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Big Hairy Ian quotes New Atlas: The wonder material graphene gets many of its handy quirks from the fact that it exists in two dimensions, as a sheet of carbon only one atom thick. But to actually make use of it in practical applications, it usually needs to be converted into a 3D form. Now, researchers have developed a new and relatively simple way to do just that, using lasers to 'forge' a threedimensional pyramid out of graphene... By focusing a laser onto a fine point on a 2D graphene lattice, the graphene at that spot is irradiated and bulges outwards. A variety of threedimensional shapes can be made by writing patterns with the laser spot, with the height of the shape controlled by adjusting the irradiation dose at each particular point. The team illustrated that technique by deforming a sheet of graphene into a 3D pyramid, standing 60 nm high. That sounds pretty tiny, but it's 200 times taller than the graphene sheet itself. "The beauty of the technique is that it's fast and
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Ars technica reports on an announcement that the kernel's longterm support releases will now be maintained for six years instead of two. "A sixyear support window will give Google, SoC Vendors, and OEMs plenty of time to develop a device and get it to market, while still leaving about four years for enduser ownership. Google currently provides two years of major OS updates on its phones and three years of security updates, but if it wanted to extend that, an announcement like this would seem like an important first step." The kernel.org releases page now shows 4.4 being maintained through February 2022.
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We completely solve the problem of classifying all onedimensional quantum potentials with nearest and nexttonearestneighbors interactions whose ground state is Jastrowlike, i.e., of Jastrow type but depending only on differences of consecutive particles. In particular, we show that these models must necessarily contain a threebody interaction term, as was the case with all previously known examples. We discuss several particular instances of the general solution, including a new hyperbolic potential and a model with elliptic interactions which reduces to the known rational and trigonometric ones in appropriate limits.
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Let $M_n$ be the topological moduli space of all parallel ncables of long framed oriented knots in 3space. We construct in a combinatorial way for each natural number $n>1$ a 1cocycle $R_n$ which represents a non trivial class in $H^1(M_n; \mathbb{Z} [x_1,x_2,...,x_1^{1},x_2^{1},...])$, where the number of variables $x_i$ depends on $n$. To each generic point in $M_n$ we associate in a canonical way an arc {\em scan} in $M_n$, such that $R_n(scan)$ is already a polynomial knot invariant. We show that $R_3(scan)$ detects the noninvertibility of the knot $8_{17}$ in a very simple way and without using the knot group. There are two wellknown canonical loops in $M_n$ for each parallel ncable of a long framed knot $K$: Gramain's loop {\em rot} and the FoxHatcher loop {\em fh}. The calculation of $R_n$ is of at most quartic complexity for these loops with respect to the number of crossings of $K$ for each fixed $n$. It follows from results of Hatcher that $K$ is not a torus knot
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We develop the theory of layered posets, and use the notion of layering to prove a new iteration theorem (Theorem 6): if $\kappa$ is weakly compact then any universal Kunen iteration of $\kappa$cc posets (each possibly of size $\kappa$) is $\kappa$cc, as long as direct limits are used sufficiently often. This iteration theorem simplifies and generalizes the various chain condition arguments for universal Kunen iterations in the literature on saturated ideals, especially in situations where finite support iterations are not possible. We also provide two applications: (1) For any $n \ge 1$, a wide variety of $<\omega_{n1}$closed, $\omega_{n+1}$cc posets of size $\omega_{n+1}$ can consistently be absorbed (as regular suborders) by quotients of saturated ideals on $\omega_n$ (see Theorem 7 and Corollary 8); and (2) For any $n \in \omega$, the Tree Property at $\omega_{n+3}$ is consistent with the Chang's Conjecture $(\omega_{n+3}, \omega_{n+1}) \twoheadrightarrow (\omega_{n+1}, \om
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In this note, we first classify all topological torus knots lying on the Heegaard torus in lens spaces, and then we study Legendrian representatives of these knots. We classify oriented positive Legendrian torus knots in the universally tight contact structures on the lens spaces up to contactomorphism.
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This work continues the study of residually wild morphisms $f\colon Y\to X$ of Berkovich curves initiated by Cohen, Temkin and Trushin in [CTT16]. The different function $\delta_f$ introduced in [CTT16] is the primary discrete invariant of such covers. When $f$ is not residually tame, it provides a nontrivial enhancement of the classical invariant of $f$ consisting of morphisms of reductions $\widetilde{f}\colon \widetilde{Y}\to\widetilde{X}$ and metric skeletons $\Gamma_f\colon \Gamma_Y\to\Gamma_X$. In this paper we interpret $\delta_f$ as the norm of the canonical trace section $\tau_f$ of the dualizing sheaf $\omega_f$, and introduce a finer reduction invariant $\widetilde{\tau}_f$, which is (loosely speaking) a section of $\omega_{\widetilde{f}}^{\rm log}$. Our main result generalizes a lifting theorem of AminiBakerBrugall\'eRabinoff from the case of residually tame morphism to the case of minimally residually wild morphisms. For such morphisms we describe all restrictions the
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Kuniba, Okado, Takagi and Yamada have found that the timeevolution of the TakahashiSatsuma boxball system can be linearized by considering rigged configurations associated with states of the boxball system. We introduce a simple way to understand the rigged configuration of $\mathfrak{sl}_2$type, and give an elementary proof of the linearization property. Our approach can be applied to a boxball system with finite carrier, which is related to a discrete modified KdV equation, and also to the combinatorial $R$matrix of $A_1^{(1)}$type. We also discuss combinatorial statistics and related fermionic formulas associated with the states of the boxball systems. A fermionictype formula we obtain for the finite carrier case seems to be new.
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From a report: U.S. stores have been closing at a faster rate in 2017 than at any time since the recession, an American phenomenon being dubbed "retail apocalypse." Though this has sofar been largely a worry for U.S. retailers, the Wall Street Journal reports that investors in Europe are worried that it is now spreading abroad.
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An anonymous reader quotes a report from Exstreamist: Comcast announced this week that they plan on rolling out their streaming service, "Xfinity Instant TV" as an option for broadbandonly customers. At our very first glance, it seemed like a pretty good deal, a livestreaming service for $18 a month, not bad right? But once we actually looked into the offering, we noticed something funny. Almost the entirety of what they're planning on charging $18 a month for could be viewed free with an antenna. According to the Wall Street Journal, the antenna as an option is apparently a long lost TV option for many consumers. Variety is reporting, "Xfinity Instant TV" intro packages, the ones that are $18, will only include a handful of broadcast channels, and a few "freebies" like the Home Shopping Network, and CSPAN So we're not exactly talking about getting access to ESPN, CNN, FX, or other more desirable channels for cord cutters, those will cost you at least $45 more a month, so basically t
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dryriver writes: Chinese researchers have taken tissue from a betathallasemia patient, created cloned embryos from that patient's cells, and used a genetic editing technique known as Base Editing to correct the gene mutation that causes betathallasemia. The embryos were not implanted in a womb, so no actual babies were created during the procedure. The BBC reports: "Precise 'chemical surgery' has been performed on human embryos to remove disease in a world first, Chinese researchers have told the BBC. The team at Sun Yatsen University used a technique called base editing to correct a single error out of the three billion 'letters' of our genetic code. They altered labmade embryos to remove the disease betathalassemia. The embryos were not implanted. The team says the approach may one day treat a range of inherited diseases. Base editing alters the fundamental building blocks of DNA: the four bases adenine, cytosine, guanine and thymine. Base editing works on the DNA bases themselv
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New submitter Rick Schumann writes about California considering a ban on internal combustion engines: The ban on internalcombustion engine automobiles would be at least 10 years away, and it's unclear at this early stage if it would ban only sales and use of new cars, or ban existing cars as well. There's also no mention of two (or three) wheeled vehicles at this stage. The California Air Resources Board (CARB) is nevertheless considering this seriously, in order to meet its ambitious emissions reduction goals. According to state data, tailpipes generate more than onethird of all greenhouse gases, and so far only a small fraction of California's motorists drive electric vehicles. The announcement was made in an interview with Bloomberg news. "I've gotten messages from the governor asking, 'Why haven't we done something already?' The governor has certainly indicated an interest in why China can do this and not California," Mary Nichols, the chairwoman of the CARB, told Bloomberg.
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Yesterday, Amazon announced six new hardware products at a surprise event in Seattle. The one that everyone is talking about though is called the Echo Spot  a little alarm clock with a camera that will probably be pointing directly at your bed. "While all the focus is on what the Echo Spot looks like, it's important to remember that Amazon is using the Spot as a very clever way of making you comfortable with having a camera in your bedroom," reports The Verge. From the report: Amazon launched its Echo Look camera earlier this year to judge your outfits. It's designed to sit in your wardrobe and offer you style advice, and it was Amazon's first Echo device with a camera. Amazon quickly followed it up with the Echo Show, a touchscreen device that sits in your kitchen and lets you watch tutorials or recipes and participate in video calls. Amazon's Look device is still only available exclusively by invitation, and in hindsight it now looks like experimental hardware to gauge the reaction
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Longtime Slashdot reader Bruce Perens writes: The U.S. Air Force has transferred control of a 10yearold orbiting satellite to AMSAT, a ham radio organization, which has enabled it for any licensed ham to use on the air, as the satellite's Air Force missions have ended. Falconsat 3's first mission was science: measuring gravity gradient, spectrometry of the plasmasphere, electronic noise in the plasmasphere, and testing threeaxis attitude control using microthrusters. Secondarily it was used to train Air Force Institute of Technology students in space operations, with close to 700 cadets obtaining ham licenses in order to operate a number of Air Force satellites using ham frequencies. Now in its third mission, control of the satellite has been transferred to AMSAT, the Radio Amateur Satellite Corporation, and all government frequencies have been disabled with only ham ones remaining. The satellite will relay APRS (position and status reporting) signals, it will operate a BBS in the
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An anonymous reader quotes a report from Bloomberg: AT&T and other broadband providers asked the U.S. Supreme Court to overturn the Obamaera "net neutrality" rule barring internet service providers from slowing or blocking rivals' content. The appeals, filed Thursday, will put new pressure on a rule enacted in 2015 when the Federal Communications Commission was under Democratic control. Filing a separate appeal from AT&T were the United States Telecom Association, a trade group, and broadband service provider CenturyLink. The embattled net neutrality rules bar internet service providers such as AT&T, Verizon and Comcast from blocking or slowing some web traffic in favor of other content  their own or a paying customer's. "The practical stakes are immense," AT&T said in its appeal of a ruling that backed the FCC. The company pointed to a dissenting opinion that said the regulation "fundamentally transforms the internet" and will have a "staggering" impact on infrastru
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According to a security guide published Wednesday, Apple recommends that children under the age of 13 do not use Face ID on the iPhone X due to the probability of a false match being significantly higher for young children. The company said this was because "their distinct facial features may not have fully developed." They also recommend that twins and siblings do not use the new feature. The Guardian reports: In all those situations, the company recommends concerned users disable Face ID and use a passcode instead. With Face ID, Apple has implemented a secondary system that exclusively looks out for attempts to fool the technology. Both the authentication and spoofing defense are based on machine learning, but while the former is trained to identify individuals from their faces, the latter is used to look for telltale signs of cheating. "An additional neural network that's trained to spot and resist spoofing defends against attempts to unlock your phone with photos or masks," the com
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future guy shares a report from New Atlas: The UAE government has announced it is building the world's largest space simulation city, and to top it off it will be designed by one of the world's flashiest architects, Bjarke Ingels, whose company is literally called BIG. The project is called the Mars Science City and will cover 1.9 million sq ft (176,516 sq m) at a cost of nearly $140 million dollars. The city will span several domes, including a space for a team to live for up to a year as part of a Mars simulation. Several scientific laboratories will be included, focusing on developing methods for a Mars colony to produce food, energy and water. A museum exhibiting great space achievements will also be incorporated into the city with the walls of the museum being 3D printed using sand from the nearby Emirati desert.
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FCC Chairman Ajit Pai pushed Apple on Friday to activate the FM radio chips in the iPhone. From a report: In the wake of three major hurricanes that have wiped out communications for millions of people over the past month, Pai issued a statement urging Apple, one of the largest makers of cellphones in the US, to "reconsider its position, given the devastation wrought by Hurricanes Harvey, Irma, and Maria." FM radios that are already included in every phone could be used to access "lifesaving information" during disasters, he said. For years the majority of smartphones sold in the US have included FM radios, but most of them have been turned off so that you couldn't use the function. Why? Mobile customers would be a lot less likely to subscribe to streaming music services if they could just listen to traditional, free broadcast radio. This incentive is especially true for Apple, which has a streaming music service. Apple said in a statement: "iPhone 7 and iPhone 8 models do not have FM
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We provide several inequalities between eigenvalues of some classical eigenvalue problems on domains with $C^2$ boundary in complete Riemannian manifolds. A key tool in the proof is the generalized Rellich identity on a Riemannian manifold. Our results in particular extend some inequalities due to Kutller and Sigillito from subsets of $\mathbb{R}^2$ to the manifold setting.
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DPcoloring of a simple graph is a generalization of list coloring, and also a generalization of signed coloring of signed graphs. It is known that for each $k \in \{3, 4, 5, 6\}$, every planar graph without $C_k$ is 4choosable. Furthermore, Jin, Kang, and Steffen \cite{JKS} showed that for each $k \in \{3, 4, 5, 6\}$, every signed planar graph without $C_k$ is signed 4choosable. In this paper, we show that for each $k \in \{3, 4, 5, 6\}$, every planar graph without $C_k$ is 4DPcolorable, which is an extension of the above results.
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Dvo\v{r}\'{a}k and Postle \cite{DP} introduced a \textit{DPcoloring} of a simple graph as a generalization of a listcoloring. They proved a Brooks' type theorem for a DPcoloring, and Bernshteyn, Kostochka and Pron \cite{BKP} extended it to a DPcoloring of multigraphs. However, detailed structure when a multigraph does not admit a DPcoloring was not specified in \cite{BKP}. In this note, we make this point clear and give the complete structure. This is also motivated by the relation to signed coloring of signed graphs.
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We consider oriented longrange percolation on a graph with vertex set $\mathbb{Z}^d \times \mathbb{Z}_+$ and directed edges of the form $\langle (x,t), (x+y,t+1)\rangle$, for $x,y$ in $\mathbb{Z}^d$ and $t \in \mathbb{Z}_+$. Any edge of this form is open with probability $p_y$, independently for all edges. Under the assumption that the values $p_y$ do not vanish at infinity, we show that there is percolation even if all edges of length more than $k$ are deleted, for $k$ large enough. We also state the analogous result for a longrange contact process on $\mathbb{Z}^d$.
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On a reduced complex space $X$, weakly holomorphic $1$forms are holomorphic $1$forms on the regular part $X_{\mathrm{reg}}$ that extend holomorphically across the exceptional set of a resolution of singularities of $X$. We show that if the sheaf of germs of weakly holomorphic $1$forms on $X$ is locally free, then $X$ is smooth.
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This work considers the problem of numerically approximating statistical moments of a Quantity of Interest (QoI) that depends on the solution of a linear parabolic partial differential equation. The geometry is assumed to be random and is parameterized by $N$ random variables. The parabolic problem is remapped to a fixed deterministic domain with random coefficients and shown to admit an extension on a well defined region embedded in the complex hyperplane. A Stochastic collocation method with an isotropic Smolyak sparse grid is used to compute the statistical moments of the QoI. In addition, convergence rates for the stochastic moments are derived and compared to numerical experiments.
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We study ancient Khmer ephemerides described in 1910 by the French engineer Faraut, in order to determine whether they rely on observations carried out in Cambodia. These ephemerides were found to be of Indian origin and have been adapted for another longitude, most likely in Burma. A method for estimating the date and place where the ephemerides were developed or adapted is described and applied.
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Using Vovk's outer measure, which corresponds to a minimal superhedging price, the existence of quadratic variation is shown for "typical price paths" in the space of c\`adl\`ag functions possessing a mild restriction on the jumps directed downwards. In particular, this result includes the existence of quadratic variation of "typical price paths" in the space of nonnegative c\`adl\`ag paths and implies the existence of quadratic variation in the sense of F\"ollmer quasi surely under all martingale measures. Based on the robust existence of the quadratic variation, a modelfree It\^o integration is developed.
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The famous MLC Conjecture states that the Mandelbrot set is locally connected, and it is considered by many to be the central conjecture in onedimensional complex dynamics. Among others, it implies density of hyperbolicity in the quadratic family $\{z^2+c\}_{c\in\mathbb{C}}$. We describe recent advances on MLC and the relations between MLC, the Density of Hyperbolicity Conjecture, the Rigidity Conjecture, the No Invariant Line Fields Conjecture, and the Triviality of Fibers Conjecture. We treat families of unicritical polynomials and rational maps as well as the exponential family and families of transcendental maps with finitely many singular values.
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We establish a variational inequality formulation that captures the transonic shock for a steady compressible potential flow. Its critical point satisfies the transonic equation; moreover the associated jump conditions across its free boundary match the usual RankineHugoniot jump conditions for a shock. By means of example we validate our formulation, and establish the necessary and sufficient condition for the existence of a transonic shock. Numerical results are also discussed.
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Common highdimensional methods for prediction rely on having either a sparse signal model, a model in which most parameters are zero and there are a small number of nonzero parameters that are large in magnitude, or a dense signal model, a model with no large parameters and very many small nonzero parameters. We consider a generalization of these two basic models, termed here a "sparse+dense" model, in which the signal is given by the sum of a sparse signal and a dense signal. Such a structure poses problems for traditional sparse estimators, such as the lasso, and for traditional dense estimation methods, such as ridge estimation. We propose a new penalizationbased method, called lava, which is computationally efficient. With suitable choices of penalty parameters, the proposed method strictly dominates both lasso and ridge. We derive analytic expressions for the finitesample risk function of the lava estimator in the Gaussian sequence model. We also provide an deviation bound fo
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Given an inner function $\theta$, the associated starinvariant subspace $K^\infty_\theta$ is formed by the functions $f\in H^\infty$ that annihilate (with respect to the usual pairing) the shiftinvariant subspace $\theta H^1$ of the Hardy space $H^1$. Assuming that $B$ is an interpolating Blaschke product with zeros $\{a_j\}$, we characterize the traces of functions from $K^\infty_B$ on the sequence $\{a_j\}$. The trace space that arises is, in general, nonideal (i.e., the sequences $\{w_j\}$ belonging to it admit no nice description in terms of $w_j$), but we point out explicit  and sharp  size conditions on $w_j$ which make it possible to solve the interpolation problem $f(a_j)=w_j$ ($j=1,2,\dots$) with a function $f\in K^\infty_B$.
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In a digital communication system, information is sent from one place to another over a noisy communication channel using binary symbols (bits). Original information is encoded by adding redundant bits, which are then used by lowdensity paritycheck (LDPC) codes to detect and correct errors that may have been introduced during transmission. Error correction capability of an LDPC code is severely degraded due to harmful structures such as small cycles in its bipartite graph representation known as Tanner graph (TG). We introduce an integer programming formulation to generate a TG for a given smallest cycle length. We propose a branchandcut algorithm for its solution and investigate structural properties of the problem to derive valid inequalities and variable fixing rules. We introduce a heuristic to obtain feasible solutions of the problem. Our computational experiments show that our algorithm can generate LDPC codes without small cycles in acceptable amount of time for practicall
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Coxeter defined the notion of frieze pattern. Conway and Coxeter proved that triangulations of polygons are in bijection with integral frieze patterns. We show a $p$angulated generalisation involving nonintegral frieze patterns. We also show that polygon dissections give rise to even more general nonintegral frieze patterns.
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Photoacoustic tomography is a hybrid imaging technique that combines high optical tissue contrast with high ultrasound resolution. Direct reconstruction methods such as filtered backprojection, time reversal and least squares suffer from curved line artefacts and blurring, especially in case of limited angles or strong noise. In recent years, there has been great interest in regularised iterative methods. These methods employ prior knowledge on the image to provide higher quality reconstructions. However, easy comparisons between regularisers and their properties are limited, since many tomography implementations heavily rely on the specific regulariser chosen. To overcome this bottleneck, we present a modular reconstruction framework for photoacoustic tomography. It enables easy comparisons between regularisers with different properties, e.g. nonlinear, higherorder or directional. We solve the underlying minimisation problem with an efficient firstorder primaldual algorithm. Conver
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This document seeks to prove there are infinitely many primes whose difference is 2, referred to as twin prime pairs. This proof's methodology involves constructing a function that approximates the number of positive integers, less than a known twin prime pair, which can be mapped to a twin prime pair greater than the known one by multiplication. This function is shown to be unbounded and less than the true count of integers it seeks to approximate for the majority of twin prime pairs. Additionally, it is shown there must be infinitely many integers that map a twin prime pair to one larger than itself without the use of the previously mentioned approximation.
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We prove a fast computable criterion that expresses nonflatness in terms of torsion: Let R be a regular algebra of finite type over a field K of characteristic zero and let F be a module finitely generated over an Ralgebra of finite type. Given a maximal ideal m in R, let S be the coordinate ring of the blowingup of Spec(R) at the closed point m. Then F is flat over R localized in m if and only if the tensor product of F with S over R is a torsionfree module over R localized in m. If K is the field of reals or complex numbers, we give a stronger criterion  without the regularity assumption on R. We also show the corresponding results in the real and complexanalytic categories.
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Korpelainen, Lozin, and Razgon conjectured that a hereditary property of graphs which is wellquasiordered by the induced subgraph order and defined by only finitely many minimal forbidden induced subgraphs is labelled wellquasiordered, a notion stronger than that of $n$wellquasiorder introduced by Pouzet in the 1970s. We present a counterexample to this conjecture. In fact, we exhibit a hereditary property of graphs which is wellquasiordered by the induced subgraph order and defined by finitely many minimal forbidden induced subgraphs yet is not $2$wellquasiordered. This counterexample is based on the widdershins spiral, which has received some study in the area of permutation patterns.
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We exhibit the simplex category $\Delta$ as an $\infty$categorical localization of the category $\Omega_\pi$ of plane rooted trees introduced by Moerdijk and Weiss. As an application we obtain an equivalence of $\infty$categories between $2$Segal simplicial spaces as introduced by Dyckerhoff and Kapranov and invertible nonsymmetric $\infty$operads. In addition, we prove analogous results where $\Delta$ is replaced by Connes' cyclic category $\Lambda$, the category of finite pointed sets or the category of nonempty finite sets; the corresponding categories of trees are given by plane trees, rooted trees and abstract trees, respectively.
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We consider in this work a system of two stochastic differential equations named the perturbed compositional gradient flow. By introducing a separation of fast and slow scales of the two equations, we show that the limit of the slow motion is given by an averaged ordinary differential equation. We then demonstrate that the deviation of the slow motion from the averaged equation, after proper rescaling, converges to a stochastic process with Gaussian inputs. This indicates that the slow motion can be approximated in the weak sense by a standard perturbed gradient flow or the continuoustime stochastic gradient descent algorithm that solves the optimization problem for a composition of two functions. As an application, the perturbed compositional gradient flow corresponds to the diffusion limit of the Stochastic Composite Gradient Descent (SCGD) algorithm for minimizing a composition of two expectedvalue functions in the optimization literatures. For the strongly convex case, such an an
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BPS coherent states closely resemble semiclassical states and they have gravity dual descriptions in terms of semiclassical geometries. The half BPS coherent states have been well studied, however less is known about quarter BPS coherent states. Here we provide a construction of quarter BPS coherent states. They are coherent states built with two matrix fields, generalizing the half BPS case. These states are both the eigenstates of annihilation operators and in the kernel of dilatation operator. Another useful labeling of quarter BPS states is by representations of Brauer algebras and their projection onto a subalgebra $\mathbb{C}[S_n\times S_m]$. Here, the SchurWeyl duality for the Walled Brauer algebra plays an important role in organizing the operators. One interesting subclass of these Brauer states are labeled by representations involving two Young tableaux. We obtain the overlap between quarter BPS Brauer states and quarter BPS coherent states, where the Schur polynomials are u
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In this paper, we prove convergence in distribution of Langevin processes in the overdamped asymptotics. The proof relies on the classical perturbed test function (or corrector) method, which is used both to show tightness in path space, and to identify the extracted limit with a martingale problem. The result holds assuming the continuity of the gradient of the potential energy, and a mild control of the initial kinetic energy.
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A projective manifold $M$ is algebraically hyperbolic if there exists a positive constant $A$ such that the degree of any curve of genus $g$ on $M$ is bounded from above by $A(g1)$. A classical result is that Kobayashi hyperbolicity implies algebraic hyperbolicity. It is known that Kobayashi hyperbolic manifolds have finite automorphism groups. Here we prove that, more generally, algebraically hyperbolic projective manifolds have finite automorphism groups.
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This paper introduces Schurconstant equilibrium distribution models of dimension n for arithmetic nonnegative random variables. Such a model is defined through the (several orders) equilibrium distributions of a univariate survival function. First, the bivariate case is considered and analyzed in depth, stressing the main characteristics of the Poisson case. The analysis is then extended to the multivariate case. Several properties are derived, including the implicit correlation and the distribution of the sum.
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In this paper we present and expand upon procedures for obtaining large d digit prime number to an arbitrary probability. We use a layered approach. The first step is to limit the pool of random number to exclude numbers that are obviously composite. We first remove any number ending in 1,3,7 or 9. We then exclude numbers whose digital root is not 3, 6, or 9. This sharply reduces the probability of the random number being composite. We then use the Prime Number Theorem to find the probability that the selected number n is prime and use primality tests to increase the probability to an arbitrarily high degree that n is prime. We apply primality tests including Euler's test based on Fermat Little theorem and the MillerRabin test. We computed these conditional probabilities and implemented it using the GNU GMP library.
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