Papers

Inspired by Nuel Belnap's "How a computer should think," which gave rise to the four-valued logic FDE, we contemplate, in this article, how a computer might think if we add a fifth value for unknowable or cannot be known. We devise two new five-valued logics, UKN1 and UKN2, called the logics of the unknowable. These are different from the five-valued logic FDEe of the FDE-family. The main difference is in the number of designated truth values. While FDEe takes two designated values, UKN1 and UKN2 have three. The four-valued reducts of these logics are also different from FDE. This is due to the fact that instead of taking one of the non-designated values as neither true nor false, as in FDE, we have interpreted this as not known yet. This value although denoted by the same letter $n$, behaves differently from its namesake in FDE.

We study the inversion enumerator of permutations whose descent set is fixed to be the set of multiples of a fixed integer $k \ge 2$. For each $n$, let $\mathfrak{S}_n^{(k)}$ denote the set of permutations of $\{1,\dots,n\}$ whose descent set is exactly $\{i : k \mid i\}$, and define the polynomial $$ a_n^{(k)}(q)=\sum_{\sigma \in \mathfrak{S}_n^{(k)}} q^{\mathrm{inv}(\sigma)}. $$ We prove that the associated $q$-exponential generating function $$ F_k(t;q)=\sum_{n\ge 0}\frac{t^n}{[n]_q!}\,a_n^{(k)}(q), $$ where $[n]_q!$ denotes the $q$-factorial, admits an explicit closed form as a ratio of two $k$-periodically truncated $q$-exponential series. The proof is purely analytic and is based on a functional equation satisfied by $F_k(t;q)$, obtained via a decomposition of the $q$-exponential series into residue classes modulo $k$. Coefficient extraction yields a convolution identity involving Gaussian binomial coefficients, which uniquely determines the inversion enumerator. This provides an analytic alternative to classical inclusion–exclusion and structural combinatorial arguments for permutation classes with periodic descent constraints.

In this paper, we obtain a complete classification of Toeplitz + Hankel operators on the vector-valued Hardy space $H^2_{\mathcal{E}}(\mathbb{D}^n)$ over the polydisc $\mathbb{D}^n$ in $\mathbb{C}^n$ for $n\geq 1$. We also characterize the paired operators on $L^2(\mathbb{T}^n)$. Furthermore, we give a complete characterization for the class of essentially Toeplitz + essentially Hankel operators on the vector-valued Hardy space $H^2_{\mathcal{E}}(\mathbb{D})$ for finite-dimensional Hilbert space $\mathcal{E}$.

We develop a finite-kernel framework for sparse extremal graph counting. The problems considered here ask for the maximum number of copies or homomorphisms of a fixed graph under sparse edge constraints. In this regime, the leading term need not be governed by a single dense block. Instead, the extremal mass may be supported on several interacting asymptotic scales. Our framework identifies these scales via a finite-dimensional linear program, separates the leading contributions through a finite state decomposition, and synchronizes or realizes them inside a finite kernel. We apply this framework in three settings. First, we prove the sparse threshold conjecture of Day and Sarkar for graphons. For every fixed graph $H$ without isolated vertices, we prove that \[ \sup_{t(K_2,W)\le \beta} t(H,W)=\beta^{|V(H)|-\alpha^*(H)}(C_T(H)+o(1)) \] as $\beta\to0$, where $\alpha^*(H)$ is the fractional independence number of $H$ and $C_T(H)$ is an explicit sharp constant attained by a three-step threshold graphon. Second, we affirmatively answer a question of Blekherman and Patel by showing that, for every graph $H$, whenever $m\to\infty$ and $m=o(n^{3/2})$, threshold graphs asymptotically maximize $\hom(H,G)$ among all graphs with at most $n$ vertices and at most $m$ edges. Third, Gerbner, Nagy, Patkós, and Vizer conjectured that, among all bipartite graphs with $n$ vertices and $m$ edges, the quasi-complete bipartite graph asymptotically maximizes the number of copies of every fixed bipartite graph $H$ whenever $m=\omega(n)$ and $m\le n^2/4$. We disprove this conjecture in the subquadratic range and give the correct order of magnitude in terms of $\kappa_H(n,m)$, a finite-kernel scale defined by a finite-dimensional variational problem.

We prove that a bounded Hilbert space operator has a nontrivial invariant subspace whenever its spectrum contains the topological boundary of its numerical range. We also give variants of this result and establish related statements. The criterion is applied to hyponormal, cohyponormal and Toeplitz operators. In these classes, convexity of the polynomial hull of the spectrum already suffices. Several examples are given that are not covered by well-known existence criteria.

A family of spanning trees of a graph is $t$-intersecting if any pair of spanning trees in the family has $t$ or more edges in common. For sufficiently large $n$ and $t \leq n/C\log_2 n$ for some absolute constant $C>0$, we give a nearly complete characterization of the extremal $t$-intersecting families of spanning trees in balanced complete bipartite graphs with parts of order $n$. In particular, for $t=1$, we give exact bounds and a full characterization of the extremal families. For $t \geq 2$, our bounds are tight up to lower-order terms, and we show that any extremal $t$-intersecting family is of the form $\mathcal{F} \cup \mathcal{S}'$ where $\mathcal{F}$ is a family of all trees containing a fixed $t$-matching, and $\mathcal{S}'$ is a distinguished set of exceptional trees of size $|\mathcal{S}'| = o(|\mathcal{F}|)$.

The paper offers the first systematic study of ordinary and essential numerical ranges of operators on $\ell_p$, $1<p<\infty$, as an atomic picture within a broader $L^p$ project. The paper begins with Banach-space foundations, including the finite-codimensional description of the essential numerical range and a Banach-space convex-hull inclusion for the essential spectrum. It then turns to finite-dimensional $\ell_p$ geometry, where one finds both positive star-shapedness phenomena and explicit $2\times2$ counterexamples. On $\ell_p$, we prove that the essential numerical range is compact and convex, identify it with the algebraic numerical range of the Calkin image, obtain a compact-perturbation formula, and show that, moreover, the closure of the numerical range is star-shaped, while points in the interior of the essential numerical range are exact star-centres of the numerical range. The paper illustrates the developed theory with sequence-space examples, covering tridiagonal Toeplitz operators and the discrete Hilbert transform, and, after relating our study to a variant of the Crouzeix inequality, closes with a brief discussion of extensions to spaces of class $(P)$ and to joint essential numerical ranges.

We present a geometric coordinate-free solution to the isotropic Schrödinger bridge problem (SBP) for the kinematic equation on the Lie group $\mathsf{SO}(2)$. We consider the angular velocity of the system as the control input and assume that the given initial and terminal state probability density functions defined on $\mathsf{SO}(2)$ in our SBP are continuous and strictly positive. We solve the SBP by proving the existence and uniqueness of a solution to the so-called Schrödinger system of equations on $\mathsf{SO}(2)$, by showing that a fixed-point recursion is contractive in a complete metric space with respect to the Hilbert's projective metric. The geometric controller thus designed only uses the intrinsic geometric structure of $\mathsf{SO}(2)$ and does not embed it in the Euclidean plane to achieve the optimal density control. The numerical simulation verifies the validity of the theoretical construction of the Schrödinger bridge. The code and animations are publicly available at \texttt{\href{https://gitlab.com/a5akhtar/sbp}{https://gitlab.com/a5akhtar/sbp}}.

We show that an oscillon can be understood as a localized discrete resonant (non-normalizable) mode. Specifically, oscillon in the vacuum arises from the threshold mode, which because of nonlinearity gets localized. Following this idea, we find wobblerons - nonlinear excitations of kinks, that is, oscillons-kink bound state. Now, the oscillon can also originate in an antibound mode, i.e., a discrete, positive energy but non-normalizable mode.

This paper presents a unified linear-algebraic framework for interpolation problems, viewing different standart problems (Lagrange polynomials, Hermite splines, and trigonometric interpolation) with the help of dual spaces. By representing interpolation conditions as linear functionals in the dual of a finite-dimensional function space, we establish a general existence and uniqueness theorem. We derive explicit formulas for basis functions dual to given interpolation schemes, enabling efficient construction of interpolants as linear combinations of these bases. The approach is extended to trigonometric interpolation, where we provide closed-form expressions for dual bases and analyze degeneracy conditions. We also address practical scenarios involving heterogeneous sensors, where function values and derivatives are specified at distinct points, and propose a corresponding interpolation scheme. The method is systematic, dimension-aware, and decouples the analysis of interpolation conditions from the choice of basis functions. We cosider examples from kinematics and periodic function interpolation demonstrate the versatility of the approach, suggesting applications in trajectory planning, signal processing, and beyond.

Adaptive stochastic convex optimization (SCO) methods face a fundamental “price of adaptivity” barrier: under the standard set of assumptions, they cannot efficiently adapt to large uncertainty in both the initial distance to optimality and the Lipschitz constant. We circumvent this barrier by requiring a small amount of additional structure common to many learning problems. Specifically, we assume that the objective decomposes into a model and a loss function, enabling us to intervene by modifying the model's output before it passes to the loss function. Under this assumption, we design a method that clips the learned model output in tail events where it deviates too much from the output of a fixed reference model. Our method matches the optimal bounds for known-parameter SCO up to logarithmic factors in the uncertainty in the distance and Lipschitz parameters, thus efficiently adapting to large uncertainty in both.

We study the polar director structure in thin planar cells filled with bent-core liquid crystals in the ferroelectric smectic-A phase (SmAP$_\text{F}$). We analyze a continuum phenomenological model proposed in the physics literature and present rigorous proofs of the existence and uniqueness of the equilibrium solutions. We further investigate the qualitative properties of nontrivial solutions and examine the effects of a bias electric field, surface anchoring, and cell thickness on the polar director configuration. Our results are consistent with previous experimental and numerical simulations reported in the physics literature. In addition, our analysis reveals new parameter-dependent behaviors supported by our numerical simulations and extends results reported from previous literature.

We formulate four-dimensional finite-proper-time QED from a gauge-covariant source functional with a retained spectral lower boundary $s_0=\Lambda^{-2}>0$ and finite physical duration $\mathcal T$. Physically, the basic $s_0$ operation is a truncated Laplace transform: the arbitrarily small-proper-time part of the spectral integral, which carries the ultraviolet singularity, is excluded without imposing a cutoff on momentum space, while finite $\mathcal T$ expresses that a physical observable is prepared and detected during a finite experiment rather than by an infinite-time measurement. The retained $s_0$ belongs to the defining source construction and controls the correlated finite corrections after mass and charge matching. The charged interval and circle sectors are generated by the same covariant heat semigroup, while the photon covariance carries the same retained lower bound at the source level. At every fixed perturbative order, gauge covariance gives the Ward–Takahashi hierarchy and an assembled graph-level quadratic-form bound establishes ultraviolet finiteness. The physical electron, positron, and transverse-photon poles retain their canonical residues, and the fixed-order cutting relation uses the matched daughter domains on the stated regulated continuation domain. Mass and charge renormalisation are replaced by finite matching, while independent observables retain correlated $s_0$ dependence. At fixed $s_0>0$, the Euclidean source has a nonperturbative definition. Its physical standing-wave response is obtained order by order from the continued FPT amplitudes with the same finite-duration channel data. An exact all-coupling physical boundary realisation requires additional channel-norm control. The limits $s_0\to0^+$, $\mathcal T\to\infty$, and exact all-coupling free-space exhaustion are separate limiting questions and are not used to define the theory.

For fixed $k\geq3$, we study the asymptotic number and typical structure of dense graphs with no induced copy of the star $K_{1,k}$. We solve the associated graphon variational problems both at fixed constant edge density $\gamma$ and for the conditioned Erdős–Rényi random graph $G(n,p)$ for constant $p$. As consequences, we obtain explicit formulas for the entropy density of induced-$K_{1,k}$-free graphs with $\Theta(n^2)$ edges and for the large deviation rate function for the event that $G(n,p)$ is induced-$K_{1,k}$-free. The entropy density exhibits a second-order phase transition at an explicit critical density $\gamma_k$, while the rate function exhibits a first-order phase transition at a critical parameter $p_k$. We completely characterize the optimizers of both variational problems. Both models have parameter values for which there are infinitely many optimal graphons, but there is always a unique graphon that represents the typical structure in cut metric. We refine the graphon-level results by giving a detailed structural description of both models. For supercritical parameters, each random graph model is the complement of a $(k-1)$-partite graph with high probability. In the subcritical regime of the fixed-density model, the typical structure is the disjoint union of the complement of a $(k-1)$-partite graph, and a sparse remainder. In the subcritical regime of the conditioned Erdős–Rényi random graph, a typical sample has $o(n^2)$ edges.

For a dynamical system $(X,T)$ we consider the induced dynamical systems $(\myper(X),T)$ and $(\hyper(X),T)$, consisting of Borel probability measures and closed non-empty subsets, respectively. We show that diam-mean equicontinuity of $(X,T)$ is equivalent to the diam-mean equicontinuity of $(\myper(X),T)$. Furthermore, we establish that $(X,T)$ is mean equicontinuous, iff $(\myper(X),T)$ is mean equicontinuous, iff $(\myper(X),T)$ is weakly-mean equicontinuous. For $(\hyper(X),T)$ the situation is different. It is not hard to see that the diam-mean equicontinuity of $(\hyper(X),T)$ implies the diam-mean equicontinuity of $(X,T)$. We provide examples for which $(X,T)$ is diam-mean equicontinuous, while $(\hyper(X),T)$ is not diam-mean equicontinuous. We prove that $(\hyper(X),T)$ is diam-mean equicontinuous, iff $(\hyper(X),T)$ is mean equicontinuous, iff $(\hyper(X),T)$ is weakly-mean equicontinuous. We present our results in the context of continuous surjective maps $T\colon X\to X$ and discuss why they also hold for actions of locally compact $\sigma$-compact amenable groups.

The vorticity distribution associated to canonical Gibbs point vortex ensembles under mean field scaling has Gaussian fluctuations for small negative inverse temperature. The perturbative argument is based on a cluster expansion of the partition function.

In this work, we introduce a family of almost hyperhermitian structures that we call semi-integrable. We subdivide them into four disjoint classes and show that each class is non-empty. Finally, we construct semi-integrable almost hyperhermitian structures on all reductive Lie algebras of compact type and dimension $4n$, with $n\geq 2$.

We give the complete solution of the one-parameter deformation of the six-vertex model on dynamical lattice introduced in [1] and dubbed gravitational seven-vertex model. The statistical model in question is mapped to a gas of self- and mutually avoiding loops on dynamical triangulations, with a temperature coupling controlling the volume not occupied by loops. The phase diagram is characterised by massive, dilute and dense critical phases, similarly to the gravitational O(n) loop model. There is however an important difference – in our model the weights of the loops are not topological but depend on the form of the loop and on the curvature defects of the lattice via lattice spin connection. The seven-vertex model on dynamical lattice is nevertheless exactly solvable after being reformulated as a large-N matrix model, which we will refer to as 7vMM, and the solution in the scaling limit was found in [1]. Here we derive the full solution in terms of Jacobi theta functions and present the (non-algebraic) spectral curve of 7vMM in a parametric form. We obtain the phase diagram in the space of the two coupling constants – the cosmological constant and the temperature – and identify the critical phases along the boundary of the physical domain. We derive the scaling solution of [1] as the asymptotic of the full solution in the vicinity of the tricritical point separating the phases of dense and massive loops.

Significant combinatorial constraints and structure on matroids can come from notions in algebraic geometry, even without the matroids themselves being representable. Let \(M\) be a loopless matroid on a finite ground set \(E\), and let \(\G\) be a building set containing the top flat \(E\). We define a tangent class \(T_{M,\G}\) in the \(K\)-ring \(K(M,\G)\), which extends the tangent bundle class of the de Concini–Procesi wonderful model from realizable matroids to arbitrary matroids with building sets. The class \(T_{M,\G}\) satisfies a matroidal Hirzebruch–Riemann–Roch package. More precisely, its Hirzebruch class specializes to the Todd class and computes the Chow polynomial of \((M,\G)\). In the realizable case, these identities agree with the usual tangent-bundle computations on the corresponding wonderful model. As an application, we prove Chern-number inequalities for \(T_{M,\G}\), including a Miyaoka–Yau type inequality with respect to the hyperplane class.

The norms of embedding operators of Sobolev spaces $\Wo^n_2[0;1]\hookrightarrow\Wo^k_\infty[0;1]$ ($0\leqslant k\leqslant n-1$) are considered. The least possible quantities $A^2_{n,k}(x)$ in the inequalities $|f^{(k)}(x)|^2\leqslant A^2_{n,k}(x)\|f^{(n)}\|^2_{L_2[0;1]}$ are studied. On the basis of the relations between the $A^2_{n,k}(x)$ and the antiderivatives of the Legendre polynomials, the properties of the maxima of the functions $A^2_{n,k}(x)$ are established. It is shown that, for all~$k$, the global maximum of the function $A^2_{n,k}$ on the closed interval $[0;1]$ is the maximum point nearest to the midpoint of the interval; in particular, for even~$k$, $x=1/2$ is such a point. For the parameter~$k$ of even order, an explicit formula for the norms of the embedding operators is obtained.

In 1984, Ogus conjectured the existence of a canonical F-isocrystal that enhances the Gauss–Manin connection, for a proper relative rigid space with analytically good reduction. We give a positive answer to this conjecture in full generality, through p-adic local systems and prismatic methods. Along the way, we introduce a prismatic refinement of the p-adic Riemann–Hilbert functor and prove a primitive purity theorem for Frobenius modules.

We prove an explicit spectral-gap lower bound for the lazy swap chain on binary matrices with prescribed row and column sums. This chain is a standard sampler for fixed-margin null models in ecology, statistics, and network analysis. Kannan, Tetali, and Vempala (KTV) conjectured that it mixes rapidly for all feasible margins \citep{kannan1997simple}. We show that for every feasible set of margins on an $m\times n$ binary matrix, the lazy swap chain has spectral gap at least $$\binom{m}{2}^{-1}\binom{n}{2}^{-1}.$$ The bound is tight in the worst case. Thus, our result proves this KTV conjecture in a stronger quantitative form. The same spectral-gap bound also verifies the Mihail–Vazirani conjecture for fixed-margin 0/1-matrix polytopes. The proof gives a new route to fixed-margin sampling that avoids stability assumptions and canonical-path constructions. We compare the swap chain with a two-row heat-bath chain and use a local-to-global spectral reduction to reduce the analysis from arbitrary $m\times n$ matrices to a three-row problem. The remaining three-row inequality is then proved by separating the scalar column-count sector from the non-scalar Johnson harmonic sectors. The proof itself was generated by ChatGPT 5.5 Pro. The author's role was to pose the problem, guide the search direction, evaluate the AI-generated arguments, rewrite the proof, and take responsibility for the final form and validity of the result. The full proof of the main theorem has been formalized in Lean, and the accompanying formalization is available at the anonymous repository https://github.com/guanyangwang/ktv-swap-lean.

Counting triangles in graphs is a fundamental operation in network analysis, underpinning metrics such as clustering coefficients and serving as a signal for community detection, link prediction, and anomaly detection. The standard approach computes the trace of the cube of the adjacency matrix, but explicitly forming $\mathbf{A}^3$ is infeasible for large graphs. Hutchinson randomized trace estimator offers an efficient alternative by approximating the trace through stochastic averaging of quadratic forms, requiring only matrix vector products with $\mathbf{A}$. However, in distributed and heterogeneous computing environments, observing all entries of these products can be costly due to communication overhead and straggler effects. To address this, we propose a new variant of Hutchinson estimator that operates under partial observation constraints, where both the number and identities of observed entries are random. We provide theoretical guarantees on unbiasedness, variance bounds, and sample complexity, and demonstrate through experiments on synthetic and real world graphs that our method achieves accurate triangle count estimates while reducing synchronization costs. This work highlights the adaptability of randomized algorithms to modern computational architectures and opens avenues for efficient motif counting in large scale network analytics.

We study a reactive Maxwell–Stefan-type membrane transport system under a finite-occupancy constraint with explicit vacancies. The admissible state space is $\mathcal{D} = \{u \in (0,1)^n : \varrho(u) := \sum_{i=1}^n u_i < 1\}$, where the vacancy fraction $1-\varrho(u)$ represents the local free volume. This bounded-occupancy geometry induces a Boltzmann–Fermi entropy and a global parametrization by entropy variables. The associated mobility is assumed to split into a composition channel, corresponding to redistribution at fixed total occupancy, and a mass channel, corresponding to variation of the filling fraction. The main structural difficulty is that the unaugmented mobility may lose coercivity in the mass channel. We show that a single rank-one augmentation of the form $\gamma_0 \mathbf{1} \otimes \mathbf{1}$ restores full coercivity while leaving the composition block unchanged. On this basis, we prove four results: a quantitative channel-wise coercivity estimate for the augmented mobility; global existence of entropy weak solutions via an implicit Rothe scheme in entropy variables; a weak–strong stability estimate in relative entropy with uniqueness in the strong class; and convergence of a fully implicit finite-volume approximation that preserves the bounded-occupancy structure and satisfies a discrete entropy inequality.

Let $E$ be an $n$-dimensional nilpotent evolution algebra of maximal nilindex over a field of characteristic zero. The Rota–Baxter operators of weights zero and one on such algebras were recently classified. In this paper we investigate the local analogue of the weight-zero case. We prove that every local Rota–Baxter operator of weight zero has a rigid diagonal part: the nonzero diagonal coefficients, when they occur, begin after the obstruction index determined by the structural matrix of $E$ and form a final geometric string governed by one scalar. In contrast, the last-column coefficients are arbitrary. This gives an explicit description of the class $\LRB_0(E)$ and shows that it is generally strictly larger than $\RB_0(\E)$. We also prove that the corresponding $s$-local notion produces no new operators. The classification is further interpreted as a finite union of quasi-affine strata, yielding a dimension comparison between $\LRB_0(E)$ and $\RB_0(E)$. Finally, we study ordinary weight-zero Rota–Baxter operators commuting with derivations and automorphisms and describe the resulting conditions in terms of the directed graph associated with $E$.

The state space of an origami crease pattern forms a flip graph, whose vertices are the flat-foldable mountain-valley assignments and whose edges join assignments differing by a single face flip. For the $m \times n$ Miura-ori, the degree sequence and diameter of this graph are known only for two rows. Each assignment maps to an integer height function on the grid, under which a vertex's degree equals its number of local extrema. In this model the vertices of each degree up to five are counted by an explicit polynomial in $m$ and $n$, valid once both exceed a bound that grows with the degree, and the height functions realizing those degrees are described explicitly. A closed-form lower bound for the diameter holds for all $m$ and $n$, and the matching upper bound reduces to an extremal inequality for $1$-Lipschitz functions on the grid, recovering the two-row distance at $m=2$. Since each invariant is read from the extrema or height differences of a grid function, the same reduction applies to any flip-graph quantity expressible in those terms.

In this paper, we study affine $\lambda$-equidistants of convex polygons with parallel opposite sides (CPPOSes), introduced by M. Craizer et al. (Polygons with Parallel Opposite Sides, Discrete & Computational Geometry 50 (2013), 474-490), and relate them to affine $\lambda$-equidistants of smooth convex curves. We show how, and under which conditions, the original CPPOS can be reconstructed from its Wigner caustic and centre symmetry set. We prove a discrete version of the improved isoperimetric inequality: $$L(\mathcal{P})^2\geqslant 8n\tan\left(\frac{\pi}{2n}\right)\cdot\big( A(\mathcal P)+2\left|A^\ast\left(\mathrm{E}_{0.5}(\mathcal P)\right)\right|\big),$$ where $\mathcal{P}$ is a CPPOS with $2n$ vertices, $L(\mathcal{P})$ is its perimeter, $A(\mathcal{P})$ is the area enclosed by $\mathcal{P}$, and $A^\ast\left(\mathrm{E}_{0.5}(\mathcal P)\right)$ denotes the oriented area of the Wigner caustic of $\mathcal{P}$. Moreover, equality holds if and only if $\mathcal{P}$ is an equiangular CPPOS of constant width. We also prove sharp area estimates for the oriented areas of the Wigner caustic and the centre symmetry set of a CPPOS. More precisely, we show that the absolute value of the oriented area of the Wigner caustic is at most one quarter of the area of $\mathcal{P}$, while the absolute value of the oriented area of the centre symmetry set is at most the area of $\mathcal{P}$. Both bounds are sharp.

We obtain upper and lower bounds for "smoothed" versions of the expected supremum of centered Gaussian processes with finite or countable index sets. These so-called soft maxima are computed in terms of expected values of random Gibbs averages at inverse temperature $\beta > 0$ and reduce to expected suprema in the zero-temperature limit $\beta \to \infty$. Our analysis builds on ideas from statistical physics and information theory, and relies crucially on the tensorization technique introduced recently by Liu (2025). The bounds retain the same multiscale structure as in the expressions for the expected supremum derived using the method of generic chaining, with a truncation term governed by the inverse temperature $\beta$. In the zero-temperature limit, we recover the majorizing measure theorem. As an illustrative example, we apply our results to the analysis of the quenched free energy in the Sherrington-Kirkpatrick model and obtain a Parisi formula in the finite system size setting.

J.E. Littlewood posed the question of how many infinite circular cylinders of unit radius can be arranged so that each touches all the others. We give a computer-free proof that one cannot find ten such cylinders. This improves a known result, namely that there are no eleven such cylinders, which was obtained making partial use of computer verification.

Sparse code multiple access (SCMA) is a promising code-domain non-orthogonal multiple access scheme which is transmitted over orthogonal frequency division multiplexing (OFDM) to exploit multicarrier diversity. In practice, however, carrier frequency offset (CFO) and phase noise (PN) may disrupt the subcarrier orthogonality in OFDM-SCMA systems. Addressing this research problem from a new SCMA codebook design angle, we propose a hardware-aware end-to-end autoencoder that embeds differentiable CFO and Wiener PN layers into the training loop. Simulations show that the proposed codebook effectively suppresses the bit error floors caused by CFO and PN without requiring real-time phase tracking.