Toroidal b-divisors and Monge–Ampère measures

We generalize the intersection theory of nef toric (Weil) b-divisors on smooth and complete toric varieties to the case of nef b-divisors on complete varieties which are toroidal with respect to a snc divisor. As a key ingredient we show the existence of a limit measure, supported on a balanced rational conical polyhedral space attached to the toroidal embedding, which arises as a limit of discrete measures defined via tropical intersection theory on the polyhedral space. We prove that the intersection theory of nef Cartier b-divisors can be extended continuously to nef toroidal Weil b-divisors and that their degree can be computed as an integral with respect to this limit measure. As an application, we show that a Hilbert–Samuel type formula holds for big and nef toroidal Weil b-divisors.

Stochastic Gene Expression Revisited

We investigate the model of gene expression in the form of Iterated Function System (IFS), where the probability of choice of any iterated map depends on the state of the phase space. Random jump times of the process mark activation periods of the gene when pre-mRNA molecules are produced before mRNA and protein processing phases occur. The main idea is inspired by the continuous-time piecewise deterministic Markov process describing stochastic gene expression. We show that for our system there exists a unique invariant limit measure. We provide full probabilistic description of the process with a comparison of our results to those obtained for the model with continuous time.

Mapping of Difficult Mathematics Topics in Vocational High Schools Based on National Examination Data

The educational achievement in Indonesia is measured by implementing the National Examination or Ujian Nasional (UN). Throughout its implementation, the UN results shown unsatisfying results, including in mathematics subjects. This fact indicated that there are mathematics topics that are considered to be difficult for students. This study aimed to investigate the difficult topics of mathematics in Vocational High School (VHS) based on UN data. This study was a descriptive research with the quantitative approach by using the UN report results in the level of VHS from Center for Educational Assessment of the Ministry of Education and Culture in the period of 2008 to 2017 (ten years) as a research data source. Data analysis was carried out in a quantitative descriptive by mapping the mathematics topics based on proportion correct answers or “daya serap”. A topic was considered difficult if the proportion correct answer was less than 50%. The results showed that there were 14 difficult mathematics topics in the UN implementation at the VHS level from 2008 to 2017. These topics include the comparison of trigonometric functions, solid figure, limit, measure of data dispersion, permutation and combination, differential, area between two curves, a measure of central data tendentious, probability, logarithm, integral, sequence and series, linear program, and polar coordinates system. The implications of the research findings for learning practice and future research opportunities are discussed.

Global Stability Properties of the Climate: Melancholia States, Invariant Measures, and Phase Transitions

<p>For a wide range of values of the incoming solar radiation, the Earth features at least two attracting states, which correspond to competing climates. The warm climate is analogous to the present one; the snowball climate features global glaciation and conditions that can hardly support life forms. Paleoclimatic evidences suggest that in past our planet flipped between these two states. The main physical mechanism responsible for such instability is the ice-albedo feedback. Following an idea developed by Eckhardt and co. for the investigation of multistable turbulent flows, we study the global instability giving rise to the snowball/warm multistability in the climate system by identifying the climatic Melancholia state, a saddle embedded in the boundary between the two basins of attraction of the stable climates. We then introduce random perturbations as modulations to the intensity of the incoming solar radiation. We observe noise-induced transitions between the competing basins of attractions. In the weak noise limit, large deviation laws define the invariant measure and the statistics of escape times. By empirically constructing the instantons, we show that the Melancholia states are the gateways for the noise-induced transitions in the weak-noise limit. In the region of multistability, in the zero-noise limit, the measure is supported only on one of the competing attractors. For low (high) values of the solar irradiance, the limit measure is the snowball (warm) climate. The changeover between the two regimes corresponds to a first order phase transition in the system. The framework we propose seems of general relevance for the study of complex multistable systems. Finally, we propose a new method for constructing Melancholia states from direct numerical simulations, thus bypassing the need to use the edge-tracking algorithm.</p><p>Refs.</p><p>V. Lucarini, T. Bodai, Edge States in the Climate System: Exploring Global Instabilities and Critical Transitions, Nonlinearity 30, R32 (2017)</p><p>V. Lucarini, T. Bodai, Transitions across Melancholia States in a Climate Model: Reconciling the Deterministic and Stochastic Points of View, Phys. Rev. Lett. 122,158701 (2019)</p>

Hand Gesture Recognition for Emoji and Text Prediction

Emoticons' are ideograms and smileys utilized in electronic messages and website pages. Emoticons exist in different classifications, including outward appearances, regular items, places and kinds of climate, and creatures. They are much similar to emojis, however emoticons are real pictures rather than typo graphics. This undertaking perceives the emoticons utilizing hand motions. We are detecting hand gestures and preparing a Convolutional Neural Network (CNN) model on a training dataset. We will make a database of hand gestures and train them. The system utilized here is a CNN. We are utilizing the SIFT filter to identify the hand and CNN for preparing the model. SIFT filter give a lot of highlights of an image that are not influenced by numerous factors, for example, object scaling and rotation. The SIFT filtering procedure comprises of two areas. The first is a procedure to identify intrigue focuses in the hand. Intrigue focuses are the points in the image in a 2D space that surpasses some limit measure and is better than straight forward edge recognition. The second segment is a procedure to make a vector like descriptor and this is the most special and prevalent part of the SIFT filter.

A JOINT ELLIOTT TYPE THEOREM FOR TWISTS OF L-FUNCTIONS OF ELLIPTIC CURVES

We consider a collection of L-functions of elliptic curves twisted by a Dirichlet character modulo q (q is a prime number), and prove for this collection a joint limit theorem for weakly convergent probability measures in the space of analytic functions as q → ∞. The limit measure is given explicitly.

Limit theorems of a two-phase quantum walk with one defect

We treat a position dependent quantum walk (QW) on the line which we assign two different time-evolution operators to positive and negative parts respectively. We call the model “the two-phase QW” here, which has been expected to be a mathematical model of the topological insulator. We obtain the stationary and time-averaged limit measures related to localization for the two-phase QW with one defect. This is the first result on localization for the two-phase QW. The analytical methods are mainly based on the splitted generating function of the solution for the eigenvalue problem, and the generating function of the weight of the passages of the model. In this paper, we call the methods “the splitted generating function method” and “the generating function method”, respectively. The explicit expression of the stationary measure is asymmetric for the origin, and depends on the initial state and the choice of the parameters of the model. On the other hand, the time-averaged limit measure has a starting point symmetry and localization effect heavily depends on the initial state and the parameters of the model. Regardless of the strong effect of the initial state and the parameters, the time-averaged limit measure also suggests that localization can be always observed for our two-phase QW. Furthermore, our results imply that there is an interesting relation between the stationary and time-averaged limit measures when the parameters of the model have specific periodicities, which suggests that there is a possibility that we can analyze localization of the two-phase QW with one defect from the stationary measure.

Limit theorems of a 3-state quantum walk and its application for discrete uniform measures

We present two long-time limit theorems of a 3-state quantum walk on the line when the walker starts from the origin. One is a limit measure which is obtained from the probability distribution of the walk at a long-time limit, and the other is a convergence in distribution for the walker’s position in a rescaled space by time. In addition, as an application of the walk, we obtain discrete uniform limit measures from the 3-state walk with a delocalized initial state.