Non-classical polynomials and the inverse theorem

In this paper we characterize when non-classical polynomials are necessary in the inverse theorem for the Gowers $U^k$ -norm. We give a brief deduction of the fact that a bounded function on $\mathbb F_p^n$ with large $U^k$ -norm must correlate with a classical polynomial when $k\le p+1$ . To the best of our knowledge, this result is new for $k=p+1$ (when $p>2$ ). We then prove that non-classical polynomials are necessary in the inverse theorem for the Gowers $U^k$ -norm over $\mathbb F_p^n$ for all $k\ge p+2$ , completely characterising when classical polynomials suffice.

Noether’s Symmetries and Its Inverse for Fractional Logarithmic Lagrangian Systems

In this paper, Noether’s theorem and its inverse theorem are proved for the fractional variational problems based on logarithmic Lagrangian systems. The Hamilton principle of the systems is derived. And the definitions and the criterions of Noether’s symmetry and Noether’s quasi-symmetry of the systems based on logarithmic Lagrangians are given. The intrinsic relation between Noether’s symmetry and the conserved quantity is established. At last an example is given to illustrate the application of the results.

Noether’s theorems of variable mass systems on time scales

This paper deals with the Noether’s theory for variable mass system on time scales. The calculus on time scales unifies and extends variable mass system continuous model and discrete model into a single theory. Firstly, Hamilton’s principle of the variable mass system on time scales is given. Secondly, based on the quasi-invariance of the Hamilton’s action under a group of infinitesimal transformations, Noether’s theorem and its inverse theorem of the variable mass system on time scales are presented. Finally, two examples are given to illustrate the applications of the results.

An inverse theorem for additive bases

Let [Formula: see text] be a finite abelian group of order [Formula: see text], and [Formula: see text] be the smallest prime dividing [Formula: see text]. Let [Formula: see text] be a sequence over [Formula: see text]. We say that [Formula: see text] is regular if for every proper subgroup [Formula: see text], [Formula: see text] contains at most [Formula: see text] terms from [Formula: see text]. Let [Formula: see text] be the smallest integer [Formula: see text] such that every regular sequence [Formula: see text] over [Formula: see text] of length [Formula: see text] forms an additive basis of [Formula: see text], i.e. [Formula: see text]. Recently, [Formula: see text] was determined for many abelian groups. In this paper, we determined [Formula: see text] for more abelian groups and characterize the structure of the regular sequence [Formula: see text] over [Formula: see text] of length [Formula: see text] and [Formula: see text].