Seifert manifolds and (1, 1)-knots

The aim of this paper is to investigate the relations between Seifert manifolds and (1, 1)-knots. In particular, we prove that each orientable Seifert manifold with invariants

$\{ Oo,0| - 1;\underbrace {(p,q),...,(p,q)}_{n times},(l,l - 1)\} $

has the fundamental group cyclically presented by G _n ((x ₁ ^q ...x _n ^{q l} x _n ^?p ) and, moreover, it is the n-fold strongly-cyclic covering of the lens space L(|nlq ? p|, q) which is branched over the (1, 1)-knot K(q, q(nl ? 2), p ? 2q, p ? q) if p ≥ 2q and over the (1, 1)-knot K(p? q, 2q ? p, q(nl ? 2), p ? q) if p< 2q.

     

Representation and character varieties of the Baumslag-Solitar groups

Representation and character varieties of the Baumslag–Solitar groups BS(p, q) are analyzed. Irreducible components of these varieties are found, and their dimension is calculated. It is proved that all irreducible components of the representation variety R_n(BS(p, q)) are rational varieties of dimension n², and each irreducible component of the character variety X_n(BS(p, q)) is a rational variety of dimension k ≤ n. The smoothness of irreducible components of the variety R_n^s (BS(p, q)) of irreducible representations is established, and it is proved that all irreducible components of the variety R_n^s (BS(p, q)) are isomorphic to A¹ {0}.     

Information of orderα and typeβ

Information I_α ^β (Q/P) of orderα and typeβ is introduced and it is shown that for every fixedβ, this information is a monotonic increasing function ofα. It is also shown that information of orderα and type 1 is non-negative when\(\sum\limits_{k = 1}^N { q_k } \geqslant \sum\limits_{k = 1}^N { p_k } \), where (q ₁,q ₂ …,q _N) and (p ₁,p ₂, …,p _N) are generalised probability distributions for Q and P respectively.     

Anisotropic Hardy-Lorentz spaces and their applications

Let p ∈(0, 1], q ∈(0, ∞] and A be a general expansive matrix on Rn. We introduce the anisotropic Hardy-Lorentz space H~(p,q)_A(R~n) associated with A via the non-tangential grand maximal function and then establish its various real-variable characterizations in terms of the atomic and the molecular decompositions, the radial and the non-tangential maximal functions, and the finite atomic decompositions. All these characterizations except the ∞-atomic characterization are new even for the classical isotropic Hardy-Lorentz spaces on Rn.As applications, we first prove that Hp,q A(Rn) is an intermediate space between H~(p1,q1)_A(Rn) and H~(p2,q2)_A(R~n) with 0 p1 p p2 ∞ and q1, q, q2 ∈(0, ∞], and also between H~(p,q1)_A(Rn) and H~(p,q2)_A(R~n) with p ∈(0, ∞)and 0 q1 q q2 ∞ in the real method of interpolation. We then establish a criterion on the boundedness of sublinear operators from H~(p,q)_A(R~n) into a quasi-Banach space; moreover, we obtain the boundedness of δ-type Calder′on-Zygmund operators from H~(p,∞)_A(R~n) to the weak Lebesgue space L~(p,∞)(R~n)(or to H~p_A(R~n)) in the ln λcritical case, from H~(p,q)_A(R~n) to L~(p,q)(R~n)(or to H~(p,q)_A(R~n)) with δ∈(0,(lnλ)/(ln b)], p ∈(1/(1+,δ),1] and q ∈(0, ∞], as well as the boundedness of some Calderon-Zygmund operators from H~(p,q)_A(R~n) to L~(p,∞)(R~n), where b := | det A|,λ_:= min{|λ| : λ∈σ(A)} and σ(A) denotes the set of all eigenvalues of A.     

Pointwise multipliers for Besov spaces of dominating mixed smoothness-Ⅱ

We continue our investigations on pointwise multipliers for Besov spaces of dominating mixed smoothness. This time we study the algebra property of the classes S_(p,q)~rB(R~d) with respect to pointwise multiplication. In addition, if p≤q, we are able to describe the space of all pointwise multipliers for S_(p,q)~rB(R~d).     

On finite alperin <Emphasis Type="Italic">p</Emphasis>-groups with homocyclic commutator subgroup

We study metabelian Alperin groups, i.e., metabelian groups in which every 2-generated subgroup has a cyclic commutator subgroup. It is known that, if the minimum number d(G) of generators of a finite Alperin p-group G is n ≥ 3, then d(G′) ≤ C _n ² for p≠ 3 and d(G′) ≤ C _n ² + C _n ³ for p = 3. The first section of the paper deals with finite Alperin p-groups G with p≠ 3 and d(G) = n ≥ 3 that have a homocyclic commutator subgroup of rank C _n ² . In addition, a corollary is deduced for infinite Alperin p-groups. In the second section, we prove that, if G is a finite Alperin 3-group with homocyclic commutator subgroup G- of rank C _n ² + C _n ³ , then G″ is an elementary abelian group.     

An Extension of a Simply Inequality Between von Neumann–Jordan and James Constants in Banach Spaces

For a non-trivial Banach space X, let J(X), CNJ(X), C_(NJ)~(p)(X) respectively stand for the James constant, the von Neumann–Jordan constant and the generalized von Neumann–Jordan constant recently inroduced by Cui et al. In this paper, we discuss the relation between the James and the generalized von Neumann–Jordan constants, and establish an inequality between them: C_(NJ)~(p)(X) ≤J(X) with p ≥ 2, which covers the well-known inequality CNJ(X) ≤ J(X). We also introduce a new constant, from which we establish another inequality that extends a result of Alonso et al.     

Asymptotic behavior of the maximal zero of a polynomial orthogonal on a segment with a nonclassical weight

Let {p _n (t)} _n=0 ^t8 be a system of algebraic polynomials orthonormal on the segment [?1, 1] with a weight p(t); let {x _n,ν ^(p) } _ν=1 ⁿ be zeros of a polynomial p _n (t) (x _x,ν ^(p) = cosθ _n,ν ^(p) ; 0 < θ _n,1 ^(p) < θ _n,2 ^(p) < ... < θ _n,n ^(p) < π). It is known that, for a wide class of weights p(t) containing the Jacobi weight, the quantities θ _n,1 ^(p) and 1 ? x _n,1 ^(p) coincide in order with n ^?1 and n ^?2, respectively. In the present paper, we prove that, if the weight p(t) has the form p(t) = 4(1 ? t ²)^?1{ln²[(1 + t)/(1 ? t)] + π ²}^?1, then the following asymptotic formulas are valid as n → ∞:

$$\theta _{n,1}^{(p)} = \frac{{\sqrt 2 }}{{n\sqrt {\ln (n + 1)} }}\left[ {1 + {\rm O}\left( {\frac{1}{{\ln (n + 1)}}} \right)} \right],x_{n,1}^{(p)} = 1 - \left( {\frac{1}{{n^2 \ln (n + 1)}}} \right) + O\left( {\frac{1}{{n^2 \ln ^2 (n + 1)}}} \right).$$

     

Arnold diffusion for a complete family of perturbations

In this work we illustrate the Arnold diffusion in a concrete example — the a priori unstable Hamiltonian system of 2 + 1/2 degrees of freedom H(p, q, I, φ, s) = p ²/2+ cos q ? 1 + I ²/2 + h(q, φ, s; ε) — proving that for any small periodic perturbation of the form h(q, φ, s; ε) = ε cos q (a ₀₀ + a ₁₀ cosφ + a ₀₁ cos s) (a ₁₀ a ₀₁ ≠ 0) there is global instability for the action. For the proof we apply a geometrical mechanism based on the so-called scattering map. This work has the following structure: In the first stage, for a more restricted case (I* ~ π/2μ, μ = a ₁₀/a ₀₁), we use only one scattering map, with a special property: the existence of simple paths of diffusion called highways. Later, in the general case we combine a scattering map with the inner map (inner dynamics) to prove the more general result (the existence of instability for any μ). The bifurcations of the scattering map are also studied as a function of μ. Finally, we give an estimate for the time of diffusion, and we show that this time is primarily the time spent under the scattering map.     

Bessel Property of the System of Root Functions of a Second-Order Singular Operator on an Interval

For the system of root functions of an operator defined by the differential operation ?u″ + p(x)u′ + q(x)u, x ∈ G = (0, 1), with complex-valued singular coefficients, sufficient conditions for the Bessel property in the space L₂(G) are obtained and a theorem on the unconditional basis property is proved. It is assumed that the functions p(x) and q(x) locally belong to the spaces L₂ and W₂^?1, respectively, and may have singularities at the endpoints of G such that q(x) = q_R(x) +q′S(x) and the functions q_S(x), p(x), q ₂ ^S (x)w(x), p²(x)w(x), and q_R(x)w(x) are integrable on the whole interval G, where w(x) = x(1 ? x).     

Two-weight inequality for operator-valued positive dyadic operators by parallel stopping cubes

We study the operator-valued positive dyadic operator

$${T_\lambda }\left( {f\sigma } \right): = \sum\limits_{Q \in D} {{\lambda _Q}} \int_Q {fd\sigma 1Q}, $$

where the coefficients {λ _Q : C → D} _Q∈D are positive operators from a Banach lattice C to a Banach lattice D. We assume that the Banach lattices C and D* each have the Hardy–Littlewood property. An example of a Banach lattice with the Hardy–Littlewood property is a Lebesgue space.

In the two-weight case, we prove that the L _C ^p (σ) → L _D ^q (ω) boundedness of the operator T _λ( · σ) is characterized by the direct and the dual L ^∞ testing conditions:

$$\left\| {{1_Q}{T_\lambda }} \right\|{\left( {{1_Q}f\sigma } \right)||_{L_D^q\left( \omega \right)}} \lesssim {\left\| f \right\|_{L_C^\infty \left( {Q,\sigma } \right)}}\sigma {\left( Q \right)^{1/p}}$$

,

$${\left\| {{1_Q}{T_\lambda }*\left( {{1_{Qg\omega }}} \right)} \right\|_{L_{C*}^{p'}\left( \sigma \right)}} \lesssim {\left\| g \right\|_{L_{D*}^\infty \left( {Q,\omega } \right)}}\omega {\left( Q \right)^{1/q'}}$$

.

Here L _C ^p (σ) and L _D ^q (ω) denote the Lebesgue–Bochner spaces associated with exponents 1 < p ≤ q < ∞, and locally finite Borel measures σ and ω.

In the unweighted case, we show that the L _C ^p (μ) → L _D ^p (μ) boundedness of the operator T _λ( · μ) is equivalent to the end-point direct L ^∞ testing condition:

$${\left\| {{1_Q}{T_\lambda }\left( {{1_Q}f\mu } \right)} \right\|_{L_D^1\left( \mu \right)}} \lesssim {\left\| f \right\|_{L_C^\infty \left( {Q,\mu } \right)}}\left( {Q,\mu } \right)\mu \left( Q \right)$$

.

This condition is manifestly independent of the exponent p. By specializing this to particular cases, we recover some earlier results in a unified way.     

Improved bounds on the Hadwiger–Debrunner numbers

Let HD _d (p, q) denote the minimal size of a transversal that can always be guaranteed for a family of compact convex sets in Rd which satisfy the (p, q)-property (p ≥ q ≥ d + 1). In a celebrated proof of the Hadwiger–Debrunner conjecture, Alon and Kleitman proved that HD _d (p, q) exists for all p ≥ q ≥ d + 1. Specifically, they prove that \(H{D_d}(p,d + 1)is\tilde O({p^{{d^2} + d}})\).We present several improved bounds: (i) For any \(q \geqslant d + 1,H{D_d}(p,d) = \tilde O({p^{d(\frac{{q - 1}}{{q - d}})}})\). (ii) For q ≥ log p, \(H{D_d}(p,q) = \tilde O(p + {(p/q)^d})\). (iii) For every ? > 0 there exists a p₀ = p₀(?) such that for every p ≥ p₀ and for every \(q \geqslant {p^{\frac{{d - 1}}{d} + \in }}\) we have p ? q + 1 ≤ HD _d (p, q) ≤ p ? q + 2. The latter is the first near tight estimate of HD _d (p, q) for an extended range of values of (p, q) since the 1957 Hadwiger–Debrunner theorem.We also prove a (p, 2)-theorem for families in R² with union complexity below a specific quadratic bound.     

Improved bounds on the generalized acyclic chromatic number

An r-acyclic edge chromatic number of a graph G, denoted by a _r ^′ r(G), is the minimum number of colors used to produce an edge coloring of the graph such that adjacent edges receive different colors and every cycle C has at least min {|C|, r} colors. We prove that a _r ^′ r(G) ≤ (4r + 1)Δ(G), when the girth of the graph G equals to max{50, Δ(G)} and 4 ≤ r ≤ 7. If we relax the restriction of the girth to max {220, Δ(G)}, the upper bound of a _r ^′ r(G) is not larger than (2r + 5)Δ(G) with 4 ≤ r ≤ 10.     

Nikol’skii inequality between the uniform norm and <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>-norm with jacobi weight of algebraic polynomials on an interval

We study the Nikol’skii inequality for algebraic polynomials on the interval [?1, 1] between the uniform norm and the norm of the space L _q ^(α,β) , 1 ≤ q < ∞, with the Jacobi weight ?(α,β)(x) = (1 ? x) ^α (1 + x) ^β , α ≥ β > ?1. We prove that, in the case α > β ≥ ?1/2, the polynomial with unit leading coefficient that deviates least from zero in the space L _q ^(α+1,,β) with the Jacobi weight ? ^(α+1,β)(x) = (1?x) ^α+1(1+x) ^β is the unique extremal polynomial in the Nikol’skii inequality. To prove this result, we use the generalized translation operator associated with the Jacobi weight. We describe the set of all functions at which the norm of this operator in the space L _q ^(α,β) for 1 ≤ q < ∞ and α > β ≥ ?1/2 is attained.     

Linear independence of values of Lerch functions

The number of linearly independent numbers among 1, Φ₁ (z, p/q), ...,Φ _a (z, p/q) is estimated depending on a natural number a, where Φ _s (z, p/q), s = 1, 2, ..., are Lerch functions.     

On the approximation of continuous functions of two variables

It is shown that if P _m ^α,β (x) (α, β > ?1, m = 0, 1, 2, …) are the classical Jaboci polynomials, then the system of polynomials of two variables {Ψ _mn ^α,β (x, y)} _m,n=0 ^r = {P _m ^α,β (x)P _n ^α,β (y)} _{m, n=0} ^r (r = m + n ≤ N ? 1) is an orthogonal system on the set Ω _N×N = ?ub;(x _i , y _i ) _i,j=0 ^N , where x _i and y _i are the zeros of the Jacobi polynomial P _n ^α,β (x). Given an arbitrary continuous function f(x, y) on the square [?1, 1]², we construct the discrete partial Fourier-Jacobi sums of the rectangular type S _{m, n, N} ^α,β (f; x, y) by the orthogonal system introduced above. We prove that the order of the Lebesgue constants ∥S _{m, n, N} ^α,β ∥ of the discrete sums S _{m, n, N} ^α,β (f; x, y) for ?1/2 < α, β < 1/2, m + n ≤ N ? 1 is O((mn) ^{q + 1/2}), where q = max?ub;α,β?ub;. As a consequence of this result, several approximate properties of the discrete sums S _{m, n, N} ^α,β (f; x, y) are considered.     

Estimation of the depth of reversible circuits consisting of NOT,CNOT and 2-CNOT gates

The paper discusses the asymptotic depth of a reversible circuits consisting of NOT, CNOT and 2-CNOT gates. The reversible circuit depth function D(n, q) is introduced for a circuit implementing a mapping f: Z₂ⁿ → Z₂ⁿ as a function of n and the number q of additional inputs. It is proved that for the case of implementation of a permutation from A(Z₂ⁿ) with a reversible circuit having no additional inputs the depth is bounded as D(n, 0) ? 2ⁿ/(3log₂n). It is also proved that for the case of transformation f: Z₂ⁿ → Z₂ⁿ with a reversible circuit having q₀ ~ 2ⁿ additional inputs the depth is bounded as D(n,q₀) ? 3n.     

Series by Haar system with monotone coefficients

Let χ = {χ _n } _n=0 ^∞ be the Haar system normalized in L ²(0, 1) and M = {M _s } _s=1 ^∞ be an arbitrary, increasing sequence of nonnegative integers. For any subsystem of χ of the form {φ _k } = χS = {χ _n } _n∈S , where S = S(M) = {n _k } _k=1 ^∞ = {n ∈ V[p]: p ∈ M}, V[0] = {1, 2} and V[p] = {2 ^p + 1, 2 ^p + 2, …, 2 ^p+1} for p = 1, 2, … a series of the form Σ _i=1 ^∞ a _i φ _i with a _i ↘ 0 is constructed, that is universal with respect to partial series in all classes L ^r (0, 1), r ∈ (0, 1), in the sense of a.e. convergence and in the metric ofL ^r (0, 1). The constructed series is universal in the class of all measurable, finite functions on [0, 1] in the sense of a.e. convergence. It is proved that there exists a series by Haar system with decreasing coefficients, which has the following property: for any ? > 0 there exists a measurable function µ(x), x ∈ [0, 1], such that 0 ≤ µ(x) ≤ 1 and |{x ∈ [0, 1], µ(x) ≠ = 1}| < ?, and the series is universal in the weighted space L _µ[0, 1] with respect to subseries, in the sense of convergence in the norm of L _µ[0, 1].     

Blow-up of <Emphasis Type="Italic">p</Emphasis>-Laplacian evolution equations with variable source power

We study the blow-up and/or global existence of the following p-Laplacian evolution equation with variable source power

$${s_j} = {\beta _j} + \overline {{\beta _{n - j}}}p$$

where Ω is either a bounded domain or the whole space ? ^N , q(x) is a positive and continuous function defined in Ω with 0 < q _? = inf q(x) ? q(x) ? sup q(x) = q+ < ∞. It is demonstrated that the equation with variable source power has much richer dynamics with interesting phenomena which depends on the interplay of q(x) and the structure of spatial domain Ω, compared with the case of constant source power. For the case that Ω is a bounded domain, the exponent p ? 1 plays a crucial role. If q+ > p ? 1, there exist blow-up solutions, while if q ₊ < p ? 1, all the solutions are global. If q _? > p ? 1, there exist global solutions, while for given q _? < p ? 1 < q ₊, there exist some function q(x) and Ω such that all nontrivial solutions will blow up, which is called the Fujita phenomenon. For the case Ω = ? ^N , the Fujita phenomenon occurs if 1 < q _? ? q ₊ ? p ? 1 + p/N, while if q _? > p ? 1 + p/N, there exist global solutions.