results for au:Jia_C in:math

- Generalized frequency division multiplexing (GFDM) is considered a non-orthogonal waveform and known to encounter difficulties when using in the spatial multiplexing mode of multiple-input-multiple-output (MIMO) scenario. In this paper, a class of GFDM prototype filters, under which the GFDM system is free from inter-subcarrier interference, is investigated, enabling frequency-domain decoupling in the processing at the GFDM receiver. An efficient MIMO-GFDM detection method based on depth-first sphere decoding is then proposed with such class of filters. Numerical results confirm a significant reduction in complexity, especially when the number of subcarriers is large, compared with existing methods presented in recent years.
- Mar 06 2018 math.PR arXiv:1803.01120v1The $L^p$ maximal inequalities for martingales are one of the classical results in probability theory. Here we establish the sharp moderate maximal inequalities for upward skip-free Markov chains, which include the $L^p$ maximal inequalities as special cases. Furthermore, we apply our theory to two specific examples and obtain their moderate maximal inequalities: the first one is the M/M/1 queue and the second one is an upward skip-free Markov chain with large death jumps. These two examples have the same total birth and death rates. However, the former exhibits a phase transition phenomenon while the latter does not.
- Nov 06 2017 math.PR arXiv:1711.00902v1The maximal inequalities for diffusion processes have drawn increasing attention in recent years. However, the existing proof of the $L^p$ maximum inequalities for the Ornstein-Uhlenbeck process was dubious. Here we give a rigorous proof of the moderate maximum inequalities for the Ornstein-Uhlenbeck process, which include the $L^p$ maximum inequalities as special cases and generalize the remarkable $L^1$ maximum inequalities obtained by Graversen and Peskir [P. Am. Math. Soc., 128(10):3035-3041, 2000]. As a corollary, we also obtain a new moderate maximal inequality for continuous local martingales, which can be viewed as a supplement of the classical Burkholder-Davis-Gundy inequality.
- Nonequilibrium fluctuation-dissipation theorems (FDTs) are one of the most important advances in stochastic thermodynamics over the past two decades. Here we provide a rigourous mathematical theory of two types of nonequilibrium FDTs for inhomogeneous diffusion processes with unbounded drift and diffusion coefficients by using the Schauder estimates for partial differential equations of parabolic type and the theory of weak generators. The FDTs proved in this paper apply to any forms of nonlinear external perturbations. Furthermore, we prove the uniqueness of the conjugate observables and clarify the precise mathematical conditions and ranges of applicability for the two types of FDTs. Examples are also given to illustrate the main results of this paper.
- Nov 22 2016 math.NT arXiv:1611.06577v1Let $f(n)$ be a multiplicative function with $|f(n)|\leq 1, q$ be a prime number and $a$ be an integer with $(a, q)=1, \chi$ be a non-principal Dirichlet character modulo $q$. Let $\varepsilon$ be a sufficiently small positive constant, $A$ be a large constant, $q^{\frac12+\varepsilon}\ll N\ll q^A$. In this paper, we shall prove that $$ \sum_n≤Nf(n)\chi(n+a)≪N\frac\log\log q\log q $$ and that $$ \sum_n≤Nf(n)\chi(n+a_1)⋯\chi(n+a_t)≪N\frac\log\log q\log q, $$ where $t\geq 2, a_1, \ldots, a_t$ are distinct integers modulo $q$.
- In this paper, we reveal a general relationship between model simplification and irreversibility based on the model of continuous-time Markov chains with time-scale separation. According to the topological structure of the fast process, we divide the states of the chain into the transient states and the recurrent states. We show that a two-time-scale chain can be simplified to a reduced chain in two different ways: removal of the transient states and aggregation of the recurrent states. Both the two operations will lead to a decrease in the entropy production rate and its adiabatic part and will keep its non-adiabatic part the same. This suggests that although model simplification can retain almost all the dynamic information of the chain, it will lose some thermodynamic information as a trade-off.
- May 12 2016 math.PR arXiv:1605.03502v1The embedding problem for Markov chains is a famous problem in probability theory and only partial results are available up till now. In this paper, we propose a variant of the embedding problem called the reversible embedding problem which has a deep physical and biochemical background and provide a complete solution to this new problem. We prove that the reversible embedding of a stochastic matrix, if it exists, must be unique. Moreover, we obtain the sufficient and necessary conditions for the existence of the reversible embedding and provide an effective method to compute the reversible embedding. Some examples are also given to illustrate the main results of this paper.
- Recent studies have shown that the entropy production rate for the master equation consists of two nonnegative terms: the adiabatic and non-adiabatic parts, where the non-adiabatic part is also known as the dissipation rate of a Boltzmann-Shannon relative entropy. In this paper, we provide some nonzero lower bounds for the relative entropy, the entropy production rate, and its adiabatic and non-adiabatic parts. These nonzero lower bounds not only reveal some novel dissipative properties for general nonequilibrium processes which are much stronger than the second law of thermodynamics, but also impose some new constraints on thermodynamic constitutive relations. Moreover, we also provide a mathematical application of these nonzero lower bounds by studying the long-time behavior of the master equation. Extensions to the Tsallis statistics are also discussed, including the nonzero lower bounds for the Tsallis-type relative entropy and its dissipation rate.
- Jul 07 2014 math.PR arXiv:1407.1263v2In probability theory, equalities are much less than inequalities. In this paper, we find a series of equalities which characterize the symmetry of the forming times of a family of similar cycles for discrete-time and continuous-time Markov chains. Moreover, we use these cycle symmetries to study the circulation fluctuations for Markov chains. We prove that the empirical circulations of a family of cycles passing through a common state satisfy a large deviation principle with a rate function which has an highly non-obvious symmetry. Finally, we discuss the applications of our work in statistical physics and biochemistry.
- May 06 2014 math.NT arXiv:1405.0643v1Let $\pi$ be an irreducible unitary cuspidal representation of $GL_m({\Bbb A}_{\Bbb Q})$ and $L(s,\,\pi)$ be the global $L-$function attached to $\pi$. If ${\rm Re}(s)>1$, $L(s,\,\pi)$ has a Dirichlet series expression. When $\pi$ is self-contragradient, all the coefficients of Dirichlet series are real. In this note, we shall give non-trivial lower bounds for the number of positive and negative coefficients respectively, which is an improvement on the recent work of Jianya Liu and Jie Wu.
- Apr 09 2014 math.NT arXiv:1404.2204v2Let $f(n)$ be a multiplicative function satisfying $|f(n)|\leq 1$, $q$ $(\leq N^2)$ be a prime number and $a$ be an integer with $(a,\,q)=1$, $\chi$ be a non-principal Dirichlet character modulo $q$. In this paper, we shall prove that $$ \sum_n≤Nf(n)\chi(n+a)≪N\over q^1\over 4\log\log(6N)+q^1\over 4N^1\over 2\log(6N)+N\over \sqrt\log\log(6N). $$ We shall also prove that \beginalign* &\sum_n≤Nf(n)\chi(n+a_1)⋯\chi(n+a_t)≪N\over q^1\over 4\log\log(6N)\\ &\quad+q^1\over 4N^1\over 2\log(6N)+N\over \sqrt\log\log(6N), \endalign* where $t\geq 2$, $a_1,\,\cdots,\,a_t$ are pairwise distinct integers modulo $q$.
- Jan 21 2014 math.NT arXiv:1401.4556v4Let $f(n)$ be a multiplicative function satisfying $|f(n)|\leq 1$, $q$ $(\leq N^2)$ be a positive integer and $a$ be an integer with $(a,\,q)=1$. In this paper, we shall prove that $$\sum_\substackn≤N\\ (n,\u2009q)=1f(n)e(a\barn\over q)≪\sqrt\tau(q)\over qN\log\log(6N)+q^1\over 4+\epsilon\over 2N^1\over 2(\log(6N))^1\over 2+N\over \sqrt\log\log(6N),$$ where $\bar{n}$ is the multiplicative inverse of $n$ such that $\bar{n}n\equiv 1\,({\rm mod}\,q),\,e(x)=\exp(2\pi ix),\,\tau(q)$ is the divisor function.
- Nov 19 2013 math.NT arXiv:1311.4041v2Let $d(n)$ be the divisor function. In 1916, S. Ramanujan stated but without proof that $$\sum_n≤xd^2(n)=xP(\log x)+E(x), $$ where $P(y)$ is a cubic polynomial in $y$ and $$ E(x)=O(x^3\over 5+\epsilon), $$ where $\epsilon$ is a sufficiently small positive constant. He also stated that, assuming the Riemann Hypothesis(RH), $$ E(x)=O(x^1\over 2+\epsilon). $$ In 1922, B. M. Wilson proved the above result unconditionally. The direct application of the RH would produce $$ E(x)=O(x^1\over 2(\log x)^5\log\log x). $$ In 2003, K. Ramachandra and A. Sankaranarayanan proved the above result without any assumption. In this paper, we shall prove $$ E(x)=O(x^1\over 2(\log x)^5). $$
- Nov 12 2013 math.PR arXiv:1311.2203v4Cyclic structure and dynamics are of great interest in both the fields of stochastic processes and nonequilibrium statistical physics. In this paper, we find a new symmetry of the Brownian motion named as the quasi-time-reversal invariance. It turns out that such an invariance of the Brownian motion is the key to prove the cycle symmetry for diffusion processes on the circle, which says that the distributions of the forming times of the forward and backward cycles, given that the corresponding cycle is formed earlier than the other, are exactly the same. With the aid of the cycle symmetry, we prove the strong law of large numbers, functional central limit theorem, and large deviation principle for the sample circulations and net circulations of diffusion processes on the circle. The cycle symmetry is further applied to obtain various types of fluctuation theorems for the sample circulations, net circulation, and entropy production rate.
- Nov 12 2013 math.PR arXiv:1311.2196v3In this paper, we consider a general class of two-time-scale Markov chains whose transition rate matrix depends on a parameter $\lambda>0$. We assume that some transition rates of the Markov chain will tend to infinity as $\lambda\rightarrow\infty$. We divide the state space of the Markov chain $X$ into a fast state space and a slow state space and define a reduced chain $Y$ on the slow state space. Our main result is that the distribution of the original chain $X$ will converge in total variation distance to that of the reduced chain $Y$ uniformly in time $t$ as $\lambda\rightarrow\infty$.
- Let ${\Bbb F}_2$ be the finite field of two elements, ${\Bbb F}_2^n$ be the vector space of dimension $n$ over ${\Bbb F}_2$. For sets $A,\,B\subseteq{\Bbb F}_2^n$, their sumset is defined as the set of all pairwise sums $a+b$ with $a\in A,\,b\in B$. Ben Green and Terence Tao proved that, let $K\geq 1$, if$A,\,B\subseteq{\Bbb F}_2^n$ and $|A+B|\leq K|A|^{1\over 2}|B|^{1\over 2}$, then there exists a subspace $H\subseteq{\Bbb F}_2^n$ with $$ |H|≫\exp(-O(\sqrtK\log K))|A| $$ and $x,\,y\in{\Bbb F}_2^n$ such that $$ |A∩(x+H)|^1\over 2|B∩(y+H)|^1\over 2≥1\over 2K|H|. $$ In this note, we shall use the method of Green and Tao with some modification to prove that if $$ |H|≫\exp(-O(\sqrtK))|A|, $$ then the above conclusion still holds true.
- Sep 06 2011 math.NT arXiv:1109.0867v1For given positive integers $n$ and $a$, let $R(n;\,a)$ denote the number of positive integer solutions $(x,\,y)$ of the Diophantine equation $$ a\over n=1\over x+1\over y. $$ Write $$ S(N;\u2009a)=\sum_\substackn≤N (n,\u2009a)=1R(n;\u2009a). $$ Recently Jingjing Huang and R. C. Vaughan proved that for $4\leq N$ and $a\leq 2N$, there is an asymptotic formula $$ S(N;\u2009a)=3\over \pi^2a\prod_p|ap-1\over p+1⋅N(\log^2N+c_1(a) \log N+c_0(a))+∆(N;\u2009a). $$ In this paper, we shall get a more explicit expression with better error term for $c_0(a)$.
- Aug 01 2011 math.NT arXiv:1107.6039v1For the positive integer $n$, let $f(n)$ denote the number of positive integer solutions $(n_1, n_2, n_3)$ of the Diophantine equation $$ 4\over n=1\over n_1+1\over n_2+1\over n_3. $$ For the prime number $p$, $f(p)$ can be split into $f_1(p)+f_2(p),$ where $f_i(p)(i=1, 2)$ counts those solutions with exactly $i$ of denominators $n_1, n_2, n_3$ divisible by $p.$ Recently Terence Tao proved that $$ \sum_p< xf_1(p)≪x\exp(c\log x\over \log\log x) $$ with other results. In this paper we shall improve it to $$ \sum_p< xf_1(p)≪x\log^5x\log\log^2x. $$
- Jul 28 2011 math.NT arXiv:1107.5394v1For the positive integer $n$, let $f(n)$ denote the number of positive integer solutions $(n_1,\,n_2,\,n_3)$ of the Diophantine equation $$ 4\over n=1\over n_1+1\over n_2+1\over n_3. $$ For the prime number $p$, $f(p)$ can be split into $f_1(p)+f_2(p),$ where $f_i(p)(i=1,\,2)$ counts those solutions with exactly $i$ of denominators$n_1,\,n_2,\,n_3$ divisible by $p.$ Recently Terence Tao proved that $$ \sum_p< xf_2(p)≪x\log^2x\log\log x $$ with other results. But actually only the upper bound $x\log^2x\log\log^2x$ can be obtained in his discussion. In this note we shall use an elementary method to save a factor $\log\log x$ and recover the above estimate.
- Jan 27 2009 math.NT arXiv:0901.3834v1In this paper we shall study the inverse problem relative to dynamics of the w function which is a special arithmetic function and shall get some results.