Bernstein <Emphasis Type="Italic">n</Emphasis>-widths of Besov embeddings on Lipschitz domains

In this paper, using an equivalent characterization of the Besov space by its wavelet coefficients and the discretization technique due to Maiorov, we determine the asymptotic degree of the Bernstein n-widths of the compact embeddings Bq0s＋t（Lp0（Ω））→Bq1s（Lp1（Ω））, t〉max{d（1/p0-1/p1）, 0}, 1 ≤ p0, p1, q0, q1 ≤∞,where Bq0s＋t（Lp0（Ω）） is a Besov space defined on the bounded Lipschitz domain Ω ？ Rd. The results we obtained here are just dual to the known results of Kolmogorov widths on the related classes of functions.     

Cyclotomic Hecke Algebras of <Emphasis Type="Italic">G</Emphasis>(<Emphasis Type="Italic">r</Emphasis>, <Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">n</Emphasis>)

In this note, we find a monomial basis of the cyclotomic Hecke algebra \({\mathcal{H}_{r,p,n}}\) of G(r,p,n) and show that the Ariki-Koike algebra \({\mathcal{H}_{r,n}}\) is a free module over \({\mathcal{H}_{r,p,n}}\), using the Gröbner-Shirshov basis theory. For each irreducible representation of \({\mathcal{H}_{r,p,n}}\), we give a polynomial basis consisting of linear combinations of the monomials corresponding to cozy tableaux of a given shape.     

<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>-Analogue of a linear transformation preserving log-convexity

In this paper, we establish the preserving log-convexity of linear transformation associated with p, q-analogue of Pascal triangle, i.e., if the sequence of nonnegative numbers {x_n}n is logconvex, then \({y_n} = {\sum\nolimits_{k = 0}^n {\left[ {\frac{n}{k}} \right]} _{pq}}{x_k}\) so is it for q ≠ p ≥ 1.     

Non-triviality of a product in the Adams <Emphasis Type="Italic">E</Emphasis> <Subscript>2</Subscript>-term

Let p be a prime greater than five and A the mod p Steenrod algebra. In this paper, we prove that \(h_n h_m \tilde \delta _{s + 4} \in Ext_A^{s + 6,t(s,n,m) + s} (Z/p,Z/p)\) is nontrivial in the Adams E₂-term when m ≥ n + 2 ≥ 7 and 0 ≤ s < p ? 4, and trivial in the Adams E₂-term when m ≥ n + 2 = 6 and 0 ≤ s < p ? 4, where \(\tilde \delta _{s + 4} \) stands for the fourth Greek letter element and t(s, n, m) = 2(p ? 1)[(s + 1) + (s + 2)p + (s + 3)p² + (s + 4)p³ + pⁿ + p^m].     

On <Emphasis Type="Italic">r</Emphasis>-recognition by prime graph of <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(3) where <Emphasis Type="Italic">p</Emphasis> is an odd prime

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p and p′ are joined by an edge if there is an element in G of order pp′. We denote by k(Γ(G)) the number of isomorphism classes of finite groups H satisfying Γ(G) = Γ(H). Given a natural number r, a finite group G is called r-recognizable by prime graph if k(Γ(G)) =  r. In Shen et al. (Sib. Math. J. 51(2):244–254, 2010), it is proved that if p is an odd prime, then B _p (3) is recognizable by element orders. In this paper as the main result, we show that if G is a finite group such that Γ(G) = Γ(B _p (3)), where p > 3 is an odd prime, then \({G\cong B_p(3)}\) or C _p (3). Also if Γ(G) = Γ(B ₃(3)), then \({G\cong B_3(3), C_3(3), D_4(3)}\), or \({G/O_2(G)\cong {\rm Aut}(^2B_2(8))}\). As a corollary, the main result of the above paper is obtained.     

Lebesgue weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-algebra on locally compact groups

Let G be a locally compact group, ω a weight function on G, and 1<p<∞. We introduce the Lebesgue weighted L ^p -space \({\mathcal{L}}_{\omega}^{1,p}(G)= L^{p}(G,\omega)\cap L^{1}(G)\) as a Banach space and introduce its dual. Furthermore, we consider this space as a Banach algebra with respect to the usual convolution and show that \({\mathcal{L}}_{\omega}^{1,p}(G)\) admits a bounded approximate identity if and only if G is discrete. In addition, we prove that amenability of this algebra implies that G is discrete and amenable. Moreover, we discuss the converse of this result.     

Periodic Groups Saturated with the Groups <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

Given an indexing set I and a finite field K_α for each α ∈ I, let ? = {L₂(K_α) | α ∈ I} and \(\mathfrak{N} = \{ SL_2 (K_\alpha )|\alpha \in I\}\). We prove that each periodic group G saturated with groups in \(\Re (\mathfrak{N})\) is isomorphic to L₂(P) (respectively SL₂(P)) for a suitable locally finite field P.     

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.     

A REALIZATION OF CERTAIN MODULES FOR THE <Emphasis Type="Italic">N</Emphasis> = 4 SUPERCONFORMAL ALGEBRA AND THE AFFINE LIE ALGEBRA <Emphasis Type="Italic">A</Emphasis><Stack><Subscript>2</Subscript><Superscript>(1)</Superscript></Stack>

We shall first present an explicit realization of the simple N = 4 superconformal vertex algebra L _c ^N?=?4 with central charge c = ?9. This vertex superalgebra is realized inside of the bcβγ system and contains a subalgebra isomorphic to the simple affine vertex algebra L _A1 \( \left(-\frac{3}{2}{\varLambda}_0\right) \). Then we construct a functor from the category of L _c ^N?=?4 -modules with c = ?9 to the category of modules for the admissible affine vertex algebra L _A1 \( \left(-\frac{3}{2}{\varLambda}_0\right) \). By using this construction we construct a family of weight and logarithmic modules for L _c ^N?=?4 and L _A1 \( \left(-\frac{3}{2}{\varLambda}_0\right) \). We also show that a coset subalgebra L _A1 \( \left(-\frac{3}{2}{\varLambda}_0\right) \) is a logarithmic extension of the W(2; 3)-algebra with c = ?10. We discuss some generalizations of our construction based on the extension of affine vertex algebra L _A1 (kΛ ₀) such that k + 2 = 1/p and p is a positive integer.     

Ordinal indices of Small Subspaces of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We calculate the ordinal L _p index defined in [3] for Rosenthal’s space X _p , \({\ell_p}\) and \({\ell_2}\). We show that an infinite-dimensional subspace of L _p \({(2 < p < \infty)}\) non-isomorphic to \({\ell_2}\) embeds in \({\ell_p}\) if and only if its ordinal index is the minimal possible. We also give a sufficient condition for a \({\mathcal{L}_p}\) subspace of \({\ell_p \oplus \ell_2}\) to be isomorphic to X _p .     

Approximation complexity of tensor product-type random fields with heavy spectrum

We consider a sequence of Gaussian tensor product-type random fields , where and are all positive eigenvalues and eigenfunctions of the covariance operator of the process X ₁, are standard Gaussian random variables, and is a subset of positive integers. For each d ∈ ?, the sample paths of X _d almost surely belong to L ₂([0, 1] ^d ) with norm ∥·∥_2,d . The tuples , are the eigenpairs of the covariance operator of X _d . We approximate the random fields X _d , d ∈ , by the finite sums X _d ⁽ⁿ⁾ corresponding to the n maximal eigenvalues λ _k , . We investigate the logarithmic asymptotics of the average approximation complexity $n_d^{pr} (\varepsilon ,\delta ): = \min \left\{ {n \in \mathbb{N}:\mathbb{P}(\left\| {X_d - X_d^{(n)} } \right\|_{2,d}^2 > \varepsilon ^2 \mathbb{E}\left\| {X_d } \right\|_{2,d}^2 ) \leqslant \delta } \right\},$ and the probabilistic approximation complexity $n_d^{avg} (\varepsilon ): = \min \left\{ {n \in \mathbb{N}:\mathbb{E}\left\| {X_d - X_d^{(n)} } \right\|_{2,d}^2 \leqslant \varepsilon ^2 \mathbb{E}\left\| {X_d } \right\|_{2,d}^2 } \right\}$ , as the parametric dimension d → ∞ the error threshold ? ∈ (0, 1) is fixed, and the confidence level δ = δ(d, ?) is allowed to approach zero. Supplementing recent results of M.A. Lifshits and E.V. Tulyakova, we consider the case where the sequence decreases regularly and sufficiently slowly to zero, which has not been previously studied.     

Spectral Properties of <Emphasis Type="Italic">k</Emphasis>-Quasi-<Emphasis Type="Italic">M</Emphasis>-hyponormal Operators

In this paper, we study the k-quasi-M-hyponormal operator and mainly prove that if T is a k-quasi-M-hyponormal operator, then \(\sigma _{ja}(T)\backslash \{0\}=\sigma _{a}(T)\backslash \{0\}\), and the spectrum is continuous on the class of all k-quasi-M-hyponormal operators; let \(d_{AB}\in B(B(H))\) denote either the generalized derivation \(\delta _{AB}= L_{A}-R_{B}\) or the elementary operator \(\Delta _{AB} =L_{A}R_{B}- I\), we show that if A and \(B^{*}\) are k-quasi-M-hyponormal operators, then \(d_{AB}\) is polaroid and generalized Weyl’s theorem holds for \(f(d_{AB})\), where f is an analytic function on \(\sigma (d_{AB})\) and f is not constant on each connected component of the open set U containing \(\sigma (d_{AB})\). In additon, we discuss the hyperinvariant subspace problem for k-quasi-M-hyponormal operators.     

<Emphasis Type="Italic">C</Emphasis>*-Homomorphisms and duality of <Emphasis Type="Italic">C</Emphasis>*-discrete quantum groups

Let

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be a C*-discrete quantum group and let

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be the discrete quantum group associated with

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. Suppose that there exists a continuous action of

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on a unital C*-algebra

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so that

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becomes a

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-algebra. If there is a faithful irreducible vacuum representation π of

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on a Hilbert space H =

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with a vacuum vector Ω, which gives rise to a

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-invariant state, then there is a unique C*-representation (θ, H) of

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supplemented by the action. The fixed point subspace of

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under the action of

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is exactly the commutant of θ(

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).

     

The maximum and minimum degree of the random <Emphasis Type="Italic">r</Emphasis>-uniform <Emphasis Type="Italic">r</Emphasis>-partite hypergraphs

In this paper we consider the random r-uniform r-partite hypergraph model H(n ₁, n ₂, ···, n _r; n, p) which consists of all the r-uniform r-partite hypergraphs with vertex partition {V ₁, V ₂, ···, V _r} where |V _i| = n _i = n _i(n) (1 ≤ i ≤ r) are positive integer-valued functions on n with n ₁ +n ₂ +···+n _r = n, and each r-subset containing exactly one element in V _i (1 ≤ i ≤ r) is chosen to be a hyperedge of H _p ∈ H (n ₁, n ₂, ···, n _r; n, p) with probability p = p(n), all choices being independent. Let

$${\Delta _{{V_1}}} = {\Delta _{{V_1}}}\left( H \right)$$

and

$${\delta _{{V_1}}} = {\delta _{{V_1}}}\left( H \right)$$

be the maximum and minimum degree of vertices in V ₁ of H, respectively;

$${X_{d,{V_1}}} = {X_{d,{V_1}}}\left( H \right),{Y_{d,{V_1}}} = {Y_{d,{V_1}}}\left( H \right)$$

,

$${Z_{d,{V_1}}} = {Z_{d,{V_1}}}\left( H \right)and{Z_{c,d,{V_1}}} = {Z_{c,d,{V_1}}}\left( H \right)$$

be the number of vertices in V ₁ of H with degree d, at least d, at most d, and between c and d, respectively. In this paper we obtain that in the space H(n ₁, n ₂, ···, n _r; n, p),

$${X_{d,{V_1}}},{Y_{d,{V_1}}},{Z_{d,{V_1}}}and{Z_{c,d,{V_1}}}$$

all have asymptotically Poisson distributions. We also answer the following two questions. What is the range of p that there exists a function D(n) such that in the space H(n ₁, n ₂, ···, n _r; n, p),

$$\mathop {\lim }\limits_{n \to \infty } P\left( {{\Delta _{{V_1}}} = D\left( n \right)} \right) = 1$$

? What is the range of p such that a.e., H _p ∈ H (n ₁, n ₂, ···, n _r; n, p) has a unique vertex in V ₁ with degree

$${\Delta _{{V_1}}}\left( {{H_p}} \right)$$

? Both answers are p = o (log n ₁/N), where

$$N = \mathop \prod \limits_{i = 2}^r {n_i}$$

. The corresponding problems on

$${\delta _{{V_i}}}\left( {{H_p}} \right)$$

also are considered, and we obtained the answers are p ≤ (1 + o(1))(log n ₁/N) and p = o (log n ₁/N), respectively.

     

Idempotent and <Emphasis Type="Italic">p</Emphasis>-potent quadratic functions: distribution of nonlinearity and co-dimension

The Walsh transform \(\widehat{Q}\) of a quadratic function \(Q:{\mathbb F}_{p^n}\rightarrow {\mathbb F}_p\) satisfies \(|\widehat{Q}(b)| \in \{0,p^{\frac{n+s}{2}}\}\) for all \(b\in {\mathbb F}_{p^n}\), where \(0\le s\le n-1\) is an integer depending on Q. In this article, we study the following three classes of quadratic functions of wide interest. The class \(\mathcal {C}_1\) is defined for arbitrary n as \(\mathcal {C}_1 = \{Q(x) = \mathrm{Tr_n}(\sum _{i=1}^{\lfloor (n-1)/2\rfloor }a_ix^{2^i+1})\;:\; a_i \in {\mathbb F}_2\}\), and the larger class \(\mathcal {C}_2\) is defined for even n as \(\mathcal {C}_2 = \{Q(x) = \mathrm{Tr_n}(\sum _{i=1}^{(n/2)-1}a_ix^{2^i+1}) + \mathrm{Tr_{n/2}}(a_{n/2}x^{2^{n/2}+1}) \;:\; a_i \in {\mathbb F}_2\}\). For an odd prime p, the subclass \(\mathcal {D}\) of all p-ary quadratic functions is defined as \(\mathcal {D} = \{Q(x) = \mathrm{Tr_n}(\sum _{i=0}^{\lfloor n/2\rfloor }a_ix^{p^i+1})\;:\; a_i \in {\mathbb F}_p\}\). We determine the generating function for the distribution of the parameter s for \(\mathcal {C}_1, \mathcal {C}_2\) and \(\mathcal {D}\). As a consequence we completely describe the distribution of the nonlinearity for the rotation symmetric quadratic Boolean functions, and in the case \(p > 2\), the distribution of the co-dimension for the rotation symmetric quadratic p-ary functions, which have been attracting considerable attention recently. Our results also facilitate obtaining closed formulas for the number of such quadratic functions with prescribed s for small values of s, and hence extend earlier results on this topic. We also present the complete weight distribution of the subcodes of the second order Reed–Muller codes corresponding to \(\mathcal {C}_1\) and \(\mathcal {C}_2\) in terms of a generating function.     

<Emphasis Type="Italic">l</Emphasis>-Groups <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">X</Emphasis>) in continuous logic

In the context of continuous logic, this paper axiomatizes both the class \(\mathcal {C}\) of lattice-ordered groups isomorphic to C(X) for X compact and the subclass \(\mathcal {C}^+\) of structures existentially closed in \(\mathcal {C}\); shows that the theory of \(\mathcal {C}^+\) is \(\aleph _0\)-categorical and admits elimination of quantifiers; establishes a Nullstellensatz for \(\mathcal {C}\) and \(\mathcal {C}^+\); shows that \(C(X)\in \mathcal {C}\) has a prime-model extension in \(\mathcal {C}^+\) just in case X is Boolean; and proves that in a sense relevant to continuous logic, positive formulas admit in \(\mathcal {C}^+\) elimination of quantifiers to positive formulas.     

Sharp <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Bound for Holomorphic Functions on the Unit Disc

For any 1 < p < ∞ and any \({X, Y\in \mathbb{R}}\) satisfying \({|X|\leq Y}\) , we determine the optimal constant C _p (X,Y) such that the following holds. If F is a holomorphic function on the unit disc satisfying ReF(0) = X and \({||{\rm Re}F||_{L^{p}(\mathbb{T})}=Y}\) , then

$$||F||_{L^p(\mathbb{T})}\geq C_p(X,Y).$$

This can be regarded as a reverse version of the classical estimates of Riesz and Essén. The proof rests on the exploitation of certain families of special subharmonic functions on the plane.

     

Restriction of representations of metaplectic GL<Subscript>2</Subscript>(<Emphasis Type="Italic">F</Emphasis>) to tori

Let F be a non-archimedean local field. We study the restriction of irreducible admissible genuine representations of the twofold metaplectic cover \({\widetilde {GL}_2}\) of GL₂(F) to the inverse image in \({\widetilde {GL}_2}\) of a maximal torus in GL₂(F).     

Distance-Regular Shilla Graphs with <Emphasis Type="Italic">b</Emphasis><Subscript>2</Subscript> = <Emphasis Type="Italic">c</Emphasis><Subscript>2</Subscript>

A Shilla graph is defined as a distance-regular graph of diameter 3 with second eigen-value θ₁ equal to a₃. For a Shilla graph, let us put a = a₃ and b = k/a. It is proved in this paper that a Shilla graph with b₂ = c₂ and noninteger eigenvalues has the following intersection array:

$$\left\{ {\frac{{{b^2}\left( {b - 1} \right)}}{2},\frac{{\left( {b - 1} \right)\left( {{b^2} - b + 2} \right)}}{2},\frac{{b\left( {b - 1} \right)}}{4};1,\frac{{b\left( {b - 1} \right)}}{4},\frac{{b{{\left( {b - 1} \right)}^2}}}{2}} \right\}$$

If Γ is a Q-polynomial Shilla graph with b₂ = c₂ and b = 2r, then the graph Γ has intersection array

$$\left\{ {2tr\left( {2r + 1} \right),\left( {2r + 1} \right)\left( {2rt + t + 1} \right),r\left( {r + t} \right);1,r\left( {r + t} \right),t\left( {4{r^2} - 1} \right)} \right\}$$

and, for any vertex u in Γ, the subgraph Γ₃(u) is an antipodal distance-regular graph with intersection array

$$\left\{ {t\left( {2r + 1} \right),\left( {2r - 1} \right)\left( {t + 1} \right),1;1,t + 1,t\left( {2r + 1} \right)} \right\}$$

The Shilla graphs with b₂ = c₂ and b = 4 are also classified in the paper.

     

A remark on the existence of entire large and bounded solutions to a (<Emphasis Type="Italic">k</Emphasis><Subscript>1</Subscript>, <Emphasis Type="Italic">k</Emphasis><Subscript>2</Subscript>)-Hessian system with gradient term

In this paper, we study the existence of positive entire large and bounded radial positive solutions for the following nonlinear system

$$\left\{ {\begin{array}{*{20}c}{S_{k_1 } \left( {\lambda \left( {D^2 u_1 } \right)} \right) + a_1 \left( {\left| x \right|} \right)\left| {\nabla u_1 } \right|^{k_1 } = p_1 \left( {\left| x \right|} \right)f_1 \left( {u_2 } \right)} & {for x \in \mathbb{R}^N ,} \\{S_{k_2 } \left( {\lambda \left( {D^2 u_2 } \right)} \right) + a_2 \left( {\left| x \right|} \right)\left| {\nabla u_2 } \right|^{k_2 } = p_2 \left( {\left| x \right|} \right)f_2 \left( {u_1 } \right)} & {for x \in \mathbb{R}^N .} \\\end{array} } \right.$$

Here \({S_{{k_i}}}\left( {\lambda \left( {{D^2}{u_i}} \right)} \right)\) is the k _i -Hessian operator, a ₁, p ₁, f ₁, a ₂, p ₂ and f ₂ are continuous functions.