Generalized modified Gold sequences

In this paper, a large family \({\mathcal{F}^k(l)}\) of binary sequences of period 2 ⁿ ? 1 is constructed for odd n = 2m + 1, where k is any integer with gcd(n, k) = 1 and l is an integer with 1 ≤ l ≤ m. This generalizes the construction of modified Gold sequences by Rothaus. It is shown that \({\mathcal{F}^k(l)}\) has family size \({2^{ln}+2^{(l-1)n}+\cdots+2^n+1}\), maximum nontrivial correlation magnitude 1 + 2^m+l. Based on the theory of quadratic forms over finite fields, all exact correlation values between sequences in \({\mathcal{F}^k(l)}\) are determined. Furthermore, the family \({\mathcal{F}^k(2)}\) is discussed in detail to compute its complete correlation distribution.     

Critical exponent for a quasilinear parabolic equation with inhomogeneous density in a cone

In this paper, we study the initial-boundary value problem of porous medium equation ρ(x)u _t  = Δu ^m  + V(x)h(t)u ^p in a cone D = (0, ∞) × Ω, where \({V(x)\,{\sim}\, |x|^\sigma, h(t)\,{\sim}\, t^s}\). Let ω ₁ denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on Ω and let l denote the positive root of l ² + (n ? 2)l = ω ₁. We prove that if \({m < p \leq 1+(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}\), then the problem has no global nonnegative solutions for any nonnegative u ₀ unless u ₀ = 0; if \({p >1 +(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}\), then the problem has global solutions for some u ₀ ≥ 0.     

An explicit height bound for the classical modular polynomial

For a prime l, let Φ _l be the classical modular polynomial, and let h(Φ _l ) denote its logarithmic height. By specializing a theorem of Cohen, we prove that \(h(\Phi_{l})\le 6l\log l+16l+14\sqrt{l}\log l\). As a corollary, we find that h(Φ _l )≤6llog?l+18l also holds. A table of h(Φ _l ) values is provided for l≤3600.     

Completely reducible super-simple designs with block size four and related super-simple packings

A design is said to be super-simple if the intersection of any two blocks has at most two elements. A super-simple design \({\mathcal{D}}\) with point set X, block set \({\mathcal{B}}\) and index λ is called completely reducible super-simple (CRSS), if its block set \({\mathcal{B}}\) can be written as \({\mathcal{B}=\bigcup_{i=1}^{\lambda} \mathcal{B}_i}\), such that \({\mathcal{B}_i}\) forms the block set of a design with index unity but having the same parameters as \({\mathcal{D}}\) for each 1 ≤ i ≤ λ. It is easy to see, the existence of CRSS designs with index λ implies that of CRSS designs with index i for 1 ≤ i ≤ λ. CRSS designs are closely related to q-ary constant weight codes (CWCs). A (v, 4, q)-CRSS design is just an optimal (v, 6, 4)_q+1 code. On the other hand, CRSS group divisible designs (CRSSGDDs) can be used to construct q-ary group divisible codes (GDCs), which have been proved useful in the constructions of q-ary CWCs. In this paper, we mainly investigate the existence of CRSS designs. Three neat results are obtained as follows. Firstly, we determine completely the spectrum for a (v, 4, 3)-CRSS design. As a consequence, a class of new optimal (v, 6, 4)₄ codes is obtained. Secondly, we give a general construction for (4, 4)-CRSSGDDs with skew Room frames, and prove that the necessary conditions for the existence of a (4, 2)-CRSSGDD of type g ^u are also sufficient except definitely for \({(g,u)\in \{(2,4),(3,4),(6,4)\}}\). Finally, we consider the related optimal super-simple (v, 4, 2)-packings and show that such designs exist for all v ≥ 4 except definitely for \({v\in \{4,5,6,9\}}\).     

A Comment on Ryser’s Conjecture for Intersecting Hypergraphs

Let \(\tau({\mathcal{H}})\) be the cover number and \(\nu({\mathcal{H}})\) be the matching number of a hypergraph \({\mathcal{H}}\). Ryser conjectured that every r-partite hypergraph \({\mathcal{H}}\) satisfies the inequality \(\tau({\mathcal{H}}) \leq (r-1) \nu ({\mathcal{H}})\). This conjecture is open for all r ≥ 4. For intersecting hypergraphs, namely those with \(\nu({\mathcal{H}}) = 1\), Ryser’s conjecture reduces to \(\tau({\mathcal{H}}) \leq r-1\). Even this conjecture is extremely difficult and is open for all r ≥ 6. For infinitely many r there are examples of intersecting r-partite hypergraphs with \(\tau({\mathcal{H}}) = r-1\), demonstrating the tightness of the conjecture for such r. However, all previously known constructions are not optimal as they use far too many edges. How sparse can an intersecting r-partite hypergraph be, given that its cover number is as large as possible, namely \(\tau({\mathcal{H}}) \ge r-1\)? In this paper we solve this question for r ≤ 5, give an almost optimal construction for r = 6, prove that any r-partite intersecting hypergraph with τ(H) ≥ r ? 1 must have at least \((3-\frac{1}{\sqrt{18}})r(1-o(1)) \approx 2.764r(1-o(1))\) edges, and conjecture that there exist constructions with Θ(r) edges.     

On the gonality sequence of smooth curves

Let C be a smooth curve of genus g. For each positive integer r the r-gonality d _r (C) of C is the minimal integer t such that there is \({L\in {\rm Pic}^t(C)}\) with h ⁰(C, L) = r + 1. Here we use nodal plane curves to construct several smooth curves C with d ₂(C)/2 < d ₃(C)/3, i.e., for which a slope inequality fails.     

On Multiple and Infinite Log-Concavity

Following Boros-Moll, a sequence (a _n ) is m-log-concave if \({\mathcal{L}^{j}(a_{n})\geqslant0}\) for all j =  0, 1, . . . , m. Here, \({\mathcal{L}}\) is the operator defined by \({\mathcal{L}(a_{n}) = a^{2}_{n}-a_{n-1}a_{n+1}}\). By a criterion of Craven-Csordas and McNamara-Sagan it is known that a sequence is ∞-log-concave if it satisfies the stronger inequality \({a^{2}_{k}\geqslant ra_{k-1}a_{k+1}}\) for large enough r. On the other hand, a recent result of Brändén shows that ∞-log-concave sequences include sequences whose generating polynomial has only negative real roots. In this paper, we investigate sequences which are fixed by a power of the operator \({\mathcal{L}}\) and are therefore ∞-log-concave for a very different reason. Surprisingly, we find that sequences fixed by the non-linear operators \({\mathcal{L}}\) and \({\mathcal{L}^{2}}\) are, in fact, characterized by a linear 4-term recurrence. In a final conjectural part, we observe that positive sequences appear to become ∞-log-concave if convoluted with themselves a finite number of times.     

On the value distribution of positive definite quadratic forms

Denote by 0 = λ ₀ < λ ₁ ≤ λ ₂ ≤ . . . the infinite sequence given by the values of a positive definite irrational quadratic form in k variables at integer points. For l ≥ 2 and an (l ?1)-dimensional interval I = I ₂×. . .×I _l we consider the l-level correlation function \({K^{(l)}_I(R)}\) which counts the number of tuples (i ₁, . . . , i _l ) such that \({\lambda_{i_1},\ldots,\lambda_{i_l}\leq R^2}\) and \({\lambda_{i_{j}}-\lambda_{i_{1}}\in I_j}\) for 2 ≤ j ≤ l. We study the asymptotic behavior of \({K^{(l)}_I(R)}\) as R tends to infinity. If k ≥ 4 we prove \({K^{(l)}_I(R)\sim c_l(Q)\,{\rm vol}(I)R^{lk-2(l-1)}}\) for arbitrary l, where c _l (Q) is an explicitly determined constant. This remains true for k = 3 under the restriction l ≤ 3.     

Non-singular Solutions of Normalized Ricci Flow on Noncompact Manifolds of Finite Volume

The main result of this paper shows that if g(t) is a complete non-singular solution of the normalized Ricci flow on a noncompact 4-manifold M of finite volume, then the Euler characteristic number χ(M)≥0. Moreover, if χ(M)≠0, there exists a sequence of times t _k →∞, a double sequence of points \(\{p_{k,l}\}_{l=1}^{N}\) and domains \(\{U_{k,l}\}_{l=1}^{N}\) with p _k,l∈U _k,l satisfying the following:

1. (i)
   \(\mathrm{dist}_{g(t_{k})}(p_{k,l_{1}},p_{k,l_{2}})\rightarrow\infty\) as k→∞, for any fixed l₁≠l₂;
    
2. (ii)
   for each l, (U _k,l,g(t _k ),p _k,l) converges in the \(C_{\mathrm{loc}}^{\infty}\) sense to a complete negative Einstein manifold (M _∞,l ,g _∞,l ,p _∞,l ) when k→∞;
    
3. (iii)
   \(\operatorname {Vol}_{g(t_{k})}(M\backslash\bigcup_{l=1}^{N}U_{k,l})\rightarrow0\) as k→∞.
    

     

Homogenization of elliptic boundary value problems in Lipschitz domains

In this paper we study the L ^p boundary value problems for \({\mathcal{L}(u)=0}\) in \({\mathbb{R}^{d+1}_+}\) , where \({\mathcal{L}=-{\rm div} (A\nabla )}\) is a second order elliptic operator with real and symmetric coefficients. Assume that A is periodic in x _d+1 and satisfies some minimal smoothness condition in the x _d+1 variable, we show that the L ^p Neumann and regularity problems are uniquely solvable for 1 < p < 2 + δ. We also present a new proof of Dahlberg’s theorem on the L ^p Dirichlet problem for 2 ? δ < p < ∞ (Dahlberg’s original unpublished proof is given in the Appendix). As the periodic and smoothness conditions are imposed only on the x _d+1 variable, these results extend directly from \({\mathbb{R}^{d+1}_+}\) to regions above Lipschitz graphs. Consequently, by localization techniques, we obtain uniform L ^p estimates for the Dirichlet, Neumann and regularity problems on bounded Lipschitz domains for a family of second order elliptic operators arising in the theory of homogenization. The results on the Neumann and regularity problems are new even for smooth domains.     

Some Hecke algebra products and corresponding random walks

Let i=1+q+???+q ^i?1. For certain sequences (r ₁,…,r _l ) of positive integers, we show that in the Hecke algebra ? _n (q) of the symmetric group \(\mathfrak{S}_{n}\), the product \((1+\boldsymbol{r}_{\boldsymbol{1}}T_{r_{1}})\cdots (1+\boldsymbol{r}_{\boldsymbol{l}}T_{r_{l}})\) has a simple explicit expansion in terms of the standard basis {T _w }. An interpretation is given in terms of random walks on \(\mathfrak{S}_{n}\).     

Continuous wavelets on compact manifolds

Let M be a smooth compact oriented Riemannian manifold, and let Δ _M be the Laplace–Beltrami operator on M. Say \({0 \neq f \in \mathcal{S}(\mathbb {R}^+)}\) , and that f (0)  =  0. For t  >  0, let K _t (x, y) denote the kernel of f (t ² Δ _M ). We show that K _t is well-localized near the diagonal, in the sense that it satisfies estimates akin to those satisfied by the kernel of the convolution operator f (t ²Δ) on \({\mathbb {R}^n}\) . We define continuous \({\mathcal {S}}\)-wavelets on M, in such a manner that K _t (x, y) satisfies this definition, because of its localization near the diagonal. Continuous \({\mathcal {S}}\)-wavelets on M are analogous to continuous wavelets on \({\mathbb {R}^n}\) in \({\mathcal {S}}\) (\({\mathbb {R}^n}\)). In particular, we are able to characterize the Hölder continuous functions on M by the size of their continuous \({\mathcal {S}}\)-wavelet transforms, for Hölder exponents strictly between 0 and 1. If M is the torus \({\mathbb T^2}\) or the sphere S ², and f (s)  =  se ^?s (the “Mexican hat” situation), we obtain two explicit approximate formulas for K _t , one to be used when t is large, and one to be used when t is small.     

Composite meromorphic functions and normal families

In this paper, we study the normality of families of meromorphic functions. We prove the result: Let α(z) be a holomorphic function and \({\mathcal{F}}\) a family of meromorphic functions in a domain D, P(z) be a polynomial of degree at least 3. If P ○ f(z) and P ○ g(z) share α(z) IM for each pair \({f(z),g(z)\in \mathcal{F}}\) and one of the following conditions holds: (1) P(z) ? α(z ₀) has at least three distinct zeros for any \({z_{0}\in D}\); (2) There exists \({z_{0}\in D}\) such that P(z) ? α(z ₀) has at most two distinct zeros and α(z) is nonconstant. Assume that β ₀ is a zero of P(z) ? α(z ₀) with multiplicity p and that the multiplicities l and k of zeros of f(z) ? β ₀ and α(z) ? α(z ₀) at z ₀, respectively, satisfy k ≠ lp, for all \({f(z)\in\mathcal{F}}\). Then \({\mathcal{F}}\) is normal in D. In particular, the result is a kind of generalization of the famous Montel criterion.     

Lie n-derivatio ns o n $ {\mathcal {}}$-subspace lattice algebras

Let \(\mathcal {L}\) be a \(\mathcal {J}\)-subspace lattice on a Banach space X over the real or complex field \(\mathbb {F}\) with dimX ≥ 3 and let n ≥ 2 be an integer. Suppose that dimK ≠ 2 for every \(K\in \mathcal {J}{(\mathcal L)}\) and \(L: \text {Alg}\, \mathcal {L}\rightarrow \text {Alg}\,\mathcal {L}\) is a linear map. It is shown that L satisfies \({\sum }_{i=1}^{n}p_{n} (A_{1}, \ldots , A_{i-1}, L(A_{i}), A_{i+1}, \ldots , A_{n})=0\) whenever p _n (A ₁,A ₂,…,A _n ) = 0 for \(A_{1},A_{2},\ldots ,A_{n}\in \text {Alg}\,\mathcal {L}\) if and only if for each \(K\in \mathcal {J}(\mathcal {L})\), there exists a bounded linear operator \(T_{K}\in \mathcal {B}(K)\), a scalar λ _K and a linear functional \(h_{K}: \text {Alg}\,\mathcal {L}\rightarrow \mathbb {F}\) such that L(A)x = (T _K A ? A T _K + λ _K A + h _K (A)I)x for all x ∈ K and all \(A\in \text {Alg}\,\mathcal {L}\). Based on this result, a complete characterization of linear n-Lie derivations on \(\text {Alg}\,\mathcal {L}\) is obtained.     

On ℤ<Subscript>2</Subscript>-graded identities of the super tensor product of <Emphasis Type="Italic">UT</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">F</Emphasis>) by the Grassmann algebra

Let F be a field of characteristic zero and E be the unitary Grassmann algebra generated over an infinite-dimensional F-vector space L. Denote by \(\mathcal{E} = \mathcal{E}^{(0)} \oplus \mathcal{E}^{(1)}\) an arbitrary ?₂-grading of E such that the subspace L is homogeneous. Given a superalgebra A = A ⁽⁰⁾ ⊕ A ⁽¹⁾, define the superalgebra \(A\hat \otimes \mathcal{E}\) by \(A\hat \otimes \mathcal{E} = (A^{(0)} \otimes \mathcal{E}^{(0)} ) \oplus (A^{(1)} \otimes \mathcal{E}^{(1)} )\). Note that when E is the canonical grading of E then \(A\hat \otimes \mathcal{E}\) is the Grassmann envelope of A. In this work we find bases of ?₂-graded identities and we describe the ?₂-graded codimension and cocharacter sequences for the superalgebras \(UT_2 (F)\hat \otimes \mathcal{E}\), when the algebra UT ₂(F) of 2 ×2 upper triangular matrices over F is endowed with its canonical grading.     

A Completion of the Spectrum for the Overlarge Sets of Pure Mendelsohn Triple Systems

A pure Mendelsohn triple system of order v, denoted by PMTS(v), is a pair \((X,\mathcal {B})\) where X is a v-set and \(\mathcal {B}\) is a collection of cyclic triples on X such that every ordered pair of X belongs to exactly one triple of \(\mathcal {B}\) and if \(\langle a,b,c\rangle \in \mathcal {B}\) implies \(\langle c,b,a\rangle \notin \mathcal {B}\). An overlarge set of PMTS(v), denoted by OLPMTS(v), is a collection \(\{(Y{\setminus }\{y_i\},{\mathcal {A}}_i)\}_i\), where Y is a \((v+1)\)-set, \(y_i\in Y\), each \((Y{\setminus }\{y_i\},{\mathcal {A}}_i)\) is a PMTS(v) and these \({\mathcal {A}}_i\)s form a partition of all cyclic triples on Y. It is shown in [3] that there exists an OLPMTS(v) for \(v\equiv 1,3\) (mod 6), \(v>3\), or \(v \equiv 0,4\) (mod 12). In this paper, we shall discuss the existence problem of OLPMTS(v)s for \(v\equiv 6,10\) (mod 12) and get the following conclusion: there exists an OLPMTS(v) if and only if \(v\equiv 0,1\) (mod 3), \(v>3\) and \(v\ne 6\).     

Sharp constant of an improved Gagliardo–Nirenberg inequality and its application

This paper contains three parts. In the first part, we determine the best constant of an improved inequality of Gagliardo–Nirenberg interpolation (Chen, in Czechoslov Math J, in press). In the second part, we use this best constant to establish a sharp criterion for the global existence and blow-up of solutions of the inhomogeneous nonlinear Schrödinger equation with harmonic potential

$ i\varphi_t=-\triangle\varphi+|x|^2\varphi-|x|^b|\varphi|^{p-1}\varphi,\quad b > 0,\quad \varphi(x,0) = \varphi_0(x) $

in the critical nonlinearity p = 1 + (4 + 2b)/N. In the third part, we use this best constant to construct an unbounded subset \({\mathcal{S}}\) of Σ and prove that the solutions exist globally in time for \({\varphi_0\in \mathcal{S}}\) and p > 1 + (4 + 2b)/N.

     

New <Emphasis Type="Italic">r</Emphasis>-Minimal Hypersurfaces via Perturbative Methods

A hypersurface in a Riemannian manifold is r-minimal if its (r+1)-curvature, the (r+1)th elementary symmetric function of its principal curvatures, vanishes identically. If n>2(r+1) we show that the rotationally invariant r-minimal hypersurfaces in ? ⁿ⁺¹ are nondegenerate in the sense that they carry no nontrivial Jacobi fields decaying rapidly enough at infinity. Combining this with a computation of the (r+1)-curvature of normal graphs and the deformation theory in weighted Hölder spaces developed by Mazzeo, Pacard, Pollack, Uhlenbeck and others, we produce new infinite dimensional families of r-minimal hypersurfaces in ? ⁿ⁺¹ obtained by perturbing noncompact portions of the catenoids. We also consider the moduli space \({\mathcal{M}}_{r,k}(g)\) of elliptic r-minimal hypersurfaces with k≥2 ends of planar type in ? ⁿ⁺¹ endowed with an ALE metric g, and show that \({\mathcal{M}}_{r,k}(g)\) is an analytic manifold of formal dimension k(n+1), with \({\mathcal{M}}_{r,k}(g)\) being smooth for a generic g in a neighborhood of the standard Euclidean metric.     

On the nilpotent residuals of all subalgebras of Lie algebras

Let \(\mathcal{N}\) denote the class of nilpotent Lie algebras. For any finite-dimensional Lie algebra L over an arbitrary field \(\mathbb{F}\), there exists a smallest ideal I of L such that L/I ∈ \(\mathcal{N}\). This uniquely determined ideal of L is called the nilpotent residual of L and is denoted by L\(\mathcal{N}\). In this paper, we define the subalgebra S(L) = ∩_H≤LI_L(H\(\mathcal{N}\)). Set S₀(L) = 0. Define S_i+1(L)/S_i(L) = S(L/S_i(L)) for i > 1. By S_∞(L) denote the terminal term of the ascending series. It is proved that L = S_∞(L) if and only if L\(\mathcal{N}\) is nilpotent. In addition, we investigate the basic properties of a Lie algebra L with S(L) = L.     

Hyperquasipolynomials for the Theta-Function

Let g be a linear combination with quasipolynomial coefficients of shifts of the Jacobi theta function and its derivatives in the argument. All entire functions f: ? → ? satisfying f(x+y)g(x?y) = α₁(x)β₁(y)+· · ·+α_r(x)β_r(y) for some r ∈ ? and α_j, β_j: ? → ? are described.