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...     

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...     

Some identities associated with mock theta functions $$\omega (q)$$ ω ( q ) and $$\nu (q)$$ ν ( q )

The Ramanujan Journal - Recently, Andrews, Dixit, and Yee defined two partition functions $$p_{\omega }(n)$$ and $$p_{\nu }(n)$$ that are related with Ramanujan’s mock theta functions...     

Congruences related to the Ramanujan/Watson mock theta functions $$\omega (q)$$ and $$\nu (q)$$ν(q)

We establish new connection formulae between Fibonacci polynomials and Chebyshev polynomials of the first and second kinds. These formulae are expressed in terms of certain values of hypergeometric functions of the type \(_2F_{1}\). Consequently, we obtain some new expressions for the celebrated Fibonacci numbers and their derivative sequences. Moreover, we evaluate some definite integrals involving products of Fibonacci and Chebyshev polynomials.     

On the Study of Processes of $$\sum (H)$$ and $$\sum _\mathrm{s}(H)$$∑s(H) Classes

In papers by Yor, a remarkable class \(({\varSigma })\) of submartingales is introduced, which, up to technicalities, are submartingales \((X_{t})_{t\ge 0}\) whose increasing process is carried by the times t such that \(X_{t}=0\). These submartingales have several applications in stochastic analysis: for example, the resolution of Skorokhod embedding problem, the study of Brownian local times and the study of zeros of continuous martingales. The submartingales of class \(({\varSigma })\) have been extensively studied in a series of articles by Nikeghbali (part of them in collaboration with Najnudel, some others with Cheridito and Platen). On the other hand, stochastic calculus has been extended to signed measures by Ruiz de Chavez (Le théorème de Paul Lévy pour des mesures signées. Séminaire de probabilités (Strasbourg). Springer, Berlin, 1984) and Beghdadi-Sakrani (Calcul stochastique pour les mesures signées. Séminaire de probabilités (Strasbourg). Springer, Berlin, 2003). In Eyi Obiang et al. (Stochastics 86(1):70–86, 2014), the authors of the present paper have extended the notion of submartingales of class \(({\varSigma })\) to the setting of Ruiz de Chavez (1984) and Beghdadi-Sakrani (2003), giving two different classes of stochastic processes named classes \(\sum (H)\) and \(\sum _\mathrm{s}(H)\) where from tools of the theory of stochastic calculus for signed measures, the authors provide general frameworks and methods for dealing with processes of these classes. In this work, we first give some formulas of multiplicative decomposition for processes of these classes. Afterward, we shall establish some representation results allowing to recover any process of one of these classes from its final value and the last time it visited the origin.     

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.     

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.     

New congruences for $$\ell $$-regular partitions for $$\ell \in \{5,6,7,49\}$$∈{5,6,7,49}

We find several new congruences for \(\ell \)-regular partitions for \(\ell \in \{5,6,7,49\}\) and also find alternative proofs of the congruences for 10- and 20-regular partitions which were proved earlier by Carlson and Webb (Ramanujan J 33:329–337, 2014) by using the theory of modular forms. We use certain p-dissections of \((q;q)_{\infty }\), \(\psi (q)\), \((q;q)_{\infty }^3\) and \(\psi (q^2)(q;q)_{\infty }^2\).     

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.     

Series representations for the Apery constant $$\zeta (3)$$ ζ ( 3 ) involving the values $$\zeta (2n)$$ ζ ( 2 n )

The Ramanujan Journal - In this note, using the well-known series representation for the Clausen function, we also provide some new representations of Apery’s constant $$\zeta (3)$$ . In...     

On hybrid moments of $$\Delta _2(x)$$ Δ 2 ( x ) and $$\Delta _3(x)$$ Δ 3 ( x )

The Ramanujan Journal - Let $$k\geqslant 2$$ be a fixed natural number and $$d_k(n)$$ denote the number of ways n can be written as a product of k positive integers. We use $$\Delta _k(x)$$ to...     

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....     

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.     

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,...     

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.     

Combinatorial proofs for identities related to generalizations of the mock theta functions $$\omega (q)$$ω(q) and $$\nu (q)$$ν(q)

The Ramanujan Journal - The two partition functions $$p_\omega (n)$$ and $$p_\nu (n)$$ were introduced by Andrews, Dixit and Yee, which are related to the third-order mock theta functions $$\omega...     

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...     

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.     

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 $$1/\pi $$ .