Local Hölder continuity of nonnegative weak solutions of inverse variation-inequality problems of non-divergence type

Yan Dong; Yan Dong

doi:10.3934/era.2024023

Electronic Research Archive

2024, Volume 32, Issue 1: 473-485. doi: 10.3934/era.2024023

Previous Article Next Article

Research article

Local Hölder continuity of nonnegative weak solutions of inverse variation-inequality problems of non-divergence type

Yan Dong ^,

Department of Basic, Shaanxi Railway Institute, Weinan 714000, China

Received: 31 July 2023 Revised: 10 October 2023 Accepted: 27 November 2023 Published: 02 January 2024

Compared to the standard variational inequalities, inverse variational inequalities are more suitable for pricing American options with indefinite payoff. This paper investigated the initial-boundary value problem of inverse variational inequalities constituted by a class of non-divergence type parabolic operators. We established the existence and Hölder continuity of weak solutions. Since the comparison principle in the case of standard variational inequalities is no longer applicable, we constructed an integral inequality using differential inequalities to determine the global upper bound of the solution. By combining it with the continuous method, we obtained the existence of weak solutions. Additionally, by employing truncation factors, we obtained the lower bound of weak solutions in the cylindrical subdomain, thereby obtaining the Hölder continuity.

Keywords:

Citation: Yan Dong. Local Hölder continuity of nonnegative weak solutions of inverse variation-inequality problems of non-divergence type[J]. Electronic Research Archive, 2024, 32(1): 473-485. doi: 10.3934/era.2024023

Related Papers:

[1]	Vladimir Edemskiy, Chenhuang Wu . On the $ k $-error linear complexity of binary sequences of periods $ p^n $ from new cyclotomy. AIMS Mathematics, 2022, 7(5): 7997-8011. doi: 10.3934/math.2022446
[2]	Sima Mashayekhi, Seyed Nourollah Mousavi . A robust numerical method for single and multi-asset option pricing. AIMS Mathematics, 2022, 7(3): 3771-3787. doi: 10.3934/math.2022209
[3]	Kenan Doğan, Murat Şahin, Oğuz Yayla . Families of sequences with good family complexity and cross-correlation measure. AIMS Mathematics, 2025, 10(1): 38-55. doi: 10.3934/math.2025003
[4]	Zongcheng Li, Jin Li . Linear barycentric rational collocation method for solving a class of generalized Boussinesq equations. AIMS Mathematics, 2023, 8(8): 18141-18162. doi: 10.3934/math.2023921
[5]	Adisorn Kittisopaporn, Pattrawut Chansangiam . Approximate solutions of the $ 2 $D space-time fractional diffusion equation via a gradient-descent iterative algorithm with Grünwald-Letnikov approximation. AIMS Mathematics, 2022, 7(5): 8471-8490. doi: 10.3934/math.2022472
[6]	Yan Wang, Ying Cao, Ziling Heng, Weiqiong Wang . Linear complexity and 2-adic complexity of binary interleaved sequences with optimal autocorrelation magnitude. AIMS Mathematics, 2022, 7(8): 13790-13802. doi: 10.3934/math.2022760
[7]	Ruiyuan Chang, Xiuli Wang, Mingqiu Wang . Corrected optimal subsampling for a class of generalized linear measurement error models. AIMS Mathematics, 2025, 10(2): 4412-4440. doi: 10.3934/math.2025203
[8]	Yannan Sun, Wenchao Qian . Fast algorithms for nonuniform Chirp-Fourier transform. AIMS Mathematics, 2024, 9(7): 18968-18983. doi: 10.3934/math.2024923
[9]	Anumanthappa Ganesh, Swaminathan Deepa, Dumitru Baleanu, Shyam Sundar Santra, Osama Moaaz, Vediyappan Govindan, Rifaqat Ali . Hyers-Ulam-Mittag-Leffler stability of fractional differential equations with two caputo derivative using fractional fourier transform. AIMS Mathematics, 2022, 7(2): 1791-1810. doi: 10.3934/math.2022103
[10]	Li-Tao Zhang, Xian-Yu Zuo, Shi-Liang Wu, Tong-Xiang Gu, Yi-Fan Zhang, Yan-Ping Wang . A two-sweep shift-splitting iterative method for complex symmetric linear systems. AIMS Mathematics, 2020, 5(3): 1913-1925. doi: 10.3934/math.2020127

Abstract

1. Introduction

The linear complexity and the $k$ -error linear complexity are important cryptographic characteristics of stream cipher sequences. The linear complexity of an $N$ -periodic sequence $s = \{s_u\}^\infty_{u = 0}$ , denoted by $LC(s)$ , is defined as the length of the shortest linear feedback shift register (LFSR) that generates it ^[1]. With the Berlekamp-Massey (B-M) algorithm ^[2], if $LC(s)\geq N/2$ , then $s$ is regarded as a good sequence with respect to its linear complexity. For an integer $k\geq0$ , the $k$ -error linear complexity $LC_k(s)$ is the smallest linear complexity that can be obtained by changing at most $k$ terms of $s$ in the first period and periodically continued ^[3]. The cryptographic background of the $k$ -error linear complexity is that some key streams with large linear complexity can be approximated by some sequences with much lower linear complexity ^[2]. For a sequence to be cryptographically strong, its linear complexity should be large enough, and its $k$ -error linear complexity should be close to the linear complexity.

The relationship between the linear complexity and the DFT of the sequence was given by Blahut in ^[4]. Let $m$ be the order of 2 modulo an odd number $N$ . For a primitive $N$ -th root $\beta\in\mathbb{F}_{2^m}$ of unity, the DFT of $s$ is defined by

$\begin{equation} \rho_i = \sum\limits_{u = 0}^{N-1}s_u\beta^{-iu}\mbox{, }0\leq i\leq N-1. \end{equation}$

(1.1)

Then

$\begin{equation} LC(s) = W_H(\rho_0, \;\rho_1, \;\cdots, \;\rho_{N-1}), \end{equation}$

(1.2)

where $W_H(A)$ is the hamming weight of the sequence $A$ . The polynomial

$\begin{equation} G(X) = \sum\limits_{i = 0}^{N-1}\rho_iX^i \in \mathbb{F}_{2^m}[X] \end{equation}$

(1.3)

is called the Mattson-Solomon polynomial (M-S polynomial) of $s$ ^[5]. It can be deduced from Eqs (1.2)and (1.3) that the linear complexity of $s$ is equal to the number of the nonzero terms of $G(X)$ , namely

$\begin{equation} LC(s) = |G(X)|. \end{equation}$

(1.4)

By the inverse DFT,

$\begin{equation} s_u = \sum\limits_{i = 0}^{N-1}\rho_i\beta^{iu} = G(\beta^u)\mbox{, }0\leq u\leq N-1. \end{equation}$

(1.5)

There are many studies about two-prime generators. In 1997–1998, Ding calculated the linear complexity and the autocorrelation values of binary Whiteman generalized cyclotomic sequences of order two ^[6,7]. In 2013, Li defined a new generalized cyclotomic sequence of order two of length $pq$ , which is based on Whiteman generalized cyclotomic classes, and calculated its linear complexity ^[8]. In 2015, Wei determined the $k$ -error linear complexity of Legendre sequences for $k = 1, 2$ ^[9]. In 2018, Hofer and Winterhof studied the 2-adic complexity of the two-prime generator of period $pq$ ^[10]. Alecu and Sălăgean transformed the optimisation problem of finding the $k$ -error linear complexity of a sequence into an optimisation problem in the DFT domain, by using Blahut's theorem in the same year ^[11]. In 2019, in terms of the DFT, Chen and Wu discussed the $k$ -error linear complexity for Legendre, Ding-Helleseth-Lam, and Hall's sextic residue sequences of odd prime period $p$ ^[12]. In 2020, Zhou and Liu presented a type of binary sequences based on a general two-prime generalized cyclotomy, and derived their minimal polynomial and linear complexity ^[13]. In 2021, the autocorrelation distribution and the 2-adic complexity of generalized cyclotomic binary sequences of order 2 with period $pq$ were determined by Jing ^[14].

This paper is organized as follows. Firstly, we present some preliminaries about Whiteman generalized cyclotomic classes and the linear complexity in Section 2. In Section 3, we give main results about the linear complexity of Whiteman generalized cyclotomic sequences of order two. In Section 4, we give the 1-error linear complexity of these sequences. At last, we conclude this paper in Section 5.

2. Preliminaries

Let $p$ and $q$ be two distinct odd primes with $\mbox{gcd}(p-1, q-1) = 2$ , and $N = pq$ , $e = (p-1)(q-1)/2$ . By the Chinese Remainder Theorem, there is a fixed common primitive root $g$ of both $p$ and $q$ such that $\mbox{ord}_N(g) = e$ . Let $x$ be an integer satisfying

$\begin{equation} x = g(\bmod p)\mbox{, }x = 1(\bmod q).\nonumber \end{equation}$

Then the set

$\begin{equation} D_i = \{g^sx^i \bmod N:s = 0, 1, \cdots, e-1\}\nonumber \end{equation}$

for $i = 0, 1$ is called a Whiteman generalized cyclotomic class of order two ^[15].

As pointed out in ^[14], the unit group of the ring $Z_N$ is

$\begin{align*} Z_N^*& = \{a(\mbox{mod }N):\mbox{gcd}(a, N) = 1\}\nonumber\\ & = \{ip+jq(\mbox{mod }N):1\leq i\leq q-1\mbox{, }1\leq j\leq p-1\}.\nonumber \end{align*}$

Let $P = \{p, 2p, \cdots, (q-1)p\}$ , $Q = \{q, 2q, \cdots, (p-1)q\}$ and $R = \{0\}$ . Then $Z_N = Z_N^*\cup P\cup Q \cup R$ . The sequence $s(a, b, c) = \{s_u\}^\infty_{u = 0}$ over $\mathbb{F}_2$ is defined by

$\begin{align*} s_u = \begin{cases} c, &\mbox{if }\;u = 0, \\ a, &\mbox{if }\;u\in P, \\ b, &\mbox{if }\;u\in Q, \\ \frac{1}{2}(1-(\frac{u}{p})(\frac{u}{q})), &\mbox{if }\;u\in Z_N^*, \\ \end{cases} \end{align*}$

where $(\frac{\cdot}{\cdot})$ denotes the Legendre symbol and $a, b, c\in \mathbb{F}_2$ ^[14].

Lemma 2.1. ^[7] $-1\in D_1$ , if $|p-q|/2$ is odd; and $-1\in D_0$ , if $|p-q|/2$ is even.

Lemma 2.2. ^[6]

$\begin{array}{l} {{\mathit{(1)} \;If \; p\equiv\pm 1(\bmod 8) , \;q\equiv\pm 1(\bmod 8) \; or \;p\equiv\pm 3(\bmod 8) , \;q\equiv\pm 3(\bmod 8) , \;then \; 2\in D_0 .}} \\ {{\mathit{(2)}\;If \; p\equiv\pm 1(\bmod 8) ,\; q\equiv\pm 3(\bmod 8) \; or \; p\equiv\pm 3(\bmod 8) , \;q\equiv\pm 1(\bmod 8) , \;then \;2\in D_1 .}} \end{array}$

Lemma 2.3. ^[6] (1) If $a\in P$ , then $aP = P$ and $aQ = R$ .

(2) If $a\in Q$ , then $aP = R$ and $aQ = Q$ .

(3) If $a\in D_i$ , then $aP = P$ , $aQ = Q$ , and $aD_j = D_{(i+j)\bmod 2}$ , where $i, j = 0, 1$ .

It was shown in ^[6] that, for a primitive $N$ -th root $\beta\in\mathbb{F}_{2^m}$ of unity, we have

$\sum\limits_{i\in P}\beta^i = 1, \;\sum\limits_{i\in Q}\beta^i = 1,$

and

$\begin{equation} \sum\limits_{i\in D_0}\beta^i+\sum\limits_{i\in D_1}\beta^i+\sum\limits_{i\in P}\beta^i+\sum\limits_{i\in Q}\beta^i = 1. \end{equation}$

(2.1)

Lemma 2.4. ^[6]

$\begin{align*} \sum\limits_{u\in D_j}\beta^{iu} = \begin{cases} \frac{p-1}{2}(\bmod 2), &\mathit{\mbox{if}}\;i\in P, \\ \frac{q-1}{2}(\bmod 2), &\mathit{\mbox{if}}\;i\in Q. \end{cases} \end{align*}$

Actually, if $p\equiv -1(\bmod 8)$ or $p\equiv 3(\bmod 8)$ , then $(p-1)/2 = 1$ ; if $p\equiv 1(\bmod 8)$ or $p\equiv -3(\bmod 8)$ , then $(p-1)/2 = 0$ . By symmetry, if $q\equiv -1(\bmod 8)$ or $q\equiv 3(\bmod 8)$ , then $(q-1)/2 = 1$ ; if $q\equiv 1(\bmod 8)$ or $q\equiv -3(\bmod 8)$ , then $(q-1)/2 = 0$ .

Lemma 2.5. Define

$D_i(X) = \sum\limits_{u\in D_i}X^u\in \mathbb{F}_2[X], \mathit{\mbox{}}i = 0, 1.$

Then for $\beta$ , we have $D_0(\beta) = 0$ and $D_1(\beta) = 1$ if $2\in D_0$ ; $D_0(\beta) = \omega$ and $D_1(\beta) = 1+\omega$ if $2\in D_1$ , where $\omega\in\mathbb{F}_4\setminus\mathbb{F}_2$ .

Proof. (1) If $2\in D_0$ , by Lemma 2.3 we have

$[D_i(\beta)]^2 = D_i(\beta^2) = \sum\limits_{2u\in D_i}\beta^{2u} = D_i(\beta)\in\mathbb{F}_2.$

(2) If $2\in D_1$ , by Lemma 2.3 we have

$\begin{align*} &[D_i(\beta)]^2 = D_i(\beta^2) = \sum\limits_{2u\in D_{i+1}}\beta^{2u} = D_{i+1}(\beta), \\ &[D_i(\beta)]^4 = [D_i(\beta)^2]^2 = [D_{i+1}(\beta)]^2 = D_{i+1}(\beta^2) = \sum\limits_{2u\in D_i}\beta^{2u} = D_i(\beta). \end{align*}$

Hence $D_i(\beta)\in\mathbb{F}_4\setminus\mathbb{F}_2$ .

And by Eq (2.1), we have $D_0(\beta)\neq D_1(\beta)$ and $D_0(\beta)+D_1(\beta) = 1$ . Assume that $D_0(\beta) = 0$ , $D_1(\beta) = 1$ for $2\in D_0$ , and $D_0(\beta) = \omega$ , $D_1(\beta) = 1+\omega$ for $2\in D_1$ , where $\omega\in\mathbb{F}_4\setminus\mathbb{F}_2$ .

3. The linear complexity of $s(a, b, c)$

Let $LC(s(a, b, c))$ be the linear complexity of $s(a, b, c)$ , and the other symbols be the same as before.

Theorem 3.1. Let $p\equiv v(\bmod 8)$ and $q\equiv w(\bmod 8)$ , where $v, w = \pm1, \pm3$ . Then the linear complexity of $s(a, b, c)$ respect to different values of $p$ and $q$ is as shown as Table 1.

Table 1. The linear complexity of

$s(a, b, c)$ .

	$s(0, 0, 0)$	$s(0, 0, 1)$	$s(0, 1, 0)$	$s(0, 1, 1)$	$s(1, 0, 0)$	$s(1, 0, 1)$	$s(1, 1, 0))$	$s(1, 1, 1)$
$(-1, -3)$ or $(3, 1)$	$pq-p$	$pq-q+1$	$pq-1$	$pq-p-q+2$	$pq-p-q+1$	$pq$	$pq-q$	$pq-p+1$
$(-1, 3)$ or $(3, -1)$	$pq-1$	$pq-p-q+2$	$pq-p$	$pq-q+1$	$pq-q$	$pq-p+1$	$pq-p-q+1$	$pq$
$(-1, 1)$ or $(3, -3)$	$\frac{pq-p+q-1}{2}$	$\frac{pq+p-q+1}{2}$	$\frac{pq+p+q-3}{2}$	$\frac{pq-p-q+3}{2}$	$\frac{pq-p-q+1}{2}$	$\frac{pq+p+q-1}{2}$	$\frac{pq+p-q-1}{2}$	$\frac{pq-p+q+1}{2}$
$(-1, -1)$ or $(3, 3)$	$\frac{pq+p+q-3}{2}$	$\frac{pq-p-q+3}{2}$	$\frac{pq-p+q-1}{2}$	$\frac{pq+p-q+1}{2}$	$\frac{pq+p-q-1}{2}$	$\frac{pq-p+q+1}{2}$	$\frac{pq-p-q+1}{2}$	$\frac{pq+p+q-1}{2}$
$(-3, -1)$ or $(1, 3)$	$pq-q$	$pq-p+1$	$pq-p-q+1$	$pq$	$pq-1$	$pq-p-q+2$	$pq-p$	$pq-q+1$
$(1, -1)$ or $(-3, 3)$	$\frac{pq+p-q-1}{2}$	$\frac{pq-p+q+1}{2}$	$\frac{pq-p-q+1}{2}$	$\frac{pq+p+q-1}{2}$	$\frac{pq+p+q-3}{2}$	$\frac{pq-p-q+3}{2}$	$\frac{pq-p+q-1}{2}$	$\frac{pq+p-q+1}{2}$

| Show Table

DownLoad: CSV

Proof. We provide the process of calculating $LC(s(0, 0, 0))$ when $v = -1$ and $w = -3$ , and can prove other cases in a similar way.

By Lemmas 2.1–2.3 and Eq (1.1), we have $-1\in D_1$ , $2\in D_1$ , then

$\begin{align*} \rho_i = \sum\limits_{u = 0}^{N-1}s_u\beta^{-iu} = \sum\limits_{u\in D_1}\beta^{-iu} = \sum\limits_{u\in D_0}\beta^{iu}, \end{align*}$

and $\rho_0 = 0$ . By Eq (1.3), we have

$\begin{align*} G(X)& = \sum\limits_{i = 0}^{N-1}\rho_iX^i = \sum\limits_{i\in D_0}\rho_iX^i+\sum\limits_{i\in D_1}\rho_iX^i+\sum\limits_{i\in P}\rho_iX^i+\sum\limits_{i\in Q}\rho_iX^i+\rho_0\\ & = \sum\limits_{i\in D_0}\sum\limits_{u\in D_0}\beta^{iu}X^i+\sum\limits_{i\in D_1}\sum\limits_{u\in D_0}\beta^{iu}X^i+\sum\limits_{i\in P}\sum\limits_{u\in D_0}\beta^{iu}X^i+\sum\limits_{i\in Q}\sum\limits_{u\in D_0}\beta^{iu}X^i.\\ \end{align*}$

Let $t = iu$ . Then by Lemmas 2.3–2.5, we have

$\begin{align*} G(X)& = \sum\limits_{i\in D_0}\sum\limits_{t\in D_0}\beta^tX^i+\sum\limits_{i\in D_1}\sum\limits_{t\in D_1}\beta^tX^i+\sum\limits_{i\in P}\frac{p-1}{2}X^i+\sum\limits_{i\in Q}\frac{q-1}{2}X^i\\ & = D_0(\beta)D_0(X)+D_1(\beta)D_1(X)+\sum\limits_{i\in P}X^i\\ & = \omega D_0(X)+(1+\omega)D_1(X)+\sum\limits_{i\in P}X^i. \end{align*}$

By Eq (1.4) we can get the linear complexity of $s(0, 0, 0)$ as

$\begin{align*} LC(s(0, 0, 0)) = |G(X)| = pq-p. \end{align*}$

Actually, the linear complexity of $s(1, 0, 0)$ was studied by Ding in ^[6] with its minimal polynomial.

4. The 1-error linear complexity of $s(a, b, c)$

Let $LC_k(s(a, b, c))$ be the $k$ -error linear complexity of $s(a, b, c)$ , $\tilde{s} = \{\tilde{s}_u\}^\infty_{u = 0}$ be the new sequence obtained by changing at most $k$ terms of $s$ , that $\tilde{s} = s+e$ , where $e = \{e_u\}^\infty_{u = 0}$ is an error sequence of period $N$ . Ding has provided in ^[2] that, the $k$ -error linear complexity of a sequence can be expressed as

$\begin{equation} LC_k(s) = \mbox{min}_{W_H(e)\leq k}\{LC(s+e)\}. \end{equation}$

(4.1)

It is clearly that $LC_0(s) = LC(s)$ and

$\begin{equation} N\geq LC_0(s)\geq LC_1(s)\geq\cdots\geq LC_l(s) = 0, \nonumber \end{equation}$

where $l = W_H(s)$ .

Let $G(X)$ , $G_k(X)$ and $\tilde{G}(X)$ be the M-S polynomials of $s$ , $e$ and $\tilde{s}$ respectively. Note that

$\begin{equation} G(X) = \sum\limits_{i = 0}^{N-1}\rho_iX^i, \mbox{ }G_k(X) = \sum\limits_{i = 0}^{N-1}\eta_iX^i, \mbox{ }\tilde{G}(X) = \sum\limits_{i = 0}^{N-1}\xi_iX^i, \end{equation}$

(4.2)

where $\rho_i$ , $\eta_i$ and $\xi_i$ are the DFTs of $s$ , $e$ and $\tilde{s}$ respectively. By Eqs (1.5), (4.1) and (4.2), we have $\tilde{G}(X) = G(X)+G_k(X)$ , then

$\begin{equation} \xi_i = \rho_i+\eta_i. \end{equation}$

(4.3)

Assume that $e_{u_0} = 1$ for $0\leq u_0\leq N-1$ and $e_u = 0$ for $u\neq u_0$ in the first period of $e$ . Then the DFT of $e$ is

$\begin{equation} \eta_i = \sum\limits_{u = 0}^{N-1}e_u\beta^{-iu} = \beta^{-iu_0}\mbox{, }0\leq i\leq N-1.\nonumber \end{equation}$

Specially, if $u_0 = 0$ , then $\eta_i = 1$ for all $0\leq i\leq N-1$ ; otherwise, $\eta_0 = 1$ and the order of $\eta_i$ is $N$ for $1\leq i\leq N-1$ .

Theorem 4.1. Let $p\equiv v(\bmod 8)$ and $q\equiv w(\bmod 8)$ , where $v, w = \pm1, \pm3$ , and the other symbols be the same as before. Then the 1-error linear complexity of $s(a, b, c)$ is as shown as Table 2.

Table 2. The 1-error linear complexity of

$s(a, b, c)$ .

	$s(0, 0, 0)$ and $s(0, 0, 1)$	$s(0, 1, 0)$ and $s(0, 1, 1)$	$s(1, 0, 0)$ and $s(1, 0, 1)$	$s(1, 1, 0))$ and $s(1, 1, 1)$
$(-1, -3)$ or $(3, 1)$	(1) $pq-p$ , if $p>q$ ; (2) $pq-q+1$ , if $p<q$ .	$pq-p-q+2$	$pq-p-q+1$	(1) $pq-p+1$ , if $p>q$ ; (2) $pq-q$ , if $p<q$ .
$(-1, 3)$ or $(3, -1)$	$pq-p-q+2$	(1) $pq-p$ , if $p>q$ ; (2) $pq-q+1$ , if $p<q$ .	(1) $pq-p+1$ , if $p>q$ ; (2) $pq-q$ , if $p<q$ .	$pq-p-q+1$
$(-1, 1)$ or $(3, -3)$	(1) $\frac{pq-p+q-1}{2}$ , if $p>q$ ; (2) $\frac{pq+p-q+1}{2}$ , if $p<q$ .	$\frac{pq-p-q+3}{2}$	$\frac{pq-p-q+1}{2}$	(1) $\frac{pq-p+q+1}{2}$ , if $p>q$ ; (2) $\frac{pq+p-q-1}{2}$ , if $p<q$ .
$(-1, -1)$ or $(3, 3)$	$\frac{pq-p-q+3}{2}$	(1) $\frac{pq-p+q-1}{2}$ , if $p>q$ ; (2) $\frac{pq+p-q+1}{2}$ , if $p<q$ .	(1) $\frac{pq-p+q+1}{2}$ , if $p>q$ ; (2) $\frac{pq+p-q-1}{2}$ , if $p<q$ .	$\frac{pq-p-q+1}{2}$
$(-3, -1)$ or $(1, 3)$	(1) $pq-p+1$ , if $p>q$ ; (2) $pq-q$ , if $p<q$ .	$pq-p-q+1$	$pq-p-q+2$	(1) $pq-p$ , if $p>q$ ; (2) $pq-q+1$ , if $p<q$ .
$(1, -1)$ or $(-3, 3)$	(1) $\frac{pq-p+q+1}{2}$ , if $p>q$ ; (2) $\frac{pq+p-q-1}{2}$ , if $p<q$ .	$\frac{pq-p-q+1}{2}$	$\frac{pq-p-q+3}{2}$	(1) $\frac{pq-p+q-1}{2}$ , if $p>q$ ; (2) $\frac{pq+p-q+1}{2}$ , if $p<q$ .

| Show Table

DownLoad: CSV

Proof. We consider the case $v = -1, w = -3$ for $LC_1(s(0, 0, 0))$ . By Lemmas 2.1–2.5 and Eq (1.1), we have $-1\in D_1$ , $2\in D_1$ and

$\begin{align*} \xi_i = \rho_i+\eta_i = \sum\limits_{u\in D_0}\beta^{iu}+\beta^{-iu_0} = \begin{cases} \omega+\beta^{-iu_0}, &\mbox{if }\;i\in D_0, \\ 1+\omega+\beta^{-iu_0}, &\mbox{if }\;i\in D_1, \\ 1+\beta^{-iu_0}, &\mbox{if }\;i\in P, \\ \beta^{-iu_0}, &\mbox{if }\;i\in Q, \\ 1, &\mbox{if }\;i = 0.\\ \end{cases} \end{align*}$

Then by Eq (4.2), we can get

$\begin{align*} \tilde{G}(X) = \sum\limits_{i = 0}^{N-1}\xi_iX^i = \sum\limits_{i\in D_0}\left(\omega+\beta^{-iu_0}\right)X^i+\sum\limits_{i\in D_1}\left(1+\omega+\beta^{-iu_0}\right)X^i+\sum\limits_{i\in P}\left(1+\beta^{-iu_0}\right)X^i+\sum\limits_{i\in Q}\beta^{-iu_0}X^i+1. \end{align*}$

According to Lemma 2.3, we can get the following results.

(1) If $u_0 = 0$ , then

$\begin{align*} &\tilde{G}(X) = \sum\limits_{i\in D_0}\left(\omega+1\right)X^i+\sum\limits_{i\in D_1}\omega X^i+\sum\limits_{i\in Q}X^i+1, \\ &|\tilde{G}(X)| = pq-q+1. \end{align*}$

(2) If $u_0\in Q$ , then

$\begin{align*} &\tilde{G}(X) = \sum\limits_{i\in D_0}\left(\omega+\beta^{-iu_0}\right)X^i+\sum\limits_{i\in D_1}\left(1+\omega+\beta^{-iu_0}\right)X^i+\sum\limits_{i\in Q}\beta^{-iu_0}X^i+1, \\ &|\tilde{G}(X)| = pq-q+1. \end{align*}$

(3) If $u_0\in D_0$ or $u_0\in D_1$ or $u_0\in P$ , then

$\begin{align*} &\tilde{G}(X) = \sum\limits_{i\in D_0}\left(\omega+\beta^{-iu_0}\right)X^i+\sum\limits_{i\in D_1}\left(1+\omega+\beta^{-iu_0}\right)X^i+\sum\limits_{i\in P}\left(1+\beta^{-iu_0}\right)X^i+\sum\limits_{i\in Q}\beta^{-iu_0}X^i+1, \\ &|\tilde{G}(X)| = pq. \end{align*}$

Compare the results of Cases (1)–(3) with $LC(s(0, 0, 0)) = pq-p$ . If $p > q$ , then $pq-p < pq-q+1 < pq$ ; if $p < q$ , then $pq-q+1 < pq-p < pq$ . Hence

$\begin{align*} LC_1(s(0, 0, 0)) = \begin{cases} pq-p, &\mbox{if }p > q, \\ pq-q+1, &\mbox{if }p < q. \end{cases} \end{align*}$

Similarly we can prove the other cases for $LC_1(s(a, b, c))$ .

All results of $LC(s(a, b, c))$ and $LC_1(s(a, b, c))$ in Sections 3 and 4 have been tested by MAGMA program.

5. Conclusions

The purpose of this paper is to determine the linear complexity and the 1-error linear complexity of $s(a, b, c)$ . In most of the cases, $s(a, b, c)$ possesses high linear complexity, namely $LC(s(a, b, c)) > N/2$ , consequently has decent stability against 1-bit error. Notice that the linear complexity of some of the sequences above is close to $N/2$ . Then the sequences can be selected to construct cyclic codes by their minimal generating polynomials with the method introduced by Ding ^[16].

Acknowledgments

This work was supported by Fundamental Research Funds for the Central Universities (No. 20CX05012A), the Major Scientific and Technological Projects of CNPC under Grant (No. ZD2019-183-008), the National Natural Science Foundation of China (Nos. 61902429, 11775306) and Shandong Provincial Natural Science Foundation of China (ZR2019MF070).

Conflict of interest

The authors declare that they have no conflicts of interest.

References

[1]	S. B. Boyana, T. Lewis, A. Rapp, Y. Zhang, Convergence analysis of a symmetric dual-wind discontinuous Galerkin method for a parabolic variational inequality, J. Comput. Appl. Math., 422 (2023), 114922. https://doi.org/10.1016/j.cam.2022.114922 doi: 10.1016/j.cam.2022.114922
[2]	D. Adak, G. Manzini, S. Natarajan, Virtual element approximation of two-dimensional parabolic variational inequalities, Comput. Math. Appl., 116 (2022), 48–70. https://doi.org/10.1016/j.camwa.2021.09.007 doi: 10.1016/j.camwa.2021.09.007
[3]	J. Dabaghi, V. Martin, M. Vohralk, A posteriori estimates distinguishing the error components and adaptive stopping criteria for numerical approximations of parabolic variational inequalities, Comput. Methods Appl. Mech. Engrg., 367 (2020), 113105. https://doi.org/10.1016/j.cma.2020.113105 doi: 10.1016/j.cma.2020.113105
[4]	T. Chen, N. Huang, X. Li, Y. Zou, A new class of differential nonlinear system involving parabolic variational and history-dependent hemi-variational inequalities arising in contact mechanics, Commun. Nonlinear Sci. Numer. Simul., 101 (2021), 105886. https://doi.org/10.1016/j.cnsns.2021.105886 doi: 10.1016/j.cnsns.2021.105886
[5]	S. Migórski, V. T. Nguyen, S. Zeng, Solvability of parabolic variational-hemivariational inequalities involving space-fractional Laplacian, Appl. Math. Comput., 364 (2020), 124668. https://doi.org/10.1016/j.amc.2019.124668 doi: 10.1016/j.amc.2019.124668
[6]	O. M. Buhrii, R. A. Mashiyev, Uniqueness of solutions of the parabolic variational inequality with variable exponent of nonlinearity, Nonlinear Anal., 70 (2009), 2325–2331. https://doi.org/10.1016/j.na.2008.03.013 doi: 10.1016/j.na.2008.03.013
[7]	M. Boukrouche, D. A. Tarzia, Existence, uniqueness, and convergence of optimal control problems associated with parabolic variational inequalities of the second kind, Nonlinear Anal. Real World Appl., 12 (2011), 2211–2224. https://doi.org/10.1016/j.nonrwa.2011.01.003 doi: 10.1016/j.nonrwa.2011.01.003
[8]	Z. Sun, Regularity and higher integrability of weak solutions to a class of non-Newtonian variation-inequality problems arising from American lookback options, AIMS Math., 8 (2023), 14633–14643. https://doi.org/10.3934/math.2023749 doi: 10.3934/math.2023749
[9]	F. Abedin, R. W. Schwab, Regularity for a special case of two-phase Hele-Shaw flow via parabolic integro-differential equations, J. Funct. Anal., 285 (2023), 110066. https://doi.org/10.1016/j.jfa.2023.110066 doi: 10.1016/j.jfa.2023.110066
[10]	V. Bögelein, F. Duzaar, N. Liao, C. Scheven, Gradient Hölder regularity for degenerate parabolic systems, Nonlinear Anal., 225 (2022), 113119. https://doi.org/10.1016/j.na.2022.113119 doi: 10.1016/j.na.2022.113119
[11]	A. Herán, Hölder continuity of parabolic quasi-minimizers on metric measure spaces, J. Differ. Equations, 341 (2022), 208–262. https://doi.org/10.1016/j.jde.2022.09.019 doi: 10.1016/j.jde.2022.09.019
[12]	D. Wang, K. Serkh, C. Christara, A high-order deferred correction method for the solution of free boundary problems using penalty iteration, with an application to American option pricing, J. Comput. Appl. Math., 432 (2023), 115272. https://doi.org/10.1016/j.cam.2023.115272 doi: 10.1016/j.cam.2023.115272
[13]	T. B. Gyulov, M. N. Koleva, Penalty method for indifference pricing of American option in a liquidity switching market, Appl. Numer. Math., 172 (2022), 525–545. https://doi.org/10.1016/j.apnum.2021.11.002 doi: 10.1016/j.apnum.2021.11.002
[14]	S. Signoriello, T. Singer, Hölder continuity of parabolic quasi-minimizers, J. Differ. Equations, 263 (2017), 6066–6114. https://doi.org/10.1016/j.jde.2017.07.008 doi: 10.1016/j.jde.2017.07.008
[15]	Y. Wang, Local Hölder continuity of nonnegative weak solutions of degenerate parabolic equations, Nonlinear Anal., 72 (2010), 3289–3302. https://doi.org/10.1016/j.na.2009.12.007 doi: 10.1016/j.na.2009.12.007

This article has been cited by:

1.	Chi Yan, Chengliang Tian, On the Stability of the Linear Complexity of Some Generalized Cyclotomic Sequences of Order Two, 2024, 12, 2227-7390, 2483, 10.3390/math12162483
2.	Chi Yan, On the Error Linear Complexity of Some Generalized Cyclotomic Sequences of Order Two of Period pq , 2024, 12, 2169-3536, 109210, 10.1109/ACCESS.2024.3438869

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

Electronic Research Archive

1.1 1.7

Metrics

Article views(3877) PDF downloads(51) Cited by(0)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Electronic Research Archive

Local Hölder continuity of nonnegative weak solutions of inverse variation-inequality problems of non-divergence type

Related Papers:

Abstract

1. Introduction

2. Preliminaries

3. The linear complexity of $s(a, b, c)$

4. The 1-error linear complexity of $s(a, b, c)$

5. Conclusions

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Catalog

Electronic Research Archive

Local Hölder continuity of nonnegative weak solutions of inverse variation-inequality problems of non-divergence type

Related Papers:

Abstract

1. Introduction

2. Preliminaries

3. The linear complexity of s(a,b,c) s(a, b, c)

4. The 1-error linear complexity of s(a,b,c) s(a, b, c)

5. Conclusions

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog

3. The linear complexity of $s(a, b, c)$

4. The 1-error linear complexity of $s(a, b, c)$