Moderate deviation principle for <Emphasis Type="Italic">m</Emphasis>-dependent random variables<Superscript>*</Superscript>

Let {X _n }_n?≥?1 be a sequence of strictly stationary m-dependent random variableswith EX₁ = 0 and \( \mathrm{E}{X}_1^2<\infty \), and let (b _n ) be an increasing sequence of positive numbers such that b _n ?↑?∞ and \( {b}_n/\sqrt{n}\downarrow 0\kern0.5em \mathrm{as}\kern0.5em n\to \infty \). We establish a moderate deviation principle of \( {\left({b}_n\sqrt{n}\right)}^{-1}{\sum}_{i=1}^n{X}_i \) under the condition

$$ \underset{n\to \infty }{\lim \sup}\frac{1}{b_n^2}\log \left[n\mathbf{P}\left(\left|{X}_1\right|>{b}_n\sqrt{n}\right)\right]=-\infty, $$

which is weaker than the classical exponential integrability condition. The results in the present paper weaken the assumptions of Chen [5] and extend partially the results of Eichelsbacher and Löwe [10].     

The first moment of central values of symmetric square <Emphasis Type="Italic">L</Emphasis>-functions in the weight aspect

We establish an asymptotic formula with arbitrary power saving for the first moment of the symmetric square L-functions \(L(s,\mathrm{sym}^2f)\) at \(s=\frac{1}{2}\) for \(f\in \mathcal {H}_k\) as even \(k\rightarrow \infty \), where \(\mathcal {H}_k\) is an orthogonal basis of weight-k Hecke eigen cusp forms for \(SL(2,\mathbb {Z})\). The approach taken allows us to extract two secondary main terms from the best-known error term \(O(k^{-\frac{1}{2}})\). Moreover, our result exhibits a connection between the symmetric square L-functions and quadratic fields, which is the main theme of Zagier’s work Modular forms whose coefficients involve zeta-functions of quadratic fields in 1977.     

A Neighborhood Union Condition for Fractional ID-[<Emphasis Type="Italic">a</Emphasis>, <Emphasis Type="Italic">b</Emphasis>]-factor-critical Graphs

Let a, b, r be nonnegative integers with \(1\leq{a}\leq{b}\) and \(r\geq2\). Let G be a graph of order n with \(n >\frac{(a+2b)(r(a+b)-2)}{b}\). In this paper, we prove that G is fractional ID-[a, b]-factor-critical if \(\delta(G)\geq\frac{bn}{a+2b}+a(r-1)\) and \(\mid N_{G}(x_{1}) \cup N_{G}(x_{2}) \cup \cdotp \cdotp \cdotp \cup N_{G}(x_{r})\mid\geq\frac{(a+b)n}{a+2b}\) for any independent subset {x₁, x₂, · · ·, x_r} in G. It is a generalization of Zhou et al.’s previous result [Discussiones Mathematicae Graph Theory, 36: 409–418 (2016)] in which r = 2 is discussed. Furthermore, we show that this result is best possible in some sense.     

Finite groups with <Emphasis Type="Italic">SS</Emphasis>-supplement

Let G be a finite group. A subgroup H of G is said to be SS-quasinormal in G if there is a subgroup K such that \(G=HK\) and \(HS=SH\), for all \(S\in \) Syl(K), where Syl(K) denotes the collection of all Sylow subgroups of K. A subgroup H of G is said to be SS-supplemented in G if there is a subgroup K such that \(G=HK\) and \(H\cap K\) is SS-quasinormal in G. In this paper, we investigate the SS-supplemented subgroups and strengthen a result of Skiba which gives a positive answer to an open question of Shemetkov.     

Existence and nonexistence of solutions to pure supercritical <Emphasis Type="Italic">p</Emphasis>-Laplacian problems

We establish multiplicity and nonexistence of solutions to the quasilinear problem

$$\begin{aligned} -\Delta _{p}v=\left| v\right| ^{q-2}v\,\,\text {in}\,\,\Omega ,\qquad v=0\text { on }{\partial {\Omega }}, \end{aligned}$$

in some bounded smooth domains \(\Omega \) in \(\mathbb {R}^{N}\), for \(1<p<N\) and some supercritical exponents \(q>p^{*}:=\frac{Np}{N-p}\). Multiplicity is established in domains arising from the Hopf maps. We show that, after a suitable change of metric, these maps become p-harmonic morphisms, i.e., they preserve the p-Laplace operator up to a factor. We use them to reduce the supercritical problem to an anisotropic quasilinear critical problem in a domain of lower dimension.

     

An <Emphasis Type="Italic">A</Emphasis>-invariant subspace for bipartite distance-regular graphs with exactly two irreducible <Emphasis Type="Italic">T</Emphasis>-modules with endpoint 2, both thin

Let \(\Gamma \) denote a bipartite distance-regular graph with vertex set X, diameter \(D \ge 4\), and valency \(k \ge 3\). Let \({{\mathbb {C}}}^X\) denote the vector space over \({{\mathbb {C}}}\) consisting of column vectors with entries in \({{\mathbb {C}}}\) and rows indexed by X. For \(z \in X\), let \({{\widehat{z}}}\) denote the vector in \({{\mathbb {C}}}^X\) with a 1 in the z-coordinate, and 0 in all other coordinates. Fix a vertex x of \(\Gamma \) and let \(T = T(x)\) denote the corresponding Terwilliger algebra. Assume that up to isomorphism there exist exactly two irreducible T-modules with endpoint 2, and they both are thin. Fix \(y \in X\) such that \(\partial (x,y)=2\), where \(\partial \) denotes path-length distance. For \(0 \le i,j \le D\) define \(w_{ij}=\sum {{\widehat{z}}}\), where the sum is over all \(z \in X\) such that \(\partial (x,z)=i\) and \(\partial (y,z)=j\). We define \(W=\mathrm{span}\{w_{ij} \mid 0 \le i,j \le D\}\). In this paper we consider the space \(MW=\mathrm{span}\{mw \mid m \in M, w \in W\}\), where M is the Bose–Mesner algebra of \(\Gamma \). We observe that MW is the minimal A-invariant subspace of \({{\mathbb {C}}}^X\) which contains W, where A is the adjacency matrix of \(\Gamma \). We show that \(4D-6 \le \mathrm{dim}(MW) \le 4D-2\). We display a basis for MW for each of these five cases, and we give the action of A on these bases.     

Beatty sequences and prime numbers with restrictions on strongly <Emphasis Type="Italic">q</Emphasis>-additive functions

We estimate exponential sums over a non-homogenous Beatty sequence with restriction on strongly q-additive functions. We then apply our result in a few special cases to obtain an asymptotic formula for the number of primes \(p=\lfloor \alpha n +\beta \rfloor \) and \(f(p)\equiv a (\mathrm{mod\,}b)\), with \(n\ge N \), where \(\alpha \), \(\beta \) are real numbers and f is a strongly q-additive function (for example, the sum of digits function in base q is a strongly q-additive function). We also prove that for any fixed integer \(k\ge 3 \), all sufficiently large \(N\equiv k (\mathrm{mod\,}2) \) could be represented as a sum of k prime numbers from a Beatty sequence with restriction on strongly q-additive functions.     

EKR Sets for Large <Emphasis Type="Italic">n</Emphasis> and <Emphasis Type="Italic">r</Emphasis>

Let \(\mathcal {A}\subset \left( {\begin{array}{c}[n]\\ r\end{array}}\right) \) be a compressed, intersecting family and let \(X\subset [n]\). Let \(\mathcal {A}(X)=\{A\in \mathcal {A}:A\cap X\ne \emptyset \}\) and \(\mathcal {S}_{n,r}=\left( {\begin{array}{c}[n]\\ r\end{array}}\right) (\{1\})\). Motivated by the Erd?s–Ko–Rado theorem, Borg asked for which \(X\subset [2,n]\) do we have \(|\mathcal {A}(X)|\le |\mathcal {S}_{n,r}(X)|\) for all compressed, intersecting families \(\mathcal {A}\)? We call X that satisfy this property EKR. Borg classified EKR sets X such that \(|X|\ge r\). Barber classified X, with \(|X|\le r\), such that X is EKR for sufficiently large n, and asked how large n must be. We prove n is sufficiently large when n grows quadratically in r. In the case where \(\mathcal {A}\) has a maximal element, we sharpen this bound to \(n>\varphi ^{2}r\) implies \(|\mathcal {A}(X)|\le |\mathcal {S}_{n,r}(X)|\). We conclude by giving a generating function that speeds up computation of \(|\mathcal {A}(X)|\) in comparison with the naïve methods.     

<Emphasis Type="Italic">q</Emphasis>-Frequent hypercyclicity in spaces of operators

We provide conditions for a linear map of the form \(C_{R,T}(S)=RST\) to be q-frequently hypercyclic on algebras of operators on separable Banach spaces. In particular, if R is a bounded operator satisfying the q-frequent hypercyclicity criterion, then the map \(C_{R}(S)=RSR^*\) is shown to be q-frequently hypercyclic on the space \(\mathcal {K}(H)\) of all compact operators and the real topological vector space \(\mathcal {S}(H)\) of all self-adjoint operators on a separable Hilbert space H. Further we provide a condition for \(C_{R,T}\) to be q-frequently hypercyclic on the Schatten von Neumann classes \(S_p(H)\). We also characterize frequent hypercyclicity of \(C_{M^*_\varphi ,M_\psi }\) on the trace-class of the Hardy space, where the symbol \(M_\varphi \) denotes the multiplication operator associated to \(\varphi \).     

Critical Sets of Full <Emphasis Type="Italic">n</Emphasis>-Latin Squares

The full n-Latin square is the \(n\times n\) array with symbols \(1,2,\dots ,n\) in each cell. In a way that is analogous to critical sets of full designs, a critical set of the full n-Latin square can be used to find a defining set for any Latin square of order n. In this paper we study the size of the smallest critical set for a full n-Latin square, showing this to be somewhere between \((n^3-2n^2+2n)/2\) and \((n-1)^3+1\). In the case that each cell is either full or empty, we show the size of a critical set in the full n-Latin square is always equal to \(n^3-2n^2-n\).     

Spectral multiplier theorems and averaged <Emphasis Type="Italic">R</Emphasis>-boundedness

Let A be a 0-sectorial operator with a bounded \(H^\infty (\Sigma _\sigma )\)-calculus for some \(\sigma \in (0,\pi ),\) e.g. a Laplace type operator on \(L^p(\Omega ),\, 1< p < \infty ,\) where \(\Omega \) is a manifold or a graph. We show that A has a \(\mathcal {H}^\alpha _2(\mathbb {R}_+)\) Hörmander functional calculus if and only if certain operator families derived from the resolvent \((\lambda - A)^{-1},\) the semigroup \(e^{-zA},\) the wave operators \(e^{itA}\) or the imaginary powers \(A^{it}\) of A are R-bounded in an \(L^2\)-averaged sense. If X is an \(L^p(\Omega )\) space with \(1 \le p < \infty \), R-boundedness reduces to well-known estimates of square sums.     

The rate of decay of stable periodic solutions for Duffing equation with <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-conditions

For a new class of g(t, x), the existence, uniqueness and stability of \({2\pi}\)-periodic solution of Duffing equation \({x'' + cx' + g(t, x) = h(t)}\) are presented. Moreover, the unique \({2\pi}\)-periodic solution is (exponentially asymptotically stable) and its rate of exponential decay c/2 is sharp. The new criterion characterizes \({g_{x}^{\prime}(t, x) - c^2/4}\) with L ^p -norms \({(p \in [1, \infty])}\), and the classical criterion employs the \({L^{\infty}}\)-norm. The advantage is that we can deal with the case that \({g_{x}^{\prime}(t, x) - c^2/4}\) is beyond the optimal bounds of the \({L^{\infty}}\)-norm, because of the difference between the L ^p -norm and the \({L^{\infty}}\)-norm.     

<Emphasis Type="Italic">b</Emphasis>-Generalized Skew Derivations on Lie Ideals

Let R be a non-commutative prime ring, Z(R) its center, Q its right Martindale quotient ring, C its extended centroid, \(F\ne 0\) an b-generalized skew derivation of R, L a non-central Lie ideal of R, \(0\ne a\in R\) and \(n\ge 1\) a fixed integer. In this paper, we prove the following two results:

1. 1.
   If R has characteristic different from 2 and 3 and \(a[F(x),x]^n=0\), for all \(x\in L\), then either there exists an element \(\lambda \in C\), such that \(F(x)=\lambda x\), for all \(x\in R\) or R satisfies \(s_4(x_1,\ldots ,x_4)\), the standard identity of degree 4, and there exist \(\lambda \in C\) and \(b\in Q\), such that \(F(x)=bx+xb+\lambda x\), for all \(x\in R\).
    
2. 2.
   If \(\mathrm{{char}}(R)=0\) or \(\mathrm{{char}}(R) > n\) and \(a[F(x),x]^n\in Z(R)\), for all \(x\in R\), then either there exists an element \(\lambda \in C\), such that \(F(x)=\lambda x\), for all \(x\in R\) or R satisfies \(s_4(x_1,\ldots ,x_4)\).
    

     

The random <Emphasis Type="Italic">k</Emphasis> cycle walk on the symmetric group

We study the random walk on the symmetric group \(S_n\) generated by the conjugacy class of cycles of length k. We show that the convergence to uniform measure of this walk has a cut-off in total variation distance after \(\frac{n}{k}\log n\) steps, uniformly in \(k = o(n)\) as \(n \rightarrow \infty \). The analysis follows from a new asymptotic estimation of the characters of the symmetric group evaluated at cycles.     

Estimates for coefficients of certain <Emphasis Type="Italic">L</Emphasis>-functions

Let \(\pi _{\varphi }\) (or \(\pi _{\psi }\)) be an automorphic cuspidal representation of \(\text {GL}_{2} (\mathbb {A}_{\mathbb {Q}})\) associated to a primitive Maass cusp form \(\varphi \) (or \(\psi \)), and \(\mathrm{sym}^j \pi _{\varphi }\) be the jth symmetric power lift of \(\pi _{\varphi }\). Let \(a_{\mathrm{sym}^j \pi _{\varphi }}(n)\) denote the nth Dirichlet series coefficient of the principal L-function associated to \(\mathrm{sym}^j \pi _{\varphi }\). In this paper, we study first moments of Dirichlet series coefficients of automorphic representations \(\mathrm{sym}^3 \pi _{\varphi }\) of \(\text {GL}_{4}(\mathbb {A}_{\mathbb {Q}})\), and \(\pi _{\psi }\otimes \mathrm{sym}^2 \pi _{\varphi }\) of \(\text {GL}_{6}(\mathbb {A}_{\mathbb {Q}})\). For \(3 \le j \le 8\), estimates for \(|a_{\mathrm{sym}^j \pi _{\varphi }}(n)|\) on average over a short interval have also been established.     

Diophantine approximation with four squares and one <Emphasis Type="Italic">k</Emphasis>th power of primes

Let k be an integer with \(k\ge 3\) and \(\eta \) be any real number. Suppose that \(\lambda _1, \lambda _2, \lambda _3, \lambda _4, \mu \) are non-zero real numbers, not all of the same sign and \(\lambda _1/\lambda _2\) is irrational. It is proved that the inequality \(|\lambda _1p_1^2+\lambda _2p_2^2+\lambda _3p_3^2+\lambda _4p_4^2+\mu p_5^k+\eta |<(\max \ p_j)^{-\sigma }\) has infinitely many solutions in prime variables \(p_1, p_2, \ldots , p_5\), where \(0<\sigma <\frac{1}{16}\) for \(k=3,\ 0<\sigma <\frac{5}{3k2^k}\) for \(4\le k\le 5\) and \(0<\sigma <\frac{40}{21k2^k}\) for \(k\ge 6\). This gives an improvement of an earlier result.     

On the semiclassical functional calculus for <Emphasis Type="Italic">h</Emphasis>-dependent functions

We study the functional calculus for operators of the form \(f_h(P(h))\) within the theory of semiclassical pseudodifferential operators, where \(\{f_h\}_{h\in (0,1]}\subset \mathrm{C^\infty _c}({{\mathbb {R}}})\) denotes a family of h-dependent functions satisfying some regularity conditions, and P(h) is either an appropriate self-adjoint semiclassical pseudodifferential operator in \(\mathrm{L}^2({{\mathbb {R}}}^n)\) or a Schrödinger operator in \(\mathrm{L}^2(M), M\) being a closed Riemannian manifold of dimension n. The main result is an explicit semiclassical trace formula with remainder estimate that is well-suited for studying the spectrum of P(h) in spectral windows of width of order \(h^\delta \), where \(0\le \delta <\frac{1}{2}\).     

Some class 1 graphs on <Emphasis Type="Italic">g</Emphasis><Subscript><Emphasis Type="Italic">c</Emphasis></Subscript>-colorings

An edge-coloring of a graph G is an assignment of colors to all the edges of G. A g _c -coloring of a graph G is an edge-coloring of G such that each color appears at each vertex at least g(v) times. The maximum integer k such that G has a g _c -coloring with k colors is called the g _c -chromatic index of G and denoted by \(\chi\prime_{g_{c}}\)(G). In this paper, we extend a result on edge-covering coloring of Zhang and Liu in 2011, and give a new sufficient condition for a simple graph G to satisfy \(\chi\prime_{g_{c}}\)(G) = δ _g (G), where \(\delta_{g}\left(G\right) = min_{v\epsilon V (G)}\left\{\lfloor\frac{d\left(v\right)}{g\left(v\right)}\rfloor\right\}\).     

Weak Type (1,1) Behavior for the Maximal Operator with <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-Dini Kernel

Let M _Ω be the maximal operator with homogeneous kernel Ω. In the present paper, we show that if Ω satisfies the L ¹-Dini condition on ?? ^n?1, then the following weak type (1,1) behaviors

$$\lim\limits _{\lambda \rightarrow 0_{+}}\lambda m(\{x\in \mathbb {R}^{n}:M_{\Omega } f(x)>\lambda \})=\frac {1}{n} \|\Omega \|_{1} \|f\|_{1},$$

$$\sup\limits_{\lambda >0}\lambda m(\{x\in \mathbb {R}^{n}:M_{\Omega } f(x)>\lambda \})\lesssim {\bigg ((\log n)\|\Omega \|_{1}+{\int }_{0}^{1/n}\frac {\tilde {\omega }_{1}(\delta )}{\delta }d\delta \bigg )}\|f\|_{1}$$

hold for the maximal operator M _Ω and \(f\in L^{1}(\mathbb {R}^{n})\), here \(\tilde {\omega }_{1}\) denotes the L ¹ integral modulus of continuity of Ω defined by translation in \(\mathbb {R}^{n}\).     

The structure of finite groups with weakly <Emphasis Type="Italic">c</Emphasis>-normal subgroups

For a subgroup of a finite group we introduce a new property called weakly c-normal. Suppose that G is a finite group and H is a subgroup of G. H is said to be weakly c-normal in G if there exists a subnormal subgroup K of G such that \(G=HK\) and \(H\cap K\) is s-quasinormally embedded in G. We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying \(1<|D|<|P|\) and study the structure of G under the assumption that every subgroup H of P with \(|H|=|D|\) is weakly c-normal in G. Some recent results are generalized and unified.