On low discrepancy sequences and low discrepancy ergodic transformations of the multidimensional unit cube

In this paper we describe a third class of low discrepancy sequences. Using a lattice Γ ? ? ^s , we construct Kronecker-like and van der Corput-like ergodic transformations T _1,Γ and T _2,Γ of [0, 1) ^s . We prove that for admissible lattices Γ, (T _ν,Γ ⁿ (x))_n≥0 is a low discrepancy sequence for all x ∈ [0, 1) ^s and ν ∈ {1, 2}. We also prove that for an arbitrary polyhedron P ? [0, 1) ^s , for almost all lattices Γ ∈ L _s = SL(s,?)/SL(s, ?) (in the sense of the invariant measure on L _s ), the following asymptotic formula

$\# \{ 0 \le n < N:T_{v,\Gamma }^n(x) \in P\} = NvolP + O({(\ln N)^{s + \varepsilon }}),N \to \infty$

holds with arbitrary small ? > 0, for all x ∈ [0, 1) ^s , and ν ∈ {1, 2}.

     

Strong approximation in the Apollonian group

The Apollonian group is a finitely generated, infinite index subgroup of the orthogonal group OQ(Z) fixing the Descartes quadratic form Q. For nonzero v∈Z⁴ satisfying Q(v)=0, the orbits Pv=Av correspond to Apollonian circle packings in which every circle has integer curvature. In this paper, we specify the reduction of primitive orbits Pv mod any integer d>1. We show that this reduction has a multiplicative structure, and that mod primes p?5 it is the full cone of integer solutions to Q(v)≡0 for v?0. This analysis is an essential ingredient in applications of the affine linear sieve as developed by Bourgain, Gamburd and Sarnak.     

HLDER CONTINUOUS SOLUTIONS OF BOUSSINESQ EQUATIONS

We show the existence of dissipative H¨older continuous solutions of the Boussinesq equations. More precise, for any β∈(0,1/5), a time interval [0, T ] and any given smooth energy profile e : [0, T ] →(0, ∞), there exist a weak solution(v, θ) of the 3 d Boussinesq equations such that(v, θ) ∈ Cβ(T~3× [0, T ]) with e(t) =′his T~3|v(x, t)|~2 dx for all t ∈ [0, T ]. Textend the result of [2] about Onsager's conjecture into Boussinesq equation and improve our previous result in [30].     

Density theorems and the mean value of arithmetical functions in short intervals

Let Γ=SL₂(?) and let Z_Γ(s) be the Selberg zeta function. Set π_Γ(P)=∑_N(P)≤P1, where P is a primitive hyperbolic class of conjugate elements in Γ and N(P) is the norm of P. It is shown that for P^1/2+θ=Q, 1≤θ≤1/2, we have $\pi _\Gamma \left( {P + Q} \right) - \pi _\Gamma \left( P \right) = \int\limits_P^{P + Q} {\frac{{du}}{{\log u}} + O_ \in \left( {QP^{ - \sigma \left( \theta \right) + \in } } \right),} $ where σ(θ)=θ²/2+O(¸₃), θ→0. Thus, a conjecture of Iwaniec (1984) is proved. Similar asymptotic formulas are obtained for the sums ∑_{Ph(-d)/ $\sqrt d $ and ∑Pr3(n)/ $\sqrt n $ , where h(?d) is the class number of the imaginary quadratic field of discriminant ?d<0 and r3(n) is the number of representations of n by the sum of three squares.}     

Ternary forms as invariants of Markoff forms and other SL2 (Z)-bundles

Consider the free group Γ = {A,B} generated by matrices A, B in SL₂(Z). We can construct a ternary form Φ(x,y,z) whose GL₃(Z) equivalence class is invariant, as it depends on Γ and not the choice of generators. If Γ is the commutator of SL₂(Z), then the generating matrices have fixed points corresponding to different fields and inequivalent Markoff forms, but they are all biuniquely determined by Φ = -z²+ y(2x+y+z) to within equivalence. When referred to transformations A, B of the upper half plane, this phenomenon is interpreted in terms of inequivalent homotopy elements which are primitive for the perforated torus.     

The partition of a strong tournament

A digraph T is strong if for every pair of vertices u and v there exists a directed path from u to v and a directed path from v to u. Denote the in-degree and out-degree of a vertex v of T by d^-(v) and d⁺(v), respectively. We define δ^-(T)=min_vV(T){d^-(v)} and δ⁺(T)=min_vV(T){d⁺(v)}. Let T₀ be a 7-tournament which contains no transitive 4-subtournament. In this paper, we obtain some conditions on a strong tournament which cannot be partitioned into two cycles. We show that a strong tournament T with n6 vertices such that TT₀ and max{δ⁺(T),δ^-(T)}3 can be partitioned into two cycles. Finally, we give a sufficient condition for a tournament to be partitioned into k cycles.     

Arithmetic expressions of Selberg's zeta functions for congruence subgroups

In [P. Sarnak, Class numbers of indefinite binary quadratic forms, J. Number Theory 15 (1982) 229-247], it was proved that the Selberg zeta function for SL₂(Z) is expressed in terms of the fundamental units and the class numbers of the primitive indefinite binary quadratic forms. The aim of this paper is to obtain similar arithmetic expressions of the logarithmic derivatives of the Selberg zeta functions for congruence subgroups of SL₂(Z). As applications, we study the Brun-Titchmarsh type prime geodesic theorem and the asymptotic formula of the sum of the class number.     

The asymptotic distribution of lattice points in hyperbolic space

Suppose x and y are two points in the upper half-plane H⁺, and suppose Γ is a discontinuous group of conformal automorphisms of H⁺ having compact fundamental domain S. Denote by N_T(x, y) the number of points of the form γy (γ?Γ) in the closed disc of hyperbolic radius T centered about x, and set $\text{Q}{}_{\mathrm{T}}\text{(x, y) = N}{}_{\mathrm{T}}\text{(x, y) ?}\text{V(T)}\text{A}\text{,}\text{where}\text{V(T)}$ is the hyperbolic area of the disc, and A is the hyperbolic area of S. The asymptotic behavior of the quantity ?_LxL(Q_T(x,y))² is estimated in terms of small eigenvalues of the Laplacian on functions automorphic under Γ.     

Root numbers and ranks in positive characteristic

For a global field K and an elliptic curve E_η over K(T), Silverman's specialization theorem implies rank(E_η(K(T)))?rank(E_t(K)) for all but finitely many t∈P¹(K). If this inequality is strict for all but finitely many t, the elliptic curve E_η is said to have elevated rank. All known examples of elevated rank for K=Q rest on the parity conjecture for elliptic curves over Q, and the examples are all isotrivial.Some additional standard conjectures over Q imply that there does not exist a non-isotrivial elliptic curve over Q(T) with elevated rank. In positive characteristic, an analogue of one of these additional conjectures is false. Inspired by this, for the rational function field K=κ(u) over any finite field κ with characteristic ≠2, we construct an explicit 2-parameter family E_c,d of non-isotrivial elliptic curves over K(T) (depending on arbitrary c,d∈κ^×) such that, under the parity conjecture, each E_c,d has elevated rank.     

PATHS AND CYCLES EMBEDDING ON FAULTY ENHANCED HYPERCUBE NETWORKS

Let Qn,k(n≥3,1≤k≤n-1) be an n-dimensional enhanced hypercube which is an attractive variant of the hypercube and can be obtained by adding some complementary edges,fv and fe be the numbers of faulty vertices and faulty edges,respectively.In this paper,we give three main results.First,a fault-free path P [u,v] of length at least 2n-2fv-1(respectively,2n-2fv-2) can be embedded on Qn,k with fv+fe≤n-1 when d Qn,k(u,v) is odd(respectively,d Qn,k(u,v) is even).Secondly,an Qn,k is(n-2) edgefault-free hyper Hamiltonian-laceable when n(≥3) and k have the same parity.Lastly,a fault-free cycle of length at least 2n-2fv can be embedded on Qn,k with fe≤n-1 and fv+fe≤2n-4.     

Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces

We show that ifG is a semisimple algebraic group defined overQ and Γ is an arithmetic lattice inG:=G _R with respect to theQ-structure, then there exists a compact subsetC ofG/Γ such that, for any unipotent one-parameter subgroup {u _t} ofG and anyg∈G, the time spent inC by the {u _t}-trajectory ofgΓ, during the time interval [0,T], is asymptotic toT, unless {g ⁻¹u_tg} is contained in aQ-parabolic subgroup ofG. Some quantitative versions of this are also proved. The results strengthen similar assertions forSL(n,Z),n≥2, proved earlier in [5] and also enable verification of a technical condition introduced in [7] for lattices inSL(3,R), which was used in our proof of Raghunathan’s conjecture for a class of unipotent flows, in [8].     

Elliptic ovoids and their rosettes in a classical generalized quadrangle of even order

Let Q₀ be the classical generalized quadrangle of order q = 2ⁿ(n≥2) arising from a non-degenerate quadratic form in a 5-dimensional vector space defined over a finite field of order q. We consider the rank two geometry \(\mathcal {X}\) having as points all the elliptic ovoids of Q₀ and as lines the maximal pencils of elliptic ovoids of Q₀ pairwise tangent at the same point. We first prove that \(\mathcal {X}\) is isomorphic to a 2-fold quotient of the affine generalized quadrangle Q?Q₀, where Q is the classical (q,q²)-generalized quadrangle admitting Q₀ as a hyperplane. Further, we classify the cliques in the collinearity graph Γ of \(\mathcal {X}\). We prove that any maximal clique in Γ is either a line of \(\mathcal {X}\) or it consists of 6 or 4 points of \(\mathcal {X}\) not contained in any line of \(\mathcal {X}\), accordingly as n is odd or even. We count the number of cliques of each type and show that those cliques which are not contained in lines of \(\mathcal {X}\) arise as subgeometries of Q defined over \(\mathbb {F}_{2}\).     

Quantitative unique continuation for the semilinear heat equation in a convex domain

In this paper, we study certain unique continuation properties for solutions of the semilinear heat equation t_∂u−△u=g(u), with the homogeneous Dirichlet boundary condition, over Ω×(0,T_∗). Ω is a bounded, convex open subset of Rd, with a smooth boundary for the subset. The function g:R→R satisfies certain conditions. We establish some observation estimates for (u−v), where u and v are two solutions to the above-mentioned equation. The observation is made over ω×{T}, where ω is any non-empty open subset of Ω, and T is a positive number such that both u and v exist on the interval [0,T]. At least two results can be derived from these estimates: (i) if ‖(u−v)(⋅,T)_‖L²(ω)=δ, then ‖(u−v)(⋅,T)_‖L²(Ω)?Cδα where constants C>0 and α∈(0,1) can be independent of u and v in certain cases; (ii) if two solutions of the above equation hold the same value over ω×{T}, then they coincide over Ω×[0,Tm). Tm indicates the maximum number such that these two solutions exist on [0,Tm).     

Bounding the diameter of a distance-transitive graph

A graph Γ is distance-transitive if for all vertices u, v, x, y such that d(u, v) = d(x, y) there is an automorphism h of Γ such that uh = x, vh = y. We show how to find a bound for the diameter of a bipartite distance-transitive graph given a bound for the order |G_α| of the stabilizer of a vertex.     

Existence of T*(3, 4,v)‐codes

A word of length k over an alphabet Q of size v is a vector of length k with coordinates taken from Q. Let Q^*₄ be the set of all words of length 4 over Q. A T^*(3, 4, v)‐code over Q is a subset C^*? Q^*₄ such that every word of length 3 over Q occurs as a subword in exactly one word of C^*. Levenshtein has proved that a T^*(3, 4, vv)‐code exists for all even v. In this paper, the notion of a generalized candelabra t‐system is introduced and used to show that a T^*(3, 4, v)‐code exists for all odd v. Combining this with Levenshtein's result, the existence problem for a T^*(3,4, v)‐code is solved completely. © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 42–53, 2005.     

Conjugated tricyclic graphs with the maximum zeroth-order general Randić index

Let G be a simple connected graph and α be a given real number. The zeroth-order general Randi? index of G is defined as ⁰ R _α (G)=∑ _v∈V(G)[d _G (v)] ^α , where d _G (v) denotes the degree of the vertex v of G. In this paper, for any α>2, we give sharp upper bounds of the zeroth-order general Randi? index ⁰ R _α of all conjugated tricyclic graphs with 2m vertices.     

Halving Steiner 2-designs

A Steiner 2-design S(2,k,v) is said to be halvable if the block set can be partitioned into two isomorphic sets. This is equivalent to an edge-disjoint decomposition of a self-complementary graph G on v vertices into K_ks. The obvious necessary condition of those orders v for which there exists a halvable S(2,k,v) is that v admits the existence of an S(2,k,v) with an even number of blocks. In this paper, we give an asymptotic solution for various block sizes. We prove that for any k?5 or any Mersenne prime k, there is a constant number v₀ such that if v>v₀ and v satisfies the above necessary condition, then there exists a halvable S(2,k,v). We also show that a halvable S(2,2ⁿ,v) exists for over a half of possible orders. Some recursive constructions generating infinitely many new halvable Steiner 2-designs are also presented.     

Trees of extremal connectivity index

The connectivity index w_α(G) of a graph G is the sum of the weights (d(u)d(v))^α of all edges uv of G, where α is a real number (α≠0), and d(u) denotes the degree of the vertex u. Let T be a tree with n vertices and k pendant vertices. In this paper, we give sharp lower and upper bounds for w₁(T). Also, for -1?α<0, we give a sharp lower bound and a upper bound for w_α(T).     

Sharp bounds of the zeroth-order general Randi? index of bicyclic graphs with given pendent vertices

Let G be a simple connected graph and α be a given real number. The zeroth-order general Randi? index of ⁰R_α(G) is defined as ∑_v∈V(G)[d_G(v)]^α, where d_G(v) denotes the degree of the vertex v of G. In this paper, for any α(≠0,1), we give sharp bounds of the zeroth-order general Randi? index ⁰R_α of all bicyclic graphs with n vertices and k pendent vertices.     

Numerical solutions with a priori error bounds for coupled self-adjoint time dependent partial differential systems

This paper is concerned with the construction of accurate continuous numerical solutions for partial self-adjoint differential systems of the type (P(t) u_t)_t = Q(t)u_xx, u(0, t) = u(d, t) = 0, u(x, 0) = f(x), u_t(x, 0) = g(x), 0 ≤ x ≤ d, t >- 0, where P(t), Q(t) are positive definite oR^r×r-valued functions such that P′(t) and Q′(t) are simultaneously semidefinite (positive or negative) for all t ≥ 0. First, an exact theoretical series solution of the problem is obtained using a separation of variables technique. After appropriate truncation strategy and the numerical solution of certain matrix differential initial value problems the following question is addressed. Given T > 0 and an admissible error ϵ > 0 how to construct a continuous numerical solution whose error with respect to the exact series solution is smaller than ϵ, uniformly in D(T) = {(x, t); 0 ≤ x ≤ d, 0 ≤ t ≤ T}. Uniqueness of solutions is also studied.