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.     

On the Gosper’s <Emphasis Type="Italic">q</Emphasis>-constant Π<Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>

Gosper introduced the functions sin_qz and cos_qz as q-analogues for the trigonometric functions sin z and cos z respectively. He stated a variety of identities involving these two q-trigonometric functions along with certain constants denoted by \({\Pi _{{q^n}}}\) (n ∈ N). Gosper noticed that all his formulas on these constants have more than two of the \({\Pi _{{q^n}}}\). So, it is natural to raise the question of establishing identities involving only two of the \({\Pi _{{q^n}}}\). In this paper, our main goal is to give examples of such formulas in only two \({\Pi _{{q^n}}}\).     

A Characterization of <Emphasis Type="Italic">PSU</Emphasis><Subscript>3</Subscript>(<Emphasis Type="Italic">q</Emphasis>) for <Emphasis Type="Italic">q</Emphasis> ≥ 5

Based on the prime graph of a finite simple group, its order is the product of its order components (see [4]). We prove that the simple groups PSU₃(q) are uniquely determined by their order components. Our result immediately implies that the Thompsons conjecture and the Wujie Shis conjecture [16] are valid for these groups.AMS Subject Classification: 20D05, 20D60     

On the equation Δ<Emphasis Type="Italic">u</Emphasis> + <Emphasis Type="Italic">q</Emphasis>(<Emphasis Type="Italic">x</Emphasis>)<Emphasis Type="Italic">u</Emphasis> = 0

Sufficient conditions for the blow-up of nontrivial generalized solutions of the interior Dirichlet problem with homogeneous boundary condition for the homogeneous elliptic-type equation Δu + q(x)u = 0, where either q(x) ≠ const or q(x) = const= λ > 0, are obtained. A priori upper bounds (Theorem 4 and Remark 6) for the exact constants in the well-known Sobolev and Steklov inequalities are established.     

An extremal problem in the Hardy space <Emphasis Type="Italic">H</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> for 0&lt;<Emphasis Type="Italic">p</Emphasis>&lt;∞

We prove that if the function determining a linear functional over the Hardy space is analytic on the disk of radius greater than 1 then the extremal function of this functional is analytic on the same disk.     

Sparse blind deconvolution and demixing through <Emphasis Type="Italic">ℓ</Emphasis><Subscript>1,2</Subscript>-minimization

This paper concerns solving the sparse deconvolution and demixing problem using ? _1,2-minimization. We show that under a certain structured random model, robust and stable recovery is possible. The results extend results of Ling and Strohmer (Inverse Probl. 31, 115002 2015), and in particular theoretically explain certain experimental findings from that paper. Our results do not only apply to the deconvolution and demixing problem, but to recovery of column-sparse matrices in general.     

The James constant for the <Emphasis Type="Italic">l</Emphasis><Subscript>3</Subscript> − <Emphasis Type="Italic">l</Emphasis><Subscript>1</Subscript> space

In this note, the exact value of the James constant for the l₃-l₁ space is obtained, J(l₃-l₁)=1.5573…. This result improves the known inequality, J(l₃-l₁)≤4/³√10, which was given by Dhompongsa, Piraisangjun and Saejung.     

Equitable total coloring of <Emphasis Type="Italic">F</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> ⋁ <Emphasis Type="Italic">W</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

The minimum number of total independent partition sets of V ∪ E of graph G(V,E) is called the total chromatic number of G denoted by χ _t (G). If the difference of the numbers of any two total independent partition sets of V ∪ E is no more than one, then the minimum number of total independent partition sets of V ∪ E is called the equitable total chromatic number of G, denoted by χ _et (G). In this paper, we obtain the equitable total chromatic number of the join graph of fan and wheel with the same order. Supported by the National Natural Science Foundation of China (No. 10771091).     

On the Crossing Number of <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">m</Emphasis></Subscript> □ <Emphasis Type="Italic">P</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

Crossing numbers of graphs are in general very difficult to compute. There are several known exact results on the crossing number of the Cartesian products of paths, cycles or stars with small graphs. In this paper we study cr(K_m □ P_n), the crossing number of the Cartesian product K_m □ P_n. We prove that for m ≥ 3,n ≥ 1 and cr(K_m □ P_n)≥ (n − 1)cr(K_m+2 − e) + 2cr(K_m+1). For m≤ 5, according to Klešč, Jendrol and Ščerbová, the equality holds. In this paper, we also prove that the equality holds for m = 6, i.e., cr(K₆ □ P_n) = 15n + 3. Research supported by NFSC (60373096, 60573022).     

On Analytic Campanato and <Emphasis Type="Italic">F</Emphasis>(<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>, <Emphasis Type="Italic">s</Emphasis>) Spaces

We investigate the relation between analytic Campanato spaces \(\mathcal {AL}_{p,s}\) and the spaces F(p, q, s), characterize the bounded and compact Riemann–Stieltjes operators from \(\mathcal {AL}_{p,s}\) to \(F(p,p-s-1,s)\). We also describe the corona theorem and the interpolating sequences for the class \(F(p,p-2,s)\), which is the Möbius invariant subspace of the analytic Besov type spaces \(B_p(s)\).     

Verifying Huppert’s Conjecture for PSp<Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>) when <Emphasis Type="Italic">q</Emphasis> > 7

Let G denote a finite group and cd (G) the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd (G) = cd (H), then G ≅ H × A, where A is an abelian group. Huppert verified the conjecture for PSp₄(q) when q = 3, 4, 5, or 7. In this paper, we extend Huppert’s results and verify the conjecture for PSp₄(q) for all q. This demonstrates progress toward the goal of verifying the conjecture for all nonabelian simple groups of Lie type of rank two.     

The critical group of <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">m</Emphasis></Subscript> × <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

In this paper, the structure of the critical group of the graph K _m × C _n is determined, where m, n ≥ 3.     

Pointwise multipliers and decomposition theorems in <Emphasis Type="Italic">F</Emphasis><Superscript><Emphasis Type="Italic">∞,q</Emphasis></Superscript><Subscript><Emphasis Type="Italic">s</Emphasis></Subscript>

In this paper we give a characterization of the pointwise multipliers of the holomorphic Triebel-Lizorkin spaces F^{, q}_s, and we apply these results to prove a corona decomposition theorem for these spaces.Mathematics Subject Classification (1991):32A37Both authors partially supported by DGICYT Grant BMF2002-04072-C02-01 and CIRIT Grant 2001-SGR-00172     

Borel equivalence relations between <Emphasis Type="Italic">ℓ</Emphasis><Subscript>1</Subscript> and <Emphasis Type="Italic">ℓ</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

In this paper, we show that, for each p 〉 1, there are continuum many Borel equivalence relations between Rω/l1 and Rω/p ordered by ≤B which are pairwise Borel incomparable.     

Weak-Polynomial Convergence on Spaces ℓ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> and <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

This paper is concerned with the study of the set P ^-1(0), when P varies over all orthogonally additive polynomials on _p and L _p spaces. We apply our results to obtain characterizations of the weak-polynomial topologies associated to this class of polynomials.     

Projecting <Emphasis Type="Italic">l</Emphasis><Subscript>∞</Subscript> onto Classical Spaces

We describe an explicit construction of a linear projection of a symmetric conical section of the n-dimensional cube onto a (1+ε)-isomorphic version of the Euclidean ball of proportional dimension, or more generally onto a (1+ε)-isomorphic image of an l _p ^m -ball. Allowing non-linear projections (of logarithmic polynomial nonlinearity) we may even project the full n-dimensional cube onto the same images. This is done by gluing together explicit projections onto two-dimensional spaces, interpreting and modifying a paper of Ben-Tal and Nemirowski on polynomial reductions of conic quadratic programming problems to linear programming problems in terms of Banach spaces.      

<Emphasis Type="Italic">q</Emphasis>-Blossoming and Hermite-Padé approximants to the <Emphasis Type="Italic">q</Emphasis>-exponential function

We derive explicit expressions for the type II Hermite-Padé approximants to the q-exponential function using the q-blossom of its partial sums. In particular, this derivation leads to an interpretation of the Padé approximants as control points of a q-Bézier curve defined in terms of the partial sums of the q-exponential function.     

Convergence in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>([0, <Emphasis Type="Italic">T</Emphasis>]) of Wavelet Expansions of <Emphasis Type="Italic">φ</Emphasis>-Sub-Gaussian Random Processes

The article presents new results on convergence in L _p ([0,T]) of wavelet expansions of φ-sub-Gaussian random processes. The convergence rate of the expansions is obtained. Specifications of the obtained results are discussed.     

The crossing number of <Emphasis Type="Italic">G</Emphasis> ▭ <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> for the graph <Emphasis Type="Italic">G</Emphasis> on six vertices

The crossing numbers of Cartesian products of paths, cycles or stars with all graphs of order at most four are known. The crossing numbers of G□C _n for some graphs G on five and six vertices and the cycle C _n are also given. In this paper, we extend these results by determining the crossing number of the Cartesian product G □ C _n , where G is a specific graph on six vertices.     

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: