Papers

The semi-random graph process is an adaptive random graph process in which an online algorithm is initially given an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the algorithm independently and uniformly at random. The algorithm then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. We also consider the offline version of the process in which the algorithm is given the entire sequence of random vertex choices before the selection takes place. For a given graph property, the objective of the algorithm is to force the graph to satisfy this property asymptotically almost surely in as few rounds as possible. In this paper, we focus on the property of being triangle-saturated and establish upper and lower bounds on the number of rounds required to construct a triangle-saturated graph in both the online and offline versions of the process.

Ice I admits cubic, hexagonal, and mixed layer stackings, but rigorous comparisons of its proton configurational entropy have focused on the two ideal endmembers. We identify every cyclic uniform-registry ice-I stacking with a word in a nonnegative layer-transfer operator and its transpose. Applying the Schatten-Hölder inequality to every even-length word proves that alternating hexagonal stacking maximizes the ice-rule configuration count at each common finite cross-section. Consequently, hexagonal ice maximizes the thermodynamic configuration constant among all periodic uniform-registry ice-I polytypes. We also construct rigorous lower and upper bounds without finite-size extrapolation. Restricting Nagle's positive even-subgraph expansion to disjoint blocks gives the lower bound; Finner's hypergraph Hölder inequality and rational Collatz-Wielandt certificates for two-replica prism operators give the upper bounds. The resulting configuration constant $w$, whose logarithm is the entropy per molecule in units of Boltzmann's constant, satisfies $1.503360 \le w \le 1.540196$, with the sharper cubic-ice ceiling $w \le 1.527699$. The same bracket controls lower and upper configuration growth rates of fixed aperiodic stackings along the specified prism-tileable exhaustions. These results establish hexagonal stacking as a possibly nonunique maximizer; equality of the cubic and hexagonal entropy constants remains unresolved.

In this paper we investigate the spectral properties of certain self-adjoint quadratic pencils, occurring at the separation of the variables in the problem of the oscillation of a semi-infinite cylinder with a pseudo isotropic elastic structure.

Let z(m,n;s,t) be the largest number of ones in an m x n zero-one matrix with no s x t all-ones submatrix. Roman's 1975 inequality remains the best general upper bound for s>=3, but it is not attained on a large part of the range just below the design threshold T=(t-1)C(m,s)/(s+1). The proof has two steps. First, for n=T-c with 1<=c<=sT/(s+2), Roman's bound equals the elementary counting bound (s+1)(T-c)+floor(2c/s), adding nothing beyond a budget inequality and convexity. Second, attainment forces all but at most one column to have size s+1 or s+2; each such column has a point lying in a number of s-sets divisible by d=gcd(s,C(s+1,2)), pinning the leftover coverage there to a single residue mu mod d, which a global count rules out. With r=c mod s and slack sigma(r) depending only on s, we prove z(m,T-c;s,t) <= Rom(m,T-c)-1 whenever 1<=sigma(r)<d and m*mu != s*sigma(r), or mu != 0 and m*mu > s*sigma(r). The first case is an odd-s phenomenon confined to one residue class, giving order-m^s values of n; the second needs mu != 0 but covers the whole interval once m exceeds a threshold depending only on s and mu. For s=4, t=2, m=28 it covers all 2730 values of n; when c=(s+1)/2 and an s-(m,s+1,t-1) design exists, z=(s+1)(T-c) exactly. Finally we relate the obstruction to linear programming: the relaxation over all 2^m subset variables collapses, under symmetrisation, to the counting bound, so no linear relaxation of the covering constraints alone can beat the bound of Chen, Horsley, and Mammoliti (arXiv:2310.12685, "Zarankiewicz numbers near the triple system threshold"). For the refined program of Davies, Gill, and Horsley, its optimum is still attained at the Roman vertex on an explicit sub-family, and we record where their program does better.

A fixed simplex, randomly projected into three dimensions and joined in Hamiltonian order, produces a rich and unusually tractable model of random stick knots. We prove that Gaussian projections and Haar-random Stiefel projections have exactly the same knot-type law, despite having different metric shapes, and that at every stick budget the model gives positive probability to precisely the knot types realizable with that many sticks. Its linear-algebraic structure yields an exact marginal distance law, an exact mean planar-crossing count, and crossing concentration, while in the first nontrivial six-stick case the complete tetrahedral sign pattern gives an exact unknot-versus-handed-trefoil classifier for every generic sample. The result is a direct bridge from random projections and finite sign geometry to the topology of random knots.

Building on the vector-Zak and cohomological framework developed by Faulhuber, Petersen, van Velthoven, and Voigtlaender in their twelve-point counterexample, we give a computer-assisted four-point counterexample with a nonzero complex-valued Schwartz window. Every symplectic triangle determinant of the explicit configuration has absolute value below one, placing it in the intrinsically subcritical regime.

This paper considers planar mathematical billiards with smooth billiard tables. We prove that through any point outside a convex smooth caustic, there passes a unique billiard table (Lemma 2). Furthermore, we describe a billiard table having a nephroid as its caustic.

Let $\alpha\models n$. We define a two-variable specialization $Z_\alpha(t,q)$ of the ribbon basis of noncommutative symmetric functions from cycle enumerators of an ordinary permutation and a rooted permutation, with $q$ recording reflection length. We realize $n!Z_\alpha(t,q)$ as the shifted bigraded Euler characteristic of an $\mathfrak S_n$-equivariant ordered-set-partition complex. When $\alpha$ has at most one odd part, each total decoration determines a set of simultaneous factorization cuts, and its fiber is the classical ribbon complex indexed by that cut set. This gives explicit nonnegative ribbon expansions for every bigraded homology representation. Organizing the complexes on labelled finite sets yields a counital differential graded comonoid in the Cauchy monoidal category of species, whose cut coproduct is compatible with the fiber decomposition. At a factorization cut, the induced map on top homology is injective; its cokernel has the near-concatenation ribbon character, and the kernel of the aggregate reduced cut coproduct on homology is $H_1$. For the zigzag compositions $\delta_n$, $Z_{\delta_n}(t,-1)$ is the order polynomial of the alternating fence. We prove \[ n!Z_{\delta_n}(t,-u)\in\mathbb N[t,u] \] using coefficientwise nonnegative recurrences derived from a Riccati equation. The same cut data define a nonnegative factorization defect $d=\lceil n/2\rceil-k-j$, which controls the homological support and makes the Euler sign constant on each defect layer. We also determine the full defect-zero edge by explicit Frobenius-character formulas.

We study the problem of lifting global and local quaternionic torus actions to principal quaternionic torus bundles. Let \(Q^k=(\operatorname{Sp}(1))^k\cong (S^3)^n\), let \(G\) be a compact Lie group acting on a connected, locally finite CW complex \(X\), and let \(Q^k\longrightarrow P\longrightarrow X\) be a principal \(Q^k\)-bundle. We first formulate a quaternionic analogue of the obstruction-theoretic framework of Hattori–Yoshida. The existence of a lifted \(G\)-action implies that the isomorphism class of \(P\) lies in the image of the restriction map induced by the Borel construction \(X_G=EG\times_GX\). In particular, the second Chern class admits an equivariant extension. Once a continuous pseudo-lift has been chosen, its failure to define a genuine action is measured by a factor set with values in the generally nonabelian gauge group \(\mathcal G(P)\cong\Gamma(\operatorname{Ad}(P))\). We obtain a necessary and sufficient lifting criterion in terms of the trivializability of this factor set, and show that, once a single lift exists, the set of all lifts modulo gauge conjugacy is classified by a pointed nonabelian \(H^1\)-set. We then apply this global theory to local quaternionic torus actions. Pulling back to the universal covering of the orbit space, untwists a local \(Q^n\)-action and produces a globally defined action on the pulled-back manifold. A preliminary lift of this global action need not be compatible with the deck transformations. We define a gauge-valued nonabelian descent defect, establish its crossed-cocycle identities and transformation law, and prove that the original local action lifts if and only if the global lifting obstruction vanishes and the descent defect is trivializable. In the abelian case, these constructions reduce to the classical obstruction theory for lifts of local torus actions.

We consider the three-dimensional stochastic wave equation (SWE) driven by a multiplicative Gaussian noise that is white in time and colored in space: \[ \frac{\partial^2 u}{\partial t^2} = \Delta u + b\bigl(u\bigr) + \sigma\bigl(u\bigr)\,\dot{W}, \] where the drift function $ b $ and diffusion coefficient $\sigma$ are assumed to be locally Lipschitz and exhibit logarithmic superlinear growth at infinity. We establish the existence and uniqueness of a global mild solution on any fixed time interval $[0,T]$ under suitable assumptions on the spatial covariance function $ f $ of the noise $\dot W(t,x)$. Our results apply, for example, to the case \[ b(u) = u (\log_+ u)^{\theta_1} \quad \text{and} \quad \sigma(u) = u (\log_+ u)^{\theta_2}, \] with parameters $\theta_1 \in (0,2)$ and $\theta_2 \in \bigl(0, \tfrac{\bar{\nu}+1}{2}\bigr)$, and $\log_+(z)=\log(z\vee e)$, where $\bar{\nu}$ is determined by the assumptions on $ f $.

Let $M_s^n$ be a non-degenerate pseudo-Riemannian submanifold of a pseudo-Euclidean space with indefinite induced metric. Restricting the contact condition to spacelike or to timelike tangent directions gives two a priori different one-sided contact numbers. We prove that, for every prescribed order, ordinary, spacelike and timelike contact are equivalent; consequently, the three contact numbers coincide. The proof is inductive. A one-sided contact hypothesis first yields higher-order orthogonality identities for the iterated covariant derivatives of the second fundamental form. Their diagonal scalar expressions are then extended from either unit pseudo-sphere to the whole tangent space by a polynomial argument, and a second induction proves causal invariance. We also construct an explicit Lorentzian surface in $\mathbb E_4^8$ whose ordinary, spacelike and timelike contact numbers are all equal to six, and place the infinite-contact limit in the semi-Riemannian theory of geodesic normal sections and helical immersions.

In 1999, two papers of Benjamini, Lyons, Peres, and Schramm showed that for Bernoulli percolation on unimodular nonamenable quasi-transitive graphs, there are no infinite clusters at criticality. Using the theory of countable Borel equivalence relations and Gaboriau's cluster graphing construction, we give a short proof of this result. We also point out other uses of the cluster graphing construction in the literature, for instance in showing nonuniqueness at $p_u$ in arXiv:1509.00247 [math.GR]. The purpose of this short note is to make folklore proofs known to experts more accessible to the wider community of probabilists and measured group theorists.

Entropy inequalities are fundamental to the well-posedness of hyperbolic conservation laws, providing the essential criterion for selecting the physically admissible solution among infinitely many weak solutions. Chen and Shu [J. Comput. Phys. 345 (2017)] proposed a unified framework for constructing high-order discontinuous Galerkin (DG) methods that satisfy entropy inequalities for any given entropy via specific numerical quadrature; however, their accompanying error analysis was limited to the truncation error level, leaving a critical gap in the rigorous convergence theory for these entropy-stable schemes. This paper closes that gap by establishing rigorous a priori error estimates for semi-discrete entropy-stable DG (ESDG) methods on general unstructured meshes for hyperbolic conservation laws. The analysis applies to both scalar equations and systems, and is built upon a finite-difference-type consistency-stability argument carried out directly at the nodal level. Under a polynomial-reconstruction hypothesis and an $L^\infty$ a priori bound, we prove an $O(h^k)$ error estimate in a quadrature-based norm, which is equivalent to the broken $L^2$ norm on the finite-dimensional reconstruction space. We further extend this framework to the entropy-stable oscillation-free DG (ESOFDG) method introduced by Liu, Lu, and Shu [SIAM J. Sci. Comput. 46 (2024)], demonstrating that the additional damping terms do not degrade the convergence order. Numerical experiments suggest that the observed convergence rates may exceed the theoretical prediction by up to half an order.

We construct a nonlinear electrodynamics (NLED) model for a static magnetic black hole. Rather than imposing a regular center, we require a Maxwell weak-field limit, finite magnetic self-energy, a positive and subluminal electromagnetic cone, the standard energy conditions, and a static exterior that satisfies known sufficient stability criteria. A positive mixture of power kernels restricts the exponent to $1/4<\gamma\leq1/2$. For the minimal two-kernel model, with $\gamma_1=1/3$ and $\gamma_2=1/2$, the field equations admit an exact solution in terms of incomplete beta functions. For the branches with nonnegative Schwarzschild mass parameter $M_0$, the metric function is strictly increasing, so there is at most one positive-radius horizon and no inner Cauchy horizon. We construct the maximal extension and show that the black-hole, horizonless, and critical branches have spacelike, timelike, and null singularities, respectively. The representative black-hole families have positive temperature and negative fixed-charge heat capacity. Electromagnetic waves split into ordinary and extraordinary optical branches. We calculate their photon spheres, shadow radii, instability rates, and illustrative Event Horizon Telescope size bands. We also construct ISCO-truncated thin-disk images. The full images remain close, but jointly normalized residual maps and radial profiles reveal a coherent branch-dependent shift near the lensed inner edge and the critical region. The extraordinary branch moves the critical curve outward and partly compensates the shadow reduction caused by magnetic charge.

We study ribbon concordances to cable knots. We formulate a conjecture predicting that any nontrivial knot admitting a ribbon concordance to a (p,q)-cable must itself be a (p,q)-cable. We prove the conjecture when the knot admitting the ribbon concordance is already a cable of the same companion, is a torus knot, or has genus one. We also verify it for a broad class of target cables. The proofs use a minimum-height invariant defined from the immersed-curve formulation of knot Floer homology. This invariant obstructs ribbon concordances and implies that any nontrivial knot admitting a ribbon concordance to a fibered cable knot is prime. We also establish genus bounds for knots admitting ribbon concordances to cable knots.

Classical likelihood-ratio tests and $\Delta$AIC exacerbate the statistical significance crisis by scaling with sample size, often flagging negligible improvements as highly significant. While causal estimands like the average treatment effect (ATE) quantify practical magnitude, their reliance on the expectation operator ties them to the data's original coordinate scale. Furthermore, existing pseudo-$R^2$ metrics are inadequate: variance-based measures ignore higher-order distributional changes, and current formulations lack invariance to monotone transformations. We resolve these limitations by introducing Entropic Variance (EV) as a rigorous, scale-independent generalization of error variance in ordinary least squares. We define the population EV-based parameter, $\rho^2_V$, which projects unbounded cross-entropy onto a standardized $[0,1]$ scale, and establish that the EV-based $F_\text{V}$ statistic asymptotically follows an $F$-distribution. Building on these distributional properties, we propose two estimators: the empirical population $R^2_{\text{SV}}$ and the out-of-sample predictive $R^2_{\text{SVP}}$. Both are derived by exponentiating per-observation cross-entropy and incorporate a degrees-of-freedom correction for training optimism. Leveraging the $F_\text{V}$-distribution, we derive refined $p$-values and confidence intervals for $\rho^2_V$ without requiring intractable Fisher information matrices. Simulation studies and a Parkinson's disease microbiome application demonstrate the superiority of variable selection via these EV-$R^2$ metrics. Notably, evaluating the $R^2_{\text{SVP}}$ of a LASSO path via data-splitting reduced false discovery rates from 80% to 6% in simulations while fully preserving signal recall.

We propose a Multi-Resonant Double-barrier Quantum-dot-in-a-Well (MR-DWELL) dosimeter with a tunable trade-off between sensitivity and radiation tolerance, realized through graphene-assisted electrostatic screening. The sensing mechanism relies on radiation-induced trapped charge that shifts and broadens resonant tunneling states in the DWELL heterostructure. The~dimensionless capacitance ratio $\eta=C_Q/C_{\mathrm{geo}}$ is introduced as the universal design parameter: it~controls the screening factor $S=\eta/(1+\eta)$ and thereby the trade-off between sensitivity and radiation hardness. For $\eta=2.09$ the effective low-dose shift coefficient is reduced threefold (from $0.14$~to $0.045~\mathrm{meV/mGy}$), while the screened architecture retains $\sim\!83\%$ of the peak tunneling current and a peak-to-valley ratio $\sim\!10$ after $1~\mathrm{MGy}$, thereby preserving the dosimetric sensitivity required for practical readout. The~second-derivative spectroscopy $d^2I/dV^2$ is identified as a robust experimental readout. A~reduced-order Landauer model calibrated against literature data is employed for the screening physics, while a self-consistent NEGF-Poisson-trap framework is formulated for future quantitative validation. Explicit design rules for three operating regimes (high sensitivity, balanced, maximum hardness) are provided, together with a three-population trap model that conceptually enables dose-rate discrimination in the FLASH radiotherapy regime. A~comparative analysis of model vintages and their methodological foundations is included to guide interpretation of numerical predictions.

The equations of fluid motion are known; retaining their physics after removing most degrees of freedom remains a fundamental problem. Closure must preserve the geometry of the dynamics it replaces. We introduce Symplectic Geometric Closure (SGC), a prognostic stochastic field theory carrying Euler's Hamiltonian, area-preserving transport into two-dimensional and $\beta$-plane turbulence. SGC couples resolved vorticity to a hidden stochastic reservoir. The coupling conserves augmented enstrophy while permitting bidirectional energy transfer. This yields a compact random attractor in the hyperviscous Navier–Stokes–$\beta$ realization. Numerically, SGC sustains jets, vortices and filaments in high-Reynolds-number turbulence with high fidelity to filtered DNS at an inertial-range cutoff, preserving geometric identities to machine precision. The induced transport folds, stretches and rearranges vorticity: cross-gradient coupling selects interactions through resolved–reservoir gradient misalignment, retaining geometric selectivity and memory. The same geometry addresses spurious sweeping decorrelation, a longstanding obstacle to Eulerian closure. The interaction vertex excludes uniform translation and is Random Galilean compatible. Eliminating the reservoir generates finite memory, stochastic backscatter and a Dyson–Volterra equation for the dressed propagator. Its one-loop, line-renormalized self-energy reproduces Kraichnan-type Direct-Interaction Approximation architecture with fourth-order infrared suppression of sweeping. Deformation-controlled memory and conservative triad constraints yield the $k^{-5/3}$ inverse-energy and $k^{-3}$ forward-enstrophy cascades under standard assumptions. SGC thus realizes Kraichnan's program through Eulerian field dynamics, opening a route to data-driven closures that learn admissible geometric interactions rather than unconstrained forces.

Problem 3.6 in the $\mathrm{K3}$ problem list of Baykur, Kirby and Ruberman asks whether every cofinal tower \[ M_0\longleftarrow M_1\longleftarrow M_2\longleftarrow\cdots \] of finite covers of a finite-volume hyperbolic $3$-manifold satisfies \[ \lim_{n\to\infty} \frac{\log|\operatorname{Tor} H_1(M_n;\mathbb Z)|} {\operatorname{vol}(M_n)} =\frac{1}{6\pi}. \] We give a negative answer. For every ideal right-angled polyhedron $P_0$, the checkerboard manifold associated to $P_0$ admits a cofinal tower all of whose levels are hyperbolic link complements in $S^3$. Thus $\operatorname{Tor} H_1(M_n;\mathbb Z)=0$ at every level, and the normalized logarithmic homology torsion is identically zero.

In this article, we prove a general cohomology vanishing theorem on arbitrary projective varieties within the framework of multigraded Castelnuovo–Mumford regularity. In particular, we apply this vaishing theorem to prove the vanishing of Koszul cohomology groups $K_{p,q}(X;F,L)$, where $L=B_1^{w_1}\otimes\cdots\otimes B_t^{w_t}$, the line bundles $B_1,\ldots,B_t$ are globally generated, and $F$ is a vector bundle. We also introduce the $K_{p,q}$-hierarchy, providing a unified perspective on the vanishing criteria for Properties $(N_{p})$ and $(M_{q})$ while shedding light on the vanishing of mixed-weight syzygies. These results generalize earlier work in \cite{HeringSchenckSmith}, \cite{GallegoPurnaprajnaII}, and \cite{Basu}. Furthermore, we give a complete description of minimal multigraded regularities of line bundles on arbitrary products of projective spaces. We also recover Green's vanishing theorem for projective space via a direct regularity argument, giving an alternative proof. Finally, we compute the complete graded Betti table of the canonical image of a hyperelliptic curve by combining our vanishing theorem with Green's duality.

We study a discrete representation of contact relations on bounded distributive lattices and some of their reducts. In the course of the paper, we apply ideas and techniques developed by Ivo Düntsch, Dimiter Vakarelov, and Michael Winter, and we show—among others—that contact relations on a distributive join semi-lattice $L$ with the bottom element correspond to reflexive and symmetric relations which are closed in the product of the Stone space of $L$.

Lewis Carroll's Pillow Problem asks for the probability that three points chosen at random in the plane form an obtuse triangle. The question has no answer until the sampling scheme is pinned down, and the cleanest way to pin it down is to put the points on a circle, which gives 3/4. This paper solves the version in which the three vertices lie on three concentric circles of radii $r_1$, $r_2$, and $r_3$. The answer is a sum of three terms, each the probability that a weighted sum of two independent arcsine variables exceeds a threshold, and it collapses to 3/4 when the radii are equal. Reading the formula gives closed forms in two subfamilies, one an arcsine and one the Legendre chi function; a Pythagorean condition deciding which vertex can carry the obtuse angle; and the bound $1/2 \leq P < 1$, with the minimum attained only when one vertex sits at the common center and the other two radii are equal. Conditioning on the radii shows that the same formula is the kernel for every independent rotationally symmetric sampling scheme, so the bound applies to all of them at once and averaging over Rayleigh radii recovers the known Gaussian value of 3/4 in closed form, supplying a link between the circular and Gaussian conventions that the literature records as missing. Simulation confirms the results throughout.

The study of codimension-one holomorphic foliations on the projective plane $\mathbb{CP}^{2}$ with a unique singular point has attracted increasing interest because of its connections with several important problems. In this paper, we focus on holomorphic foliations of degree $d$ having a singular point of maximal multiplicity $d$. We describe explicitly the foliations in this class and characterize those having a unique singular point. We also state some results concerning Hilbert-Mumford stability for these foliations.

We study the finite time blow up phenomenon for the quintic, mass critical gKdV equation. We prove the existence of a class of solutions $U$ with infinite point, finite time blow up behavior, at the particular blow up rate $$\|\partial_x U(t)\|_{L^2} \sim (T-t)^{-\nu}\quad \text{as} \quad t \uparrow T,$$ where $\nu = \frac 12$, $T$ is the blow up time and where the travel speed of the blow up bubble is logarithmic. Therefore, we call this behaviour slowly travelling infinite point blow up. The special blow up rate $\nu =\frac 12$ is a threshold which separates finite and infinite point bubbling. In a previous work arXiv:2511.13538, the author constructed other infinite point blow up solutions for the continuum of rates $\nu\in(\frac 12,1)$, using polynomial tails in the space variable and extending the results in arXiv:1209.2510, restricted to $\nu> \frac{11}{13}$. However, that work suggested a change of the tail for the threshold case. In the present paper, we consider an exponentially decaying tail on the right in space. As in arXiv:2511.13538, the initial data can be taken arbitrarily close to the ground state in $H^1$. From a technical perspective, in addition to the change of tail, we have to adapt the energy-virial functional to the presence of the exponential tail, by modifying a scaling term used to control the right-hand side of the blow up solution.

We develop a one-shot (finite-blocklength) channel-coding framework based on the pairwise error probability (PEP) of a decoder with randomized tie-breaking. The tie-breaking rule yields a probability-integral-transform identity: the induced error spectrum describes both random-coding achievability and exact fixed-code converse statements, for an arbitrary decoding metric. We derive two variational identities for metric-weighted tail functionals of the PEP, one through the Neyman-Pearson $\beta$-functional and one through a reverse channel, valid for an arbitrary metric. Under matched maximum-likelihood decoding they specialize to representations of the spectrum itself, which is then jointly convex in the testing level and the input prior; combined with the reverse-channel representation, this gives a linear program for the prior-optimized minimax meta-converse – finite-dimensional in general and, for memoryless channels with fixed alphabets, of size polynomial in the blocklength after a type reduction. Prior optimization of the random-coding bound is formulated as a concave program over input distributions with an explicit gradient, solved by a direct first-order method. The framework recovers several classical one-shot bounds, including the random-coding union bound and minimax meta-converse of Polyanskiy-Poor-Verdu, the information-spectrum bounds of Han-Verdu, and the linear-programming converse of Matthews. Numerical examples on the AWGN and binary Z-channels illustrate the achievability-converse comparison and the effect of prior optimization.

We prove universality of limiting local eigenvalue statistics for random matrices over $\mathbb{Z}_p$. In previous work of the author and Van Peski (arXiv:2601.06283), the limiting eigenvalue correlation functions of additive Haar random matrices were studied in arbitrary finite extensions of $\mathbb{Q}_p$. The same Haar random matrix model plays a central role in the Ellenberg-Jain-Venkatesh heuristic for zeros of $p$-adic $L$-functions. We show that its limiting local eigenvalue statistics are unchanged for a broad class of random matrices with independent entries satisfying a mild non-concentration condition. Thus the random matrix predictions underlying the Ellenberg-Jain-Venkatesh heuristic are not artifacts of the particular Haar ensemble, but instead reflect universal limiting eigenvalue statistics. In this sense, our results provide additional theoretical support for the robustness of their random matrix heuristic. Our proof is based on a new framework, which we call the resultant distribution method. The method recovers limiting laws and root statistics of $p$-adic polynomials from the distributions of their resultant valuations against fixed test polynomials, together with suitable degree estimates. As a second application, we consider random $p$-adic polynomials with independent coefficients satisfying a mild non-concentration condition. Caruso (arXiv:2110.03942) determined the joint root correlation functions of the Haar coefficient model over finite extensions of $\mathbb{Q}_p$. We prove that, for roots of absolute value one, these limiting correlation functions are universal and persist for a broad class of independent coefficient distributions.

This paper studies the Shannon capacity of lexicographic products of finite simple graphs, together with the Lovász theta function and the fractional Haemers number. The Shannon capacity is proved to be supermultiplicative under lexicographic products in either order, and these products are compared with the strong product. We explicitly construct three countably infinite families of lexicographic powers based on the Schläfli graph, the McLaughlin graph, and its second subconstituent; in each family, pairing each member with its complement yields strict supermultiplicativity and arbitrarily large multiplicative gaps. Bounds and exact-capacity criteria for lexicographic products are derived, and the resulting upper bounds are shown to be incomparable. The capacities of lexicographic products involving Kneser graphs, their complements, and $q$-analogues of Kneser graphs are determined. It is also shown that a lexicographic product with a complete outer factor preserves the Shannon capacity of an arbitrary inner factor. The capacities of iterated lexicographic powers are determined, including those of self-complementary graphs that are vertex-transitive or strongly regular. Elementary, self-contained proofs are also given for three known results: the multiplicativity of the Lovász theta function and the fractional Haemers number under lexicographic products, and the equality of the fractional and ordinary Lovász theta functions. Finally, an open problem concerning the Shannon capacities of lexicographic and strong products is posed.

The quantum lattice Boltzmann scheme of Succi and Dellar advances a four-component Dirac spinor on a lattice by a fixed sequence of local, exactly norm-preserving operations: a basis rotation, a collision, a streaming shift, and the inverse rotation. While this correspondence has been noted qualitatively, it has remained unclear whether it can be realized exactly and unitarily as a gate-model quantum circuit. This unitarity is a structural property of the scheme, which should, in principle, enable an operator-by-operator port onto quantum hardware. We give a gate-level construction of every operation of the three-dimensional Dirac QLB scheme: the fixed rotation gates, the collision gate, the streaming shift as a controlled increment on a position register, the position-dependent potential as a phase oracle, and periodic and reflecting boundary conditions as unitary circuits. Together, these constructions provide the first complete demonstration that the scheme is exactly representable in the gate model without any approximation beyond the classical method itself. We then compose them into single-axis, two- and three-dimensional time steps. On a state-vector emulator the resulting circuits reproduce the classical QLB solver to machine precision, thus verifying our quantum implementation. Our aim is representability and correctness rather than computational speedup; questions of state preparation, asymptotic cost, and hardware noise are identified as open avenues for future work. Appendices restore physical units, derive the scheme's Zitterbewegung in closed form and compare it with a trapped-ion simulation, and execute a reduced two-site instance on a Rigetti superconducting processor, where choosing the qubits from the live calibration lowers the state-preparation error. All operators, circuits, and figures are reproducible from the open-source quantumKineticMethods library.

We consider the ruin problem for an insurance company investing its whole reserve in a risky asset whose parameters depend on a Markov random environment. Using the Green's function method we prove $C^2$-smoothness of the survival probability for annuity payments under minimal assumptions-the jump distribution need only be a probability measure on the positive half-line. For two-sided jumps smoothness holds whenever, in each regime, the jump distribution has no atoms on the negative half-line or the survival probability vanishes at the origin. This condition is sharp: an isolated atom combined with a positive value at the origin makes the second derivative discontinuous, and the size of the discontinuity is computed explicitly.

Machine learning and optimization have advanced together, with practical demands motivating new theory and theoretical breakthroughs enabling new applications. Modern large-scale training relies on classical optimization principles, but the constraints of distributed systems require these foundations to be reconsidered. This thesis addresses seven challenges at the intersection of theory and practice, focusing on key bottlenecks in federated learning and distributed optimization. First, we introduce ProxSkip and prove that local gradient steps can accelerate communication, providing a theoretical foundation for this widely used heuristic. Second, we develop Variance Reduced ProxSkip, which eliminates the neighborhood error of stochastic local updates while balancing communication and local computation. Third, we show that local steps retain their communication acceleration under partial client participation. Fourth, we prove that server-side stepsizes and sampling without replacement improve convergence in heterogeneous settings. Fifth, for Random Reshuffling, we demonstrate that compressing gradient differences rather than gradients yields better theoretical and practical performance. Sixth, we establish that Byzantine robustness and partial participation can be achieved simultaneously using gradient-difference clipping. Finally, we develop the first theoretical framework for low-rank adaptation based on randomized asymmetric chains, providing new insights into fine-tuning large models. Across these contributions, we introduce novel algorithmic frameworks, establish sharp guarantees under realistic assumptions, and support the theory with numerical experiments.