Papers

We prove Sylvester's conjecture, originating in his 1879 study of ternary cubic equations, that every prime $p\equiv4,7,8\pmod9$ is a sum of two rational cubes. Elkies announced a proof for the classes $4$ and $7$ in 1994, and Yin recently supplied a complete proof. For the remaining class $p\equiv8\pmod9$, we prove that the elliptic curve $E_p:y^2=x^3+p^2/4$ has analytic rank one, as predicted by the Birch and Swinnerton-Dyer conjecture, and so $p$ is a sum of two rational cubes. The proof begins by adapting the auxiliary Rankin–Selberg construction from the authors' work on the rank one converse for CM elliptic curves. The Rankin–Selberg $L$-function factors as the $L$-function of $E_p$ times a complementary $L$-function. Chan's $3$-isogeny descent and the rank zero converse show that the complementary central $L$-value is non-zero, and so it suffices to prove that a cubic component of the associated Heegner point is non-torsion. A basic difficulty is that the unweighted Hecke trace of the underlying CM orbit vanishes. Our decisive idea is to take $\lambda$-division before taking the trace, where $\lambda=1-\omega$ and $\omega$ is a primitive cube root of unity. We prove that the resulting division boundary is non-zero by analysing Frobenius at $p$. The Galois action on the CM orbit and ramification theory then transfer this non-vanishing to the cubic component.

In this paper, we derive an explicit solution of the stationary prediction with expert advice PDE for five experts. The formula is given in three regions. In the first two regions, it is the four-expert solution plus a single integral with an elementary positive density. In the third region, it is a finite sum of hyperbolic products whose coefficients are determined by one scalar quadrature. Our formula establishes that the direction $(1,0,1,0,0)$ is optimal throughout the ordered sector, and that the COMB strategy $(1,0,1,0,1)$ is optimal only on a lower dimensional subset of the sector (where $x_1=x_2$ and $x_3=x_4$). This disproves the COMB optimality conjecture of Gravin, Peres and Sivan. The verification of the Hamiltonian inequalities is a tedious task, part of which is completed with a computer assisted proof. The verification reduces to 21 scalar inequalities, which we prove using 147 exact rational Bernstein polynomial certificates. The exact certificates and their independent arithmetic checks are included in a supplement to this paper, and a Lean 4 formalization machine-checks the verification and both main theorems, apart from the viscosity characterization.

We initiate a joining rigidity theory for rational maps on the Riemann sphere $\mathbb P^1=\mathbb P^1(\mathbb C)$. Let $f_1,f_2\colon\mathbb P^1\to\mathbb P^1$ be rational maps of degree at least $2$, and $\mu_1,\mu_2$ their respective measures of maximal entropy, whose supports are the Julia sets $J(f_1)$ and $J(f_2)$. We study ergodic joinings of the systems $(J(f_1),f_1,\mu_1)$ and $(J(f_2),f_2,\mu_2)$, namely ergodic probability measures on $J(f_1)\times J(f_2)$ which are invariant under $f_1\times f_2$ and whose marginals are $\mu_1$ and $\mu_2$. Our main theorem shows that a positive-mass local holomorphic relation forces algebraic rigidity. More precisely, if the joining charges the graph of a local biholomorphism, then that local relation globalizes to an invariant algebraic curve and yields either a finite cycle of rational graph or transpose-graph relations, or a genuinely multi-valued invariant algebraic correspondence. If no local biholomorphic graph has positive joining measure, then the joining generates a compact non-discrete family of local holomorphic relations. The proof introduces normalized inverse branch transfer maps and studies their cluster limits. Starting from a local biholomorphic graph of positive joining measure, recurrence and contraction of inverse branches produce recurrent local intertwining relations. These are promoted to an algebraic relation by a local-to-global rigidity argument in the non-Lattès case and by affine uniformization in the Lattès case. In the absence of any positive-mass local biholomorphic graph, the cluster family must be infinite, and its non-discrete closure gives the second alternative.

We prove the superlinear gradient growth for the one dimensional Vlasov-Poisson equation on the torus. We show them by first constructing the stationary solution which has a hyperbolic flow (locally). Then we perturb the solution and use the $L^\infty$ stability of the characteristic velocity field to show the superlinear growth. This proof was inspired by the work of Denisov.

We study front propagation in a time-fractional Amari neural field equation with a Caputo derivative and a Heaviside firing-rate function. For a class of nonnegative even connectivity kernels and strictly decreasing $C^1$ front-like initial data, we reduce the Cauchy problem to a nonlinear Volterra equation for the threshold location. We prove the existence and uniqueness of a global front solution, determine the short-time behavior of its threshold location, and identify its asymptotic speed. We also construct the associated asymptotic traveling-wave profile and establish uniqueness of the speed and profile, up to translation, in the nonzero-speed case. Using a rigidity argument, we prove the asymptotic linearity of the local increments of the threshold location and obtain uniform convergence to the profile in the threshold-centered frame. Finally, we characterize the kernel-dependent tail asymptotics of the wave profile.

We adapt the renormalized energy method developed in \cite{duerinckx2020mean} in order to prove the monokinetic mean field limit for relativistic Newtonian dynamics. The resulting monokinetic PDE is the relativistic Euler-Poisson equation. Since the position evolves relativistically in comparison to the momentum, the kinetic part of the modulated energy has to adjusted, leading to further obstructions that are not present in the non-relativistic settings. A weak-strong stability principle for the relativistic Euler-Poisson equation is investigated, followed by a renormalization procedure leading to the mean-field limit. Our main result constitute the first mean-field limit for relativistic Coulomb flows.

Hyperbolic band theory (HBT) has recently unveiled a wealth of exotic physical phenomena in negatively curved spaces. Concurrently, the quantum metric has emerged as a fundamental tensor governing quantum geometry and topological phases. In this Letter, we bridge these two frontiers by investigating the quantum metric in HBT. Because conventional Euclidean metric extraction protocols fundamentally fail in these non-commutative geometries, we introduce holonomy shaking, a novel, experimentally viable technique utilizing periodic lattice driving to dynamically extract the metric. Using the {8,3} regular hyperbolic and kagome-like lattices as concrete examples, we demonstrate the high-fidelity dynamical extraction of the quantum metric. Our results establish a robust blueprint for probing hyperbolic quantum geometries, paving the way for the discovery of exotic topological phenomena in hyperbolic quantum matter.

We introduce a generalized amnesic elephant random walk in which, at each time, the direction of an underlying amnesic elephant random walk selects one of two increment distributions, while the actual increment is sampled from the corresponding independent sequence. The selected distribution does not need to determine the sign of the increment. This construction combines memory reinforcement, amnesia, and randomness in jump sizes. Assuming finite second moments, we establish central limit theorems and functional limit theorems, identifying diffusive, critical, and superdiffusive regimes determined by the memory and amnesia parameters. We further prove a Gaussian fluctuation theorem around the random superdiffusive limit, with an explicit limiting variance that captures the contributions of both the underlying amnesic walk and the random jump magnitudes. For the special case without amnesia, we also establish stable limit theorems and their functional counterparts when the increments belong to the domain of attraction of a non-Gaussian stable law, in the regime where heavy-tailed jumps dominate the memory contribution.

Let $R$ be a commutative ring with nonzero identity and let $R^\#$ denote the set of its nonzero nonunits. We extend the divisor topology $D(R)$, previously studied for integral domains, to arbitrary commutative rings and introduce the generalized divisor topology $GD(R)$ on $EC(R^\#)$. Its basic open sets are \[ B_a=\{[b]\in EC(R^\#): b\mid a^n \text{ for some }n\geq 1\}. \] The relation \[ [b]\in B_a \quad\Longleftrightarrow\quad \sqrt{aR}\subseteq\sqrt{bR} \] shows that $GD(R)$ records radical divisibility among principal ideals. We prove that $GD(R)$ is an Alexandrov space and identify its Kolmogorov quotient with the poset of radicals of nonzero proper principal ideals. This description yields characterizations of the $T_0$ and discrete properties and of the equality $GD(R)=D(R)$. We also determine the isolated points of $GD(R)$. Further, we characterize nestedness, compactness, the Lindelöf property, and Noetherianity in terms of the order structure of radicals of principal ideals. In particular, for an integral domain $R$, $GD(R)$ is compact if and only if $R$ is a $G$-domain, while for a UFD the Lindelöf and Noetherian properties are determined by the number of nonassociate prime elements. Finally, we study the interaction of $GD(R)$ with multiplication and describe the behavior of its Kolmogorov quotient under surjective homomorphisms with nil kernel.

We study the iterated action of the Lie derivative on the curvature tensor of a Riemannian manifold with rank-one anisotropic curvature, whose Riemann tensor is expressed via the Kulkarni-Nomizu product as $R = \lambda (\xi^\flat\otimes\xi^\flat)\owedge g$. First, we examine the conditions under which such a manifold admits a Ricci soliton structure and demonstrate that this property implies the almost $\theta$-Yamabe soliton structure. Furthermore, we show that if the associated potential vector field $X$ is a symmetry of the Ricci tensor of a fixed order $k$ (i.e., $\mathcal{L}_X^k \operatorname{Ric} = 0$), the geometric problem reduces to solving a partial differential equation of order $k+1$ along the flow. Finally, under the assumption that $X$ is a conformal vector field ($\mathcal{L}_X g = 2\varphi g$) whose infinitesimal flow preserves the line distribution $\mathcal{D}=\operatorname{Span}\{\xi\}$ (with $[X,\xi]=a\xi$ for $a\in\mathbb{R}$), we prove that several key geometric problems (such as establishing the relation $\mathcal{L}_X^k R = R$, determining the minimal order $k$ for $X$ to be a Lie curvature symmetry, or satisfying $\mathcal{L}_X^{k+1}R = f \mathcal{L}_X^k R$ for a continuous function $f$) are equivalent to a scalar differential problem governed by the operator $D_X = X + 6\varphi + 2a$.

We study the minimum coefficient degree $N_{k,\F}(G)$ of a Nullstellensatz certificate for Bayer's $k$-coloring equations, where the characteristic of $\F$ does not divide $k$. If $J$ is a \HJ\ join of non-$k$-colorable graphs $G,H$ and $m=\max\{N_{k,\F}(G),N_{k,\F}(H)\}$, then $N_{k,\F}(J)\leq m+k$. When deletion of the selected edge makes each input $k$-colorable, we also have $N_{k,\F}(J)\geq m$; the degree congruence then gives $N_{k,\F}(J)\in\{m,m+k\}$. This partially answers a question of Li, Lowenstein, and Omar. For three-coloring over $\F_2$, we construct an infinite $4$-critical family of exact degree seven, attaining the bound at input degree four. In contrast, every graph constructed from $K_4$ solely by \HJ\ joins has degree $O(\log n)$ and a certificate with polynomially many terms: joins preserve treewidth at most three, and balanced separators yield low-degree certificates. Additional vertex identifications are excluded from this obstruction. We classify all single identifications of the $25$-vertex base graph; exactly $36$ preserve degree seven, producing $24$-vertex $4$-critical graphs of treewidth four. A compressed self-join at adjacent true twins prevents degree loss and gives a repeatable rule adding four vertices per round. The rule does not establish degree amplification or preservation of criticality. Exact witnesses and standalone verification programs accompany the finite results.

We give geometric characterizations of analytically linearizable nondegenerate Siegel foliations in $\C^2$ using complex tangencies with shrinking strictly pseudoconvex hypersurfaces. Our principal result treats a fixed bounded smoothly bounded strictly pseudoconvex Reinhardt domain: in coordinates compatible with the eigendirections of the linear part, two-dimensional tangency loci along a shrinking sequence characterize analytic Siegel linearizability. For round spheres we prove a stronger statement: no a priori alignment of the Euclidean coordinate axes with the eigendirections is required; the tangency hypothesis itself forces the linear part to be unitarily diagonalizable with negative real eigenvalue ratio. We also establish a polynomial Shilov-boundary criterion on a round sphere and a logarithmic one-boundary criterion valid for arbitrary smooth strictly pseudoconvex domains, provided the embedded closed bidisc lies on the pseudoconvex side. Finally, we study the transverse-holomorphic dynamics of smooth tangency tori, including a rigidity theorem in the periodic case.

Any objects in space may encounter singularity due to the inverse-square law of the gravitational force, hindering the analysis and propagation of particle motion in space. Regularization in celestial mechanics removes such gravitational singularities in equations of motion by transforming the time and coordinates. This paper addresses a class of regularization methods known as global regularization for restricted three-body problems (R3BPs), which simultaneously removes the two singularities (e.g., Earth and Moon in cislunar space). It enables analysis, propagation, and prediction of particle motion throughout the domain of R3BPs, which is in stark contrast to regularizing each singularity individually. In particular, this paper develops a formulation for global regularization of the spatial elliptic R3BP (ER3BP) under external perturbations. The presented formulation leverages mathematical tools from quaternionic (hyper-complex) algebra and Hamiltonian mechanics. This approach enables us to (i) elegantly handle the four-dimensional coordinates arising from the dimension-lifting operation for spatial regularization, (ii) systematically derive the coordinate-momentum transformation while preserving the canonical structure and incorporating perturbing forces, and (iii) provide geometric insights for analyzing the regularity of the transformation, resulting Hamiltonian, and dynamical flow. Numerical examples demonstrate the validity of the mathematical results as well as significant numerical benefits of the regularization formulation in perturbed Earth-Moon ER3BP.

We study the real three-dimensional unital algebra with basis $\{1,i,j\}$ and products $i^2=j^2=-1,\quad ij=0,\quad ji=i+j$. Its quadratic and power-associative elements form the union of exactly three complex subalgebras; their nonzero elements are precisely those with a two-sided inverse, which is unique. Every nonzero subalgebra contains the unit, and every element outside these three planes generates the whole algebra. We classify one-sided inverses, including the singular cases, and show that two-sided invertibility does not imply cancellation. The multiplication algebra is $M_3(\mathbb{R})$, so the algebra is absolutely central simple. Its derivations vanish, its automorphism group is trivial, and its commutator Lie algebra is $\mathbb{R}\oplus\mathfrak{aff}(1)$. Its unique antiautomorphism is involutive, and its complexification admits no complex-linear structurable involution. Finally, replacing $ji=i+j$ by $ji=t(i+j)$ gives real algebras $\mathcal{A}_t$ with $\mathcal{A}_s\cong\mathcal{A}_t$ exactly when $s^2=t^2$. We describe the exceptional parameters and thereby exhibit nontrivial deformations of the original algebra.

The generalized source integral equation (GSIE)approach is extended to enhance the compressibility of moment matrices for essentially-convex homogeneous penetrable scatterers. In the proposed indirect surface integral equation (SIE) formulation, the scattered field representations use a single distribution of generalized sources that radiate only weakly into the scatterer. A separate distribution of conventional sources is used for representing the interior fields. The wave interactions between surface subdomains via the generalized kernel are of a reduced effective dimensionality. The corresponding exterior integral operator moment matrix blocks exhibit enhanced rank deficiency. For lossy scatterers, this secures the low-rank (LR) compressibility of hierarchical matrix structures for the full system. For scatterers in a much slower background medium, significant savings with very delayed asymptotic scaling of the costs can be achieved. The GSIE is formulated for the transverse magnetic (TM) scattering problem. It is validated for representative examples and its superior compressibility is demonstrated.

We construct two families of compact complex threefolds fibred over $\mathbb P^1$ by two-dimensional complex tori. For each positive integer $r$ and each parameter $c$ in a fixed half-plane, the families $Y_{r,c}$ and $Z_{r,c}$ have two multiple fibres of multiplicities $(4,6)$ and $(3,6)$, respectively, and one reduced normal-crossings fibre whose normalization consists of $r$ del Pezzo surfaces of degree six. Their fundamental groups are $\mathbb Z/2$ and $\mathbb Z/3$, respectively, and both have Euler characteristic $2r$. We compute their integral homology; the index-one members are rational homology six-spheres. Every member has algebraic dimension one, lies outside Fujiki's class $\mathcal C$, and has connected automorphism group $\mathbb C^*$. For fixed type and index, two members are biholomorphic exactly when their parameters differ by an element of $(6/r)\mathbb Z$. Each underlying smooth six-manifold therefore supports uncountably many pairwise non-biholomorphic complex structures. The arithmetic input classifies finite monodromy pairs and their integral lifts to a parabolic Jacobi group acting on a rank-four lattice. A quadratic form controls integral lifting and, on the rank-two square-zero locus, computes the monodromy saturation index. We prove that the relevant invariant tensors persist under finite-index restriction, which extends the construction to arbitrary compact base curves. We also construct translation twists after base change, establish a local periodic toroidal quotient theorem in arbitrary dimension, and determine the additive Bockstein spectral sequences of the original threefolds.

We propose a localized core–tail Fourier–Laguerre frame method for approximation and model problems on the real line. The domain is decomposed into a finite core and two semi-infinite tails. Local Fourier extension is used in the core to resolve nonperiodic, oscillatory, and locally nonsmooth structures, while modulated Laguerre frames are used in the tails so that the Laguerre functions approximate only slowly varying decaying envelopes. The local Fourier extension component also provides two data-driven mechanisms: coefficient-energy indicators for internal edge detection and local frequency indicators for selecting tail modulation centers. We derive error estimates that separate the core approximation error, the modulated Laguerre envelope error, the effect of frequency mismatch, and the finite-tail truncation error. The analysis shows that the tail complexity is governed mainly by the residual phase after modulation rather than by the original carrier frequency. Numerical experiments demonstrate high accuracy for oscillatory, multi-frequency, and derivative-discontinuous functions, and a decaying model problem illustrates that the representation can be combined with differential operators through exact interface constraints.

We study singularities of rational inner functions (RIFs) $\phi$ in three and more variables. We focus on curve and higher-dimensional singularities of $\phi$ in $\mathbb{T}^n$ and we show that, under certain conditions, derivative integrability conditions of partial derivatives of $\phi=\frac{q}{p}$ stay constant along irreducible components of the zero set of $p$, and that the presence of vertical line singularities does not affect global integrability. This answers Question 2 in arXiv:1906.10913. Those results are then generalized to $n$ variables. Finally, we consider slice matrices, which we use to obtain derivative integrability conditions of unidirectional compositions $\phi^N$. We present several new examples, including an RIF with two vertical line singularities and one with vertical surface singularities.

Let $X$ be a compact connected Riemann surface equipped with two nonconstant meromorphic projections $q,z:X\to\mathbb{P}^1$. Their differentials determine a canonical meromorphic relative differential morphism $Z:T_q=q^*T\mathbb{P}^1\to T_z=z^*T\mathbb{P}^1$, whose local coefficient in coordinate frames is $Z=z_q\,\partial_z\otimes dq$. Its divisor is $\operatorname{Div}(Z)=R_z-R_q$, and its local logarithmic derivative defines a rank-one logarithmic connection on $L=\operatorname{Hom}(T_q,T_z)$. The general theory of logarithmic connections is classical. The main point of the present paper is that, in the dual conformal-net setting, the Levi–Civita curvature data of the two nets realize this logarithmic connection by the symmetric formula $H^{-1}\widetilde{\omega}\,dz-H\omega\,dq =2i\,d\log z_q =2i\,Z^{-1}dZ$, where the last expression is understood through local trivializations of $L$. Thus the two dual conformal connection differentials occur with opposite signs, and their difference is the logarithmic Maurer–Cartan form of the relative differential morphism. Its residues recover the relative ramification multiplicities, $\operatorname{Res}_p\Theta=2i(r_z(p)-r_q(p))$, and globally $\sum_{p\in X}\operatorname{Res}_p\Theta =4i(\operatorname{deg}z-\operatorname{deg}q)$. Thus local conformal connection data determine both relative ramification and the difference of the degrees of the two projections. We also discuss the dual morphism, the square-root issue, and a hyperelliptic example.

We classify the generators of stability-preserving semigroups on polynomial spaces with bounded coordinate degrees. The generators have differential order at most two, with principal coefficients characterized by low-degree nonnegativity conditions. In disk coordinates, positive degree damping gives strict zero-freeness and a sharp spectral gap, extending the Hermitian gap of Bravyi, Gosset, Liu, and Wong to complex generators. The resulting analytic domain yields deterministic classical algorithms for ground energies and stable product-state queries for bounded-degree Suzuki-Fisher Hamiltonians with bounded local strength and fixed positive fields. Field perturbation gives zero-field energy-value approximation schemes for bounded-degree unweighted EPR and bipartite Quantum MaxCut. We also prove that, up to scalars, the Hermitian qubit Hamiltonians whose full Gibbs tensors are Lee-Yang at every temperature in a fixed basis are precisely the edgewise phase-rotated Suzuki-Fisher family. With strictly positive longitudinal fields, these tensors are zero-free on a polydisk of radius greater than one at each positive inverse temperature. These two statements answer questions of Wong, Bravyi, Gosset, and Liu. Degree-preserving, coefficient-positive stability semigroups have concave sector growth rates, yielding token-graph concavity. We also give a counterexample to a proposed explicit ground-state radius.

We study the monic polynomials $\Sob{n}$ orthogonal with respect to a Sobolev-type modification of the discrete Krawtchouk inner product in which a single point mass at an exterior point $\alpha<0$, outside the support $\{0,1,\dots,N\}$ of the binomial measure, acts through the $j$-th forward difference $\dif^{j}$ (here $\Nzero=\{0,1,2,\dots\}$, $\Z_{+}=\{1,2,\dots\}$, $\R_{+}=(0,\infty)$). In the joint scaling $n,N\to\infty$, $n/N\to r$ with $0<p<r<1$ (the reciprocal-Gamma regime of Dominici's classification), we prove the Mehler–Heine formula. The exterior mass survives the large-degree limit exactly through the linear factor $z-\alpha$, with the explicit universal constant $C_{\ast}$, and the limit is independent of both $\lambda>0$ and $j\ge1$; the same conclusion holds in the Uvarov case $j=0$ (Corollary~\ref{cor:uvarov}). The derivation is pointwise and self-contained: it uses only a rank-one connection formula (Theorem~\ref{thm:connection}), the reproducing-kernel recursion, Dominici's pointwise limit and the classical three-term recurrence; the normalised Sobolev correction obeys an exact affine recursion at fixed $N$, controlled by a triangular contraction argument over two backward windows with an explicit boundary estimate, and the contraction ratio $\varrho<1$ is exactly the condition $r>p$. We also discuss the scale of the theorem: in this normalisation the classical term is asymptotically negligible, and the independence of $\lambda$ concerns each fixed $\lambda>0$, not the continuity at $\lambda=0$. Multiprecision computations illustrate the convergence and the exact value $C_{\ast}=2.8$ for $p=0.3$, $r=3/5$, $\alpha=-2.5$, $j=2$, $\lambda=5$.

The decimal reversals 1729 and 9271 have multiplicative orders of $2$ equal to 36 and 63. Motivated by this symmetry, we classify the affine commutation equation $R_B(hx+1)=hR_B(x)+1$, where $R_B$ reverses base-$B$ digits, for every $B\ge3$, $2\le h<B$ and positive integer $x$ such that neither $x$ nor $hx+1$ ends in zero. Comparing the two carry sequences separates the solutions by their digit lengths. Equal lengths force an odd number of digits alternating between two explicit intervals. When the output gains a digit, coprimality of $h$ and $B+1$ forces a repdigit input with palindromic output. In the noncoprime case, all inputs instead have even length and form a finite family. A slack-variable bijection gives the sharp maximum length, binomial counts at each length, and totals expressed through Fibonacci numbers or powers of two. For reversal pairs of primes already known to share an order index $2\le h<B$, the classification gives an exact digit criterion for reversal of their orders. Finally, an exhaustive computation with independently checkable primality certificates proves that $\{1729,9271\}$ is the unique pair of distinct odd decimal reversals whose orders of $2$ are distinct two-digit reversals of each other.

We show that there exist infinitely many polynomials $F\in\mathbb{Q}[x]$ with exactly $m$ nonzero coefficients such that every irreducible factor of \(F\) over \(\mathbb{Q}\) has \(\exp\) \(\!\bigl(\Omega(\sqrt{m/\log m})\bigr)\) nonzero coefficients. This places multiplication in a sharply different category from powers, composition, and other algebraic operations for which reverse-sparsity principles are known. It also gives an unconditional, degree-free obstruction to sparse factorization, complementing positive factor-sparsity results that impose hypotheses on coefficient height, exponent positions, reciprocal structure, or degree bounds.

We prove that every infinite introenumerable set contains an infinite uniformly introreducible subset, answering both parts of Question 1.7 of Greenberg, Harrison-Trainor, Patey, and Turetsky affirmatively. Moreover, a single procedure computes the original set from every infinite subset of the selected set. The proof combines the uniformization and refinement results of these authors with Ramsey-theoretic methods and a canonical reconstruction argument. We also obtain refinements controlling the ordinals and hyperjumps of the selected subsets.

We study a cut-and-paste operation in which torus neighborhoods in the author's symplectic building blocks $Y_K$ and $X_K$, associated to a genus-one fibered knot $K$, are replaced by marked exteriors of tori in $S^4$ with infinite cyclic complement group and two null peripheral slopes. We also recall the exact Luttinger-surgery realization of $M_K\times S^1$ from $\Sigma_g\times T^2$, keeping the knot-surgery/fiber-sum and Luttinger-surgery viewpoints in the same framework. For the trefoil block $Y_K$, two marked replacements give a simply connected manifold with intersection form $H$, hence a manifold homeomorphic to $S^2\times S^2$. A one-exterior gluing gives a smooth homotopy $4$-sphere. For the rank-six construction we use the identity double of two copies of $Y_K\setminus\nu\Sigma_2$, rather than the involutive gluing defining the original $X_K$. Two marked replacements along the surviving rim tori give a simply connected manifold with $e=8$ and $\sigma=0$. An explicit geometric basis has intersection form $3H$, so the resulting manifold is homeomorphic to $\#_3(S^2\times S^2)$. The Case II small-perturbation Seiberg–Witten invariant vanishes. The Seiberg–Witten invariant of the identity-glued rank-six family also vanishes; in particular, these manifolds are nonsymplectic.

In a previous article doi.org/10.3390/geometry2030012, the author investigated a class of elementary plane geometry problems closely related to the theme of this work. Here, we prove a weaker version of a previous conjecture by demonstrating that there are infinitely many maximally circlable (MAC) numbers – positive integers $n$ for which there exists a largest circle enclosing exactly $n$ interior lattice points. Furthermore, by extending numerical computations to $n \le 2700$, we identify two counterexamples to a conjecture in loc cit. regarding the symmetry of the largest circle enclosing a strong MAC number (a MAC number $n$ where $n+1$ is non-MAC). We also propose a potential infinite family of strong MAC numbers derived from Pythagorean triples; the existence of this family would imply the infinity of non-MAC numbers, as conjectured by Zhao. Throughout this paper, we provide extensive data characterizing both MAC and strong MAC numbers alongside their corresponding largest enclosing circles.

The binary length function $\ell_2(r,R)$ is the smallest length of a binary linear code with codimension (redundancy) $r$ and covering radius $R$. Let $s_2(N,\rho)$ be the smallest size of a $\rho$-saturating set in the binary projective space $\mathrm{PG}(N,2)$. It is known that the binary length function satisfies $\ell_2(r,R)=s_2(r-1,R-1)$. We obtain the following new upper bounds on $\ell_2(r,R)$, improving the previously known upper bounds by $\Delta(r,R)$: $R=2,r=2t\ge28$, and $r=10,18,20$, $\ell_2(r,2)=s_2(r-1,1)\le51\cdot2^{r/2-5}-1,\Delta(r,2)=2^{r/2-5}$; $R=3,r=3t-1\ge41$ and $r=26$, $\ell_2(r,3)=s_2(r-1,2)\le818\cdot2^{(r-26)/3}-1$, $\Delta(41,3)=63,\Delta(r,3)=2^{(r-26)/3}$ if $r\ge44$. To obtain these bounds, we construct new infinite code families, using distinct versions of the $q^m$-concatenating constructions of covering codes, including new versions proposed in this paper. We also obtain new useful partitions of column sets of parity check matrices of some codes. The new code families provide the asymptotic covering-density bounds $\overline{\mu}(2)\le2601/2048<1.27002$, $\overline{\mu}(3)\le1.35935$, improving the previously known bounds and giving rise to the new upper bounds $f(2)\le2601/2048<1.27002$, $f(3)\le1.35935$ on the constants $f(2),f(3)$ of the Green's Open Problem 40.

We study self-referential multiple-choice tests in which the answer choices are positive integers and the question asks how many answer choices are correct. These tests quickly lead to several kinds of ambiguity and paradox. The paper presents the ideas through a dialogue between Queen Tanya and her ten sages, developing a framework for studying these immensely questionable tests.

This paper investigates the completeness properties of the fuzzy order on the fuzzy number space $\mathbb{E}^1$ within the framework of $[0,1]$-enriched categories. \(S\)-convergence under the three fundamental continuous t-norms is characterized, and inclusion relations among the induced open ball topologies are established. It is shown that $(\mathbb{E}^1,P)$ is Cauchy complete if and only if the t-norm is not isomorphic to the Łukasiewicz t-norm, whereas $(\mathbb{E}^1,P)$ fails to be Smyth complete for every continuous t-norm. Consequently, the symmetrization $(\mathbb{E}^1,S)$ is Smyth complete if and only if the underlying continuous t-norm is not isomorphic to the Łukasiewicz t-norm.

We construct hyperbolic renormalization horseshoes for analytic generalized interval exchange transformations (GIETs) of bounded type and with bounded geometry, provided their boundaries are small (but not necessarily zero). As a consequence, we describe the manifold structure of smooth conjugacy classes of GIETs.