The Tate conjecture for K3 surfaces over finite fields

Artin’s conjecture states that supersingular K3 surfaces over finite fields have Picard number 22. In this paper, we prove Artin’s conjecture over fields of characteristic p≥5. This implies Tate’s conjecture for K3 surfaces over finite fields of characteristic p≥5. Our results also yield the Tate conjecture for divisors on certain holomorphic symplectic varieties over finite fields, with some restrictions on the characteristic. As a consequence, we prove the Tate conjecture for cycles of codimension 2 on cubic fourfolds over finite fields of characteristic p≥5.     

The Integration of Math and Science Via Centroids

Science problems enhance and promote math functions to establish some formulas to solve them; conversely, many math results give the explanation and the development of science phenomena and their related situation. From Archimedes’ Law of the Lever, together with some properties of vector's representation, the geometrical construction of the weighted centroid of gravity of finite particles is given, the new proving of Ceva's and Menelaus's results is explored, and a related result to spacial shape is set up. These presentations are important in math, physics, chemistry, statistics, and engineering. The ideas are significant to these fields for the integration of multifarious curricula.     

Conceptual issues in understanding the inner logic of statistical inference: Insights from two teaching experiments

We report on a sequence of two classroom teaching experiments that investigated high school students’ understandings as they explored connections among the ideas comprising the inner logic of statistical inference—ideas involving a core image of sampling as a repeatable process, and the organization of its outcomes into a distribution of sample statistics as a basis for making inferences. Students’ responses to post-instruction test questions indicate that despite understanding various individual components of inference—a sample, a population, and a distribution of a sample statistic—their abilities to coordinate and compose these into a coherent and well-connected scheme of ideas were usually tenuous. We argue that the coordination and composition required to assemble these component ideas into a coherent scheme is a major source of difficulty in developing a deep understanding of inference.     

The number of flags in finite vector spaces: asymptotic normality and Mahonian statistics

We study the generalized Galois numbers which count flags of length r in N-dimensional vector spaces over finite fields. We prove that the coefficients of those polynomials are asymptotically Gaussian normally distributed as N becomes large. Furthermore, we interpret the generalized Galois numbers as weighted inversion statistics on the descent classes of the symmetric group on N elements and identify their asymptotic limit as the Mahonian inversion statistic when r approaches ∞. Finally, we apply our statements to derive further statistical aspects of generalized Rogers–Szeg? polynomials, reinterpret the asymptotic behavior of linear q-ary codes and characters of the symmetric group acting on subspaces over finite fields, and discuss implications for affine Demazure modules and joint probability generating functions of descent-inversion statistics.     

Remark on the norm of random Hankel matrices

In recent years, the asymptotic properties of structured random matrices have attracted the attention of many experts involved in probability theory. In particular, R. Adamczak (J. Theor. Probab., Vol. 23, 2010) proved that, under fairly weak conditions, the squared spectral norms of large square Hankel matrices generated by independent identically distributed random variables grow with probability 1, as Nln(N), where N is the size of a matrix. On the basis of these results, by using the technique and ideas of Adamczak’s paper cited above, we prove that, under certain constraints, the squared spectral norms of large rectangular Hankel matrices generated by linear stationary sequences grow almost certainly no faster than Nln(N), where N is the number of different elements in a Hankel matrix. Nekrutkin (Stat. Interface, Vol. 3, 2010) pointed out that this result may be useful for substantiating (by using series of perturbation theory) so-called “signal subspace methods,” which are often used for processing time series. In addition to the main result, the paper contains examples and discusses the sharpness of the obtained inequality.     

Elliptic logarithms,diophantine approximation and the Birch and Swinnerton-Dyer conjecture

Most, if not all, unconditional results towards the abc-conjecture rely ultimately on classical Baker’s method. In this article, we turn our attention to its elliptic analogue. Using the elliptic Baker’s method, we have recently obtained a new upper bound for the height of the S-integral points on an elliptic curve. This bound depends on some parameters related to the Mordell-Weil group of the curve. We deduce here a bound relying on the conjecture of Birch and Swinnerton-Dyer, involving classical, more manageable quantities. We then study which abc-type inequality over number fields could be derived from this elliptic approach.     

Validation of regression metamodels in simulation: Bootstrap approach

Simulation experiments are often analyzed through a linear regression model of their input/output data. Such an analysis yields a metamodel or response surface for the underlying simulation model. This metamodel can be validated through various statistics; this article studies (1) the coefficient of determination (R-square) for generalized least squares, and (2) a lack-of-fit F-statistic originally formulated by Rao [Biometrika 46 (1959) 49], who assumed multivariate normality. To derive the distributions of these two validation statistics, this paper shows how to apply bootstrapping—without assuming normality. To illustrate the performance of these bootstrapped validation statistics, the paper uses Monte Carlo experiments with simple models. For these models (i) R-square is a conservative statistic (rejecting a valid metamodel relatively rarely), so its power is low; (ii) Rao’s original statistic may reject a valid metamodel too often; (iii) bootstrapping Rao’s statistic gives only slightly conservative results, so its power is relatively high.     

A Cantorian potential theory for describing dynamical systems on El Naschie’s space–time

In this paper we analyze classical systems, in which motion is not on a classical continuous path, but rather on a Cantorian one. Starting from El Naschie’s space–time we introduce a mathematical approach based on a potential to describe the interaction system-support. We study some relevant force fields on Cantorian space and analyze the differences with respect to the analogous case on a continuum in the context of Lagrangian formulation. Here we confirm the idea proposed by the first author in dynamical systems on El Naschie’s ϵ^(∞) Cantorian space–time that a Cantorian space could explain some relevant stochastic and quantum processes, if the space acts as an harmonic oscillating support, such as that found in Nature. This means that a quantum process could sometimes be explained as a classical one, but on a nondifferential and discontinuous support. We consider the validity of this point of view, that in principle could be more realistic, because it describes the real nature of matter and space. These do not exist in Euclidean space or curved Riemanian space–time, but in a Cantorian one. The consequence of this point of view could be extended in many fields such as biomathematics, structural engineering, physics, astronomy, biology and so on.     

A remark on Gagola’s Theorem

In this paper, we generalize Gagola’s Theorem [1]. Firstly we obtain several new identities. With the help of these identities, we prove a conclusion similar with Gagola’s under some more general conditions. Finally, we get a result regarding the control of p-transfer by Tate’s Theorem.     

Monotone convergence of Newton-like methods for M-matrix algebraic Riccati equations

For the algebraic Riccati equation whose four coefficient matrices form a nonsingular M-matrix or an irreducible singular M-matrix K, the minimal nonnegative solution can be found by Newton’s method and the doubling algorithm. When the two diagonal blocks of the matrix K have both large and small diagonal entries, the doubling algorithm often requires many more iterations than Newton’s method. In those cases, Newton’s method may be more efficient than the doubling algorithm. This has motivated us to study Newton-like methods that have higher-order convergence and are not much more expensive each iteration. We find that the Chebyshev method of order three and a two-step modified Chebyshev method of order four can be more efficient than Newton’s method. For the Riccati equation, these two Newton-like methods are actually special cases of the Newton–Shamanskii method. We show that, starting with zero initial guess or some other suitable initial guess, the sequence generated by the Newton–Shamanskii method converges monotonically to the minimal nonnegative solution.We also explain that the Newton-like methods can be used to great advantage when solving some Riccati equations involving a parameter.     

Conditional Probability on the Kopka＇s D-posets

K?pka’s D-poset is a very important notion in quantum structures. In this paper the conditional probability on the K?pka’s D-posets is studied. The notion of conditional probability is introduced and the basic properties of conditional probability are proved.     

Confirmation,Increase in Probability,and the Likelihood Ratio Measure: a Reply to Glass and McCartney

Bayesian confirmation theory is rife with confirmation measures. Zalabardo (2009) focuses on the probability difference measure, the probability ratio measure, the likelihood difference measure, and the likelihood ratio measure. He argues that the likelihood ratio measure is adequate, but each of the other three measures is not. He argues for this by setting out three adequacy conditions on confirmation measures and arguing in effect that all of them are met by the likelihood ratio measure but not by any of the other three measures. Glass and McCartney (2015), hereafter “G&M,” accept the conclusion of Zalabardo’s argument along with each of the premises in it. They nonetheless try to improve on Zalabardo’s argument by replacing his third adequacy condition with a weaker condition. They do this because of a worry to the effect that Zalabardo’s third adequacy condition runs counter to the idea behind his first adequacy condition. G&M have in mind confirmation in the sense of increase in probability: the degree to which E confirms H is a matter of the degree to which E increases H’s probability. I call this sense of confirmation “IP.” I set out four ways of precisifying IP. I call them “IP1,” “IP2,” “IP3,” and “IP4.” Each of them is based on the assumption that the degree to which E increases H’s probability is a matter of the distance between p(H | E) and a certain other probability involving H. I then evaluate G&M’s argument (with a minor fix) in light of them.     

Holography and Local Fields

This paper, which is a summary (in which considerable creative license has been taken) of the author’s talk at the sixth international conference on p-adic mathematical physics and its applications (CINVESTAV, Mexico City, October 2017), reviews some recent work connecting field theories defined on the p-adic numbers and ideas from the AdS/CFT correspondence. Some results are included, along with general discussion of the utility and interest of p-adic analogues of Lagrangian field theories, at least from the author’s perspective. A few challenges, shortcomings, and ideas for future work are also discussed.     

Short-term financial planning with uncertain receipts and disbursements

The problem of short-term financial planning is to determine an optimal credit mix to meet the short-term cash needs and an optimal investment plan for excess cash. A number of linear optimization models have been developed to solve this problem, some of which are in practical use. The purpose of this paper is to generalize the assumptions of these models concerning the available information about future receipts and disbursements. It is presupposed that the financial officer has some idea as to the amount involved which, however, cannot be specified by a probability distribution. On the contrary, we assume that these ideas only permit qualitative probability statements such as the following:“That the difference between disbursements and receipts in a certain period lies in an interval I₁ is no less probable than that it lies in an interval I₂”.For this level of information we formulate a model for short-term financial planning, and we develop a solution procedure to determine the optimum financial alternatives. Finally, the entire procedure is demonstrated by a medium sized example.     

Stationary Increments Harmonizable Stable Fields: Upper Estimates on Path Behaviour

Studying sample path behaviour of stochastic fields/processes is a classical research topic in probability theory and related areas such as fractal geometry. To this end, many methods have been developed for a long time in Gaussian frames. They often rely on some underlying “nice” Hilbertian structure and can also require finiteness of moments of high order. Therefore, they can hardly be transposed to frames of heavy-tailed stable probability distributions. However, in the case of some linear non-anticipative moving average stable fields/processes, such as the linear fractional stable sheet and the linear multi-fractional stable motion, rather new wavelet strategies have already proved to be successful in order to obtain sharp moduli of continuity and other results on sample path behaviour. The main goal of our article is to show that, despite the difficulties inherent in the frequency domain, such kind of a wavelet methodology can be generalized and improved, so that it also becomes fruitful in a general harmonizable stable setting with stationary increments. Let us point out that there are large differences between this harmonizable setting and the moving average stable one. The real-valued harmonizable stable stochastic field X on which we focus is defined on \(\mathbb {R}^d\) through an arbitrary spectral density belonging to a general and wide class of functions. First, we introduce a wavelet-type random series representation of X and express it as the finite sum \(X=\sum _\eta X^\eta \), where the fields \(X^\eta \) are called the \(\eta \)-frequency parts, since they extend the usual low-frequency and high-frequency parts. Moreover, we show the continuity of the sample paths of the \(X^\eta \)’s and X; also, we discuss the existence and continuity of their partial derivatives of an arbitrary order. Thereafter, we obtain several almost sure upper estimates related to: (a) the anisotropic behaviour of generalized directional increments of the \(X^\eta \)’s and X, on an arbitrary fixed compact cube of \(\mathbb {R}^d\); (b) the behaviour at infinity of the \(X^\eta \)’s, of X, and of their partial derivatives, when they exist. We mention that all the results on sample paths obtained in the article are valid on the same event of probability 1; furthermore, this event is “universal”, in the sense that it does not depend, in any way, on the spectral density associated with X.     

Power functions with low uniformity on odd characteristic finite fields

In this paper, we give some new low differential uniformity of some power functions defined on finite fields with odd characteristic. As corollaries of the uniformity, we obtain two families of almost perfect nonlinear functions in GF(3 ⁿ ) and GF(5 ⁿ ) separately. Our results can be used to prove the Dobbertin et al.’s conjecture.