Interior Differentiability Results for Nonlinear Variational Parabolic Systems with Nonlinearity 1 <?q < 2

In this work we prove that the solutions

Seas of squares with sizes from a Π<Stack><Subscript>1</Subscript><Superscript>0</Superscript></Stack> set

A family of sets is union-free if there are no three distinct sets in the family such that the union of two of the sets is equal to the third set. Kleitman proved that every union-free family has size at most (1+o(1))( _n/2 ⁿ ). Later, Burosch–Demetrovics–Katona–Kleitman–Sapozhenko asked for the number α(n) of such families, and they proved that \({2^{\left( {\begin{array}{*{20}{c}} n \\ {n/2} \end{array}} \right)}} \leqslant \alpha \left( n \right) \leqslant {2^{2\sqrt 2 \left( {\begin{array}{*{20}{c}} n \\ {n/2} \end{array}} \right)\left( {1 + o\left( 1 \right)} \right)}}\) They conjectured that the constant \(2\sqrt 2 \) can be removed in the exponent of the right-hand side. We prove their conjecture by formulating a new container-type theorem for rooted hypergraphs.     

Combined ℓ<Subscript>2</Subscript> data and gradient fitting in conjunction with ℓ<Subscript>1</Subscript> regularization

We are interested in minimizing functionals with ℓ₂ data and gradient fitting term and ℓ₁ regularization term with higher order derivatives in a discrete setting. We examine the structure of the solution in 1D by reformulating the original problem into a contact problem which can be solved by dual optimization techniques. The solution turns out to be a ’smooth’ discrete polynomial spline whose knots coincide with the contact points while its counterpart in the contact problem is a discrete version of a spline with higher defect and contact points as knots. In 2D we modify Chambolle’s algorithm to solve the minimization problem with the ℓ₁ norm of interacting second order partial derivatives as regularization term. We show that the algorithm can be implemented efficiently by applying the fast cosine transform. We demonstrate by numerical denoising examples that the ℓ₂ gradient fitting term can be used to avoid both edge blurring and staircasing effects.      

Conformally flat Lorentzian hypersurfaces in ℝ<Stack><Subscript>1</Subscript><Superscript>4</Superscript></Stack> with three distinct principal curvatures

A three dimensional Lorentzian hypersurface x: M ₁ ³ → ? ₁ ⁴ is called conformally flat if its induced metric is conformal to the flat Lorentzian metric, and this property is preserved under the conformal transformation of ? ₁ ⁴ . Using the projective light-cone model, for those whose shape operators have three distinct real eigenvalues, we calculate the integrability conditions by constructing a scalar conformal invariant and a canonical moving frame in this paper. Similar to the Riemannian case, these hypersurfaces can be determined by the solutions to some system of partial differential equations.     

A Hilbert-type theorem for spacelike surfaces with constant Gaussian curvature in ℍ<Superscript>2</Superscript> × ℝ<Subscript>1</Subscript>

There are examples of complete spacelike surfaces in the Lorentzian product ℍ² × ℝ₁ with constant Gaussian curvature K ≤ −1. In this paper, we show that there exists no complete spacelike surface in ℍ² × ℝ₁ with constant Gaussian curvature K > −1.     

Verifiable conditions of <Emphasis Type="Italic">ℓ</Emphasis><Subscript>1</Subscript>-recovery for sparse signals with sign restrictions

We propose necessary and sufficient conditions for a sensing matrix to be “s-semigood” – to allow for exact ℓ ₁-recovery of sparse signals with at most s nonzero entries under sign restrictions on part of the entries. We express error bounds for imperfect ℓ ₁-recovery in terms of the characteristics underlying these conditions. These characteristics, although difficult to evaluate, lead to verifiable sufficient conditions for exact sparse ℓ ₁-recovery and thus efficiently computable upper bounds on those s for which a given sensing matrix is s-semigood. We examine the properties of proposed verifiable sufficient conditions, describe their limits of performance and provide numerical examples comparing them with other verifiable conditions from the literature.     

On threefolds of general type with <Emphasis Type="Italic">χ</Emphasis> = 1

For a canonical threefold of general type, we know that the pluri–canonical map is stably birational for a sufficiently large n. This paper aims to find the lower bound of n for such kind of threefolds with χ = 1. To prove our main result, we will estimate the lower bound of plurigenus. L. Zhu is supported by Fudan Graduate Students’ Innovation Projects (EYH5928004).     

Toeplitz operators on hardy spaceH p (S) with 0<p≤1

LetB ⁿ be the unit ball inC ⁿ ,S is the boundary ofB ⁿ . We letL ^p (S) denote the usual Lebesgue spaces overS with respect to the normalized surface measure,H ^p (B ⁿ ) is its usua holomorphic subspace.H ^p (S) denotes the atomic Hardy spaces defined in [GL]. LetPL ² (S)H ²(B ⁿ ) denote the orthogonal projection. For eachfL (S), we useM _f L ^p (S)L ^p (S) to denote the multiplication operator, and we define the Toeplitz operatorT _f =PM _f . The paper gives a characterization theorem onf such that the Toeplitz operatorsT _f and are bounded fromH ^p (S)H ^p (B ⁿ ) with 0<p1. Also several equivalent conditions are given.     

New symplectic 4–manifolds with <Emphasis Type="Italic">b</Emphasis><Subscript>+</Subscript>=1

In 1995 Dusa McDuff and Dietmar Salamon conjectured the existence of symplectic 4–manifolds (X,ω) which satisfy b₊=1, K²=0, K·ω>0, and which fail to be of Lefschetz type. This is equivalent to finding a symplectic, homology T²×S² manifold with nontorsion canonical class and a cohomology ring which is not isomorphic to the cohomology ring of T²×S². They needed such examples to complete a list of possible symplectic 4–manifolds with b₊=1. In that same year Tian-Jun Li and Ai-ko Liu, working from a different point of view, questioned whether there existed symplectic 4–manifolds with b₊=1 with Seiberg- Witten invariants that did not depend on the chamber structure of the moduli space. The purpose of this paper is to construct an infinite number of examples which satisfy both requirements. The author was partially supported by NSF grant DMS-0406021.     

On Fourier multipliers between Besov spaces with 0 <p o ≤ min{1,p 1}

, 0<0 min{1,p1}.     

Hamiltonian Gromov–Witten invariants on ℂ<Superscript><Emphasis Type="Italic">n</Emphasis>+1</Superscript> with <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript>-action

Consider a Hamiltonian action of S1 on (C ⁿ⁺¹, ω _std), we shown that the Hamiltonian Gromov–Witten invariants of it are well-defined. After computing the Hamiltonian Gromov–Witten invariants of it, we construct a ring homomorphism from \(H_{{S^1},CR}^*\left( {X,R} \right)\) to the small orbifold quantum cohomology of X// _τ S ¹ and obtain a simpler formula of the Gromov–Witten invariants for weighted projective space.     

Contraction Principles for Vector Valued Martingales with Respect to Random Variables Having Exponential Tail with Exponent 2<α<∞

We prove a contraction principle for vector valued martingales with respect to random variables having exponential tail behaviour with exponent 2<<.     

Every Banach space with a <Emphasis Type="Italic">w</Emphasis><Superscript>*</Superscript>-separable dual has a 1+<Emphasis Type="Italic">ɛ</Emphasis>-equivalent norm with the ball covering property

A normed space is said to have ball-covering property if its unit sphere can be contained in the union of countably many open balls off the origin. This paper shows that for every ɛ > 0 every Banach space with a w ^*-separable dual has a 1+ɛ-equivalent norm with the ball covering property.     

Sparse recovery in probability via l_q-minimization with Weibull random matrices for 0 < q ≤ 1

Although Gaussian random matrices play an important role of measurement matrices in compressed sensing, one hopes that there exist other random matrices which can also be used to serve as the measurement matrices. Hence, Weibull random matrices induce extensive interest. In this paper, we first propose the l_(2,q)robust null space property that can weaken the D-RIP, and show that Weibull random matrices satisfy the l_(2,q) robust null space property with high probability. Besides, we prove that Weibull random matrices also possess the l_q quotient property with high probability. Finally, with the combination of the above mentioned properties,we give two important approximation characteristics of the solutions to the l_q-minimization with Weibull random matrices, one is on the stability estimate when the measurement noise e ∈ R~n needs a priori ‖e‖_2 ≤ε, the other is on the robustness estimate without needing to estimate the bound of ‖e‖_2. The results indicate that the performance of Weibull random matrices is similar to that of Gaussian random matrices in sparse recovery.     

Eisenstein series associated with Γ<Subscript>0</Subscript>(2)

In this paper, we define the normalized Eisenstein series ℘, e, and associated with Γ₀(2), and derive three differential equations satisfied by them from some trigonometric identities. By using these three formulas, we define a differential equation depending on the weights of modular forms on Γ₀(2) and then construct its modular solutions by using orthogonal polynomials and Gaussian hypergeometric series. We also construct a certain class of infinite series connected with the triangular numbers. Finally, we derive a combinatorial identity from a formula involving the triangular numbers.      

Distance-Regular Graph with <Emphasis Type="Italic">c</Emphasis><Subscript>2</Subscript> > 1 and <Emphasis Type="Italic">a</Emphasis><Subscript>1</Subscript> = 0 < <Emphasis Type="Italic">a</Emphasis><Subscript>2</Subscript>

Let Γ be a distance-regular graph of diameter d ≥ 3 with c ₂ > 1. Let m be an integer with 1 ≤ m ≤ d − 1. We consider the following conditions:

Differentiability of Mappings of the Sobolev Space <Emphasis Type="Italic">W</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis>−1</Subscript><Superscript>1</Superscript> with Conditions on the Distortion Function

We define two scales of the mappings that depend on two real parameters p and q, with n?1 ≤ q ≤ p < ∞, as well as a weight function θ. The case q = p = n and θ ≡ 1 yields the well-known mappings with bounded distortion. The mappings of a two-index scale are applied to solve a series of problems of global analysis and applications. The main result of the article is the a.e. differentiability of mappings of two-index scales.     

On (2<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript> + 1)-variable symmetric Boolean functions with submaximum algebraic immunity 2<Superscript><Emphasis Type="Italic">m</Emphasis>−1</Superscript>

All (2 ^m + 1)-variable symmetric Boolean functions with submaximal algebraic immunity 2 ^m−1 are described and constructed. The total number of such Boolean functions is 3² · 2^2m−3 +3 ^m−2 · 2⁴ − 2 for m ⩾ 2. This work was supported by the Major State Basic Research Development Program of China (Grant No. 2004CB3180004) and National Natural Science Foundation of China (Grant No. 60433050)     

The Algorithm Z and Ramanujan’s <Subscript>1</Subscript><Emphasis Type="Italic">ψ</Emphasis><Subscript>1</Subscript> summation

We use the Algorithm Z on partitions due to Zeilberger, in a variant form, to give a combinatorial proof of Ramanujan’s ₁ ψ ₁ summation formula.