Hankel Determinants of Random Moment Sequences

For \(t \in [0,1]\) let \(\underline{H}_{2\lfloor nt \rfloor } = (m_{i+j})_{i,j=0}^{\lfloor nt \rfloor }\) denote the Hankel matrix of order \(2\lfloor nt \rfloor \) of a random vector \((m_1,\ldots ,m_{2n})\) on the moment space \(\mathcal {M}_{2n}(I)\) of all moments (up to the order 2n) of probability measures on the interval \(I \subset \mathbb {R}\). In this paper we study the asymptotic properties of the stochastic process \(\{ \log \det \underline{H}_{2\lfloor nt \rfloor } \}_{t\in [0,1]}\) as \(n \rightarrow \infty \). In particular weak convergence and corresponding large deviation principles are derived after appropriate standardization.     

Uniqueness of self-shrinkers to the degree-one curvature flow with a tangent cone at infinity

Given a smooth, symmetric and homogeneous of degree one function \(f\left( \lambda _{1},\ldots ,\lambda _{n}\right) \) satisfying \(\partial _{i}f>0\quad \forall \,i=1,\ldots , n\), and a properly embedded smooth cone \({\mathcal {C}}\) in \({\mathbb {R}}^{n+1}\), we show that under suitable conditions on f, there is at most one f self-shrinker (i.e. a hypersurface \(\Sigma \) in \({\mathbb {R}}^{n+1}\) satisfying \(f\left( \kappa _{1},\ldots ,\kappa _{n}\right) +\frac{1}{2}X\cdot N=0\), where \(\kappa _{1},\ldots ,\kappa _{n}\) are principal curvatures of \(\Sigma \)) that is asymptotic to the given cone \({\mathcal {C}}\) at infinity.     

Complete weight enumerators of a class of linear codes

Let \(\mathbb {F}_{q}\) be the finite field with \(q=p^{m}\) elements, where p is an odd prime and m is a positive integer. For a positive integer t, let \(D\subset \mathbb {F}^{t}_{q}\) and let \({\mathrm {Tr}}_{m}\) be the trace function from \(\mathbb {F}_{q}\) onto \(\mathbb {F}_{p}\). In this paper, let \(D=\{(x_{1},x_{2},\ldots ,x_{t}) \in \mathbb {F}_{q}^{t}\setminus \{(0,0,\ldots ,0)\} : {\mathrm {Tr}}_{m}(x_{1}+x_{2}+\cdots +x_{t})=0\},\) we define a p-ary linear code \(\mathcal {C}_{D}\) by

$$\begin{aligned} \mathcal {C}_{D}=\{\mathbf {c}(a_{1},a_{2},\ldots ,a_{t}) : (a_{1},a_{2},\ldots ,a_{t})\in \mathbb {F}^{t}_{q}\}, \end{aligned}$$

where

$$\begin{aligned} \mathbf {c}(a_{1},a_{2},\ldots ,a_{t})=({\mathrm {Tr}}_{m}(a_{1}x^{2}_{1}+a_{2}x^{2}_{2}+\cdots +a_{t}x^{2}_{t}))_{(x_{1},x_{2},\ldots ,x_{t}) \in D}. \end{aligned}$$

We shall present the complete weight enumerators of the linear codes \(\mathcal {C}_{D}\) and give several classes of linear codes with a few weights. This paper generalizes the results of Yang and Yao (Des Codes Cryptogr, 2016).

     

Nonlinear $$\ast$$-Lie-type derivations on von Neumann algebras

Let \({\mathcal{H}}\) be a complex Hilbert space, \({\mathcal{B(H)}}\) be the algebra of all bounded linear operators on \({\mathcal{H}}\) and \({\mathcal{A} \subseteq \mathcal{B(H)}}\) be a von Neumann algebra without nonzero central abelian projections. Let \({p_n(x_1,x_2 ,\ldots ,x_n)}\) be the commutator polynomial defined by n indeterminates \({x_1, \ldots , x_n}\) and their skew Lie products. It is shown that a mapping \({\delta \colon \mathcal{A} \longrightarrow \mathcal{B(H)}}\) satisfies

$$\delta(p_n(A_1, A_2 ,\ldots , A_n))=\sum_{k=1}^np_n(A_1 ,\ldots , A_{k-1}, \delta(A_k), A_{k+1} ,\ldots , A_n)$$

for all \({A_1, A_2 ,\ldots , A_n \in \mathcal{A}}\) if and only if \({\delta}\) is an additive *-derivation. This gives a positive answer to Conjecture 4.2 of [14].

     

Study of LG-Holling type III predator–prey model with disease in predator

Let \(\mathbb F_{q}\) be a finite field with \(q=p^{m}\) elements, where p is an odd prime and m is a positive integer. In this paper, let \(D=\{(x_{1},x_{2},\ldots ,x_{n})\in \mathbb F_{q}^{n}\backslash \{(0,0,\ldots )\}: Tr(x_{1}^{p^{k_{1}}+1}+x_{2}^{p^{k_{2}}+1}+\cdots +x_{n}^{p^{k_{n}}+1})=c\}\), where \(c\in \mathbb F_p\), Tr is the trace function from \(\mathbb F_{q}\) to \(\mathbb F_{p}\) and each \(m/(m,k_{i})\) ( \(1\le i\le n\) ) is odd. we define a p-ary linear code \(C_{D}=\{c(a_{1},a_{2},\ldots ,a_{n}):(a_{1},a_{2},\ldots ,a_{n})\in \mathbb F_{q}^{n}\}\), where \(c(a_{1},a_{2},\ldots ,a_{n})=(Tr(a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}))_{(x_{1},x_{2},\ldots ,x_{n})\in D}\). We present the weight distributions of the classes of linear codes which have at most three weights.     

Inequalities for sine sums with more variables

A result of Vietoris states that if the real numbers \(a_1,\ldots ,a_n\) satisfy

$$\begin{aligned} \text{(*) } \qquad a_1\ge \frac{a_2}{2} \ge \cdots \ge \frac{a_n}{n}>0 \quad \text{ and } \quad a_{2k-1}\ge a_{2k} \quad (1\le k\le n/2), \end{aligned}$$

then, for \(x_1,\ldots ,x_m>0\) with \(x_1+\cdots +x_m <\pi \),

$$\begin{aligned} \begin{aligned} \text{(**) } \qquad \sum _{k=1}^n a_k \frac{\sin (k x_1) \cdots \sin (k x_m)}{k^m}>0. \end{aligned} \end{aligned}$$

We prove that \((**)\) (with “\(\ge \)” instead of “>”) holds under weaker conditions. It suffices to assume, instead of \((*)\), that

$$\begin{aligned} \sum _{k=1}^N a_k \frac{\sin (kt)}{k}>0 \quad (N=1,\ldots ,n; \, 0<t<\pi ), \end{aligned}$$

and, moreover, \((**)\) is valid for a larger region, namely, \(x_1,\ldots ,x_m\in (0,\pi )\).

     

The convolution algebra

For L a complete lattice L and \(\mathfrak {X}=(X,(R_i)_I)\) a relational structure, we introduce the convolution algebra \(L^{\mathfrak {X}}\). This algebra consists of the lattice \(L^X\) equipped with an additional \(n_i\)-ary operation \(f_i\) for each \(n_i+1\)-ary relation \(R_i\) of \(\mathfrak {X}\). For \(\alpha _1,\ldots ,\alpha _{n_i}\in L^X\) and \(x\in X\) we set \(f_i(\alpha _1,\ldots ,\alpha _{n_i})(x)=\bigvee \{\alpha _1(x_1)\wedge \cdots \wedge \alpha _{n_i}(x_{n_i}):(x_1,\ldots ,x_{n_i},x)\in R_i\}\). For the 2-element lattice 2, \(2^\mathfrak {X}\) is the reduct of the familiar complex algebra \(\mathfrak {X}^+\) obtained by removing Boolean complementation from the signature. It is shown that this construction is bifunctorial and behaves well with respect to one-one and onto maps and with respect to products. When L is the reduct of a complete Heyting algebra, the operations of \(L^\mathfrak {X}\) are completely additive in each coordinate and \(L^\mathfrak {X}\) is in the variety generated by \(2^\mathfrak {X}\). Extensions to the construction are made to allow for completely multiplicative operations defined through meets instead of joins, as well as modifications to allow for convolutions of relational structures with partial orderings. Several examples are given.     

循环群$Z_{2}$的模向量不变式

Let G be the finite cyclic group Z_2 and V be a vector space of dimension 2n with basis x_1,...,x_n,y_1,...,y_n over the field F with characteristic 2.If σ denotes a generator of G,we may assume that σ（x_i）= ayi,σ（y_i）= a～-1x_i,where a ∈ F.In this paper,we describe the explicit generator of the ring of modular vector invariants of F[V]～G.We prove that F[V]～G = F[l_i = x_i ＋ ay_i,q_i = x_iy_i,1 ≤ i ≤ n,M_I = X_I ＋ a～-I-Y_I],where I∈An = {1,2,...,n},2 ≤-I-≤ n.     

Monochromatic Solutions for Multi-Term Unknowns

Given integers \(k\ge 2\), \(n \ge 2\), \(m \ge 2\) and \( a_1,a_2,\ldots ,a_m \in {\mathbb {Z}}{\backslash }{\{0\}}\), and let \(f(z)= \sum _{j=0}^{n}c_jz^j\) be a polynomial of integer coefficients with \(c_n>0\) and \((\sum _{i=1}^ma_i)|f(z)\) for some integer z. For a k-coloring of \([N]=\{1,2,\ldots ,N\}\), we say that there is a monochromatic solution of the equation \(a_1x_1+a_2x_2+\cdots +a_mx_m=f(z)\) if there exist pairwise distinct \(x_1,x_2,\ldots ,x_m\in [N]\) all of the same color such that the equation holds for some \(z\in \mathbb {Z}\). Problems of this type are often referred to as Ramsey-type problems. In this paper, it is shown that if \(a_i>0\) for \(1\le i\le m\), then there exists an integer \(N_0=N(k,m,n)\) such that for \(N\ge N_0\), each k-coloring of [N] contains a monochromatic solution \(x_1,x_2,\ldots ,x_m\) of the equation \(a_1x_1+a_2x_2+ \cdots +a_mx_m= f(z)\). Moreover, if n is odd and there are \(a_i\) and \(a_j\) such that \(a_ia_j<0\) for some \(1 \le i\ne j\le m\), then the assertion holds similarly.     

A characterization of affinely regular polygons

In 1970, Coxeter gave a short and elegant geometric proof showing that if \(p_1, p_2, \ldots , p_n\) are vertices of an n-gon P in cyclic order, then P is affinely regular if, and only if there is some \(\lambda \ge 0\) such that \(p_{j+2}-p_{j-1} = \lambda (p_{j+1}-p_j)\) for \(j=1,2,\ldots , n\). The aim of this paper is to examine the properties of polygons whose vertices \(p_1,p_2,\ldots ,p_n \in \mathbb {C}\) satisfy the property that \(p_{j+m_1}-p_{j+m_2} = w (p_{j+k}-p_j)\) for some \(w \in \mathbb {C}\) and \(m_1,m_2,k \in \mathbb Z\). In particular, we show that in ‘most’ cases this implies that the polygon is affinely regular, but in some special cases there are polygons which satisfy this property but are not affinely regular. The proofs are based on the use of linear algebraic and number theoretic tools. In addition, we apply our method to characterize polytopes with certain symmetry groups.     

The generalized Roper-Suffridge extension operator in Banach spaces (III)

Suppose that X is a complex Banach space with the norm ‖·‖ and n is a positive integer with dim X ⩾ n ⩾ 2. In this paper, we consider the generalized Roper-Suffridge extension operator $ \Phi _{n,\beta _2 ,\gamma _2 , \ldots ,\beta _{n + 1} ,\gamma _{n + 1} } (f) $ \Phi _{n,\beta _2 ,\gamma _2 , \ldots ,\beta _{n + 1} ,\gamma _{n + 1} } (f) on the domain $ \Omega _{p_1 ,p_2 , \ldots ,p_{n + 1} } $ \Omega _{p_1 ,p_2 , \ldots ,p_{n + 1} } defined by

Wandering Subspaces of the Bergman Space and the Dirichlet Space Over {{\mathbb{D}^{n}}}

Doubly commuting invariant subspaces of the Bergman space and the Dirichlet space over the unit polydisc \({\mathbb{D}^n}\) (with \({n \geq 2}\) ) are investigated. We show that for any non-empty subset \({\alpha=\{\alpha_1,\ldots,\alpha_k\}}\) of \({\{1,\ldots,n\}}\) and doubly commuting invariant subspace \({\mathcal{S}}\) of the Bergman space or the Dirichlet space over \({\mathbb{D}^n}\) , restriction of the multiplication operator tuple on \({\mathcal{S}, M_{\alpha}|_\mathcal{S}:=(M_{z_{\alpha_1}}|_\mathcal{S},\ldots, M_{z_{\alpha_k}}|_\mathcal{S})}\) , always possesses generating wandering subspace of the form $$\bigcap_{i=1}^k(\mathcal{S}\ominus z_{\alpha_i}\mathcal{S})$$ .     

Space–Time Fractional Equations and the Related Stable Processes at Random Time

In this paper, we consider the general space–time fractional equation of the form \(\sum _{j=1}^m \lambda _j \frac{\partial ^{\nu _j}}{\partial t^{\nu _j}} w(x_1, \ldots , x_n ; t) = -c^2 \left( -\varDelta \right) ^\beta w(x_1, \ldots , x_n ; t)\), for \(\nu _j \in \left( 0,1 \right] \) and \(\beta \in \left( 0,1 \right] \) with initial condition \(w(x_1, \ldots , x_n ; 0)= \prod _{j=1}^n \delta (x_j)\). We show that the solution of the Cauchy problem above coincides with the probability density of the n-dimensional vector process \(\varvec{S}_n^{2\beta } \left( c^2 \mathcal {L}^{\nu _1, \ldots , \nu _m} (t) \right) \), \(t>0\), where \(\varvec{S}_n^{2\beta }\) is an isotropic stable process independent from \(\mathcal {L}^{\nu _1, \ldots , \nu _m}(t)\), which is the inverse of \(\mathcal {H}^{\nu _1, \ldots , \nu _m} (t) = \sum _{j=1}^m \lambda _j^{1/\nu _j} H^{\nu _j} (t)\), \(t>0\), with \(H^{\nu _j}(t)\) independent, positively skewed stable random variables of order \(\nu _j\). The problem considered includes the fractional telegraph equation as a special case as well as the governing equation of stable processes. The composition \(\varvec{S}_n^{2\beta } \left( c^2 \mathcal {L}^{\nu _1, \ldots , \nu _m} (t) \right) \), \(t>0\), supplies a probabilistic representation for the solutions of the fractional equations above and coincides for \(\beta = 1\) with the n-dimensional Brownian motion at the random time \(\mathcal {L}^{\nu _1, \ldots , \nu _m} (t)\), \(t>0\). The iterated process \(\mathfrak {L}^{\nu _1, \ldots , \nu _m}_r (t)\), \(t>0\), inverse to \(\mathfrak {H}^{\nu _1, \ldots , \nu _m}_r (t) =\sum _{j=1}^m \lambda _j^{1/\nu _j} \, _1H^{\nu _j} \left( \, _2H^{\nu _j} \left( \, _3H^{\nu _j} \left( \ldots \, _{r}H^{\nu _j} (t) \ldots \right) \right) \right) \), \(t>0\), permits us to construct the process \(\varvec{S}_n^{2\beta } \left( c^2 \mathfrak {L}^{\nu _1, \ldots , \nu _m}_r (t) \right) \), \(t>0\), the density of which solves a space-fractional equation of the form of the generalized fractional telegraph equation. For \(r \rightarrow \infty \) and \(\beta = 1\), we obtain a probability density, independent from t, which represents the multidimensional generalization of the Gauss–Laplace law and solves the equation \(\sum _{j=1}^m \lambda _j w(x_1, \ldots , x_n) = c^2 \sum _{j=1}^n \frac{\partial ^2}{\partial x_j^2} w(x_1, \ldots , x_n)\). Our analysis represents a general framework of the interplay between fractional differential equations and composition of processes of which the iterated Brownian motion is a very particular case.     

The fundamental theory of abstract majorization inequalities

Using the axiomatic method,abstract concepts such as abstract mean, abstract convex function and abstract majorization are proposed. They are the generalizations of concepts of mean, convex function and majorization, respectively. Through the logical deduction, the fundamental theorems about abstract majorization inequalities are established as follows: for arbitrary abstract mean Σ and Σ , and abstract Σ → Σ strict convex function f(x) on the interval I, if xi, yi ∈ I (i = 1, 2, . . . , n) satisfy that (x1...     

两类广泛递归序列的动力特性

We investigate the dynamics of two extensive classes of recursive sequences:xn+1=c∑ k ∑xn-ioxn-i1…xn-i2j+f(xn-io,xn-i1,…,xn-i2k)j=0(i0,i1,…,i2j)∈A2j/c∑ k ∑xn-ioxn-i1…xn-i2j-1+c+f(xn-io,xn-i1,…,xn-i2k)j=1(i0,i1,…,i2j)∈A2j-1 and xn+1=c∑ k ∑xn-ioxn-i1…xn-i2j-1+c+f(xn-io,xn-i1,…,xn-i2k)j=1(i0,i1,…,i2j)∈A2j-1/c∑ k ∑xn-ioxn-i1…xn-i2j+f(xn-io,xn-i1,…,xn-i2k)j=0(i0,i1,…,i2j)∈A2j We prove that their unique positive equilibrium x = 1 is globally asymptotically stable.And a new access is presented to study the theory of recursive sequences.     

Distributions and the Global Operator of Slice Monogenic Functions

In this paper we consider functions \(f\) defined on an open set \(U\) of the Euclidean space \(\mathbb{R }^{n+1}\) and with values in the Clifford Algebra \(\mathbb{R }_n\) . Slice monogenic functions \(f: U \subseteq \mathbb{R }^{n+1} \rightarrow \mathbb{R }_n\) belong to the kernel of the global differential operator with non constant coefficients given by \( \mathcal{G }=|{\underline{x}}|^2\frac{\partial }{\partial x_0} \ + \ {\underline{x}} \ \sum _{j=1}^n x_j\frac{\partial }{\partial x_j}. \) Since the operator \(\mathcal{G }\) is not elliptic and there is a degeneracy in \( {\underline{x}}=0\) , its kernel contains also less smooth functions that have to be interpreted as distributions. We study the distributional solutions of the differential equation \(\mathcal{G }F(x_0,{\underline{x}})=G(x_0,{\underline{x}})\) and some of its variations. In particular, we focus our attention on the solutions of the differential equation \( ({\underline{x}}\frac{\partial }{\partial x_0} \ - E)F(x_0,{\underline{x}})=G(x_0,{\underline{x}}), \) where \(E= \sum _{j=1}^n x_j\frac{\partial }{\partial x_j}\) is the Euler operator, from which we deduce properties of the solutions of the equation \( \mathcal{G }F(x_0,{\underline{x}})=G(x_0,{\underline{x}})\) .     

Existence and uniqueness for a class of multi-term fractional differential equations

In this paper, we consider the initial value problem for the nonlinear fractional differential equations

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} D^\alpha u(t)=f(t,u(t),D^{\beta _1}u(t),\ldots ,D^{\beta _N}u(t)), \quad &{}t\in (0,1],\\ D^{\alpha -k}u(0)=0, \quad &{}k=1,2,\ldots ,n, \end{array} \right. \end{aligned}$$

where \(\alpha >\beta _1>\beta _2>\cdots \beta _N>0\), \(n=[\alpha ]+1\) for \(\alpha \notin \mathbb {N}\) and \(\alpha =n\) for \(\alpha \in \mathbb {N}\), \(\beta _j<1\) for any \(j\in \{1,2,\ldots ,N\}\), D is the standard Riemann–Liouville derivative and \(f:[0,1]\times \mathbb {R}^{N+1}\rightarrow \mathbb {R}\) is a given function. By means of Schauder fixed point theorem and Banach contraction principle, an existence result and a unique result for the solution are obtained,respectively. As an application, some examples are presented to illustrate the main results.

     

Group divisible designs with large block sizes

For positive integers n, k with \(3\le k\le n\), let \(X=\mathbb {F}_{2^n}\setminus \{0,1\}\), \({\mathcal {G}}=\{\{x,x+1\}:x\in X\}\), and \({\mathcal {B}}_k=\left\{ \{x_1,x_2,\ldots ,x_k\}\!\subset \!X:\sum \limits _{i=1}^kx_i=1,\ \sum \limits _{i\in I}x_i\!\ne \!1\ \mathrm{for\ any}\ \emptyset \!\ne \!I\!\subsetneqq \!\{1,2,\ldots ,k\}\right\} \). Lee et al. used the inclusion–exclusion principle to show that the triple \((X,{\mathcal {G}},{\mathcal {B}}_k)\) is a \((k,\lambda _k)\)-GDD of type \(2^{2^{n-1}-1}\) for \(k\in \{3,4,5,6,7\}\) where \(\lambda _k=\frac{\prod _{i=3}^{k-1}(2^n-2^i)}{(k-2)!}\) (Lee et al. in Des Codes Cryptogr,  https://doi.org/10.1007/s10623-017-0395-8, 2017). They conjectured that \((X,{\mathcal {G}},{\mathcal {B}}_k)\) is also a \((k,\lambda _k)\)-GDD of type \(2^{2^{n-1}-1}\) for any integer \(k\ge 8\). In this paper, we use a similar construction and counting principles to show that there is a \((k,\lambda _k)\)-GDD of type \((q^2-q)^{(q^{n-1}-1)/(q-1)}\) for any prime power q and any integers k, n with \(3\le k\le n\) where \(\lambda _k=\frac{\prod _{i=3}^{k-1}(q^n-q^i)}{(k-2)!}\). Consequently, their conjecture holds. Such a method is also generalized to yield a \((k,\lambda _k)\)-GDD of type \((q^{\ell +1}-q^{\ell })^{(q^{n-\ell }-1)/(q-1)}\) where \(\lambda _k=\frac{\prod _{i=3}^{k-1}(q^n-q^{\ell +i-1})}{(k-2)!}\) and \(k+\ell \le n+1\).     

Some remarks on energy inequalities for harmonic maps with potential

We call the \({\delta}\)-vector of an integral convex polytope of dimension d flat if the \({\delta}\)-vector is of the form \({(1,0,\ldots,0,a,\ldots,a,0,\ldots,0)}\), where \({a \geq 1}\). In this paper, we give the complete characterization of possible flat \({\delta}\)-vectors. Moreover, for an integral convex polytope \({\mathcal{P}\subset \mathbb{R}^N}\) of dimension d, we let \({i(\mathcal{P},n)=|n\mathcal{P}\cap \mathbb{Z}^N|}\) and \({i^*(\mathcal{P},n)=|n(\mathcal{P} {\setminus}\partial \mathcal{P})\cap \mathbb{Z}^N|}\). By this characterization, we show that for any \({d \geq 1}\) and for any \({k,\ell \geq 0}\) with \({k+\ell \leq d-1}\), there exist integral convex polytopes \({\mathcal{P}}\) and \({\mathcal{Q}}\) of dimension d such that (i) For \({t=1,\ldots,k}\), we have \({i(\mathcal{P},t)=i(\mathcal{Q},t),}\) (ii) For \({t=1,\ldots,\ell}\), we have \({i^*(\mathcal{P},t)=i^*(\mathcal{Q},t)}\), and (iii) \({i(\mathcal{P},k+1) \neq i(\mathcal{Q},k+1)}\) and \({i^*(\mathcal{P},\ell+1)\neq i^*(\mathcal{Q},\ell+1)}\).     

Sharp Inequalities Involving $$(n!)^{1/n}$$

We prove that, for all integers \(n\ge 1\),

$$\begin{aligned} \Big (\sqrt{2\pi n}\Big )^{\frac{1}{n(n+1)}}\left( 1-\frac{1}{n+a}\right) <\frac{\root n \of {n!}}{\root n+1 \of {(n+1)!}}\le \Big (\sqrt{2\pi n}\Big )^{\frac{1}{n(n+1)}}\left( 1-\frac{1}{n+b}\right) \end{aligned}$$

and

$$\begin{aligned} \big (\sqrt{2\pi n}\big )^{1/n}\left( 1-\frac{1}{2n+\alpha }\right) <\left( 1+\frac{1}{n}\right) ^{n}\frac{\root n \of {n!}}{n}\le \big (\sqrt{2\pi n}\big )^{1/n}\left( 1-\frac{1}{2n+\beta }\right) , \end{aligned}$$

with the best possible constants

$$\begin{aligned}&a=\frac{1}{2},\quad b=\frac{1}{2^{3/4}\pi ^{1/4}-1}=0.807\ldots ,\quad \alpha =\frac{13}{6} \\&\text {and}\quad \beta =\frac{2\sqrt{2}-\sqrt{\pi }}{\sqrt{\pi }-\sqrt{2}}=2.947\ldots . \end{aligned}$$