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.

Boundary growth of Sobolev functions of monotone type for double phase functionals

Our aim in this paper is to deal with boundary growth of spherical means of Sobolev functions of monotone type for the double phase functional \(\Phi_{p,q}(x,t) = t^{p} + (b(x) t)^{q}\) in the unit ball B of \(\mathbb{R}^n\), where \(1 < p < q < \infty\) and \(b(\cdot)\) is a non-negative bounded function on B which is Hölder continuous of order \(\theta \in (0,1]\).

An existence level for the residual sum of squares of the power-law regression with an unknown location parameter

In a recent paper [JUKIĆ, D.: A necessary and sufficient criterion for the existence of the global minima of a continuous lower bounded function on a noncompact set, J. Comput. Appl. Math. 375 (2020)], a new existence level was introduced and then was used to obtain a necessary and sufficient criterion for the existence of the global minima of a continuous lower bounded function on a noncompact set. In this paper, we determined that existence level for the residual sum of squares of the power-law regression with an unknown location parameter, and so we obtained a necessary and sufficient condition which guarantee the existence of the least squares estimate.

Global Existence of Solutions to a Class of Reaction–Diffusion Systems on Rn

We prove in this work the existence of a unique global nonnegative classical solution to the class of reaction–diffusion systems uttx=aΔutx−guvm,vttx=dΔvtx+λtxguvm, where a>0, d>0, t>0,x∈Rn, n,m∈N∗, λ is a nonnegative bounded function with λt.∈BUCRn for all t∈R+, the initial data u0, v0∈BUCRn, g:BUCRn→BUCRn is a of class C1,dgudu∈L∞R, g0=0 and gu≥0 for all u≥0. The ideas of the proof is inspired from the work of Collet and Xin who proved the same result in the particular case d>a=1, λ=1,gu=u. Moreover, they showed that the L∞-norm of v can not grow faster than Olnlnt for any space dimension.

On some smooth approximations on thick periodic multi-structures

In this paper we study the approximation properties of measurable and square-integrable functions. In particular we show that any $L^2$-bounded function can be approximated in $L^2$-norm by smooth functions defined on a highly oscillating boundary of thick multi-structures in ${\mathbb{R}}^n$. We derive the norm estimates for the approximating functions and study their asymptotic behaviour.

Solvability of the abstract evolution equations in L s -spaces with critical temporal weights

This paper deals with the abstract evolution equations in L s {L}^{s} -spaces with critical temporal weights. First, embedding and interpolation properties of the critical L s {L}^{s} -spaces with different exponents s s are investigated, then solvability of the linear evolution equation, attached to which the inhomogeneous term f f and its average Φ f \Phi f both lie in an L 1 / s s {L}_{1\hspace{-0.08em}\text{/}\hspace{-0.08em}s}^{s} -space, is established. Based on these results, Cauchy problem of the semi-linear evolution equation is treated, where the nonlinear operator F ( t , u ) F\left(t,u) has a growth number ρ ≥ s + 1 \rho \ge s+1 , and its asymptotic behavior acts like α ( t ) / t \alpha \left(t)\hspace{-0.1em}\text{/}\hspace{-0.1em}t as t → 0 t\to 0 for some bounded function α ( t ) \alpha \left(t) like ( − log t ) − p {\left(-\log t)}^{-p} with 2 ≤ p < ∞ 2\le p\lt \infty .

Sublacunary sets and interpolation sets for nilsequences

<p style='text-indent:20px;'>A set <inline-formula><tex-math id="M1">\begin{document}$ E \subset \mathbb{N} $\end{document}</tex-math></inline-formula> is an interpolation set for nilsequences if every bounded function on <inline-formula><tex-math id="M2">\begin{document}$ E $\end{document}</tex-math></inline-formula> can be extended to a nilsequence on <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{N} $\end{document}</tex-math></inline-formula>. Following a theorem of Strzelecki, every lacunary set is an interpolation set for nilsequences. We show that sublacunary sets are not interpolation sets for nilsequences. Here <inline-formula><tex-math id="M4">\begin{document}$ \{r_n: n \in \mathbb{N}\} \subset \mathbb{N} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M5">\begin{document}$ r_1 &lt; r_2 &lt; \ldots $\end{document}</tex-math></inline-formula> is <i>sublacunary</i> if <inline-formula><tex-math id="M6">\begin{document}$ \lim_{n \to \infty} (\log r_n)/n = 0 $\end{document}</tex-math></inline-formula>. Furthermore, we prove that the union of an interpolation set for nilsequences and a finite set is an interpolation set for nilsequences. Lastly, we provide a new class of interpolation sets for Bohr almost periodic sequences, and as a result, obtain a new example of interpolation set for <inline-formula><tex-math id="M7">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-step nilsequences which is not an interpolation set for Bohr almost periodic sequences.</p>

APROKSIMASI FUNGSI KONTINU TERBATAS DENGAN KONVOLUSI

Convolution is a mathematical operation on two functions that produces a new function that can be seen as a modified version of one of its original functions. The convolution operator has no identity element. However, it has an approximate identity. It can be found as a sequence of gk such that convolution of f and gk converges to f for k→∞. It implies that convolution can be used to approximate a function. In this article, we have proven basic theorems about approximation function by convolution for a bounded function in C(Rd). Konvolusi adalah suatu operasi pada dua fungsi dan menghasilkan suatu fungsi baru yang dapat dipandang sebagai versi modifikasi dari salah satu fungsi aslinya. Operasi konvolusi tidak memiliki unsur identitas. Namun, operasi konvolusi memiliki identitas hampiran, yakni dapat ditemukannya suatu barisan fungsi gk sehingga konvolusi dari f dan gk konvergen ke f untuk k→∞. Hal ini mengakibatkan konvolusi dapat digunakan untuk aproksimasi fungsi. Pada artikel ini dibuktikan teorema-teorema yang mendasari aproksimasi fungsi dengan konvolusi bagi fungsi terbatas di C(Rd) .