz. We also show that the same is true for both collapsed and uncollapsed Hrushovski's “ab initio” constructions, and discuss some connections to Zilber's trichotomy principle. For strong Erdős-Hajnal, we show that in fact it holds in a more general class of $1$-semi-equational theories.

We show that if $L$ is an extremal Type II lattice of rank $72$, then $L$ is generated by its vectors of norm $10$. The proof determines the full inner product distribution between the shells of norms $8$ and $10$ using the spherical $11$-design property of the norm-$10$ shell.

We study degree-$2$ Siegel theta series of extremal even unimodular lattices from the genus-$2$ viewpoint initiated by Ozeki. Using Igusa's structure theorem, we define a depth filtration on genus-$2$ cusp forms, measured by the total degree in $\chi_{10}$ and $\chi_{12}$, and relate it to the vanishing of low Fourier–Jacobi coefficients forced by extremality. In ranks $48$, $72$, and $96$, this interaction closes exactly and yields a direct genus-$2$ proof that the degree-$2$ theta series is uniquely determined by extremality (conditional on existence in rank $96$). In rank $120$ (again conditional on existence), the same argument leaves a one-dimensional residual line spanned by $\chi_{10}^6$: with $\chi_{10}$ in the standard integral normalization, any two such degree-$2$ theta series differ by an integer multiple of $\chi_{10}^6$.

In this paper, we propose a finite transposition method to solve backward stochastic differential equations (BSDEs, for short). Based on the transposition solution theory for BSDEs, our method offers a promising way of efficiently computing solutions, which can be regarded as an analogous method for BSDEs as the classical finite element method for partial differential equations. Our method has the advantage of easily computable conditional expectations.

The prediction-powered inference (PPI) proposed by Angelopoulos et al. (2023) is a popular method that leverages a small number of labeled samples and machine learning predictions for semi-supervised inference. While several variants of PPI have appeared in the literature, its rigorous statistical theory has not been fully developed. In this paper, we study the statistical optimality of PPI. Our contributions span both foundational theory and new methodology. First, we frame PPI as an M-estimation problem, revealing a link between the bias-corrected PPI estimating equation and the ideal full-data estimating equation. This connection leads to the consistency and asymptotic normality of the PPI estimator under simple random sampling without replacement. Next, we identify the efficient influence function and prove that PPI can attain the semiparametric efficiency lower bound when the predictor is score-calibrated, that is, when the predictor's output aligns with the true conditional expectation of the estimating function. Finally, for learned prediction rules, we develop asymptotic theory for cross-fitting and for a single-fit variant with variance correction in the special case of semiparametric mean estimation. Simulation experiments and a real-data application support these findings.

Many classically studied function classes are known to be approximated optimally by superpositional methods, i.e. with approximants constructed as the linear combination of elements in some dictionary. Here optimality means that the uniform approximation error viewed as a function of the number of parameters used has polynomial decay of the highest order achievable by any parametrized method whose parameters can be encoded as a bit string of length proportional, up to logarithmic factors, to the number of parameters. While compositional methods like neural networks are structurally different, their approximation rates can be made comparable by imposing constraints that ensure such a proportional bit string encoding. In this work we study function classes exhibiting structural properties that limit superpositional approximation rates to be strictly lower than compositional approximation rates. In particular, we construct explicit examples for which there is an arbitrarily large gap.

Recently the poset of ranked cactuses $(\mathfrak{P}(X),\preceq)$ was introduced. For a finite set $X$, this poset consists of a set $\mathfrak{P}(X)$ of certain collections of ordered pairs of subsets of $X$ together with an ordering $\preceq$ that is similar to the refinement ordering of partitions of a finite set. In addition, the maximal chains in this poset correspond to binary ranked cactuses, a fact which can be used to construct the so-called space of equidistant cactuses. In this paper, we show that the poset of ranked cactuses is EL-shellable. As a consequence we also show that the proper part of the link of the origin of the space of equidistant cactuses has the homotopy type of a wedge of spheres.

We make use of the free data formalism developed in \cite{CK25} to construct solutions of the Einstein vacuum constraint equations by integrating in the forward direction. This, together with a new gauge condition based on effective uniformization, allows us to construct general solutions with limited decay at spacelike infinity. In particular, we construct solutions with minimal and even borderline decay, as considered in \cite{Shen23}, \cite{Shen24} in connection with the stability of the Minkowski space. In a forthcoming paper, we make use of the techniques we develop here to identify and construct a general class of short-pulse Cauchy data that lead to the formation of trapped surfaces, extending the well-known result of \cite{LiYu}.

Let \(V\) be an \(n\)-dimensional vector space over the finite field \(\Fq\), and let \(\Gr{V}{k}\) denote the family of all \(k\)-dimensional subspaces of \(V\). A family \(\cF\subseteq\Gr{V}{k}\) is called intersecting if \(\dim(F\cap F')\ge1\) for all \(F,F'\in\cF\). For a point \(P\le V\), let \(d_P(\cF)\) denote the number of members of \(\cF\) that contain \(P\), and order the point-degrees as \(d_1(\cF)\ge d_2(\cF)\ge\cdots\ge d_{\points{n}}(\cF)\), where \(\points{m}=(q^m-1)/(q-1)\) is the number of points in an \(m\)-dimensional subspace. Recent work of Frankl and Wang~\cite{FW2025} and of Huang and Rao~\cite{HR2026} established that for \(k\)-uniform intersecting families \(\cF\subseteq\binom{[n]}{k}\) with \(n\ge2k+1\), the bound \(\binom{n-2}{k-2}\) governs the order statistic \(d_{2k+1}(\cF)\). We prove that every intersecting family \(\cF\subseteq\Gr{V}{k}\) with \(n\ge2k+1\) satisfies \(d_{\points{k}^{2}}(\cF)\le\qbinom{n-2}{k-2}\). The naive \(q\)-analog of the Huang–Rao \((k+2)\)-th degree theorem fails, as a vector-space Hilton–Milner construction has \(\points{k+1}\) points of degree larger than \(\qbinom{n-2}{k-2}\); we prove the corrected bound \(d_{1+\points{k+1}}(\cF)\le\qbinom{n-2}{k-2}\) for fixed \(q\), sufficiently large \(k\), and \(n>3k\), using a structural theorem of Ihringer and Kupavskii~\cite{IK2026}. For larger degree indices \(i\), a saturated Frankl–Hilton–Milner family of Ihringer and Kupavskii identifies the conjectural sharp bound on \(d_{\points{i+1}}(\cF)\), and we prove two necessary conditions that any strict counterexample to this conjecture must satisfy.

This note announces exact Fourier dimension formulas for canonical Mandelbrot cascade measures under the minimal Kahane Peyriere integrability condition and records the canonical b adic extension on cubes. In the dyadic interval setting, the theorem is proved in a balanced vector weight model allowing dependence between sibling weights. Almost surely on non extinction, the Fourier, energy, and L2 dimensions all equal the energy exponent. The scalar specialization gives the canonical Mandelbrot Kahane Fourier dimension formula under the minimal integrability condition. On the circle, the endpoint formula is given by the endpoint lower local dimension exponent. For the b adic Mandelbrot cascade on cubes, the Fourier dimension is the minimum of 2 and the energy exponent, with the universal Fourier barrier at dimension two providing the high dimensional obstruction.