Some Estimates of Higher Order Riesz Transform Related to Schrödinger Type Operators

In this paper we consider the Schrödinger type operators \(H_2=(-\Delta)^2 +V^2\), where the nonnegative potential V belongs to the reverse Hölder class \(B_{q_{_1}}\) for \(q_{_1}\geq \frac{n}{2}, n\geq 5\). The L ^p and weak type (1, 1) estimates of higher order Riesz transform \(\nabla^2H^{-\frac{1}{2}}_2 \) related to Schrödinger type operators H ₂ are obtained. In particular, \(\nabla^2H^{-\frac{1}{2}}_2 \) is a Calderón-Zygmund operator if V?∈?B _2n or \(V\in B_\frac{n}{2}\) and there exists a constant C such that V(x)?≤?Cm(x,V)².     

Map determined by rank-$$\varvec{}$$ matrice for relatively mall $$\varvec{}$$

Let n and s be integers such that \(1\le s<\frac{n}{2}\), and let \(M_n(\mathbb {K})\) be the ring of all \(n\times n\) matrices over a field \(\mathbb {K}\). Denote by \([\frac{n}{s}]\) the least integer m with \(m\ge \frac{n}{s}\). In this short note, it is proved that if \(g:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K})\) is a map such that \(g\left( \sum _{i=1}^{[\frac{n}{s}]}A_i\right) =\sum _{i=1}^{[\frac{n}{s}]}g(A_i)\) holds for any \([\frac{n}{s}]\) rank-s matrices \(A_1,\ldots ,A_{[\frac{n}{s}]}\in M_n(\mathbb {K})\), then \(g(x)=f(x)+g(0)\), \(x\in M_n(\mathbb {K})\), for some additive map \(f:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K})\). Particularly, g is additive if \(char\mathbb {K}\not \mid \left( [\frac{n}{s}]-1\right) \).     

Characterization of finite $$\varvec{}$$-grous by their Schur multilier

Let G be a finite p-group of order \(p^n\) and M(G) be its Schur multiplier. It is a well known result by Green that \(|M(G)|= p^{\frac{1}{2}n(n-1)-t(G)}\) for some \(t(G) \ge 0\). In this article, we classify non-abelian p-groups G of order \(p^n\) for \(t(G)=\log _p(|G|)+1\).     

On the gaps between non-zero Fourier coefficients of cusp forms of higher weight

We show that if a modular cuspidal eigenform f of weight 2k is 2-adically close to an elliptic curve \(E/\mathbb {Q}\), which has a cyclic rational 4-isogeny, then n-th Fourier coefficient of f is non-zero in the short interval \((X, X + cX^{\frac{1}{4}})\) for all \(X \gg 0\) and for some \(c > 0\). We use this fact to produce non-CM cuspidal eigenforms f of level \(N>1\) and weight \(k > 2\) such that \(i_f(n) \ll n^{\frac{1}{4}}\) for all \(n \gg 0\).     

Stable CLTs and Rates for Power Variation of <Emphasis Type="Italic">α</Emphasis>-Stable Lévy Processes

In a central limit type result it has been shown that the pth power variations of an α-stable Lévy process along sequences of equidistant partitions of a given time interval have \(\frac{\alpha}{p}\)-stable limits. In this paper we give precise orders of convergence for the distances of the approximate power variations computed for partitions with mesh of order \(\frac{1}{n}\) and the limiting law, measured in terms of the Kolmogorov-Smirnov metric. In case 2α?<?p the convergence rate is seen to be of order \(\frac{1}{n}\), in case α?<?p?<?2α the order is \(n^{1-\frac{p}{\alpha}}.\)     

Edge Roman Domination on Graphs

An edge Roman dominating function of a graph G is a function \(f:E(G) \rightarrow \{0,1,2\}\) satisfying the condition that every edge e with \(f(e)=0\) is adjacent to some edge \(e'\) with \(f(e')=2\). The edge Roman domination number of G, denoted by \(\gamma '_R(G)\), is the minimum weight \(w(f) = \sum _{e\in E(G)} f(e)\) of an edge Roman dominating function f of G. This paper disproves a conjecture of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi and Sadeghian Sadeghabad stating that if G is a graph of maximum degree \(\Delta \) on n vertices, then \(\gamma _R'(G) \le \lceil \frac{\Delta }{\Delta +1} n \rceil \). While the counterexamples having the edge Roman domination numbers \(\frac{2\Delta -2}{2\Delta -1} n\), we prove that \(\frac{2\Delta -2}{2\Delta -1} n + \frac{2}{2\Delta -1}\) is an upper bound for connected graphs. Furthermore, we provide an upper bound for the edge Roman domination number of k-degenerate graphs, which generalizes results of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi and Sadeghian Sadeghabad. We also prove a sharp upper bound for subcubic graphs. In addition, we prove that the edge Roman domination numbers of planar graphs on n vertices is at most \(\frac{6}{7}n\), which confirms a conjecture of Akbari and Qajar. We also show an upper bound for graphs of girth at least five that is 2-cell embeddable in surfaces of small genus. Finally, we prove an upper bound for graphs that do not contain \(K_{2,3}\) as a subdivision, which generalizes a result of Akbari and Qajar on outerplanar graphs.     

Size of Monochromatic Double Stars in Edge Colorings

We show that in every r-coloring of the edges of K _n there is a monochromatic double star with at least \(\frac{n(r+1)+r-1}{r^2}\) vertices. This result is sharp in asymptotic for r = 2 and for r≥ 3 improves a bound of Mubayi for the largest monochromatic subgraph of diameter at most three. When r-colorings are replaced by local r-colorings, our bound is \(\frac{n(r+1)+r-1}{r^2+1}\).     

Counting polynomial subset sums

Let D be a subset of a finite commutative ring R with identity. Let \(f(x)\in R[x]\) be a polynomial of degree d. For a nonnegative integer k, we study the number \(N_f(D,k,b)\) of k-subsets S in D such that

$$\begin{aligned} \sum _{x\in S} f(x)=b. \end{aligned}$$

In this paper, we establish several bounds for the difference between \(N_f(D,k, b)\) and the expected main term \(\frac{1}{|R|}{|D|\atopwithdelims ()k}\), depending on the nature of the finite ring R and f. For \(R=\mathbb {Z}_n\), let \(p=p(n)\) be the smallest prime divisor of n, \(|D|=n-c \ge C_dn p^{-\frac{1}{d}}\,+\,c\) and \(f(x)=a_dx^d +\cdots +a_0\in \mathbb {Z}[x]\) with \((a_d, \ldots , a_1, n)=1\). Then

$$\begin{aligned} \left| N_f(D, k, b)-\frac{1}{n}{n-c \atopwithdelims ()k}\right| \le {\delta (n)(n-c)+(1-\delta (n))\left( C_dnp^{-\frac{1}{d}}+c\right) +k-1\atopwithdelims ()k}, \end{aligned}$$

answering an open question raised by Stanley (Enumerative combinatorics, 1997) in a general setting, where \(\delta (n)=\sum _{i\mid n, \mu (i)=-1}\frac{1}{i}\) and \(C_d=e^{1.85d}\). Furthermore, if n is a prime power, then \(\delta (n) =1/p\) and one can take \(C_d=4.41\). Similar and stronger bounds are given for two more cases. The first one is when \(R=\mathbb {F}_q\), a q-element finite field of characteristic p and f(x) is general. The second one is essentially the well-known subset sum problem over an arbitrary finite abelian group. These bounds extend several previous results.

     

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.     

On a conjecture of Stein

Stein (Pac J Math 59:567–575, 1975) proposed the following conjecture: if the edge set of \(K_{n,n}\) is partitioned into n sets, each of size n, then there is a partial rainbow matching of size \(n-1\). He proved that there is a partial rainbow matching of size \(n(1-\frac{D_n}{n!})\), where \(D_n\) is the number of derangements of [n]. This means that there is a partial rainbow matching of size about \((1- \frac{1}{e})n\). Using a topological version of Hall’s theorem we improve this bound to \(\frac{2}{3}n\).     

Induced Cycles in Graphs

The maximum number vertices of a graph G inducing a 2-regular subgraph of G is denoted by \(c_\mathrm{ind}(G)\). We prove that if G is an r-regular graph of order n, then \(c_\mathrm{ind}(G) \ge \frac{n}{2(r-1)} + \frac{1}{(r-1)(r-2)}\) and we prove that if G is a cubic, claw-free graph on order n, then \(c_\mathrm{ind}(G) > \frac{13}{20}n\) and this bound is asymptotically best possible.     

$$L^p$$ Christoffel–Minkowski problem: the case $$1< p<k+1$$1<p<k+1

We consider a fully nonlinear partial differential equation associated to the intermediate \(L^p\) Christoffel–Minkowski problem in the case \(1<p<k+1\). We establish the existence of convex body with prescribed k-th even p-area measure on \(\mathbb S^n\), under an appropriate assumption on the prescribed function. We construct examples to indicate certain geometric condition on the prescribed function is needed for the existence of smooth strictly convex body. We also obtain \(C^{1,1}\) regularity estimates for admissible solutions of the equation when \( p\ge \frac{k+1}{2}\).     

New lower bounds for the Shannon capacity of odd cycles

The Shannon capacity of a graph G is defined as \(c(G)=\sup _{d\ge 1}(\alpha (G^d))^{\frac{1}{d}},\) where \(\alpha (G)\) is the independence number of G. The Shannon capacity of the cycle \(C_5\) on 5 vertices was determined by Lovász in 1979, but the Shannon capacity of a cycle \(C_p\) for general odd p remains one of the most notorious open problems in information theory. By prescribing stabilizers for the independent sets in \(C_p^d\) and using stochastic search methods, we show that \(\alpha (C_7^5)\ge 350\), \(\alpha (C_{11}^4)\ge 748\), \(\alpha (C_{13}^4)\ge 1534\), and \(\alpha (C_{15}^3)\ge 381\). This leads to improved lower bounds on the Shannon capacity of \(C_7\) and \(C_{15}\): \(c(C_7)\ge 350^{\frac{1}{5}}> 3.2271\) and \(c(C_{15})\ge 381^{\frac{1}{3}}> 7.2495\).     

Power Laws Variance Scaling of Boolean Random Varieties

Long fibers or stratified media show very long range correlations. These media can be simulated by models of Boolean random varieties and their iteration. They show non standard scaling laws with respect to the volume of domains K for the variance of the local volume fraction: on a large scale, the variance of the local volume fraction decreases according to power laws of the volume of K. The exponent γ is equal to \(\frac {n-k}{n}\) for Boolean varieties with dimension k in the space \( \mathbb {R}^{n}\): \(\gamma =\frac {2}{3}\) for Boolean fibers in 3D, and \(\gamma =\frac {1}{3}\) for Boolean strata in 3D. When working in 2D, the scaling exponent of Boolean fibers is equal to \(\frac {1}{2}\). From the results of numerical simulations, these scaling laws are expected to hold for the prediction of the effective properties of such random media.     

Eichler’s commutation relation and some other invariant subspaces of Hecke operators

Let k be an odd positive integer, L a lattice on a regular positive definite k-dimensional quadratic space over \(\mathbb {Q}\), \(N_L\) the level of L, and \(\mathscr {M}(L)\)  be the linear space of \(\theta \)-series attached to the distinct classes in the genus of L. We prove that, for an odd prime \(p|N_L\), if \(L_p=L_{p,1}\,\bot \, L_{p,2}\), where \(L_{p,1}\) is unimodular, \(L_{p,2}\) is (p)-modular, and \(\mathbb {Q}_pL_{p,2}\) is anisotropic, then \(\mathscr {M}(L;p):=\) \(\mathscr {M}(L)\) \(+T_{p^2}.\) \(\mathscr {M}(L)\)  is stable under the Hecke operator \(T_{p^2}\). If \(L_2\) is isometric to \(\left( \begin{array}{ll}0&{}\frac{1}{2}\\ \frac{1}{2}&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle \varepsilon \rangle \) or \(\left( \begin{array}{ll}0&{}\frac{1}{2}\\ \frac{1}{2}&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle 2\varepsilon \rangle \) or \(\left( \begin{array}{ll}0&{}1\\ 1&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle \varepsilon \rangle \) with \(\varepsilon \in \mathbb {Z}_2^{\times }\) and \(\kappa :=\frac{k-1}{2}\), then \(\mathscr {M}(L;2):=T_{2^2}.\mathscr {M}(L)+T_{2^2}^2.\,\mathscr {M}(L)\) is stable under the Hecke operator \(T_{2^2}\). Furthermore, we determine some invariant subspaces of the cusp forms for the Hecke operators.     

Power-normed spaces

To each power-norm \(((E^n, \Vert \cdot \Vert _n):n\in {\mathbb N})\) based on a given Banach space E, we associate two maximal symmetric sequence spaces \(L_\Phi ^E\) and \(L_\Psi ^E\) whose norms \(\Vert (z_k)\Vert _{L_\Phi ^E}\) and \(\Vert (z_k)\Vert _{L_\Psi ^E}\) are defined by \(\sup \{ \Vert (z_1x,\ldots ,z_nx)\Vert _n: \Vert x\Vert =1, n\in {\mathbb N}\}\) and \(\sup \{ \Vert \sum _{k=1}^n z_kx_k\Vert : \Vert (x_1,\ldots ,x_n)\Vert _n=1, n\in {\mathbb N}\}\) respectively. For each \(1\le p\le \infty \), we introduce and study the p-power-norms as those power-norms for which \(L_\Phi ^E=\ell ^p\) and \(L_\Psi ^E=\ell ^{p'}\), where \(1/p+1/p'=1\). As a special cases of p-power-norms we introduce certain smaller class, to be called the class of \(\ell ^p\)-power-norms, which is shown to contain the p-multi-norms defined in (Dales et al., Multi-norms and Banach lattices, 2016), and to coincide with the multi-norms and dual-multi-norms defined in (Dales and Polyakov, Diss Math 488, 2012) in the cases \(p=\infty \) and \(p=1\) respectively. We give several procedures to construct examples of such p-power and \(\ell ^p\)-power-norms and show that the natural formulations of the (p, q)-summing, (p, q)-concave, Rademacher power norms, t-standard power norms among others are examples in these classes. In particular, for instance the Rademacher power norm is a 2-power norm and the (p, q)-summing power-norm is a \(\ell ^r\)-power-norm for \(p>q\) with \(\frac{1}{r}=\frac{1}{q}-\frac{1}{p}\).     

Determination of cusp forms by central values of Rankin–Selberg <Emphasis Type="BoldItalic">L</Emphasis>-functions<Superscript>*</Superscript>

Let f be a fixed holomorphic Hecke eigen cusp form of weight k for \( SL\left( {2,{\mathbb Z}} \right) \), and let \( {\mathcal U} = \left\{ {{u_j}:j \geqslant 1} \right\} \) be an orthonormal basis of Hecke–Maass cusp forms for \( SL\left( {2,{\mathbb Z}} \right) \). We prove an asymptotic formula for the twisted first moment of the Rankin–Selberg L-functions \( L\left( {s,f \otimes {u_j}} \right) \) at \( s = \frac{1}{2} \) as u _j runs over \( {\mathcal U} \). It follows that f is uniquely determined by the central values of the family of Rankin–Selberg L-functions \( \left\{ {L\left( {s,f \otimes {u_j}} \right):{u_j} \in {\mathcal U}} \right\} \).     

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.     

Rainbow Spanning Subgraphs of Small Diameter in Edge-Colored Complete Graphs

Let s(n, t) be the maximum number of colors in an edge-coloring of the complete graph \(K_n\) that has no rainbow spanning subgraph with diameter at most t. We prove \(s(n,t)={\left( {\begin{array}{c}n-2\\ 2\end{array}}\right) }+1\) for \(n,t\ge 3\), while \(s(n,2)={\left( {\begin{array}{c}n-2\\ 2\end{array}}\right) }+\left\lfloor {\frac{n-1}{2}}\right\rfloor \) for \(n\ne 4\) (and \(s(4,2)=2\)).     

Partial permutation decoding for binary linear and $$Z_4$$-linear Hadamard codes

In this paper, s-\({\text {PD}}\)-sets of minimum size \(s+1\) for partial permutation decoding for the binary linear Hadamard code \(H_m\) of length \(2^m\), for all \(m\ge 4\) and \(2 \le s \le \lfloor {\frac{2^m}{1+m}}\rfloor -1\), are constructed. Moreover, recursive constructions to obtain s-\({\text {PD}}\)-sets of size \(l\ge s+1\) for \(H_{m+1}\) of length \(2^{m+1}\), from an s-\({\text {PD}}\)-set of the same size for \(H_m\), are also described. These results are generalized to find s-\({\text {PD}}\)-sets for the \({\mathbb {Z}}_4\)-linear Hadamard codes \(H_{\gamma , \delta }\) of length \(2^m\), \(m=\gamma +2\delta -1\), which are binary Hadamard codes (not necessarily linear) obtained as the Gray map image of quaternary linear codes of type \(2^\gamma 4^\delta \). Specifically, s-PD-sets of minimum size \(s+1\) for \(H_{\gamma , \delta }\), for all \(\delta \ge 3\) and \(2\le s \le \lfloor {\frac{2^{2\delta -2}}{\delta }}\rfloor -1\), are constructed and recursive constructions are described.