Abstract We present a novel method to analyze extreme events of flows over manifolds called Peaks Over Manifold (POM). Here we show that under general and realistic hypotheses, the distribution of affectation measures converges to a Generalized Pareto Distribution (GPD). The method is applicable to floods,ice cover extent, extreme rainfall or marine heatwaves. We present an appli cation to a synthetic data set on tide height and to real ice cover data in An tartica.

Abstract We provide the derivation of a new formula for the approximation of an integral Markov process arising in the approximation of stochastic differen tial equations. This formula extends an existing formula derived in [1]. We have shown numerically that the leading order approximation of the differen tial equation with noise by solving an associated averaged problem and esti mating the difference between them and the result is illustrated through some examples.

Abstract This paper introduces how to use geometric figures to represent integers, and successfully proves Goldbach’s conjecture by using the mapping relationship between the internal angles of circles and sectors and the number of integers. It is also explained and proved that w(n) is the function that calculates the lower limit of the number of prime pairs. A very effective new method is found to solve this kind of integer problems.

Abstract This paper presents a graphical procedure, using an unmarked straightedge and compass only, for trisecting an arbitrary acute angle. The procedure, when applied to the 30˚ angle that has been “proven” to be not trisectable, produced a construction having the identical angular relationship with Archimedes’ Construction, in which the required trisection angle was found to be exactly one-third of the given angle (or ∠E'MA = 1/3∠E'CG = 10˚), as shown in Figure 1(D) and Figure 1(E) and Section 4 PROOF in this paper. Hence, based on this identical angular relationship between the construction presented and Archimedes’ Construction, one can only conclude that geometric requirements for arriving at an exact trisection have been met, notwithstanding the theoretical proofs of Wantzel, Dudley, and others.

Abstract The finite field q has q elements, where k q p = for prime p and k ∈ . Then q [ x] is a unique factorization domain and its polynomials can be bijectively associated with their unique (up to order) factorizations into irreducibles. Such a factorization for a polynomial of degree n can be viewed as conforming to a specific template if we agree that factors with higher degree will be written before those with lower degree, and factors of equal degree can be written in any order. For example, a polynomial f x( ) of degree n may factor into irreducibles and be written as (a)(b)(c), where deg deg deg abc ≥ ≥ . Clearly, the various partitions of n correspond to the templates available for these canonical factorizations and we identify the templates with the possible partitions. So if f x( ) is itself irreducible over Fq , it would belong to the template [n], and if f x( ) split over q , it would belong to the template [1,1, ,1 ] . Our goal is to calculate the cardinalities of the sets of polynomials corresponding to available templates for general q and n. With this information, we characterize the associated probabilities that a randomly selected member of Fq [ x] belongs to a given template. Software to facilitate the investigation of various cases is available upon request from the authors.

Abstract It is known that the prime-number-formula at any distance from the origin has a systematic error. It is shown that this error is proportional to the square of the number of primes present up to the square root of the distance. The proposed completion of the prime-number-formula in the present paper eliminates this systematic error. This is achieved by using a quickly converging recursive formula. The remaining error is reduced to a symmetric dispersion of the effective number of primes around the completed prime-number-formula. The standard deviation of the symmetric dispersion at any distance is proportional to the number of primes present up to the square root of the distance. Therefore, the absolute value of the dispersion, relative to the number of primes is approaching zero and the number of primes resulting from the prime-number formula represents the low limit of the number of primes at any distance.

Abstract The purpose of this study is to prove the Collatz conjecture using a theorem proving system. First, the division sequence is defined as an alignment of the number of times division by 2 is performed in the Collatz operation. Then, the star conversion is defined, which is a mapping from a specific division sequence to a division sequence. Here it is important to map to some division sequence, not which division sequence. The important point is that the finite length of the division sequence does not change before and after the star conversion. In theorem proving system, we considered two parallel methods: main-proof is a claim to a computer proposition that has the same meaning as the Collatz conjecture. Theorem proving support system “Idris” was used.Moreover, we sub-proved that the 12 “extended star conversion” are closed to the “Collatz operation”. Egison’s computer algebra system is used for proof. The results of the two methods achieved the goal of proving the Collatz conjecture using a theorem proving system.

Abstract In this paper, we propose a robust mixture regression model based on the skew scale mixtures of normal distributions (RMR-SSMN) which can accommodate asymmetric, heavy-tailed and contaminated data better. For the variable selection problem, the penalized likelihood approach with a new combined penalty function which balances the SCAD and 2l penalty is proposed. The adjusted EM algorithm is presented to get parameter estimates of RMRSSMN models at a faster convergence rate. As simulations show, our mixture models are more robust than general FMR models and the new combined penalty function outperforms SCAD for variable selection. Finally, the proposed methodology and algorithm are applied to a real data set and achieve reasonable results.

Abstract In this paper, we mainly study the orbital graphs of primitive groups with the socle A A 7 7 × which acts by diagonal action. Firstly, we calculate the element conjugate classes of A7 , then we discuss the stabilizer of two points in A7 . Finally, according to the relation between suborbit and orbital, we obtain the orbitals, and determine the orbital graphs.

Abstract Möbius transformations, which are one-to-one mappings of  onto have remarkable geometric properties susceptible to be visualized by drawing pictures. Not the same thing can be said about m-Möbius transformations mf mapping m  onto  . Even for the simplest entity, the pre-image by mf of a unique point, there is no way of visualization. Pre-images by mf of figures from  are like ghost figures in m  . This paper is about handling those ghost figures. We succeeded in doing it and proving theorems about them by using their projections onto the coordinate planes. The most important achievement is the proof in that context of a theorem similar to the symmetry principle for Möbius transformations. It is like saying that the images by m-Möbius transformations of symmetric ghost points with respect to ghost circles are symmetric points with respect to the image circles. Vectors in m  are well known and vector calculus in m  is familiar, yet the pre image by mf of a vector from  is a different entity which materializes by projections into vectors in the coordinate planes. In this paper, we study the interface between those entities and the vectors in m  . Finally, we have shown that the uniqueness theorem for Möbius transformations and the property of preserving the cross-ratio of four points by those transformations translate into similar theorems for m-Möbius transformations.

Abstract The union of the straight and over the point of reflection—reflected series of the arithmetic progression of primes results in the double density of occupation of integer positions. It is shown that the number of free positions left by the double density of occupation has a lower limit function, which is growing to infinity. The free positions represent equidistant primes to the point of reflection: in case the point of reflection is an even number, they satisfy Goldbach’s conjecture. The double density allows proving as well that at any distance from the origin large enough—the distance between primes is smaller, than the square root of the distance to the origin. Therefore, the series of primes represent a continuum and may be integrated. Furthermore, it allows proving that the largest gap between primes is growing to infinity with the distance and that the number of any two primes, with a given even number as the distance between them, is unlimited. Thus, the number of twin primes is unlimited as well.

Abstract This work shows, after a brief introduction to Riemann zeta function ζ (z) ,the demonstration that all non-trivial zeros of this function lies on the so-called “critical line”, ℜ = (z) 1 2 , the one Hardy demonstrated in his famous work that infinite countable zeros of the above function can be found on it. Thus, out of this strip, the only remaining zeros of this function are the so-called “trivial ones” z n = −2 , n ∈  . After an analytical introduction reminding the existence of a germ from a generic zero lying in ℜ = (z) 1 2 , we show through a Weierstrass-Hadamard representation approach of the above germ that nontrivial zeros out of ℜ = (z) 1 2 cannot be found.

Abstract In this paper, we consider the general ordinary quasi-differential expression τ of order n with complex coefficients and its formal adjoint τ+ on the interval [a b, ) . We shall show in the case of one singular end-point and under suitable conditions that all solutions of a general ordinary quasi-differential equation (τ λ − = w u wf ) are in the weighted Hilbert space ( ) 2 , L ab w provided that all solutions of the equations (τ λ − = w u) 0 and its adjoint (τ λ w v) 0 + − = are in ( ) 2 , L ab w . Also, a number of results concerning the location of the point spectra and regularity fields of the operators generated by such expressions may be obtained. Some of these results are extensions or generalizations of those in the symmetric case, while the others are new.