Papers

Let M be an imaginary quadratic field and A be an abelian variety of odd dimension g defined over Q for which there is a unitary algebra embedding of M in End^0(A_M) of signature ((g+1)/2,(g-1)/2). Extending a result by Fité and Goodman, we show the existence of an elliptic curve E over Q with complex multiplication by M such that H_et^1(E_Qbar, Q_l) is a G_Q-subrepresentation of H_et^g ( A_Qbar, Q_l((g-1)/2) ). The main results of the article consist on explicit descriptions of E for A in two different families. First, when A is the Jacobian of the curve y^2 = xh(x^2), where h in Z[x] is a degree g monic separable polynomial with nonzero h(0), we show that E is given by y^2 = x^3 + h(0)x. Second, when A is the Jacobian of y^3=f(x), where h in Z[x] is a degree 4 monic separable polynomial, we show that E is given by y^2 = x^3 + 16 disc(f). This latter result builds on work by Laga and Shnidman in the bielliptic case, of which we give an alternative proof by means of a new p-adic method. As an application, we derive a practical average polynomial time algorithm to compute the Zeta functions of the reductions of a Picard curve.

This paper studies several first-order methods for low-rank matrix minimization in the Burer–Monteiro form, where the target matrix is represented as the product of a nonnegative factor matrix and its transpose. Such a factorization destroys convexity of the objective function and global Lipschitz continuity of the gradient. We prove that for a quadratic objective function that is strongly convex only on the subspace of symmetric matrices, the Burer-Monteiro factorized objective possesses generalized smoothness with a fractional exponent 2/3. For a quadratic objective, the generalized smoothness property requires positive definiteness of the operator A only on the subspace of symmetric matrices, which relaxes the assumptions of Dragomir et al. On this basis, three methods are proposed: an adaptive gradient descent method with Armijo backtracking; an unconstrained SafeAdam method combining the Adam direction with a descent-angle certificate and a fallback to the normalized gradient; and a certified projected version for the nonnegativity constraint. Global convergence and explicit convergence rates for the proposed methods have been proved. The obtained convergence rate is better than the convergence rate of ordinary unaccelerated gradient descent in the case of a sufficiently large gradient. In the small-gradient regime, the convergence guarantee is of square-root type, and the corresponding complexity scales as the inverse square of the desired accuracy. In the large-gradient regime, the convergence guarantee is of three-quarters type, and the corresponding complexity is proportional to the inverse of the accuracy raised to the power of 4/3. Experiments on SymNMF using the TDT2, Reuters and CBCL datasets, which are standard benchmarks for document and image clustering, show that the methods match or outperform Dyn-NoLips without tuning the relative-smoothness kernel parameters.

Compartment models of infectious diseases are often linear in coefficients that are physical parameters or products or ratios of them. We exploit this with closed-form least squares and equip the estimates with distribution-free uncertainty. Physics-informed regression (PIR) fits the parameters to finite-difference derivatives of the data; we introduce physics-informed weak regression (PIWR), which integrates the model in time instead, needs no derivatives of the data, estimates the initial state jointly and is high-order accurate. To quantify uncertainty we wrap both estimators in split conformal prediction, with a calibration population generated by the physics-based model itself, giving simultaneous intervals for all parameters with a finite-sample coverage guarantee. A residual-adaptive factor, computed without ground truth, scales the intervals to each data set (PIWR-RACC). On the eleven benchmark models of disease-informed neural networks (DINNs), PIWR or PIR-FD6 is one to four orders of magnitude more accurate than the reported DINN results wherever the sampling resolves the dynamics, in about a millisecond per fit instead of minutes to hours of training, and the 90% intervals achieve 87-92% coverage in all 88 settings. On the S3I3R model of the original PIR study, PIWR lowers the reported PIR errors by about three orders of magnitude. On real data from the 1978 boarding-school influenza outbreak, a variable-projection variant gives calibrated 95% intervals that overlap those of a Bayesian analysis. On a time-varying transmission benchmark, a B-spline expansion of beta(t) keeps PIR and PIWR linear; they are more accurate than physics-informed neural networks in three of four cases and comparable in the noisiest, with simultaneous bands for beta(t). The complete conformal calibration takes less than two minutes for every benchmark.

Pediatric vaccine procurement for low- and lower-middle-income countries (LICs and LMICs) requires long-term coordination across heterogeneous vaccine markets, competing stakeholder objectives, and uncertain demand, yet tenders are currently scheduled as a series of independent renewals. We extend the Hypothetically Coordinated Vaccine Market (HCVM) framework to schedule multi-year tenders with flexible lengths and all-unit quantity discounts, and compare three objectives: a balanced coordinatingentity model (UG), a social-benefit model that minimizes missed doses (SB), and a producer profit-maximizing model (MP). Using UNICEF ten-year demand forecasts and 30 demand replications across six discount scenarios, we find that the procurement objective, not demand variability, drives cost-to-tender and schedule complexity. Quantity discounts favor large producers, concentrating the market and increasing supply chain vulnerability. Discount effectiveness depends on market maturity rather than valency: pentavalent and hexavalent vaccines are already near their price floor, and among monovalent vaccines only rotavirus responds to discounts, whereas HPV and PCV remain too expensive to compete. Removing pentavalent and hexavalent vaccines from the procurement pool sharply increases missed doses. These results give planners practical guidance on objective selection, discount design, and protecting access to combination vaccines.

Because its binary optimization landscape contains exponentially many local optima, MAX-CUT is highly sensitive to initialization. We study Particle Polynomial Relaxation (PPR), a moment-based nonconvex continuous reformulation possessing an advantageous optimization landscape. We warm-start PPR with Combinatorial Multigrid (CMG), a spectral graph-coarsening method that produces structure-aware initial solutions in near-linear time without training data. On t2g20 ($n=400$), benchmark instance with known exact optimum, the combined CMG+PPR pipeline reaches $97.5\%$ of optimum on average and $98.4\%$ on the best run. An extended variant with discrete local-search refinement reaches the published optimum on all sixty Biq Mac instances evaluated by a recent neural branch-and-bound method. Using a sparse CSR representation, the same approach scales to G-set instances with up to $20{,}000$ vertices reducing memory requirements by up to $3333\times$ while maintaining a consistent improvement over the CMG warm start. Gains are largest on sparse weighted graphs; on dense graphs we complement the pipeline with spectral sparsification.

We study online task assignment with reusable resources. This problem arises in peer reviewer assignment, skill-based call routing, rideshare matching, emergency dispatch, and crowdsourcing: Tasks arrive according to a geometric process and must be immediately assigned to servers that become temporarily busy, yielding rewards that depend on the task-server pair. The goal is to maximize long-run average reward. We cast the problem as a contextual restless multi-armed bandit (CRMAB). We derive an occupancy-measure LP (OCC-LP) that upper-bounds the reward of any work-conserving policy. From its dual we obtain Lag, a closed-form index policy. We introduce a scaling regime in which the number of copies of each server and the number of arrivals per step grow together. This regime yields the first deterministic fluid limit for a contextual bandit. In it, Lag is asymptotically optimal. Finally, we evaluate our algorithms on peer review datasets and call center routing. The Lagrangian index achieves 83-99% of the LP upper bound across all load levels. Moreover, the Lagrangian index policy significantly outperforms the natural greedy assignment scheme, improving reward by up to 12.3% on journal data and 36% on structured synthetic instances.

We study reversible Higgs bundles on a compact Riemann surface $\Sigma$ of genus $g\geq 2$, that is, $G$-Higgs bundles $(E,\Phi)$ with $(E,\Phi)\cong(E,-\Phi)$. For a complex reductive group $G$ realized as a self-adjoint subgroup of $GL(n,\mathbb{C})$, we show that if $(E,\Phi)$ corresponds to a representation $\rho$ of $\pi_1(\Sigma)$, then $(E,-\Phi)$ corresponds to $(\overline{\rho}^{\,t})^{-1}$. As a consequence, the moduli space of reversible Higgs bundles is the fixed locus of an involution of the character variety and carries the structure of a real affine algebraic set. For a connected semisimple group and every topological type, we prove that the reversible locus in the smooth moduli space is a $(B,A,A)$-brane. We then describe reversibility through spectral data: in even rank, for a smooth spectral curve, reversible Higgs bundles correspond to line bundles invariant under the involution $\eta\mapsto-\eta$; in odd rank, under natural genericity assumptions for which the spectral curve is nodal, we give a gluing criterion together with an explicit polystability condition. Finally, for $SL(2n,\mathbb{C})$ we construct a relative Fourier–Mukai transform over the regular locus of the Hitchin base and prove that it maps the structure sheaf of the reversible locus to an object whose support is exactly the regular degree-zero $PSp(2n,\mathbb{C})$-locus in the Langlands dual $PGL(2n,\mathbb{C})$-moduli space. This support is of type $(B,B,B)$.

Motivated by the standard 2-dimensional 3-segment snake model, leading to the famous $(2,3,5)$ geometry, we discuss a series of 3-dimensional models of similar snakes. In particular, we seek for rules allowing to steer them exclusively by the two joints, which expects 4-dimensional control freedom. This leads to interesting $(4,9)$ and $(4,10)$ models. Finally, we discuss modifications of the previous snakes which lead to $(4,8)$ models. All of the models open further questions and should be a good baseline for further research.

Chance-constrained stochastic programs (CCSPs) require solutions that satisfy uncertain constraints with high probability. Standard approaches either embed scenarios directly into optimization by considering sample average approximation (SAA)-based reformulations, growing prohibitively expensive and often failing to find feasible solutions; or employ conservative reformulations that guarantee feasibility but sacrifice objective quality by design. We propose ACCESS (Adaptive Chance-Constrained Exploration via Structured Sampling), a structured sampling framework that decouples optimization from feasibility evaluation. First, high-quality candidate solutions are generated by solving many inexpensive single-scenario subproblems, avoiding sampling directly in the constrained high-dimensional solution space. Second, feasibility is assessed separately via large-scale batched simulation over a massive number of scenarios, which is naturally amenable to GPU parallelism. That reduces the search from a high-dimensional solution space to a two-dimensional parameter space controlling conservatism and scenario diversity; while concentrating on candidates near the feasibility boundary, where high-quality feasible solutions lie. We also introduce a variant, ACCESS($\delta$), which provides certified feasibility guarantees with probability $1-\delta$. Experiments across three benchmark classes and five scenario distributions show that ACCESS matches or exceeds the best objective found by any method in nearly every setting, while ACCESS($\delta$) is empirically feasible on every instance. The conservative reformulations remain several percent worse in objective, and SAA is frequently infeasible while taking an order of magnitude longer on the harder benchmarks.

Let $\Gamma$ be a countable discrete group acting on a tracial von Neumann algebra $(Q,\tau_Q)$ in a trace-preserving way. We study invariant von Neumann subalgebras of $Q\rtimes\Gamma$ and of $L(\Gamma)\bar\otimes Q$, where the latter is equipped with the diagonal action. We show that, under the pointwise weak mixing assumption, an invariant subalgebra whose image under the canonical conditional expectation onto $Q$ is scalar is necessarily contained in $L(\Gamma)$. Further, under relative weak mixing and central freeness assumptions, we determine the structure of invariant subalgebras of the crossed product with non-trivial intersection with $Q$. For p.m.p. actions of groups with trivial amenable radical and the invariant subalgebras rigidity property, pointwise relative weak mixing gives a complete classification of invariant subalgebras in the tensor product case. We construct essentially free, prime, pointwise weakly mixing p.m.p. actions of $\mathbb{F}_n$ for every $2\leq n\leq\infty$, as well as non-prime examples with explicitly described factors.

We analyze exponential mixing for analytic observables and show that mixing at a double exponential rate occurs for genuinely nonlinear systems. Neither its presence nor its sharp exponent is stable under arbitrarily small area-preserving perturbations in the real-analytic topology. We construct two families of area-preserving Anosov Blaschke product diffeomorphisms that converge in the real-analytic topology to the same hyperbolic toral automorphism. For nonzero parameters, both deformations are topologically conjugate to the linear model but not $C^1$-conjugate. For all nonzero parameter values, the maps in the first family are mixing at a double exponential rate with the same sharp exponent, strictly smaller than that of the linear map. The sharp exponent is therefore not lower semicontinuous at the linear map. In the second family, mixing at a double exponential rate is lost altogether, although the maps remain exponentially mixing. We also establish double exponential bounds for every ergodic toral endomorphism and uniform bounds of the same type for totally ergodic higher-rank toral actions. Our method combines the Fourier escape argument with Koopman operator estimates from a Hilbert space of analytic functions to its dual space of hyperfunctions.

An internal partition of a graph is a partition of its vertex set into two nonempty parts such that every vertex has at least half of its neighbors in its own part. DeVos asked whether, for every fixed positive integer $r$, all but finitely many $r$-regular graphs admit an internal partition. This was previously known to hold only for $r\in\{3,4,6\}$. In this paper, we confirm this for $r\in\{5,8\}$ by proving that every $r$-regular graph on at least $10^7$ vertices has an internal partition.

This is my written Laudatio for Graeme Segal on receiving the Chern Medal at the 2026 International Congress of Mathematicians. I make some general remarks about Segal's qualities as a mathematician. Then I discuss the origins of his work on classifying spaces of categories and Gamma-spaces. In the last section I describe his axiom system for Wick-rotated quantum field theory and some of his subsequent work in the area. Of course, this is necessarily only a small slice of Segal's total research output.

For integers $\alpha,d \geqslant1$, we determine the best constants in the inequalities $||(a_j)_{j=1}^{d}||_{\ell^1} \leqslant C_{\alpha,d} ||((\Delta^\alpha a)_j)_{j=1}^{d+\alpha}||_{\ell^2},$ where $\Delta$ is the difference operator. We find that these constants correspond to the $\ell^2$ norm of the discrete Chebyshev polynomials, and show that the extremal sequences may be obtained by appropriate sampling of the discrete Landau polynomials. From beginning to end, this project was carried out without use of AI.

In this paper, we investigate the exponential stability of coupled slow–fast stochastic systems with a small delay. Under an averaged Lyapunov condition, we establish practical $p$th-moment exponential stability for $0<p<2$ using a cutoff-function approach to handle singular Lyapunov derivatives near the origin. Exponential stability in the $p$th moment and almost surely is further obtained through two distinct mechanisms: homogeneous dissipation of the fast dynamics or vanishing sensitivity of the slow coefficients to the fast state near the origin. We extend these results to locally Lipschitz systems, establishing exponential stability with arbitrarily high probability for bounded sets of initial data, thereby broadening the applicability of stability analysis for stochastic systems with delayed feedback. Numerical examples illustrate our results.

Symmetries of a quantum field theory or a quantum lattice model are expected to be described not by a group but by a homotopy type, whose homotopy groups are the higher symmetry groups and whose Postnikov invariants encode 't Hooft anomalies. We propose a homotopy-theoretic definition of 't Hooft anomalies based on the structure of localized symmetries in the operator-algebraic approach. The input is a crossed precosheaf: a collection of groups attached to regions of space, equipped with pairings recording that the commutator of two transformations localized in two regions is localized in their intersection. To a crossed precosheaf and a finite family of regions we attach a pointed connected homotopy type, the Cech-Loday classifying space. As the family of regions ranges over finite covers of space, the classifying spaces form a pro-object in the homotopy category. There is a canonical map from the ordinary classifying space of the symmetry group to this pro-type. Composing this map with a group action gives an anomaly whose nontriviality obstructs gauging. In favorable cases, the Cech-Loday space is essentially independent of the cover. In such cases, the homotopy fiber of the canonical map is a candidate for the classifying space of the higher group of symmetries. We give examples of crossed precosheaves arising from gauge transformations in continuum field theory and from quantum cellular automata and finite-depth circuits. For quantum spin systems in one and two dimensions, our construction recovers the Else-Nayak index and its generalizations to two dimensions.

We ask when the law of an infinite exchangeable sequence is uniquely determined by a finite-dimensional marginal. For mixtures of a probability kernel, a weak localization condition makes three coordinates determine the directing law. Completeness alone makes four coordinates sufficient. Neither order can be lowered in general. For exponential kernels with a nonnegative sufficient statistic, a Müntz-Szász support condition implies three coordinates suffice. In the scalar case, this result is sharp. We also characterize when completeness holds simultaneously for all scalar mixing laws by a weighted Müntz density condition and show that statistic support alone need not decide completeness. These methods yield sharp finite-dimensional refinements of Freedman's theorem on rotatable sequences, namely that spherical symmetry of three coordinates makes an infinite exchangeable sequence rotatable while symmetry of the first two coordinates is in general insufficient. Finally, for energy-radial densities we give a complete-monotonicity criterion for infinite extendibility and determine the exact uniqueness thresholds for $\ell_q$-norm exponential laws.

We construct giant graviton expansions of the thermal and $(-1)^F$-graded partition functions of $\mathcal{N}=4$ and $\mathcal{N}=1$ super Yang-Mills theories at zero 't Hooft coupling. These observables are generically non-BPS and do not admit a direct bulk interpretation in a supergravity regime. Instead, we exploit the corresponding self-dual theories, which arise as chiral zero-coupling limits and admit twistor string holographic duals. We apply a quiver generalization of Murthy's expansion motivated by twistor strings. Notably, we obtain expansions at both finite and infinite $N$. We reproduce the terms in these expansions from bulk closed string and D-brane sectors, identifying the nonperturbative corrections with D-instanton/anti-D-instanton pairs. At infinite $N$, resumming the D-instanton sectors reproduces the Hagedorn temperatures of the free gauge theories, providing a microscopic bulk origin for their Hagedorn singularities.

Chiral topological order is incompatible with strictly two-dimensional local commuting-projector Hamiltonians on finite-dimensional tensor-product Hilbert spaces. Invertible subalgebras (ISAs) provide an alternative framework. We construct an ISA supporting chiral semions, together with its anti-semion commutant, on a square lattice with one qubit per edge. We give explicit non-Pauli generators for both algebras and prove that together they generate all local operators on the Hilbert space. Starting from the $\mathbb{Z}_4$ toric code, we condense the boson $e^2m^2$ and identify the semion and anti-semion algebras. We then decompose each $\mathbb{Z}_4$ qudit into two qubits and use a finite-depth circuit to disentangle one qubit per edge fixed by the condensation constraints. Pumping the semion ISA along the $z$ direction defines an explicit three-dimensional non-Clifford quantum cellular automaton (QCA). We show that pumping eight times is equivalent to a translation followed by a finite-depth quantum circuit with ancillary qubits. We also give an explicit finite-depth circuit relating this construction to the semion QCA previously obtained from topological quantum field theory. A similar condensation construction yields a $U(1)_4$ ISA on a square lattice. Together with the semion and earlier odd-prime ISAs, finite stacking realizes every pointed Witt class of bosonic Abelian anyon theories. Finally, we extend both constructions to all spatial dimensions $d=4k{-}2$, with one qubit or $\mathbb{Z}_4$ qudit on each $(2k{-}1)$-cell for the semion or $U(1)_4$ ISA, respectively.

Let $X:M^3\to\R^4$ be a complete, connected self-shrinker without boundary, and suppose that the squared norm $S$ of its second fundamental form is constant. We prove that $S=0$ or $S=1$. Accordingly, the image is a hyperplane through the origin or a generalized cylinder $\mathbb S^k(\sqrt{k})\times\R^{3-k}$, $1\le k\le3$, and the immersion is a Riemannian covering onto its image.

For every fixed integer $r\ge3$, we prove that the Turán density of every $r$-uniform hypergraph $F$ with $m\ge1$ edges satisfies $\pi(F)\le1-c_rm^{-(r-1)/r}$, where $c_r>0$ depends only on $r$. This improves Sidorenko's general bound, whose gap from $1$ is of order $m^{-1}$. Complete $r$-uniform hypergraphs show that the exponent $(r-1)/r$ is optimal. The proof combines a potential argument for partial embeddings with the Lovász Local Lemma.

In this paper, the relationship is studied between the Lehmer index and the reciprocal sum of the prime factors of a Lehmer number. Some upper bounds and a lower bound are proposed for the Lehmer index. Another constraint on Lehmer index is also proposed if a Lehmer number has 3 as its prime factor.

Generalized Pellet's theorem is extended to analytic matrix-valued functions using Descartes' rule of signs for infinite series. Cauchy's estimate is incorporated to allow numerical computation of the bounds. To address the issue that Pellet's theorem and its generalizations may not always guarantee the existence of bounds, for the case of matrix size one, we employ the Dandelin- Lobachevsky-Graeffe method, which ensures the existence of the bounds for the roots and further improves them as the order of the transform increases. Without the need for any a priori information on the function, these theorems allow us to compute mathematically verified bounds for the roots in a certain region using an interval arithmetic implementation of the function. Numerical examples and an implementation are provided to illustrate the effectiveness of the proposed method.

Motivated by the 3d-3d correspondence, we introduce an index $\mathcal{I}_g^\phi$ for a mutation loop $\phi$ of a quiver and an integer $g \geq 0$. We prove its integrality $\mathcal{I}_g^\phi \in \mathbb{Z}$ under certain conditions, which extends the integrality conjecture of Gang–Kim–Yoon for $3$-manifolds to the broader framework of mutation loops in cluster algebras. Our index also has an interpretation as the genus-$g$ partition function of a $(1+1)$-dimensional TQFT. We provide explicit computations for mutation loops of period one. In particular, in the case of the generalized Kronecker quivers, we give an explicit formula expressing the generating function of the indices for all genera $g$ as a ratio of polynomials. Altogether, we provide a new insight to the interplay between the geometry of $3$-manifolds and cluster algebra.

This paper studies Wong–Zakai convergence and Fourier neural operator learning for stochastic Volterra integral equations (SVIEs) describing random systems with memory. A Wong–Zakai theorem establishes convergence of pathwise solutions driven by piecewise linear approximations of Brownian motion to the corresponding SVIE solution. This convergence provides a foundation for using mirror-padded Fourier neural operators to learn the solution operator mapping initial conditions and driving noise to entire solution trajectories. Numerical experiments demonstrate competitive prediction accuracy and generalization to temporal grids finer than those used for training, while assessing the influence of padding and computational cost. As a follow-up to \citet{LKP:26}, this study extends their operator-learning framework to SVIEs through separate convergence and approximation analyses tailored to convolutional memory.

We study PAC learning of binary classifiers when labels arrive after random delays and are censored beyond an observation window $W$. How many samples does this observation model require? We propose a distribution-dependent answer, the Delay Complexity Dimension (DCD). The DCD combines the VC dimension with the exact label-information rate of the observation channel. Every learner needs $\Omega(\mathrm{DCD}/\varepsilon^2)$ samples. An oracle pseudo-label ERM matches this rate on the instance family of the lower bound, up to constants and logarithmic factors. On that family the DCD is within a constant factor of the classical censored-ERM complexity, and whether the DCD rate is achievable where the two separate remains open. The theory also yields a quantization scheme under which delay information lowers the residual label variance, an expedition policy with a $(1-1/e)$ guarantee for its version-space objective, and a phenomenon with no analogue in the no-delay setting. Under a fixed collection horizon, a moderate window can strictly beat immediate labels. Experiments on synthetic data and the Criteo conversion logs support these predictions. A budget-matched test of a companion labeler-diversity prediction is reported as a negative result.

This paper presents a conceptual, cognitive-constructivist model for teaching the half-derivative. The model starts from a student's question and frames it as the problem of generalizing the familiar operation of differentiation. On a suitably chosen space of monomials, under the assumptions of linearity and of lowering the exponent by one half, the requirement that the operator applied twice give the ordinary derivative produces a recurrence for the coefficients. The freedom of normalization becomes visible, and an additional convention yields a formula compatible with the monomial rule of the Riemann-Liouville half-derivative. The domain of the operator and its relation to the Riemann-Liouville and Caputo operators are treated separately. The mathematical results rest on the author's earlier preprints; the contribution of this paper is their transformation into an instructional pathway organized in phases, with diagnostic questions, an analysis of typical errors, and comparative problems. The model is a design proposal: its effectiveness has not yet been established by systematic empirical research. The interactive Socratic implementation and the detailed diagnostics are the subject of Part II.

In this paper, we investigate the impact of antenna beam pattern models on the beam placement problem in Low Earth Orbit (LEO) satellite networks. Unlike conventional beam placement studies that often rely on a fixed antenna pattern or a predefined half-power beamwidth, this work compares four representative gain models, namely the Bessel $J_1$ model, the combined Bessel $J_1+J_3$ model, the Gaussian model, and a 3GPP-TR 37.842 model. For each antenna pattern, the corresponding half-power angle is obtained and used to construct a user compatibility graph. The beam placement problem is then transformed into a Minimum Clique Cover problem and solved using a graph coloring-based algorithm. Numerical results compare the considered antenna models in terms of the number of active beams, load balancing, statistical channel gain-to-noise ratio (SCGNR), and signal-to-interference-plus-noise ratio (SINR). We analyze the impact of increasing the number of active beams on SINR-based spectral efficiency to explain the importance of carefully controlling the number of active beams. The results show that antenna beam pattern models affect both beam resource usage and link quality, highlighting the need for antenna-pattern-aware beam placement in LEO satellite networks.

We introduce a spectrum of dimensions of measures, the packing intermediate dimensions $\dim_{P,\theta}\mu$, $\theta\in(0,1]$, of a Borel probability measure $\mu$ on $\mathbb{R}^n$ and the quasi-Hausdorff dimension $\dim_{qH}\mu$. We show that this spectrum encodes the information on $\mu$ that governs the packing dimension of its projections and Hölder images. First, $\dim_{qH}\mu$ is the exact threshold for the packing dimension profiles of Falconer and Howroyd: for every real $0<s\leq n$, $\dim_P^s\mu=s$ if and only if $s\leq\dim_{qH}\mu$. Second, we prove a lower bound for the profiles in terms of $\dim_H\mu$, $\dim_{P,\theta}\mu$ and the Assouad dimension of $\operatorname{supp}\mu$, which improves the bound of Falconer and Mattila, the best previously known general lower bound for the packing dimension of projected measures. The improvement is strict whenever $\dim_H\mu<\dim_P\mu$ and $\dim_A(\operatorname{supp}\mu)<n$, and when $\dim_A(\operatorname{supp}\mu)=n$ it occurs exactly when $\dim_{P,\theta}\mu$ exceeds an explicit universal lower bound for some $\theta$ in an explicit range. We present examples of measures with $\dim_H\mu<\dim_P\mu<\dim_A(\operatorname{supp}\mu)=n$ for which our bound gives the exact value $\dim_P\mu_V=\min\{\dim_P\mu,m\}$ for almost every $m$-dimensional subspace $V$, whereas the bound of Falconer and Mattila is strictly smaller. As applications, for a large family of random $\alpha$-Hölder maps $f_\omega:\mathbb{R}^n\to\mathbb{R}^m$, we prove that, for compactly supported $\mu$, $\dim_P\mu_{f_\omega}=m$ almost surely if and only if $\alpha m\leq\dim_{qH}\mu$, and that the packing analogue of Kahane's formula holds for fractional Brownian motions if and only if $\dim_{qH}\mu=\dim_P\mu$. Analogous results are obtained for analytic sets.

We study the distribution and oscillatory properties of prime-supported Fourier coefficients attached to a Hecke–Maass form for $\mathrm{SL}(3, \mathbb{Z})$ twisted by additive characters. For linear phases involving irrationals of finite type, we construct an automorphic variant of Vaughan's identity and exploit bilinear cancellation to obtain an unconditional power-saving estimate. For non-linear phases $e(\alpha n^\theta)$ with $0 < \theta < 1$, we refine the Type~II bilinear forms via Weyl differencing and modern exponent pair methods. Combined with the analysis of untwisted $\mathrm{GL}(3)$ sums, this framework breaks the $2/3$-barrier of Jiang and Lü and widens the admissible range of non-trivial cancellation.