共找到 20 条结果
If $p$ is prime, a sequence of prime numbers $\{p, 2p+1, 4p+3,...,2^{n-1}(p+1)-1\}$ is called a Cunningham chain. These are finite sequences of prime numbers, for which each element but the last is a Sophie Germain prime. It is conjectured that there are arbitrarily large such Cunningham chains, and these chains form an essential part of the study of Sophie Germain primes. In this paper, we aim to significantly improve existing bounds for the length of Cunningham chains by considering their behaviour in the framework of what we will define as $\textit{rogueness}$.
Let $p$ be a prime number. A chain $\{p,2p+1,4p+3,\cdots,(p+1)2^{l(p)-1}-1\}$ is called the Cunningham chain generated by $p$ if all elements are prime number and $(p+1)2^{l(p)}-1$ is composite. Then $l(p)$ is called the length of the Cunningham chain. It is conjectured by Bateman and Horn in 1962 that the number of prime $p\leq N$ such that $l(p)\geq k$ is asymptotically equal to $B_k N/(\log N)^k$ with a real $B_k>0$ for all natural number $k$. This suggests that $l(p)=Ω(\log p/\log\log p)$. However, so far no good estimation is known. It has not even been proven whether $\limsup_{p\to\infty} l(p)$ is infinite or not. All we know is that $l(p)=5$ if $p=2$ and $l(p)<p$ for odd $p$ by Fermat's little theorem. Let $α\geq3$ be an integer. In this article, a generalized Fibonacci sequence $\mathcal{F}_α=\{F_n\}_{n=0}^\infty$ is defined as $F_0=0,F_1=1, F_{n+2}=αF_{n+1}+F_n (n\geq0)$, and ${}_{\mathcal{F}_α}σ(n)=\sum_{d\mid n, 0<d\in\mathcal{F}_α}d$ is called a divisor function on $\mathcal{F}_α$. Then we obtain an interesting relation between the iteration of ${}_{\mathcal{F}_α}σ$ and the length of Cunningham chains. For two primes $p$ and $q$, the fact $p=2q+1$ or $2q-1$ is
Let $ε\in \{-1,1\}$. A sequence of prime numbers $p_1, p_2, p_3, ...$, such that $p_i=2p_{i-1}+ε$ for all $i$, is called a {\it Cunningham chain} of the first or second kind, depending on whether $ε=1$ or -1 respectively. If $k$ is the smallest positive integer such that $2p_k+ε$ is composite, then we say the chain has length $k$. Although such chains are necessarily finite, it is conjectured that for every positive integer $k$, there are infinitely many Cunningham chains of length $k$. A sequence of polynomials $f_1(x), f_2(x), ...$, such that $f_i(x)\in \Z[x]$, $f_1(x)$ has positive leading coefficient, $f_i(x)$ is irreducible in $\Q[x]$, and $f_i(x)=xf_{i-1}(x)+ε$ for all $i$, is defined to be a {\it polynomial Cunningham chain} of the first or second kind, depending on whether $ε=1$ or -1 respectively. If $k$ is the least positive integer such that $f_{k+1}(x)$ is reducible over $\Q$, then we say the chain has length $k$. In this article, for chains of each kind, we explicitly give infinitely many polynomials $f_1(x)$, such that $f_{k+1}(x)$ is the only term in the sequence $\{f_i(x)\}_{i=1}^{\infty}$ that is reducible. As a first corollary, we deduce that there exist infinitel
In this paper we give an exponential lower bound for Cunningham's least recently considered (round-robin) rule as applied to parity games, Markhov decision processes and linear programs. This improves a recent subexponential bound of Friedmann for this rule on these problems. The round-robin rule fixes a cyclical order of the variables and chooses the next pivot variable starting from the previously chosen variable and proceeding in the given circular order. It is perhaps the simplest example from the class of history-based pivot rules. Our results are based on a new lower bound construction for parity games. Due to the nature of the construction we are also able to obtain an exponential lower bound for the round-robin rule applied to acyclic unique sink orientations of hypercubes (AUSOs). Furthermore these AUSOs are realizable as polytopes. We believe these are the first such results for history based rules for AUSOs, realizable or not. The paper is self-contained and requires no previous knowledge of parity games.
In the matroid intersection problem, we are given two matroids of rank $r$ on a common ground set $E$ of $n$ elements and the goal is to find the maximum set that is independent in both matroids. In this note, we show that Cunningham's algorithm for matroid intersection can be implemented to use $O(nr\log^2(r))$ independent oracle calls.
暂无摘要，请点击原文查看详情
In adiabatic quantum computing the aim is to track an eigenstate as the Hamiltonian changes. In the usual setup this is achieved using the natural time-dependent Hamiltonian evolution of the system and the main technical tool is the adiabatic theorem. We propose several alternative processes that achieve the same goal, but can easily be implemented on a gate-based quantum computer without the overhead of simulating time-dependent Hamiltonian evolution. We give a general framework for deriving `adiabatic' theorems for these processes. As an application, we give various algorithms for solving the Quantum Linear Systems Problem (QLSP) with optimal scaling in the condition number. One of these algorithms was previously developed in [Cunningham, Roland 2024] and another can be seen as a randomised version of the discrete adiabatic algorithm of [Costa et al. 2022]. We also describe versions of Trotterisation in our framework, which allows several results from [An et al. 2025] to be reproduced in a randomised setting. In particular, bounds on the Trotter error in terms of the fidelity are obtained that are asymptotically better than the standard bounds.
We present a practical system for privacy-aware large language model (LLM) inference that splits a transformer between a trusted local GPU and an untrusted cloud GPU, communicating only intermediate activations over the network. Our system addresses the unique challenges of autoregressive LLM decoding over high-latency wide-area networks (WANs), contributing: (1) an asymmetric layer split where embedding and unembedding layers remain local, ensuring raw tokens never leave the trusted device; (2) the first application of lookahead decoding to split inference over WANs, amortizing network round-trip latency across multiple tokens per iteration; (3) an empirical inversion attack evaluation showing that split depth provides a tunable privacy-performance tradeoff -- an attacker can recover ~59%% of tokens at a 2-layer split but only ~35%% at an 8-layer split, with minimal throughput impact; (4) ablation experiments showing that n-gram speculation accepts 1.2-1.3 tokens per decoding step on average (peak of 7 observed on code), with acceptance rates consistent across model scales; (5) formal verification that lookahead decoding produces token-identical output to sequential decoding under
We show that for a generic real or complex-valued compactly supported potential, the corresponding Schroedinger operator achieves maximal resonance density, in the sense that its integrated resonance counting function achieves the optimal asymptotic upper bound. For odd dimensions this follows from results of Dinh-Vu once we adapt an argument of Christiansen Hislop. The proof for even dimensions constitutes the bulk of the paper, and we prove several new results on resonances which have analogues in the odd dimensional case. This includes a sharp upper bound on the integrated resonance counting function for any compactly support potential, a proof that the characteristic function of a ball has resonance counting function which achieves the optimal upper bound, and an even-dimensional analogue of the result of Dinh-Vu on asymptotics of the resonance counting functions for complements of pluripolar subsets of analytic families of potentials. We use the characterization of resonances as zeros of certain Fredholm determinant functions related to the scattering matrix, allowing us to apply techniques and results from the theories of one and several complex variables. Our proof that the
The theory of simplicial complexes is a cornerstone of topology, offering a sophisticated tool for computing invariants. We present a formalization of abstract simplicial complexes and stellar subdivisions in the Lean proof assistant. We adopt a purely combinatorial framework in order to provide a cohesive foundation for studying the theory of stellar subdivisions as seen in many contexts of combinatorial topology. In particular, we provide formalizations of morphisms between abstract simplicial complexes; several crucial constructions and operations on complexes, such as links and joins; and perform a comprehensive study of how stellar subdivisions interact with these operations. We state and prove a number of identities commonly used in the study of triangulated manifolds, such as deriving equivalences between links in an abstract simplicial complex $K$ and in a stellar subdivision $σ_s K$, including results with no references in the standard literature. To our knowledge, this is the first formalization of stellar subdivisions in any proof assistant.
We prove a new fractal Weyl upper bound for the high-energy distribution of resonances of convex co-compact hyperbolic surfaces which matches the improved spectral gap given by Fourier decay. This improves upon the fractal Weyl bound of Dyatlov which matches the Patterson-Sullivan spectral gap. We also give a new resolvent estimate improving the ones given by Dyatlov-Zahl and Dyatlov. Analogous results are obtained for quantum open baker's maps, improving an estimate of Dyatlov-Jin, where we also give an improved fractal Weyl bound matching a spectral gap given by additive energy estimates. We refine known methods for proving fractal Weyl bounds which reduce the problem to an estimate of a certain determinant function; however, we use a different determinant function which allows us to make sharper estimates by applying the methods of proof of the fractal uncertainty principle in each setting.
There is a gap between the theoretical foundations of disentanglement and the practice of modern representation learning. Existing theoretical frameworks, particularly Independent Component Analysis (ICA) and its nonlinear variants, assume a generative model with statistically independent latent variables underlying the data so that disentanglement amounts to identifying the latents that could have generated the data. This generative framework is interpretable and theoretically justified, but its strong assumptions make it difficult to apply to modern representation learning. Modern pretrained encoders often learn features that exhibit disentangled properties without making generative assumptions, yet there is no general theory for interpreting these features as independent factors of variation. We take a step toward such a theory by introducing Riemannian ICA (RICA), which replaces ICA's global generative model with local geometric structure. RICA is founded on the observation that in ICA, the factors of variation underlying a data point can be understood through radial curves emanating from the point that map to axis-aligned lines in the latent space. We formalize this perspectiv
The highest rank of a string C-group representation of the alternating group $A_n$ is known for each $n$, but no self-dual representations attaining this highest rank are known when $n > 12$. Motivated by computational results for alternating groups of small degree, we examine a vertex-gluing construction for permutation representation graphs. We establish conditions under which gluing two string C-groups produces another string C-group, and use this construction to obtain infinite families of self-dual representations of alternating groups. In particular, for every $n = 4m+3 \geq 15$, we construct $\left \lfloor \frac{n+9}{8} \right \rfloor$ distinct self-dual string C-groups of rank $2m$ isomorphic to $A_{n}$. These representations have rank one below the maximum possible rank of string C-group representations for $A_n$, and to the authors' knowledge are the highest-rank self-dual representations currently known for alternating groups.
The mix of two maniplexes is the minimal maniplex that covers both. This construction has many important applications, such as finding the smallest regular cover of a maniplex. If one of the maniplexes is an abstract polytope, a natural question to ask is whether the mix is also a polytope. We describe here a general criterion for the polytopality of the mix which generalizes several previously-known polytopality criteria.
For a graph $Γ$ and group $G$, $G^Γ$ is the subgroup of $G^{|Γ|}$ generated by elements with $g$ in the coordinates corresponding to $v$ and its neighbors in $Γ$. There is a natural epimorphism $G^Γ\to (G/[G,G])^Γ$ with kernel $[G,G]^n \cap G^Γ$. When $[G,G]^n \leq G^Γ$, the structure of $G^Γ$ is easily described from $(G/[G,G])^Γ$. Fixing $Γ$, if $[G,G]^{|Γ|} \leq G^Γ$ for all $G$, we say that $Γ$ is RA (reducible to abelian). We showed in [2] that wide classes of graphs are RA, including graphs of girth 5 or more. The key tool is the RA matrix $C_Γ$, and we showed that $Γ$ is RA if and only if the row space $Row(C_Γ) = \mathbb Z^{|Γ|}$. Here, we study the possibilities for the elementary divisors of $C_Γ$; the more nontrivial elementary divisors we get, the further $Γ$ is from being RA (and the harder $G^Γ$ is to describe). We show that while many graphs, including those of girth 4, cartesian products, and most tensor products have at most one nontrivial elementary divisor, one can construct a graph of girth 3 with any prescribed set of elementary divisors and $\mathbb Z$-nullity.
Building on a beyond-GW many-body framework that incorporates higher-order vertex effects in the self-energy -- giving rise to T-matrix and second-order exchange contributions -- this approach is extended to now include the vertex derived in that work to the kernel in the Bethe-Salpeter Equation (BSE) for the reducible polarization function. This results in a frequency-dependent interaction kernel that naturally captures random phase approximation (RPA) effects, dynamical excitonic interactions, and the correlated propagation of multiple correlated electron-hole pairs that model multi- (including bi- and tri-) excitonic effects, relevant for nonlinear optics and high harmonic generation. These processes emerge from including the functional derivatives of the screening and vertex with respect to the Green's function in the vertex, enabling a fully abinitio, time-dependent treatment of correlation effects. By focusing on the reducible rather than irreducible polarization function, this approach provides a computationally viable framework for capturing complex many-body interactions for calculating the self-energy, optical spectra and EELS. The resulting interaction kernel is relative
There has been some concern about the impact of predatory publishers on scientific research for some time. Recently, publishers that might previously have been considered `predatory' have established their bona fides, at least to the extent that they are included in citation impact scores such as the field-weighted citation impact (FWCI). These are sometimes called `grey' publishers (MDPI, Frontiers, Hindawi). In this paper, we show that the citation landscape for these grey publications is significantly different from the mainstream landscape and that affording publications in these venues the same status as publications in mainstream journals may significantly distort metrics such as the FWCI.
The advancement of science relies on the exchange of ideas across disciplines and the integration of diverse knowledge domains. However, tracking knowledge flows and interdisciplinary integration in rapidly evolving, multidisciplinary fields remains a significant challenge. This work introduces a novel network analysis framework to study the dynamics of knowledge transfer directly from citation data. By applying dynamic community detection to cumulative, time-evolving citation networks, we can identify research areas as groups of papers sharing knowledge sources and outputs. Our analysis characterises the life-cycles and knowledge transfer patterns of these dynamic communities over time. We demonstrate our approach through a case study of eXplainable Artificial Intelligence (XAI) research, an emerging interdisciplinary field at the intersection of machine learning, statistics, and psychology. Key findings include: (i) knowledge transfer between these important foundational topics and the contemporary topics in XAI research is limited, and the extent of knowledge transfer varies across different contemporary research topics; (ii) certain application domains exist as isolated "knowle
Starting with Hedins equations, simple expressions for the irreducible self-energy are derived. The derivation with vertex effects included in the self-energy results in a number of terms beyond GW such as second-order screened exchange (the term that also gives rise to vertex/excitonic effects in the polarisation) and an infinite series describing correlations between the added-particle(removed-hole) and the excited electrons and holes: the screened T-matrix channels. Two-body correlated propagation is considered, with the third propagating freely, however, 3-body correlations are discussed and these can be added hierarchically to the method. For electron-hole propagation the reducible polarisation is calculated, which results in an expression for the self-energy that can be solved analytically without the need for widely used expensive numerical methods for frequency integration or having to adopt the plasmon-pole approximation. The method requires diagonalisation of the usual particle-hole matrix - derived from the Bethe-Salpter equation - and also a particle-particle and hole-hole matrix that has a similar structure; with the second-order exchange present in all channels. The m
The energetic stability of positron di-anion systems [A$^-;e^+;$A$^-$] is studied via many-body theory, where $A^-$ includes H$^{-}$, F$^{-}$, Cl$^{-}$ and the molecular anions (CN)$^{-}$ and (NCO)$^{-}$. Specifically, the energy of the system as a function of ionic separation is determined by solving the Dyson equation for the positron in the field of the two anions, using a positron-anion self energy as constructed in [J. Hofierka, B. Cunningham, C. M. Rawlins, C. H. Patterson and D. G. Green, \emph{Nature} {\bf 606} 688 (2022)] that accounts for correlations including polarization, screening, and virtual-positronium formation. Calculations are performed for a positron interacting with H$_{2}^{2-}$, F$_{2}^{2-}$, and Cl$_{2}^{2-}$, and are found to be in good agreement with previous theory. In particular, we confirm the presence of two minima in the potential energy of the [H$^-;e^+$;H$^-$] system with respect to ionic separation: one a positronically-bonded [H$^-;e^+$;H$^-$] local minimum at ionic separations $r\sim3.4$~Å\phantom{}, and a global minimum at smaller ionic separations $r\lesssim1.6$~Å\phantom{} that gives overall instability of the system with respect to dissociati