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In this paper we determine two asymptotic results for Jack measures on partitions, a model defined by two specializations of Jack polynomials proposed by Borodin-Olshanski in [European J. Combin. 26.6 (2005): 795-834]. Assuming these two specializations are the same, we derive limit shapes and Gaussian fluctuations for the anisotropic profiles of these random partitions in three asymptotic regimes associated to diverging, fixed, and vanishing values of the Jack parameter. To do so, we introduce a generalization of Motzkin paths we call "ribbon paths", show for general Jack measures that certain joint cumulants are weighted sums of connected ribbon paths on $n$ sites with $n-1+g$ pairings, and derive our two results from the contributions of $(n,g)=(1,0)$ and $(2,0)$, respectively. Our analysis makes use of Nazarov-Sklyanin's spectral theory for Jack polynomials. As a consequence, we give new proofs of several results for Schur measures, Plancherel measures, and Jack-Plancherel measures. In addition, we relate our weighted sums of ribbon paths to the weighted sums of ribbon graphs of maps on non-oriented real surfaces recently introduced by Chapuy-Dol\k{e}ga., Comment: Several results in this paper first appeared in the author's unpublished monograph arXiv:1508.03063. Version 2: revised and accepted for publication in International Mathematics Research Notices (IMRN)

We revisit Rozansky’s construction of Khovanov homology for links in S 2 × S 1 , extending it to define the Khovanov homology Kh ( L ) for links L in M r = # r ( S 2 × S 1 ) for any r . The graded Euler characteristic of Kh ( L ) can be used to recover WRT invariants at certain roots of unity and also recovers the evaluation of L in the skein module S ( M r ) of Hoste and Przytycki when L is null-homologous in M r . The construction also allows for a clear path toward defining a Lee’s homology Kh ' ( L ) and associated s -invariant for such L , which we will explore in an upcoming paper. We also give an equivalent construction for the Khovanov homology of the knotification of a link in S 3 and show directly that this is invariant under handle-slides, in the hope of lifting this version to give a stable homotopy type for such knotifications in a future paper.

In this paper we investigate Ramanujan’s inequality concerning the prime counting function, asserting that π ( x 2 ) < e x log x π ( x e ) \pi \left( {{x^2}} \right) < {{ex} \over {\log x}}\pi \left( {{x \over e}} \right) for x sufficiently large. First, we study its sharpness by giving full asymptotic expansions of its left and right hand sides expressions. Then, we discuss the structure of Ramanujan’s inequality, by replacing the factor x log x {x \over {\log x}} on its right hand side by the factor x log x - h {x \over {\log x - h}} for a given h, and by replacing the numerical factor e by a given positive α. Finally, we introduce and study inequalities analogous to Ramanujan’s inequality.

Property $$(UW {\scriptstyle \Pi })$$ for a bounded linear operator $$T\in L(X)$$ on a Banach space X means that the points $$\lambda $$ of the approximate point spectrum for which $$\lambda I-T$$ is upper semi-Weyl are exactly the poles of the resolvent of T. In this paper we give some new results on this property and relate it with some other variants of Browder’s theorem, as property (gaz) and property $$(Z{\scriptstyle \Pi _a})$$ . In the last part the theory is applied to Toeplitz operators $$T_\phi $$ with symbol $$\phi $$ on Hardy spaces.

In this paper, we consider a class of quasilinear Schrodinger-Poisson problems of the form $$\left\{ {\matrix{{ - \left( {a + b\int_{{^N}} {{{\left| {\nabla u} \right|}^2}dx} } \right)\Delta u + V(x)u + \phi u - {1 \over 2}u\Delta ({u^2}) - \lambda {{\left| u \right|}^{p - 2}}u = 0} \hfill & {{\rm{in}}\,\,{\mathbb{R}^N},} \hfill \cr { - \Delta \phi = {u^2},\,\,\,\,\,u(x) \to 0,\,\,\,\,\,\left| x \right| \to \infty } \hfill & {{\rm{in}}\,\,{\mathbb{R}^N},} \hfill \cr {\int_{{\mathbb{R}^N}} {{{\left| u \right|}^p}dx = 1,} } \hfill & {} \hfill \cr } } \right.$$ where a > 0, b ⩾ 0, N ⩾ 3, λ appears as a Lagrangian multiplier, and $$4 < p < 2 \cdot {2^\ast} = {{4N} \over {N - 2}}$$ . We deal with two different cases simultaneously, namely lim∣x∣→∞V(x) = ∞ and lim∣x∣→∞V(x) = V∞. By using the method of invariant sets of the descending flow combined with the genus theory, we prove the existence of infinitely many sign-changing solutions. Our results extend and improve some recent work.

This paper concerns the reconstruction of a function f in the Hardy space of the unit disc $$\mathbb{D}$$ by using a sample value f (a) and certain n-intensity measurements $$\left| {\left\langle {f,{E_{{a_1} \cdots {a_n}}}} \right\rangle} \right|$$ , where a1, ⋯, $${a_n} \in \mathbb{D}$$ , and $${E_{{a_1} \cdots {a_n}}}$$ is the n-th term of the Gram-Schmidt orthogonalization of the Szego kernels $${k_{{a_1}}}, \cdots ,{k_{{a_n}}}$$ , or their multiple forms. Three schemes are presented. The first two schemes each directly obtain all the function values f (z). In the first one we use Nevanlinna’s inner and outer function factorization which merely requires the 1-intensity measurements equivalent to know the modulus |f (z)|. In the second scheme we do not use deep complex analysis, but require some 2- and 3-intensity measurements. The third scheme, as an application of AFD, gives sparse representation of f (z) converging quickly in the energy sense, depending on consecutively selected maximal n-intensity measurements $$\left| {\left\langle {f,{E_{{a_1} \cdots {a_n}}}} \right\rangle} \right|$$ .

The notion of best approximation mapping (BAM) with respect to a closed affine subspace in finite-dimensional space was introduced by Behling, Bello Cruz and Santos to show the linear convergence of the block-wise circumcentered-reflection method. The best approximation mapping possesses two critical properties of the circumcenter mapping for linear convergence. Because the iteration sequence of BAM linearly converges, the BAM is interesting in its own right. In this paper, we naturally extend the definition of BAM from closed affine subspace to nonempty closed convex set and from $\mathbb{R}^{n}$ to general Hilbert space. We discover that the convex set associated with the BAM must be the fixed point set of the BAM. Hence, the iteration sequence generated by a BAM linearly converges to the nearest fixed point of the BAM. Connections between BAMs and other mappings generating convergent iteration sequences are considered. Behling et al.\ proved that the finite composition of BAMs associated with closed affine subspaces is still a BAM in $\mathbb{R}^{n}$. We generalize their result from $\mathbb{R}^{n}$ to general Hilbert space and also construct a new constant associated with the composition of BAMs. This provides a new proof of the linear convergence of the method of alternating projections. Moreover, compositions of BAMs associated with general convex sets are investigated. In addition, we show that convex combinations of BAMs associated with affine subspaces are BAMs. Last but not least, we connect BAM with circumcenter mapping in Hilbert spaces., 31 pages and 2 figures

A natural variant of Lehmer’s conjecture that the Ramanujan $$\tau $$ -function never vanishes asks whether, for any given integer $$\alpha $$ , there exist any $$n \in \mathbb {Z}^+$$ such that $$\tau (n) = \alpha $$ . A series of recent papers excludes many integers as possible values of the $$\tau $$ -function using the theory of primitive divisors of Lucas numbers, computations of integer points on curves, and congruences for $$\tau (n)$$ . We synthesize these results and methods to prove that if $$0< \left| \alpha \right| < 100$$ and $$\alpha \notin T := \{2^k, -24,-48, -70,-90, 92, -96\}$$ , then $$\tau (n) \ne \alpha $$ for all $$n > 1$$ . Moreover, if $$\alpha \in T$$ and $$\tau (n) = \alpha $$ , then n is square-free with prescribed prime factorization. Finally, we show that a strong form of the Atkin-Serre conjecture implies that $$\left| \tau (n) \right| > 100$$ for all $$n > 2$$ .

Let $${\mathbb{D}_r}: = \{ z = x + {\rm{i}}y \in \mathbb{C}:\left| z \right| < r\} ,\,r \le 1$$ . For a normalized analytic function f in the unit disk $$\mathbb{D}: = {\mathbb{D}_1}$$ , estimating the Dirichlet integral $$\Delta (r,f) = \int {\,\int_{{\mathbb{D}_r}} {{{\left| {f\prime (z)} \right|}^2}dxdy} } $$ , z = x +iy, is an important classical problem in complex analysis. Geometrically, Δ(r, f) represents the area of the image of $${\mathbb{D}_r}$$ under f counting multiplicities. In this paper, our main objective is to estimate areas of images of $${\mathbb{D}_r}$$ under non-vanishing analytic functions of the form (z/f)μ, μ > 0, in principal powers, when f ranges over certain classes of analytic and univalent functions in $$\mathbb{D}$$ .

Consider an integer n≥2, m∈[n,+∞) and put α‾:=m−1n−1. Moreover: ∙ Let γx:(0,+∞)→Rn be the line through x=(x1,…,xn)∈Rn defined as γx(t):=(tx1,tα‾x2,…,tα‾xn); ∙ If T is any motion of Rn, then let γx(T) be the line through x=(x1,…,xn)∈Rn defined as γx(T):=T∘γT−1(x); ∙ Let H1 denote the one-dimensional Hausdorff measure in Rn. The main goal of this paper is to prove the following property: If x0 is an m-density point of a Lebesgue measurable set E and T is an arbitrary motion of Rn mapping the origin to x0, then we have lim supt→0+ H1(E∩γx(T)((0,t])) H1(γx(T)((0,t]))=1 for almost every x∈T({1}×Rn−1). An application of this result to locally finite perimeter sets is provided.

As the generalization of the digraph, the directed hypergraph (dihypergraph), denoted by $${H}$$ , is considered in this paper. First, we introduce the notions of the zeta function of $${H}$$ and dihypergraph coverings over $${H}$$ . Next, several expressions for the zeta function of $${H}$$ are given, and all dihypergraph coverings over $${H}$$ are generated by permutation voltage assignments on the incidence bipartite digraph or the edge-colored digraph of $${H}$$ . Final, we derive two explicit decomposition formulae for the zeta function of any dihypergraph covering $$\overline{{H}}$$ over $${H}$$ , which turn out that the zeta function of $${H}$$ divides the zeta function of $$\overline{{H}}$$ .

Given a sequence $(X_n)$ of real or complex random variables and a sequence of numbers $(a_n)$, an interesting problem is to determine the conditions under which the series $\sum_{n=1}^\infty a_n X_n$ is almost surely convergent. This paper extends the classical Menshov--Rademacher theorem on the convergence of orthogonal series to general series of dependent random variables and derives interesting sufficient conditions for the almost everywhere convergence of trigonometric series with respect to singular measures whose Fourier transform decays to 0 at infinity with positive rate.

Let λf (n) be the normalized n-th Fourier coefficient of holomorphic eigenform f for the full modular group and ℙc(x): = {p ≤ x ∣ [pc]prime}, c ∈ ℝ+. In this paper, we show that for all 0 < c < 1 the mean value of λf(n) in ℙc(x) is ≪x log−Ax assuming the Riemann Hypothesis. Unconditionally, in the sense of Lebesgue measure, it holds for almost all c ∈ (e, 1 − e).

Let $\Omega \subset {\mathbb{R}}^d $, $d \geq 2$, be a bounded convex domain and $f\colon \Omega \to{\mathbb{R}}$ be a non-negative subharmonic function. In this paper, we prove the inequality $$\begin{equation*} \frac{1}{|\Omega|}\int_{\Omega} f(x)\, \textrm{d}x \leq \frac{d}{|\partial\Omega|}\int_{\partial\Omega} f(x)\, \textrm{d}\sigma(x)\,. \end{equation*}$$Equivalently, the result can be stated as a bound for the gradient of the Saint Venant torsion function. Specifically, if $\Omega \subset{\mathbb{R}}^d$ is a bounded convex domain and $u$ is the solution of $-\Delta u =1$ with homogeneous Dirichlet boundary conditions, then $$\begin{equation*} \|\nabla u\|_{L^\infty(\Omega)} < d\frac{|\Omega|}{|\partial\Omega|}\,. \end{equation*}$$Moreover, both inequalities are sharp in the sense that if the constant $d$ is replaced by something smaller there exist convex domains for which the inequalities fail. This improves upon the recent result that the optimal constant is bounded from above by $d^{3/2}$ due to Beck et al. [2].

Xu and Wu proved that if every 5-cycle of a planar graph G is not simultaneously adjacent to 3-cycles and 4-cycles, then G is 4-choosable. In this paper, we improve this result as follows. If G is a planar graph without pairwise adjacent 3-,4-,5-, and 6-cycle, then G is 4-choosable., Comment: 17 pages and 2 figures

The Ewens–Pitman sampling model (EP-SM) is a distribution for random partitions of the set {1,…,n}, with n∈N, which is indexed by real parameters α and θ such that either α∈[0,1) and θ&gt, −α, or α&lt, 0 and θ=−mα for some m∈N. For α=0, the EP-SM is reduced to the Ewens sampling model (E-SM), which admits a well-known compound Poisson perspective in terms of the log-series compound Poisson sampling model (LS-CPSM). In this paper, we consider a generalisation of the LS-CPSM, referred to as the negative Binomial compound Poisson sampling model (NB-CPSM), and we show that it leads to an extension of the compound Poisson perspective of the E-SM to the more general EP-SM for either α∈(0,1), or α&lt, 0. The interplay between the NB-CPSM and the EP-SM is then applied to the study of the large n asymptotic behaviour of the number of blocks in the corresponding random partitions—leading to a new proof of Pitman’s α diversity. We discuss the proposed results and conjecture that analogous compound Poisson representations may hold for the class of α-stable Poisson–Kingman sampling models—of which the EP-SM is a noteworthy special case.

In this paper we introduce and study a new generalization of Fibonacci polynomials which generalize Fibonacci, Jacobsthal and Narayana numbers, simultaneously. We give a graph interpretation of these polynomials and we obtain a binomial formula for them. Moreover by modification of Pascal’s triangle, which has a symmetric structure, we obtain matrices generated by coefficients of generalized Fibonacci polynomials. As a consequence, the direct formula for generalized Fibonacci polynomials was given. In addition, we determine matrix generators for generalized Fibonacci polynomials, using the symmetric matrix of initial conditions.

Let p be a prime, $$\chi $$ denote a Dirichlet character modulo p. For any integer x ( $$1\le x\le p-1$$ ), $${\bar{x}}$$ denotes the integer inverse of x such that $$x{\bar{x}}\equiv 1(\bmod \, p)$$ , we study the following mean value of a kind of character sums with generalized Dirichlet L-functions $$\begin{aligned} {\mathop {\mathop {\sum }\limits _{\chi (-1)=1}}\limits _{\chi \ne \chi _0}} \left| \sum _{x=1}^{p-1}\chi (x+{\bar{x}})\right| ^2|L(1,\chi ,a)|^2, \end{aligned}$$ where $$\chi _0$$ is the principal character modulo p, and $$L(1,\chi ,a)$$ is the generalized Dirichlet L-functions. In this paper, we will use the analytic method and get a sharp asymptotic formula.

A proper total k-coloring ϕ of G with ∑z∈EG(u)∪{u}ϕ(z)≠∑z∈EG(v)∪{v}ϕ(z) for each uv∈E(G) is called a total neighbor sum distinguishing k-coloring, where EG(u)={uv|uv∈E(G)}. Pilśniak and Woźniak conjectured that every graph with maximum degree Δ exists a total neighbor sum distinguishing (Δ+3)-coloring. In this paper, we proved that any IC-planar graph with Δ≥12 satisfies this conjecture, which improves the result of Song and Xu.

In this paper, we study the constrained minimization problem (1.1) of the Kirchhoff–Schrodinger energy functional under an $$L^{2}$$ -subcritical perturbation. The existence and nonexistence of constraint minimizers are completely classified in terms of the $$L^{2}$$ -subcritical exponent q. Especially for $$q\in (\frac{4}{3},\frac{8}{3})$$ , we prove that there exists a critical value $$\beta ^{*}$$ such that (1.1) has no minimizer if the coefficient $$\beta $$ of $$L^{2}$$ -critical term satisfies $$\beta =\beta ^{*}$$ . For $$q\in (\frac{4}{3},\frac{8}{3})$$ , the blow-up behavior of minimizers as $$\beta \nearrow \beta ^{*}$$ are also analyzed rigorously if the coefficient $$\lambda $$ of $$L^{2}$$ -subcritical term satisfies $$\lambda >\lambda _{0}$$ , where $$\lambda _{0}$$ is a positive constant.

Let G be a connected reductive complex algebraic group with a maximal torus T. We denote by Λ the coweight lattice of T. Let Λ + ⊂ Λ be the submonoid of dominant coweights. For λ ∈ Λ + , μ ∈ Λ , μ ⩽ λ , in [7] , authors defined a generalized transversal slice W ‾ μ λ . This is an algebraic variety of the dimension 〈 2 ρ ∨ , λ − μ 〉 , where 2 ρ ∨ is the sum of positive roots of G. In this paper, we construct an isomorphism W ‾ μ λ ≃ W ‾ μ + λ × A 〈 2 ρ ∨ , μ + − μ 〉 for μ ∈ Λ such that 〈 α ∨ , μ 〉 ⩾ − 1 for any positive root α ∨ , here μ + ∈ W μ is the dominant representative in the Weyl group orbit of μ. We consider the example when λ is minuscule, μ ∈ W λ and describe natural coordinates, Poisson structure on W ‾ μ λ ≃ A 〈 2 ρ ∨ , λ − μ 〉 and its T × C × -character. We apply these results to compute T × C × -characters of tangent spaces at fixed points of convolution diagrams W ˜ μ λ _ with minuscule λ i . We also apply our results to construct open coverings by affine spaces of convolution diagrams W ˜ μ λ _ over slices with μ such that 〈 α ∨ , μ 〉 ⩾ − 1 for any positive root α ∨ and minuscule λ i and to compute Poincare polynomials of such convolution diagrams W ˜ μ λ _ .

In this paper, we introduce and investigate an ideal-based dot total graph of commutative ring R with nonzero unity. We show that this graph is connected and has a small diameter of at most two. Furthermore, its vertex set is divided into three disjoint subsets of R. After that, connectivity, clique number, and girth have also been studied. Finally, we determine the cases when it is Eulerian, Hamiltonian, and contains a Eulerian trail.

In this paper we use a graph interpretation of distance Fibonacci polynomials to get a new generalization of Lucas polynomials in the distance sense. We give a direct formula, a generating function and we prove some identities for generalized Lucas polynomials. We present Pascal-like triangles with left-justified rows filled with coefficients of these polynomials, in which one can observe some symmetric patterns. Using a general Q-matrix and a symmetric matrix of initial conditions we also define matrix generators for generalized Lucas polynomials.