On Some Properties of a Class of Fractional Stochastic Heat Equations

We consider nonlinear parabolic stochastic equations of the form \(\partial _t u=\mathcal {L}u + \lambda \sigma (u)\dot{\xi }\) on the ball \(B(0,\,R)\), where \(\dot{\xi }\) denotes some Gaussian noise and \(\sigma \) is Lipschitz continuous. Here \(\mathcal {L}\) corresponds to a symmetric \(\alpha \)-stable process killed upon exiting B(0, R). We will consider two types of noises: space-time white noise and spatially correlated noise. Under a linear growth condition on \(\sigma \), we study growth properties of the second moment of the solutions. Our results are significant extensions of those in Foondun and Joseph (Stoch Process Appl, 2014) and complement those of Khoshnevisan and Kim (Proc AMS, 2013, Ann Probab, 2014).     

Blockwise bootstrap of the estimated empirical process based on $$\psi $$-weakly dependent observations

The distributional convergence of the bootstrapped estimated empirical process is shown and bootstrap consistency in the \(\sup \)-norm for test statistics based on that process. Bootstrapping the estimated empirical process has up to now been considered by assuming independence of the observations, where we give up this assumption now and allow the observations to be \(\psi \)-weakly dependent in the sense of Doukhan and Louhichi (Stoch Proc Appl 84:313–342, 1999). Due to the fact that no model assumptions on the original process are made, a nonparametric blockwise bootstrap procedure is used, which has previously been used in empirical process theory based on mixing observations. Here, we succeeded in proving that assuming \(l=o(n)\) and \(l\rightarrow \infty \) as only conditions for the blocklength is sufficient to show convergence of the bootstrap process to the same limit as for the original process under \({\mathcal {H}}_0\), which is the weakest condition that has been imposed in that context up to now.     

A Multivariate CLT for Bounded Decomposable Random Vectors with the Best Known Rate

We prove a multivariate central limit theorem with explicit error bound in a non-smooth function distance for sums of bounded decomposable \(d\)-dimensional random vectors. The decomposition structure is similar to that of Barbour et al. (J Combin Theory Ser 47:125–145, 1989) and is more general than the local dependence structure considered in Chen and Shao (Ann Probab 32:1985–2028, 2004). The error bound is of the order \(d^{\frac{1}{4}} n^{-\frac{1}{2}}\), where \(d\) is the dimension and \(n\) is the number of summands. The dependence on \(d\), namely \(d^{\frac{1}{4}}\), is the best known dependence even for sums of independent and identically distributed random vectors, and the dependence on \(n\), namely \(n^{-\frac{1}{2}}\), is optimal. We apply our main result to a random graph example.     

Weighted Poincaré Inequalities for Non-local Dirichlet Forms

Let V be a locally bounded measurable function on \({\mathbb {R}}^d\) such that \(\mu _V(\mathrm{d}x)=C_V \mathrm{e}^{-V(x)}\,\mathrm{d}x\) is a probability measure. Explicit criteria are presented for weighted Poincaré inequalities of the following non-local Dirichlet form

$$\begin{aligned} \hat{D}_{\rho ,V}(f,f)=\iint _{\{|x-y|>1\}}(f(y)-f(x))^2\rho (|y-x|)\,\mathrm{d}y\, \mu _V(\mathrm{d}x). \end{aligned}$$

Taking \(\rho (r)={\mathrm{e}^{-\delta r}}{r^{-(d+\alpha )}}\) with \(0<\alpha <2\) and \(\delta \geqslant 0\), we get new conclusions for (exponentially) tempered fractional Dirichlet forms, which not only complete our recent work (Chen and Wang in Stoch Process Their Appl 124:123–153, 2014; Wang and Wang in J Theor Probab 28:423–448, 2015), but also improve the main result in Mouhot et al. (J Math Pures Appl 95:72–84, 2011).

     

No Outliers in the Spectrum of the Product of Independent Non-Hermitian Random Matrices with Independent Entries

We consider products of independent square random non-Hermitian matrices. More precisely, let \(n\ge 2\) and let \(X_1,\ldots ,X_n\) be independent \(N\times N\) random matrices with independent centered entries (either real or complex with independent real and imaginary parts) with variance \(N^{-1}\). In Götze and Tikhomirov (On the asymptotic spectrum of products of independent random matrices, 2011. arXiv:1012.2710) and O’Rourke and Soshnikov (Electron J Probab 16(81):2219–2245, 2011) it was shown that the limit of the empirical spectral distribution of the product \(X_1\cdots X_n\) is supported in the unit disk. We prove that if the entries of the matrices \(X_1,\ldots ,X_n\) satisfy uniform subexponential decay condition, then the spectral radius of \(X_1\cdots X_n\) converges to 1 almost surely as \(N\rightarrow \infty \).     

New constructions of permutation polynomials of the form $$x^rh\left( x^{q-1}\right) $$ over $${\mathbb F}_{q^2}$$Fq2

Permutation polynomials over finite fields have been studied extensively recently due to their wide applications in cryptography, coding theory, communication theory, among others. Recently, several authors have studied permutation trinomials of the form \(x^rh\left( x^{q-1}\right) \) over \({\mathbb F}_{q^2}\), where \(q=2^k\), \(h(x)=1+x^s+x^t\) and \(r, k>0, s, t\) are integers. Their methods are essentially usage of a multiplicative version of AGW Criterion because they all transformed the problem of proving permutation polynomials over \({\mathbb F}_{q^2}\) into that of showing the corresponding fractional polynomials permute a smaller set \(\mu _{q+1}\), where \(\mu _{q+1}:=\{x\in \mathbb {F}_{q^2} : x^{q+1}=1\}\). Motivated by these results, we characterize the permutation polynomials of the form \(x^rh\left( x^{q-1}\right) \) over \({\mathbb F}_{q^2}\) such that \(h(x)\in {\mathbb F}_q[x]\) is arbitrary and q is also an arbitrary prime power. Using AGW Criterion twice, one is multiplicative and the other is additive, we reduce the problem of proving permutation polynomials over \({\mathbb F}_{q^2}\) into that of showing permutations over a small subset S of a proper subfield \({\mathbb F}_{q}\), which is significantly different from previously known methods. In particular, we demonstrate our method by constructing many new explicit classes of permutation polynomials of the form \(x^rh\left( x^{q-1}\right) \) over \({\mathbb F}_{q^2}\). Moreover, we can explain most of the known permutation trinomials, which are in Ding et al. (SIAM J Discret Math 29:79–92, 2015), Gupta and Sharma (Finite Fields Appl 41:89–96, 2016), Li and Helleseth (Cryptogr Commun 9:693–705, 2017), Li et al. (New permutation trinomials constructed from fractional polynomials, arXiv: 1605.06216v1, 2016), Li et al. (Finite Fields Appl 43:69–85, 2017) and Zha et al. (Finite Fields Appl 45:43–52, 2017) over finite field with even characteristic.     

Weakly horospherically convex hypersurfaces in hyperbolic space

In Bonini et al. (Adv Math 280:506–548, 2015), the authors develop a global correspondence between immersed weakly horospherically convex hypersurfaces \(\phi :M^n \rightarrow \mathbb {H}^{n+1}\) and a class of conformal metrics on domains of the round sphere \(\mathbb {S}^n\). Some of the key aspects of the correspondence and its consequences have dimensional restrictions \(n\ge 3\) due to the reliance on an analytic proposition from Chang et al. (Int Math Res Not 2004(4):185–209, 2004) concerning the asymptotic behavior of conformal factors of conformal metrics on domains of \(\mathbb {S}^n\). In this paper, we prove a new lemma about the asymptotic behavior of a functional combining the gradient of the conformal factor and itself, which allows us to extend the global correspondence and embeddedness theorems of Bonini et al. (2015) to all dimensions \(n\ge 2\) in a unified way. In the case of a single point boundary \(\partial _{\infty }\phi (M)=\{x\} \subset \mathbb {S}^n\), we improve these results in one direction. As an immediate consequence of this improvement and the work on elliptic problems in Bonini et al. (2015), we have a new, stronger Bernstein type theorem. Moreover, we are able to extend the Liouville and Delaunay type theorems from Bonini et al. (2015) to the case of surfaces in \(\mathbb {H}^{3}\).     

Bumpy metrics on spheres and minimal index growth

The existence of two geometrically distinct closed geodesics on an n-dimensional sphere \(S^n\) with a non-reversible and bumpy Finsler metric was shown independently by Duan and Long [7] and the author [25]. We simplify the proof of this statement by the following observation: If for some \(N \in \mathbb {N}\) all closed geodesics of index \(\le \)N of a non-reversible and bumpy Finsler metric on \(S^n\) are geometrically equivalent to the closed geodesic c, then there is a covering \(c^r\) of minimal index growth, i.e.,

$$\begin{aligned} \mathrm{ind}(c^\mathrm{rm})=m \,\mathrm{ind}(c^r)-(m-1)(n-1), \end{aligned}$$

for all \(m \ge 1\) with \(\mathrm{ind}\left( c^\mathrm{rm}\right) \le N.\) But this leads to a contradiction for \(N =\infty \) as pointed out by Goresky and Hingston [13]. We also discuss perturbations of Katok metrics on spheres of even dimension carrying only finitely many closed geodesics. For arbitrarily large \(L>0\), we obtain on \(S^2\) a metric of positive flag curvature carrying only two closed geodesics of length \(<L\) which do not intersect.

     

Nonexistence of two classes of generalized bent functions

We obtain some new nonexistence results of generalized bent functions from \({\mathbb {Z}}^n_q\) to \({\mathbb {Z}}_q\) (called type [n, q]) in the case that there exist cyclotomic integers in \( {\mathbb {Z}}[\zeta _{q}]\) with absolute value \(q^{\frac{n}{2}}\). This result generalizes two previous nonexistence results \([n,q]=[1,2\times 7]\) of Pei (Lect Notes Pure Appl Math 141:165–172, 1993) and \([3,2\times 23^e]\) of Jiang and Deng (Des Codes Cryptogr 75:375–385, 2015). We also remark that by using a same method one can get similar nonexistence results of GBFs from \({\mathbb {Z}}^n_2\) to \({\mathbb {Z}}_m\).     

Vanishing theorems for constructible sheaves on abelian varieties over finite fields

Let \(\kappa \) be a field, finitely generated over its prime field, and let k denote an algebraically closed field containing \(\kappa \). For a perverse \(\overline{\mathbb {Q}}_\ell \)-adic sheaf \(K_0\) on an abelian variety \(X_0\) over \(\kappa \), let K and X denote the base field extensions of \(K_0\) and \(X_0\) to k. Then, the aim of this note is to show that the Euler–Poincare characteristic of the perverse sheaf K on X is a non-negative integer, i.e. \(\chi (X,K)=\sum _\nu (-1)^\nu \dim _{\overline{\mathbb {Q}}_\ell }(H^\nu (X,K))\ge 0\). This generalizes the result of Franecki and Kapranov [9] for fields of characteristic zero. Furthermore we show that \(\chi (X,K)=0\) implies K to be translation invariant. This result allows to considerably simplify the proof of the generic vanishing theorems for constructible sheaves on complex abelian varieties of [11]. Furthermore it extends these vanishing theorems to constructible sheaves on abelian varieties over finite fields.     

Central Limit Theorems for Supercritical Superprocesses with Immigration

In this paper, we establish a central limit theorem for a large class of general supercritical superprocesses with immigration with spatially dependent branching mechanisms satisfying a second moment condition. This central limit theorem extends and generalizes the results obtained by Ren et al. (Stoch Process Appl 125:428–457, 2015). We first give laws of large numbers for supercritical superprocesses with immigration since there are few convergence results on immigration superprocesses, then based on these results, we establish the central limit theorem.     

Improved Upper Bounds on the Domination Number of Graphs With Minimum Degree at Least Five

An algorithmic upper bound on the domination number \(\gamma \) of graphs in terms of the order n and the minimum degree \(\delta \) is proved. It is demonstrated that the bound improves best previous bounds for any \(5\le \delta \le 50\). In particular, for \(\delta =5\), Xing et al. (Graphs Comb. 22:127–143, 2006) proved that \(\gamma \le 5n/14 < 0.3572 n\). This bound is improved to 0.3440 n. For \(\delta =6\), Clark et al. (Congr. Numer. 132:99–123, 1998) established \(\gamma <0.3377 n\), while Biró et al. (Bull. Inst. Comb. Appl. 64:73–83, 2012) recently improved it to \(\gamma <0.3340 n\). Here the bound is further improved to \(\gamma < 0.3159n\). For \(\delta =7\), the best earlier bound 0.3088n is improved to \(\gamma < 0.2927n\).     

A Distributional Equality for Suprema of Spectrally Positive Lévy Processes

Let \(Y\) be a spectrally positive Lévy process with \({\mathbb {E}}Y_1\!<\!0\) and \(C\) an independent subordinator with finite expectation, and let \(X\!=\!Y\!+\!C\). A curious distributional equality proved in Huzak et al. (Ann Appl Probab 14:1278–1397, 2004) states that if \({\mathbb {E}}X_1<0\), then \(\sup _{0\le t <\infty }Y_t\) and the supremum of \(X\) just before the first time its new supremum is reached by a jump of \(C\) have the same distribution. In this paper, we give an alternative proof of an extension of this result and offer an explanation why it is true.     

Functional Convergence of Linear Processes with Heavy-Tailed Innovations

We study convergence in law of partial sums of linear processes with heavy-tailed innovations. In the case of summable coefficients, necessary and sufficient conditions for the finite dimensional convergence to an \(\alpha \)-stable Lévy Motion are given. The conditions lead to new, tractable sufficient conditions in the case \(\alpha \le 1\). In the functional setting, we complement the existing results on \(M_1\)-convergence, obtained for linear processes with nonnegative coefficients by Avram and Taqqu (Ann Probab 20:483–503, 1992) and improved by Louhichi and Rio (Electr J Probab 16(89), 2011), by proving that in the general setting partial sums of linear processes are convergent on the Skorokhod space equipped with the \(S\) topology, introduced by Jakubowski (Electr J Probab 2(4), 1997).     

A Maxtrimmed St. Petersburg Game

Let \(S_n\), \(n\ge 1\), describe the successive sums of the payoffs in the classical St. Petersburg game. The celebrated Feller weak law states that \(\frac{S_n}{n\log _2 n}\mathop {\rightarrow }\limits ^{p}1\) as \(n\rightarrow \infty \). It is also known that almost sure convergence fails. However, Csörg? and Simons (Stat Probab Lett 26:65–73, 1996) have shown that almost sure convergence holds for trimmed sums, that is, for \(S_n-\max _{1\le k\le n}X_k\). Since our actual distribution is discrete there may be ties. Our main focus in this paper is on the “maxtrimmed sum”, that is, the sum trimmed by the random number of observations that are equal to the largest one. We prove an analog of Martin-Löf’s (J Appl Probab 22:634–643, 1985) distributional limit theorem for maxtrimmed sums, but also for the simply trimmed ones, as well as for the “total maximum”. In a final section, we interpret these findings in terms of sums of (truncated) Poisson random variables.     

Convergence of empirical spectral distributions of large dimensional quaternion sample covariance matrices

In this paper, we establish the limit of empirical spectral distributions of quaternion sample covariance matrices. Motivated by Bai and Silverstein (Spectral analysis of large dimensional random matrices, Springer, New York, 2010) and Mar?enko and Pastur (Matematicheskii Sb, 114:507–536, 1967), we can extend the results of the real or complex sample covariance matrix to the quaternion case. Suppose \(\mathbf X_n = ({x_{jk}^{(n)}})_{p\times n}\) is a quaternion random matrix. For each \(n\), the entries \(\{x_{ij}^{(n)}\}\) are independent random quaternion variables with a common mean \(\mu \) and variance \(\sigma ^2>0\). It is shown that the empirical spectral distribution of the quaternion sample covariance matrix \(\mathbf S_n=n^{-1}\mathbf X_n\mathbf X_n^*\) converges to the Mar?enko–Pastur law as \(p\rightarrow \infty \), \(n\rightarrow \infty \) and \(p/n\rightarrow y\in (0,+\infty )\).     

Remarks on curvature dimension conditions on graphs

We show a connection between the \(CDE'\) inequality introduced in Horn et al. (Volume doubling, Poincaré inequality and Gaussian heat kernel estimate for nonnegative curvature graphs. arXiv:1411.5087v2, 2014) and the \(CD\psi \) inequality established in Münch (Li–Yau inequality on finite graphs via non-linear curvature dimension conditions. arXiv:1412.3340v1, 2014). In particular, we introduce a \(CD_\psi ^\varphi \) inequality as a slight generalization of \(CD\psi \) which turns out to be equivalent to \(CDE'\) with appropriate choices of \(\varphi \) and \(\psi \). We use this to prove that the \(CDE'\) inequality implies the classical CD inequality on graphs, and that the \(CDE'\) inequality with curvature bound zero holds on Ricci-flat graphs.     

Inverse ambiguous functions on some finite non-abelian groups

In Schmitz (Aequ Math 91:373–389, 2017), the first author defines an “inverse ambiguous function” on a group G to be a bijective function \(f:G \rightarrow G\) satisfying the functional equation \(f^{-1}(x) = (f(x))^{-1}\) for all \(x \in G\). Using a simple criterion involving the number of elements in G not equal to their own inverse, the classification of finite abelian groups admitting inverse ambiguous functions is achieved. In this paper we aim to extend the results from (2017) to determine the existence of inverse ambiguous functions on members of certain families of non-abelian groups, namely the symmetric groups \(S_n\), the alternating groups \(A_n\), and the general linear groups GL(2, q) over a finite field \(\mathbb {F}_q\).     

Games with the total bandwagon property meet the Quint–Shubik conjecture

This paper revisits the total bandwagon property (TBP) introduced by Kandori and Rob (Games Econ Behav 22:30–60, 1998). With this property, we characterize the class of two-player symmetric \(n\times n\) games, showing that a game has TBP if and only if the game has \(2^{n}-1\) symmetric Nash equilibria. We extend this result to bimatrix games by generalizing TBP. This sheds light on the (wrong) conjecture of Quint and Shubik (Int J Game Theory 26:353–359, 1997) that any nondegenerate \(n\times n\) bimatrix game has at most \(2^{n}-1\) Nash equilibria. We also provide an equilibrium selection criterion to two subclasses of games with TBP.     

Nonuniform Multiresolution Analysis on Local Fields of Positive Characteristic

A generalization of Mallat’s classic theory of multiresolution analysis (MRA) on local fields of positive characteristic was considered by Jiang et al. (J Math Anal Appl 294:523–532, 2004). In this paper, we present a notion of nonuniform MRA on local field \(K\) of positive characteristic. The associated subspace \(V_0\) of \(L^2(K)\) has an orthonormal basis, a collection of translates of the scaling function \(\varphi \) of the form \(\{ \varphi (x-\lambda ) \}_{ \lambda \in \Lambda }\) where \(\Lambda = \{ 0,r/N \}+ \mathcal{Z}, \,N \ge 1\) is an integer and \(r\) is an odd integer such that \(r\) and \(N\) are relatively prime and \(\mathcal{Z}=\{u(n): n\in \mathbb {N}_{0}\}\). We obtain the necessary and sufficient condition for the existence of associated wavelets and present an algorithm for the construction of nonuniform MRA on local fields starting from a low-pass filter \(m_{0}\) with appropriate conditions.