A primal-dual homotopy algorithm for $$\ell _{1}$$-minimization with $$\ell _{\infty }$$ℓ∞-constraints

In this paper we propose a primal-dual homotopy method for \(\ell _1\)-minimization problems with infinity norm constraints in the context of sparse reconstruction. The natural homotopy parameter is the value of the bound for the constraints and we show that there exists a piecewise linear solution path with finitely many break points for the primal problem and a respective piecewise constant path for the dual problem. We show that by solving a small linear program, one can jump to the next primal break point and then, solving another small linear program, a new optimal dual solution is calculated which enables the next such jump in the subsequent iteration. Using a theorem of the alternative, we show that the method never gets stuck and indeed calculates the whole path in a finite number of steps. Numerical experiments demonstrate the effectiveness of our algorithm. In many cases, our method significantly outperforms commercial LP solvers; this is possible since our approach employs a sequence of considerably simpler auxiliary linear programs that can be solved efficiently with specialized active-set strategies.     

Dedekind $$\sigma $$ σ -complete $$\ell $$ ℓ -groups and Riesz spaces as varieties

Positivity - We prove that the category of Dedekind $$\sigma $$ -complete Riesz spaces is an infinitary variety, and we provide an explicit equational axiomatization. In fact, we show that finitely...     

Congruences for $$\ell $$-regular overpartitions and Andrews’ singular overpartitions

Let \(\overline{A}_{\ell }(n)\) be the number of overpartitions of n into parts not divisible by \(\ell \). In a recent paper, Shen calls the overpartitions enumerated by the function \(\overline{A}_{\ell }(n)\) as \(\ell \)-regular overpartitions. In this paper, we find certain congruences for \(\overline{A}_{\ell }(n)\), when \(\ell =4, 8\), and 9. Recently, Andrews introduced the partition function \(\overline{C}_{k, i}(n)\), called singular overpartition, which counts the number of overpartitions of n in which no part is divisible by k and only parts \(\equiv \pm i\pmod {k}\) may be over-lined. He also proved that \(\overline{C}_{3, 1}(9n+3)\) and \(\overline{C}_{3, 1}(9n+6)\) are divisible by 3. In this paper, we prove that \(\overline{C}_{3, 1}(12n+11)\) is divisible by 144 which was conjectured to be true by Naika and Gireesh.     

Quantitative weakly compact sets and Banach-Saks sets in $$\ell _1$$ ℓ 1

Archiv der Mathematik - In this paper, we show a quantitative version of the theorem stating that relatively weakly compact sets in $$\ell _1$$ coincide with those having the Banach-Saks property....     

Rigidity theorems for complete $$\lambda $$ λ -hypersurfaces

Archiv der Mathematik - In this article, we study hypersurfaces $$\Sigma \subset {\mathbb {R}}^{n+1}$$ with constant weighted mean curvature, also known as $$\lambda $$ -hypersurfaces. Recently,...     

A $$\tau $$ τ -Conjecture for Newton Polygons

Foundations of Computational Mathematics - One can associate to any bivariate polynomial $$P(X,Y)$$ its Newton polygon. This is the convex hull of the points $$(i,j)$$ such that the monomial $$X^i...     

Extending Euler’s product for $$\sqrt $$

In this note we present some sundry infinite products for \(\sqrt 2 \) and six general formulas for some rational powers of 2.     

Bases of Feigin–Stoyanovsky’s type subspaces for $$C_\ell ^{(1)}$$

In this paper, we construct combinatorial bases of Feigin–Stoyanovsky’s type subspaces of standard modules for level k affine Lie algebra \(C_\ell ^{(1)}\). We prove spanning by using annihilating field \(x_\theta (z)^{k+1}\) of standard modules. In the proof of linear independence, we use simple currents and intertwining operators whose existence is given by fusion rules.     

Optimal portfolio selections via $$\ell _{1, 2}$$ ℓ 1 , 2 -norm regularization

Computational Optimization and Applications - There has been much research about regularizing optimal portfolio selections through $$\ell _1$$ norm and/or $$\ell _2$$ -norm squared. The common...     

On $$\gamma $$ γ - and local $$\gamma $$ γ -vectors of the interval subdivision

Journal of Algebraic Combinatorics - We show that the $$\gamma $$ -vector of the interval subdivision of a simplicial complex with a nonnegative and symmetric h-vector is nonnegative. In...     

Certain Somos’s $$\varvec{}$$–$$\varvec{Q}$$Q type Dedekind $$\varvec{\eta }$$η-function identities

In this paper, we provide a new proof for the Dedekind \(\eta \)-function identities discovered by Somos. During this process, we found two new Dedekind \(\eta \)-function identities. Furthermore, we extract interesting partition identities from some of the \(\eta \)-function identities.     

Fourier Spectrum Characterizations of $$H^{p}$$ Spaces on Tubes Over Cones for $$1\le p \le \infty $$1≤p≤∞

We give Fourier spectrum characterizations of functions in the Hardy \(H^p\) spaces on tubes for \(1\le p \le \infty .\) For \(F\in L^p(\mathbb {R}^n), \) we show that F is the non-tangential boundary limit of a function in a Hardy space, \(H^{p}(T_\Gamma ),\) where \(\Gamma \) is an open cone of \(\mathbb {R}^n\) and \(T_\Gamma \) is the related tube in \(\mathbb {C}^n,\) if and only if the classical or the distributional Fourier transform of F is supported in \(\Gamma ^*,\) where \(\Gamma ^*\) is the dual cone of \(\Gamma .\) This generalizes the results of Stein and Weiss for \(p=2\) in the same context, as well as those of Qian et al. in one complex variable for \(1\le p\le \infty .\) Furthermore, we extend the Poisson and Cauchy integral representation formulas to the \(H^p\) spaces on tubes for \(p\in [1, \infty ]\) and \(p\in [1,\infty ),\) with, respectively, the two types of representations.     

On the $$\mathbb {}$$-Riemann integral and Hermite–Hadamard inequalities for $$\mathbb {}$$-convex functions

In the present paper we introduce a notion of the \(\mathbb {K}\)-Riemann integral as a natural generalization of a usual Riemann integral and study its properties. The aim of this paper is to extend the classical Hermite–Hadamard inequalities to the case when the usual Riemann integral is replaced by the \(\mathbb {K}\)-Riemann integral and the convexity notion is replaced by \(\mathbb {K}\)-convexity.     

On difference sets with small $$\lambda $$ λ

Journal of Algebraic Combinatorics - In a 1989 paper, Arasu (Arch Math 53:622–624, 1989) used an observation about multipliers to show that no (352, 27, 2) difference set...     

Hidden q-analogues of Ramanujan-like $$\pi $$ π -series

The Ramanujan Journal - By examining known q-series identities, we derive q-analogues of several $$\pi $$ -related infinite series, including some of Ramanujan’s series for $$1/\pi $$ .     

The Dedekind $$\eta $$-function,a Hauptmodul for $$\Gamma _0(13)$$Γ0(13), and invariant theory

We give an explicit formula for the Hauptmodul \(\left( \frac{\eta (\tau )}{\eta (13 \tau )}\right) ^2\) of the level-13 Hecke modular group \(\Gamma _0(13)\) as a quotient of theta constants, together with some related explicit formulas. Similar results for primes \(p=2, 3, 5, 7\) (the other p for which \(\Gamma _0(p)\) has genus zero) are well known, and date back to Klein and Ramanujan. Moreover, we find an exotic modular equation, i.e., it has the same form as Ramanujan’s modular equation of degree 13, but with different kinds of modular parameterizations.     

ON HEYDE’S THEOREM ON THE GROUP $$\mathbb{R}$$ × $$\mathbb{T}$$

Doklady Mathematics - According to the well-knows Heyde theorem the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of...     

General fractional differential equations of order $$\alpha \in (1,2)$$ and Type $$\beta \in [0,1]$$β∈[0,1] in Banach spaces

In this paper, we are concerned with general fractional differential equations of order \(\alpha \in (1,2)\) and type \(\beta \in [0,1]\) in Banach spaces. We define and develop a theory of general fractional sine functions and show that they are essentiality equivalent to a general fractional resolvent. We use such theory to study the well-posedness of the above general fractional differential equations. An illustration example is presented.     

$$I_0$$ and combinatorics at $$\lambda ^+$$λ+

We investigate the compatibility of \(I_0\) with various combinatorial principles at \(\lambda ^+\), which include the existence of \(\lambda ^+\)-Aronszajn trees, square principles at \(\lambda \), the existence of good scales at \(\lambda \), stationary reflections for subsets of \(\lambda ^{+}\), diamond principles at \(\lambda \) and the singular cardinal hypothesis at \(\lambda \). We also discuss whether these principles can hold in \(L(V_{\lambda +1})\).     

Asymptotic behavior of ground state radial solutions for problems involving the $$\Phi $$ Φ -Laplacian

Positivity - We are concerned with the existence of positive solutions to the following boundary value problem in $$(0,\infty ),$$ $$\begin{aligned} \frac{1}{A}\left( A\phi \left( \left| u^{\prime...