A Note on “New Characterizations of P-Supersolubility of Finite Groups”

In this note, two propositions and two examples are given to indicate that one main result in “New Characterizations of p-Supersolubility of Finite Groups” is not correct.     

Note on the Paper “A Note on the SVD of Matrices”

Let F be a field, R be an F-algebra having SVD property['], R+ = {a e Rla = a}. In[2], we proved thatLemma 1 Let R be an F-algebra which having SVD proPerty Then R+ is a sub6e1d inF, R+ is a fOrmaIly rea1 6eId, and R+ is Pythagorean with no zerChdivisor.In the proof of Lernma 1 of [2], thereis this step: "FOr any a E R+ l clearly, there ekistsd E R+ n F such that a2 = ad = a', thus a = la E F, is fOllOws that R+ C F".Recently, paper [11 shows that "The proof of this step is no…     

A brief note on analyticity and causality,and the “levels of physics”

We argue that the causality of physics has its roots in the analyticity within the two dimensions of a fractal quantum path and further show how this picture has convergence with quantum superstrings.     

A remark on the size of πq (Sn)

Classical results of homotopy theory are used to prove that dim for all primes p. Simultaneously we obtain bounds on the order of the p-torsion subgroup of _q(Sⁿ).partially supported by Studienstiftung des deutschen Volkes     

A New Lower Bound Based on Gromov’s Method of Selecting Heavily Covered Points

Boros and Füredi (for d=2) and Bárány (for arbitrary d) proved that there exists a positive real number c _d such that for every set P of n points in R ^d in general position, there exists a point of R ^d contained in at least $c_{d}\binom{n}{d+1}$ d-simplices with vertices at the points of P. Gromov improved the known lower bound on c _d by topological means. Using methods from extremal combinatorics, we improve one of the quantities appearing in Gromov??s approach and thereby provide a new stronger lower bound on c _d for arbitrary d. In particular, we improve the lower bound on c ₃ from 0.06332 to more than 0.07480; the best upper bound known on c ₃ being 0.09375.     

A Note on the Rate of Convergence of the Euler–Maruyama Method for Stochastic Differential Equations

Stochastic processes are approximated by wavelet operators. For the stochastic process and the scale function satisfying some conditions, a new method is given to estimate the degree of approximation.     

A Certain Regular Property of the Method I Construction and Packing Measure

Let τ be a premeasure on a complete separable metric space and let τ＊ be the Method I measure constructed from τ. We give conditions on T such that τ＊ has a regularity as follows： Every τ＊-measurable set has measure equivalent to the supremum of premeasures of its compact subsets. Then we prove that the packing measure has this regularity if and only if the corresponding packing premeasure is locally finite.     

Proximal Point Method on Finslerian Manifolds and the “Effort–Accuracy” Trade-off

In this paper, we consider minimization problems with constraints. We show that, if the set of constraints is a Finslerian manifold of non-positive flag curvature, and the objective function is differentiable and satisfies the Kurdyka-Lojasiewicz property, then the proximal point method can be naturally extended to solve this class of problems. We prove that the sequence generated by our method is well defined and converges to a critical point. We show how tools of Finslerian geometry, specifically non-symmetrical metrics, can be used to solve non-convex constrained problems in Euclidean spaces. As an application, we give one result regarding decision-making speed and costs related to change.     

A Note on the Range of the Elementary Operator △_(AC)

ＡＮｏｔｅｏｎｔｈｅＲａｎｇｅｏｆｔｈｅＥｌｅｍｅｎｔａｒｙＯｐｅｒａｔｏｒ△_（ＡＣ）ＳｕｎＨｏｎｇｗｅｉ（孙宏伟）ＬｕＳｈｉｊｉｅ（鲁世杰）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＺｈｅｊｉａｎｇＵｎｉｖｅｒｓｉｔｙ，Ｈａｎｇｚｈｏｕ，...     

Hierarchical Elements,Local Mappings and the $h$-$p$ Version of the Finite Element Method (Ⅰ)

This is the first half of an article which develops a theory the hierarchical elements for the $h$-$p$ version of the finite element method in the two-dimensional case. The approximation properties of the hierarchical elements are discussed. The second part will address the convergence rate when geometric meshes are used.     

Hierarchical Elements,Local Mappings and the $H$-$P$ Version of the Finite Element Method (Ⅱ)

This is the second half of the article. The rate of convergence for the $h$-$p$ version with geometric meshes is discussed.