arXiv subjects
Publications and source records attributed to Yuhua Sun.
We investigate the non-existence and existence of positive solutions to biharmonic elliptic inequalities on manifolds. Using Green function and volume growth conditions, we establish the critical exponent for biharmonic problem.
We study nonexistence and existence of nontrivial positive solutions to the following semilinear elliptic inequality involving gradient terms \[ Δu+u^p\left|\nabla u\right|^q\leq0, \] on weighted graphs, where $(p,q)\in\mathbb{R}^2$. We give a complete classification of $(p,q)$ under which sharp volume growth assumptions are established.
We are concerned with the nonexistence of sign-changing global weak solutions for a class of semilinear parabolic differential inequalities with convection terms in exterior domains. A weight function of the form $t^α|x|^σ$ is considered in front of the power nonlinearity. Two types of non-homogeneous boundary conditions are investigated: Neumann-type and Dirichlet-type boundary conditions. Using a unified approach, for each case, we establish sufficient criteria for the nonexistence of global weak solutions. When $α=0$, the critical exponent in the sense of Fujita is obtained. This exponent is bigger than that found previously by Zheng and Wang (2008) in the case of homogeneous Neumann and Dirichlet boundary conditions.
Let $(V,μ)$ be an infinite, connected, locally finite weighted graph. We study the problem of existence or non-existence of positive solutions to a semi-linear elliptic inequality \begin{equation*} Δu+u^σ\leq0\quad \text{in}\,\,V, \end{equation*} where $Δ$ is the standard graph Laplacian on $V$ and $σ>0$. For $σ\in(0,1]$, the inequality admits no nontrivial positive solution. For $σ>1$, assuming condition \textbf{($p_0$)} on $(V,μ)$, we obtain a sharp condition for nonexistence of positive solutions in terms of the volume growth of the graph, that is \begin{equation*} μ(o,n)\lesssim n^{\frac{2σ}{σ-1}}(\ln n)^{\frac{1}{σ-1}} \end{equation*} for some $o\in V$ and all large enough $n$. For any $\varepsilon>0$, we can construct an example on a homogeneous tree $\mathbb T_N$ with $μ(o,n)\approx n^{\frac{2σ}{σ-1}}(\ln n)^{\frac{1}{σ-1}+\varepsilon}$, and a solution to the inequality on $(\mathbb T_N,μ)$ to illustrate the sharpness of $\frac{2σ}{σ-1}$ and $\frac{1}{σ-1}$.
Quantum synchronizable codes (QSCs) are special quantum error-correcting codes which can be used to correct the effects of quantum noise on qubits and misalignment in block synchronization. In this paper, a new class of quantum synchronizable codes of length $2^n$ are constructed by using the cyclotomic cosets, whose synchronization capabilities always reach the upper bound $2^n$.
In this paper, the 2-adic complexity of a class of balanced Whiteman generalized cyclotomic sequences of period $pq$ is considered. Through calculating the determinant of the circulant matrix constructed by one of these sequences, we derive a lower bound on the 2-adic complexity of the corresponding sequence, which further expands our previous work (Zhao C, Sun Y and Yan T. Study on 2-adic complexity of a class of balanced generalized cyclotomic sequences. Journal of Cryptologic Research,6(4):455-462, 2019). The result shows that the 2-adic complexity of this class of sequences is large enough to resist the attack of the rational approximation algorithm(RAA) for feedback with carry shift registers(FCSRs), i.e., it is in fact lower bounded by $pq-p-q-1$, which is far larger than one half of the period of the sequences. Particularly, the 2-adic complexity is maximal if suitable parameters are chosen.
Conflict-avoiding codes (CACs) have been used in multiple-access collision channel without feedback. The size of a CAC is the number of potential users that can be supported in the system. A code with maximum size is called optimal. The use of an optimal CAC enables the largest possible number of asynchronous users to transmit information efficiently and reliably. In this paper, a new upper bound on the maximum size of arbitrary equi-difference CAC is presented. Furthermore, three optimal constructions of equi-difference CACs are also given. One is a generalized construction for prime length $L=p$ and the other two are for two-prime length $L=pq$.
We investigate the nonexistence and existence of nontrivial positive solutions to $Δ_m u+u^p|\nabla u|^q\leq0$ on noncompact geodesically complete Riemannian manifolds, where $m>1$, and $(p,q)\in \mathbb{R}^2$. According to classification of $(p, q)$, we establish different volume growth conditions to obtain Liouville's theorems for the above quasilinear differential inequalities, and we also show these volume growth conditions are sharp in most cases. Moreover, the results are completely new for $(p, q)$ of negative pair, even in the Euclidean space.
In 2008, a class of binary sequences of period $N=4(2^k-1)(2^k+1)$ with optimal autocorrelation magnitude has been presented by Yu and Gong based on an $m$-sequence, the perfect sequence $(0,1,1,1)$ of period $4$ and interleaving technique. In this paper, we study the 2-adic complexities of these sequences. Our results show that they are larger than $N-2\lceil\mathrm{log}_2N\rceil+4 $ (which is far larger than $N/2$) and could attain the maximum value $N$ if suitable parameters are chosen, i.e., the 2-adic complexity of this class of interleaved sequences is large enough to resist the Rational Approximation Algorithm.
LSB (Least Significant Bit) sequences are widely used as the initial inputs in some modern stream ciphers, such as the ZUC algorithm-the core of the 3GPP LTE International Encryption Standard. Therefore, analyzing the statistical properties (for example, autocorrelation, linear complexity and 2-adic complexity) of these sequences becomes an important research topic. In this paper, we first reduce the autocorrelation distribution of the LSB sequence of a $p$-ary $m$-sequence with period $p^n-1$ for any order $n\geq2$ to the autocorrelation distribution of a corresponding Costas sequence with period $p-1$, and from the computing of which by computer, we obtain the explicit autocorrelation distribution of the LSB sequence for each prime $p<100$. In addition, we give a lower bound on the 2-adic complexity of each of these LSB sequences for all primes $p < 20$, which proves to be large enough to resist the analysis of RAA (Rational Approximation Algorithm) for FCSRs (Feedback with Carry Shift Registers). In particular, for a Mersenne prime $p=2^k-1$ (i.e., $k$ is a prime such that $p$ is also a prime), our results hold for all its bit-component sequences since they are shift equivalent to the LSB sequence.
We reinvestigate nonexistence and existence of global positive solutions to heat equation with a potential term on Riemannian manifolds. Especially, we give a very natural sharp condition only in terms of the volume of geodesic ball to obtain nonexistence results.
Let $M$ be a complete non-compact Riemannian manifold and let $σ$ be a Radon measure on $M$. We study the problem of existence or non-existence of positive solutions to a semilinear elliptic inequaliy \begin{equation*} -Δu\geq σu^{q}\quad \text{in}\,\,M, \end{equation*} where $q>1$. We obtain necessary and sufficent criteria for existence of positive solutions in terms of Green function of $Δ$. In particular, explicit necessary and sufficient conditions are given when $M$ has nonnegative Ricci curvature everywhere in $M$, or more generally when Green's function satisfies the 3G-inequality.
Recently, a class of binary sequences with optimal autocorrelation magnitude has been presented by Su et al. based on interleaving technique and Ding-Helleseth-Lam sequences (Des. Codes Cryptogr., https://doi.org/10.1007/s10623-017-0398-5). And its linear complexity has been proved to be large enough to resist the B-M Algorighm (BMA) by Fan (Des. Codes Cryptogr., https://doi.org/10.1007/s10623-018-0456-7). In this paper, we study the 2-adic complexity of this class of binary sequences. Our result shows that the 2-adic complexity of this class of sequence is no less than one half of its period, i.e., its 2-adic complexity is large enough to resist the Rational Aproximation Algorithm (RAA).
Let $p,q$ be distinct primes satisfying $\mathrm{gcd}(p-1,q-1)=d$ and let $D_i$, $i=0,1,\cdots,d-1$, be Whiteman's generalized cyclotomic classes with $Z_{pq}^{\ast}=\cup_{i=0}^{d-1}D_i$. In this paper, we give the values of Gauss periods based on the generalized cyclotomic sets $D_0^{\ast}=\sum_{i=0}^{\frac{d}{2}-1}D_{2i}$ and $D_1^{\ast}=\sum_{i=0}^{\frac{d}{2}-1}D_{2i+1}$. As an application, we determine a lower bound on the 2-adic complexity of modified Jacobi sequence. Our result shows that the 2-adic complexity of modified Jacobi sequence is at least $pq-p-q-1$ with period $N=pq$. This indicates that the 2-adic complexity of modified Jacobi sequence is large enough to resist the attack of the rational approximation algorithm (RAA) for feedback with carry shift registers (FCSRs).
Let $p$ be an odd prime, $n$ a positive integer and $g$ a primitive root of $p^n$. Suppose $D_i^{(p^n)}=\{g^{2s+i}|s=0,1,2,\cdots,\frac{(p-1)p^{n-1}}{2}\}$, $i=0,1$, is the generalized cyclotomic classes with $Z_{p^n}^{\ast}=D_0\cup D_1$. In this paper, we prove that Gauss periods based on $D_0$ and $D_1$ are both equal to 0 for $n\geq2$. As an application, we determine a lower bound on the 2-adic complexity of a class of Ding-Helleseth generalized cyclotomic sequences of period $p^n$. The result shows that the 2-adic complexity is at least $p^n-p^{n-1}-1$, which is larger than $\frac{N+1}{2}$, where $N=p^n$ is the period of the sequence.
Pseudo-random sequences with good statistical property, such as low autocorrelation, high linear complexity and large 2-adic complexity, have been applied in stream cipher. In general, it is difficult to give both the linear complexity and 2-adic complexity of a periodic binary sequence. Cai and Ding \cite{Cai Ying} gave a class of sequences with almost optimal autocorrelation by constructing almost difference sets. Wang \cite{Wang Qi} proved that one type of those sequences by Cai and Ding has large linear complexity. Sun et al. \cite{Sun Yuhua} showed that another type of sequences by Cai and Ding has also large linear complexity. Additionally, Sun et al. also generalized the construction by Cai and Ding using $d$-form function with difference-balanced property. In this paper, we first give the detailed autocorrelation distribution of the sequences was generalized from Cai and Ding \cite{Cai Ying} by Sun et al. \cite{Sun Yuhua}. Then, inspired by the method of Hu \cite{Hu Honggang}, we analyse their 2-adic complexity and give a lower bound on the 2-adic complexity of these sequences. Our result show that the 2-adic complexity of these sequences is at least $N-\mathrm{log}_2\sqrt{N+1}$ and that it reach $N-1$ in many cases, which are large enough to resist the rational approximation algorithm (RAA) for feedback with carry shift registers (FCSRs).
Generalized cyclotomic sequences of period pq have several desirable randomness properties if the two primes p and q are chosen properly. In particular,Ding deduced the exact formulas for the autocorrelation and the linear complexity of these sequences of order 2. In this paper, we consider the generalized sequences of order 4. Under certain conditions, the linear complexity of these sequences of order 4 is developed over a finite field F_l. Results show that in many cases they have high linear complexity.
Cyclic codes have wide applications in data storage systems and communication systems. Employing two-prime Whiteman generalized cyclotomic sequences of order 6, we construct several classes of cyclic codes over the finite field GF}(q) and give their generator polynomials. And we also calculate the minimum distance of some cyclic codes and give lower bounds of the minimum distance for some other cyclic codes.