Miracle-dz/UniFAR-FTUs huggingface dataset:
Exact Sampling from Perfect Matchings of Dense Nearly Regular Bipartite Graphs

We present the first algorithm for generating random variates exactly uniformly from the set of perfect matchings of a bipartite graph with a polynomial expected running time over a nontrivial set of graphs. Previous Markov chain approaches obtain approximately uniform variates for arbitrary graphs in polynomial time, but their general running time is $\Theta(n^{26} (\ln n)^2). $ Our algorithm employs acceptance/rejection together with a new upper limit on the permanent of a form similar to Bregman's Theorem. For a graph with $2n$ nodes where the degree of every node is nearly $\gamma n$ for a constant $\gamma$, the expected running time is $O(n^{1.5 + .5/\gamma})$. Under these conditions, Jerrum and Sinclair showed that a Markov chain of Broder can generate approximately uniform variates in $\Theta(n^{4.5 + .5/\gamma} \ln n)$ time, making our algorithm significantly faster on this class of graph. With our approach, approximately counting the number of perfect matchings (equivalent to finding the permanent of a 0-1 matrix and so $\sharp P$ complete) can be done without use of selfreducibility.
On the Construction of Gröbner Bases with Coefficients in Quotient Rings

Let $\Lambda$ be a commutative Noetherian ring, and let $I$ be a proper ideal of $\Lambda$, $R=\Lambda /I$. Consider the polynomial rings $T=\Lambda [x_1,. x_n]$ and $A=R[x_1,. ,x_n]$. Suppose that linear equations are solvable in $\Lambda$. It is shown that linear equations are solvable in $R$ (thereby theoretically Gr\"obner bases for ideals of $A$ are well defined and constructible) and that practically Gr\"obner bases in $A$ with respect to any given monomial ordering can be obtained by constructing Gr\"obner bases in $T$, and moreover, all basic applications of a Gr\"obner basis at the level of $A$ can be realized by a Gr\"obner basis at the level of $T$. Typical applications of this result are demonstrated respectively in the cases where $\Lambda=D$ is a PID, $\Lambda =D[y_1,. ,y_m]$ is a polynomial ring over a PID $D$, and $\Lambda =K[y_1,. ,y_m]$ is a polynomial ring over a field $K$.
Some applications of matrix inequalities in R\'enyi entropy

The R{\'e}nyi entropy is one of the important information measures that generalizes Shannon's entropy. The quantum R{\'e}nyi entropy has a fundamental role in quantum information theory, therefore, bounding this quantity is of vital importance. Another important quantity is R{\'e}nyi relative entropy on which R{\'e}nyi generalization of the conditional entropy, and mutual information are defined based. Thus, finding lower bound for R{\'e}nyi relative entropy is our goal in this paper. We use matrix inequalities to prove new bounds on the entropy of type $\beta$, R{\'e}nyi entropy.
Alternative approach to derive q-potential measures of refracted spectrally L\'evy processes

For a refracted L\'evy process driven by a spectrally negative L\'evy process, we use a different approach to derive expressions for its q-potential measures without killing. Unlike previous methods whose derivations depend on scale functions which are defined only for spectrally negative L\'evy processes, our approach is free of scale functions. This makes it possible to extend the result here to a quite general refracted L\'evy process by applying the approach presented below.
The equivariant local $\epsilon$-constant conjecture for unramified twists of $\mathbb{Z}_p(1)$

We study the equivariant local epsilon constant conjecture, denoted by $C_{EP}^{na}(N/K,V)$, as formulated in various forms by Kato, Benois and Berger, Fukaya and Kato and others, for certain 1-dimensional twists $T=\mathbb{Z}_p(\chi^{nr})(1)$ of $\mathbb{Z}_p(1)$. Following ideas of recent work of Izychev and Venjakob we prove that for $T=\mathbb{Z}_p(1)$ a conjecture of Breuning is equivalent to $C_{EP}^{na}(N/K,V)$. As our main result we show the validity of $C_{EP}^{na}(N/K,V)$ for certain wildly and weakly ramified abelian extensions $N/K$. A crucial step in the proof is the construction of an explicit representative of $R\Gamma(N,T)$.
NEW RESULTS ON SUM‐PRODUCT TYPE GROWTH OVER FIELDS

We prove a range of new sum-product type growth estimates over a general field $\mathbb{F}$, in particular the special case $\mathbb{F}=\mathbb{F}_p$. They are unified by the theme of "breaking the $3/2$ threshold", epitomising the previous state of the art. These estimates stem from specially suited applications of incidence bounds over $\mathbb{F}$, which apply to higher moments of representation functions. We establish the estimate $|R[A]| \gtrsim |A|^{8/5}$ for cardinality of the set $R[A]$ of distinct cross-ratios defined by triples of elements of a (sufficiently small if $\mathbb{F}$ has positive characteristic, similarly for the rest of the estimates) set $A\subset \mathbb{F}$, pinned at infinity. The cross-ratio naturally arises in various sum-product type questions of projective nature and is the unifying concept underlying most of our results. It enables one to take advantage of its symmetry properties as an onset of growth of, for instance, products of difference sets. The geometric nature of the cross-ratio enables us to break the version of the above threshold for the minimum number of distinct triangle areas $Ouu'$, defined by points $u,u'$ of a non-collinear point set $P\subset \mathbb{F}^2$. Another instance of breaking the threshold is showing that if $A$ is sufficiently small and has additive doubling constant $M$, then $|AA|\gtrsim M^{-2}|A|^{14/9}$. This result has a second moment version, which allows for new upper bounds for the number of collinear point triples in the set $A\times A\subset \mathbb{F}^2$, the quantity often arising in applications of geometric incidence estimates.
Separability criteria for mixed three-qubit states

We study the noisy GHZ-W mixture (where GHZ denotes Greenberger-Horne-Zeilinger). We demonstrate some necessary but not sufficient criteria for different classes of separability of these states. It turns out that the partial transposition criterion of Peres [Phys. Rev. Lett. 77, 1413 (1996)] and the criteria of Guehne and Seevinck [New J. Phys. 12, 053002 (2010)] dealing with matrix elements are the strongest ones for different separability classes of this two-parameter state. As a result, we determine a set of entangled states of positive partial transpose.
The free boundary problem of American butterfly option

This article aims to characterize behaviors of the free boundary arising from American butterfly option pricing. We prove the existence and uniqueness of the solution and free boundary. And then, we classify the shape of the free boundary under different conditions. The main contribution of this paper is to prove that the free boundary of evolutionary problem converges to the free boundary of stationary problem. Copyright © 2013 John Wiley & Sons, Ltd.
Scattering of charge carriers by point defects in bilayer graphene

A theory of scattering of massive chiral fermions in bilayer graphene by radial symmetric potential is developed. It is shown that in the case when the electron wavelength is much larger than the radius of the potential the scattering cross section is proportional to the electron wavelength. This leads to the mobility independent on the electron concentration. In contrast with the case of single-layer, neutral and charged defects are, in general, equally relevant for the resistivity of the bilayer graphene.
A bifurcational geometric approach to the problem of chaos transition in the classical Lorenz system

The classical Lorenz system is considered. For many years, this system has been the subject of study by numerous authors. However, until now the structure of the Lorenz attractor is not clear completely yet, and the most important question at present is to understand the bifurcation scenario of chaos transition in this system. Using some numerical results and our bifurcational geometric approach, we present a new scenario of chaos transition in the classical Lorenz system.
Relativity theory and the creation of matter

It is shown that the continuous creation of matter can be treated by the standard methods of general relativity provided that the possibility of a ‘zero-point’ stress is allowed. The matter ‘created’ is the mass-equivalent of the work done by this stress in the expansion of the universe. According to relativity theory this stress also makes a contribution to the density of gravitational mass. When these two effects are taken into account it is shown that the Newtonian analogues of relativistic models of the universe discovered by Milne and the writer can be extended to cover all relativistic models. This result makes it possible to obtain a unified physical interpretation of Hoyle’s treatment of the cosmological problem. It is suggested that the interpretation in terms of a zero-point stress may lead to a connexion with the quantum theory of fields.
On the Effectiveness of the Eddy Straining Mechanism for the Maintenance of Blocking Flows.

In connection with the maintenance of blocking flows by the migrating synoptic disturbances, we examine the effectiveness of the eddy straining mechanism proposed by Shutts (1983) in an equivalent barotropic β-channel model. The model used here is identical to that in Haines and Marshall (1987) (hereafter referred to as HM) except with a channel twice as wide. This model possesses two stationary solutions accompanying isolated structures in prescribed uniform westerlies when a vorticity forcing associated with the analytical modon solution is assumed: “blocking solution” similar to the modon solution and “zonal flow solution” characterized by dominant zonal flows. The infinitesimal transient eddies which mimic synoptic disturbances are generated by a wavemaker forcing located far upstream of the diffluence associated with the basic flow prescribed by stable stationary solutions. The effectiveness of the eddy straining mechanism is evaluated from the resemblance between the basic flow and the second-order flow induced by the time-averaged eddy potential vorticity (PV) flux divergence. The distribution of the time-averaged PV divergence for the blocking solution in our model is the same as in HM in a sense that there is a PV north/south, divergence/convergence dipole upstream of the diffluence of the basic flow. However, the computed second-order flow has a quadruple structure which tends to shift the blocking dipole downstream instead of the dipole structure enforcing the blocking as shown in HM. On the other hand, the second-order flow tends to maintain a weak diffluence associated with the zonal solution. Thus, the effectiveness of the eddy straining mechanism depends on the basic flow. The second-order flow for the blocking solution is also drastically deformed by a negligible distortion of the PV divergence field associated with a small change in the property of imposed eddies. The effectiveness of the eddy straining mechanism depends also on the property of synoptic eddies. Finally, this study suggests that the PV north/south, divergence/convergence dipole pattern does not necessarily maintain the blocking dipole. This remark is especially important for the observational study to assess the role of synoptic disturbances in the maintenance of blocking flows.
Castelnuovo-Mumford Regularity and GV-Sheaves on Irregular Varieties

Inspired by Beauville's recent construction of Ulrich sheaves on abelian surfaces, we pose the question of whether a torsion-free sheaf on a polarized smooth projective variety with Castelnuovo-Mumford regularity 1 is a GV (generic vanishing) sheaf, and present evidence that this question is governed by the positivity of cycles on generalized Brill-Noether loci. We prove that it has an affirmative answer for natural polarizations on many well-known irregular surfaces, as well as some polarizations on ruled threefolds over a curve.
Random potentials for pinning models with \nabla and \Delta interactions

We consider two models for biopolymers, the $\nabla$ interaction and the $\Delta$ one, both with the Gaussian potential in the random environment. A random field $\varphi:{0,1,. ,N}\rightarrow \Bbb{R}^d$ represents the position of the polymer path. The law of the field is given by $\exp(-\sum_i\frac{|\nabla\varphi_i|^2}{2})$ where $\nabla$ is the discrete gradient, and by $\exp(-\sum_i\frac{|\Delta\varphi_i|^2}{2})$ where $\Delta$ is the discrete Laplacian. For every Gaussian potential $\frac{|\cdot|^2}{2}$, a random charge is added as a factor: $(1+\beta\omega_i)\frac{|\cdot|^2}{2}$ with $\Bbb{P}(\omega_i=\pm 1)=1/2$ or $\exp(\beta\omega_i)\frac{|\cdot|^2}{2}$ with $\omega_i$ obeys a normal distribution. The interaction with the origin in the random field space is considered. Each time the field touches the origin, a reward $\epsilon\geq 0$ is given. Although these models are quite different from the pinning models studied in Giacomin (2007), the result about the gap between the annealed critical point and the quenched critical point stays the same.
Vector bundles on plane cubic curves and the classical Yang-Baxter equation

In this article, we develop a geometric method to construct solutions of the classical Yang-Baxter equation, attaching to the Weierstrass family of plane cubic curves and a pair of coprime positive integers, a family of classical r-matrices. It turns out that all elliptic r-matrices arise in this way from smooth cubic curves. For the cuspidal cubic curve, we prove that the obtained solutions are rational and compute them explicitly. We also describe them in terms of Stolin's classification and prove that they are degenerations of the corresponding elliptic solutions.
$p$-adic multiple zeta values at roots of unity and $p$-adic pro-unipotent harmonic actions - IV-1 : $p$-adic multiple zeta values at roots of unity extended to sequences of integers of any sign

This work is a study of $p$-adic multiple zeta values at roots of unity ($p$MZV$\mu_{N}$'s), the $p$-adic periods of the crystalline pro-unipotent fundamental groupoid of $(\mathbb{P}^{1} - \{0,\mu_{N},\infty\})/ \mathbb{F}_{q}$. The main tool is new objects which we call $p$-adic pro-unipotent harmonic actions. In this part IV we define and study $p$-adic analogues of some elementary complex analytic functions which interpolate multiple zeta values at roots of unity such as the multiple zeta functions. The indices of $p$MZV$\mu_{N}$'s involve sequences of positive integers ; in this IV-1, by considering an operation which we call localization (inverting certain integration operators) in the pro-unipotent fundamental groupoid of $\mathbb{P}^{1} - \{0,\mu_{N},\infty\}$, and by using $p$-adic pro-unipotent harmonic actions, we extend the definition of $p$MZV$\mu_{N}$'s to indices for which these integers can be negative, and we study these generalized $p$MZV$\mu_{N}$'s.
Technical Notes on "A new approach to the d-MC problem"

System reliability is the probability of the maximum flow in a stochastic-flow network from the source node to the sink node being more than a demand level d. There are several approaches to compute system reliability using upper boundary points, called d-MinCuts (d-MCs). Search for all the d-MCs in a stochastic-flow network is an NP-hard problem. Here, a work proposed by Yeh [Yeh WC. A new approach to the d-MC problem. Reliab Eng and Syst Saf 2002; 77(2): 201-206. ] for determining all the d-MCs is investigated. Two results (Lemma 3 and Theorem 5) are shown to be incorrect and their correct versions are established. Also, the complexity result (Theorem 6) is shown to be incorrect and the correct count is provided.
Global convergence for ill-posed equations with monotone operators: the dynamical systems method

Consider an operator equation F ( u ) = 0 in a real Hilbert space. Let us call this equation ill-posed if the operator F ′ ( u ) is not boundedly invertible, and well-posed otherwise. If F is monotone C 2 loc ( H ) operator, then we construct a Cauchy problem, which has the following properties: 1) it has a global solution for an arbitrary initial data, 2) this solution tends to a limit as time tends to inﬁnity, 3) the limit is the minimum norm solution to the equation F ( u ) = 0 . Example of applications to linear ill-posed operator equation is given.
Factors of Pisot tiling spaces and the coincidence rank conjecture

We consider the structure of Pisot substitution tiling spaces, in particular, the structure of those spaces for which the translation action does not have pure discrete spectrum. Such a space is always a measurable m-to-one cover of an action by translation on a group called the maximal equicontinuous factor. The integer m is the coincidence rank of the substitution and equals one if and only if translation on the tiling space has pure discrete spectrum. By considering factors intermediate between a tiling space and its maximal equicontinuous factor, we establish a lower bound on the cohomology of a one-dimensional Pisot substitution tillng space with coincidence rank two and dilation of odd norm. The Coincidence Rank Conjecture, for coincidence rank two, is a corollary.
Convergence Rates for Hierarchical Gibbs Samplers

We establish some results for the rate of convergence in total variation of a Gibbs sampler to its equilibrium distribution. This sampler is motivated by a hierarchical Bayesian inference construction for a gamma random variable. Our results apply to a wide range of parameter values in the case that the hierarchical depth is 3 or 4, and are more restrictive for depth greater than 4. Our method involves showing a relationship between the total variation of two ordered copies of our chain and the maximum of the ratios of their respective co-ordinates. We construct auxiliary stochastic processes to show that this ratio does converge to 1 at a geometric rate.
Dynamic shadow mask technique for device fabrication in UHV

Clean structures are fabricated in ultrahigh vacuum conditions by evaporation through a shadow mask, avoiding contamination by resist, chemicals or exposure to air. Moving the shadow mask with nanometer precision during the growth of the structures gives additional freedom in determining the lateral shape and the thickness profile. Different materials can easily be combined and chosen from a large range of metals, semiconductors or insulators. In-situ treated surfaces or, conversely, ex-situ pre-fabricated samples can be used as substrates.
Moser's mathemagical work on the equation 1^k+2^k+...+(m-1)^k=m^k

If the equation of the title has an integer solution with k>=2, then m>10^{10^6}. Leo Moser showed this in 1953 by amazingly elementary methods. With the hindsight of more than 50 years his proof can be somewhat simplified. We give a further proof showing that Moser's result can be derived from a von Staudt-Clausen type theorem. Based on more recent developments concerning this equation, we derive a new result using the divisibility properties of numbers in the sequence 2^{2e+1}+1, e=0,1,2,. In the final section we show that certain Erdos-Moser type equations arising in a recent paper of Kellner can be solved completely.
Classification of log del Pezzo surfaces of index $\le 2$

This is an expanded version of our work [AN88], 1988, in Russian. We classify del Pezzo surfaces over C with log terminal singularities of index \le 2. By classification, we understand a description of the intersection graph of all exceptional curves on an appropriate (somewhat stronger than minimal) resolution of singularities together with the subgraph of the curves which are contracted to singular points. The final results are similar to classical results about classification of non-singular del Pezzo surfaces and use the usual finite root systems. However, the intermediate considerations use the theory of K3 surfaces (especially of K3 surfaces with non-symplectic involutions) and the theory of reflection groups in hyperbolic spaces (especially or reflection groups of hyperbolic lattices). As an ``elementary'' application, our results permit one to classify sextics in P^2 with simple singularities and a component of geometric genus \ge 2.
Categorification of Hopf algebras of rooted trees

We exhibit a monoidal structure on the category of finite sets indexed by P-trees for a finitary polynomial endofunctor P. This structure categorifies the monoid scheme (over Spec ℕ) whose semiring of functions is (a P-version of) the Connes-Kreimer bialgebra H of rooted trees (a Hopf algebra after base change to ℤ and collapsing H0). The monoidal structure is itself given by a polynomial functor, represented by three easily described set maps; we show that these maps are the same as those occurring in the polynomial representation of the free monad on P.
Minimum codegree threshold for $(K_4^3-e)$-factors

Given hypergraphs H and F, an F-factor in H is a spanning subgraph consisting of vertex disjoint copies of F. Let K_4^3-e denote the 3-uniform hypergraph on 4 vertices with 3 edges. We show that for \gamma>0 there exists an integer n_0 such that every 3-uniform hypergraph $H$ of order n > n_0 with minimum codegree at least (1/2+\gamma)n and 4|n contains a (K_4^3-e)-factor. Moreover, this bound is asymptotically the best possible and we further give a conjecture on the exact value of the threshold for the existence of a (K_4^3-e)-factor. Therefore, all minimum codegree thresholds for the existence of F-factors are known asymptotically for 3-uniform hypergraphs F on 4 vertices.
A stochastic approach to enhanced diffusion : Lévy walks

We study anomalous transport in one-dimension through Levy walks and concentrate on the superlinear diffusion regime. The problem is formulated in terms of continuous-time random walks (CTRW) with kernels that couple space and time. We calculate P(r, t), the probability of being at position r at time t, and r2(t), the mean-squared displacement for which we present approximate asymptotic forms. We examine the relationship between Levy walks and Levy distributions and discuss the validity range of the asymptotic (heuristic) relation P(0, t) ~ r2(t)-1/2.
Historical Changes in the Concepts of Number, Mathematics and Number Theory

This essay traces the history of three interconnected strands. Firstly, changes in the concept of number, secondly, the study of the qualities of number, which evolved into number theory, and thirdly, the nature of mathematics itself, from early Greek mathematics to the 20th century. These were embedded in philosophical shifts, from the classical Greek ontologies through increasing pragmatism to formalism and logical positivism. Given that Goedel demonstrated the limitations of the latter as a foundation for mathematics, this essay explores phenomenology and Lakatosian ideas, which together offer a more sound epistemological and ontological basis for mathematics and a methodology for mathematical development.The question also then arises of the possible resurrection of earlier, neglected mathematical projects, including widening the domain of number theory to include integer qualities revealed in the growth of mathematics in general, which has predominantly been the growth of quantitative mathematics or logistike, the complement of arithmetike in classical Greece.
On the Siegel-Sternberg linearization theorem

We establish a general version of the Siegel-Sternberg linearization theorem for ultradiffentiable maps which includes the analytic case, the smooth case and the Gevrey case. It may be regarded as a small divisior theorem without small divisor conditions. Along the way we give an exact characterization of those classes of ultradifferentiable maps which are closed under composition, and reprove regularity results for solutions of ode’s and pde’s. This will open up new directions in kam-theory and other applications of ultradifferentiable functions. We consider the problem of linearizing a map g in the neighbourhood of a fixed point. Placing this fixed point at the origin we write
Uniqueness of singular self-similar solutions of a semilinear parabolic equation

We study the uniqueness of singular radial (forward and backward) self-similar positive solutions of the equation $u_t-\Delta u = u^p, \quad x\in{\mathbb R}^n,\ t>0,$ where $p\geq(n+2)/(n-2)_+$.