Towards a Lanczos’ $$\tau $$-Method Toolkit for Differential Problems

The aim of this work is to build a numerical software library based on the \(\tau \)-method to solve differential problems using MATLAB. The \(\tau \)-method can be very effective in the solution of certain type of these problems, and therefore, the existence of a numerical library for its dissemination is of major importance. Furthermore, the method has been used for the solution of particular problems but has not yet been explored as a general technique. Focus will be on stability issues, namely those issued from the solution of algebraic linear systems required for the process. Additionally, preconditioners for the solution with the \(\tau \)-method will be tackled, with emphasizes on incomplete LU factorizations and (block) Jacobi preconditioners. We also propose an iterative approach, build upon an LU factorization over a moderate initial size, generating better approximations and providing a priori error estimate at each iteration. Numerical results enlightening the efficiency of the proposed methods will be presented.     

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 \(\psi (q^2)(q;q)_{\infty }^2\).     

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,...     

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.     

On the snake-in-the-box codes for rank modulation under Kendall’s $$\tau $$-metric

For exactly and efficiently representing and storing data in flash memories, the rank modulation scheme has been presented. In this scheme, Gray codes over the permutations are important, which are used to represent information in flash memories. For a Gray code, two consecutive codewords are obtained using one “push-to-the-top” operation. Specially, a snake-in-the-box code under the Kendall’s \(\tau \)-metric is a Gray code, which is capable of detecting one Kendall’s \(\tau \)-error. In this paper, we consider only the Kendall’s \(\tau \)-metric on the permutations. And we answer one open problem proposed by Horovitz and Etzion. That is, we prove that the length of the longest snake in \(S_{2n+2}\) is longer than the length of the longest snake in \(S_{2n+1}\).     

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.     

n-APR tilting and $$tau $$ τ -mutations

Journal of Algebraic Combinatorics - APR tilts for path algebra kQ can be realized as the mutation of the quiver Q in $${mathbb Z}Q$$ with respect to the translation. In this paper, we show that...     

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...     

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...     

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.     

A control theorem for p-adic automorphic forms and Teitelbaum’s $$\mathcal {L}$$ L -invariant

The Ramanujan Journal - We extend the results of Kaneko–Zagier and Baba–Granath on relations of supersingular polynomials and solutions of certain second-order modular differential...     

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.     

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.     

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 $$1/\pi $$ .     

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...     

New bounds of permutation codes under Hamming metric and Kendall’s $$\tau $$-metric

Permutation codes are widely studied objects due to their numerous applications in various areas, such as power line communications, block ciphers, and the rank modulation scheme for flash memories. Several kinds of metrics are considered for permutation codes according to their specific applications. This paper concerns some improvements on the bounds of permutation codes under Hamming metric and Kendall’s \(\tau \)-metric respectively, using mainly a graph coloring approach. Specifically, under Hamming metric, we improve the Gilbert–Varshamov bound asymptotically by a factor n, when the minimum Hamming distance d is fixed and the code length n goes to infinity. Under Kendall’s \(\tau \)-metric, we narrow the gap between the known lower bounds and upper bounds. Besides, we also obtain some sporadic results under Kendall’s \(\tau \)-metric for small parameters.     

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.     

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 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...     

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...