Finiteness of $$\pi _1$$-sensitive Hofer–Zehnder capacity and equivariant loop space homology

We prove that if M is a closed, connected, oriented, rationally inessential manifold, then the Hofer–Zehnder capacity of the unit disk bundle of the cotangent bundle of M is finite.     

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.     

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.     

An Itô formula for a family of stochastic integrals and related Wong–Zakai theorems

The aim of this paper is to generalize two important results known for the Stratonovich and Itô integrals to any stochastic integral obtained as limit of Riemann sums with arbitrary evaluating point: the ordinary chain rule for certain nonlinear functions of the Brownian motion and the Wong–Zakai approximation theorem. To this scope we begin by introducing a new family of products for smooth random variables which reduces for specific choices of a parameter to the pointwise and to the Wick products. We show that each product in that family is related in a natural way to a precise choice of the evaluating point in the above mentioned Riemann sums and hence to a certain notion of stochastic integral. Our chain rule relies on a new probabilistic representation for the solution of the heat equation while the Wong–Zakai type theorem follows from a reduction method for quasi-linear SDEs together with a formula of Gjessing’s type.     

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.     

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...     

The Iwasawa invariant $$\mu$$ μ vanishes for $$\mathbb{Z}_{2}$$ Z 2 -extensions of certain real biquadratic fields

Acta Mathematica Hungarica - For a real biquadratic field, we denote by $$\lambda$$ , $$\mu$$ and $$\nu$$ the Iwasawa invariants of cyclotomic $$\mathbb{Z}_{2}$$ -extension of $$k$$ . We give...     

Complete classification of finite semigroups for which the inverse monoid of local automorphisms is a $$\varDelta $$ Δ -semigroup

Semigroup Forum - A semigroup S is called a $$\varDelta $$ -semigroup if the lattice of its congruences forms a chain relative to the inclusion. A local automorphism of a semigroup S is defined as...     

A simpler proof of a Katsurada’s theorem and rapidly converging series for $$\zeta {(2n+1)}$$ ζ ( 2 n + 1 ) and $$\beta {(2n)}$$ β ( 2 n )

Annali di Matematica Pura ed Applicata (1923 -) - In a recent work on Euler-type formulae for even Dirichlet beta values, i.e., $$\beta {(2n)}$$ , I have derived an exact closed-form expression for...     

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...     

Estimates of the levy constant for\sqrt p and class number one criterion for ℚ(\sqrt p )

Let p ≡ = 3 (mod 4) be a prime, let ?( $\sqrt p $ ) be the length of the period of the expansion of $\sqrt p $ into a continued fraction, and let h(4p) be the class number of the field ?( $\sqrt p $ ). Our main result is as follows. For p > 91, h(4p)=1 if and only if ?( $\sqrt p $ ) > 0.56 $\sqrt p $ L_4p(1), where L_4p(1) is the corresponding Dirichlet series. The proof is based on studying linear relations between convergents of the expansion of $\sqrt p $ into a continued fraction. Bibliography: 13 titles.     

A proof of the Anderson–Badawi $$ \mathbf{rad}\varvec{(I)}^{\varvec{n}} \varvec{\subseteq } {\varvec{I}}$$ formula for $$\varvec{n}$$n-absorbing ideals

In [1], Anderson and Badawi conjectured that \(\mathrm{rad}(I)^n \subseteq I\) for every n-absorbing ideal I of a commutative ring. In this article, we prove their conjecture. We also prove related conjectures for radical ideals.     

Relations among complementary and supplementary pairings of Saalschützian $${}_4F_3(1)$$ series

We investigate sums \(J(\vec {x})\) and \(L(\vec {x})\) of pairs of normalized Saalschützian \({}_4F_3(1)\) hypergeometric series, and develop a theory of relations among these J and L functions. The function \(L(\vec {x})\) has been studied extensively in the literature and has been shown to satisfy a number of two-term and three-term relations with respect to the variable \(\vec {x}\). More recent works have framed these relations in terms of Coxeter group actions on \(\vec {x}\) and have developed a similar theory of two-term and three-term relations for \(J(\vec {x})\). In this article, we derive “mixed” three-term relations, wherein any one of the L (respectively, J) functions arising in the above context may be expressed as a linear combination of two of the above J (respectively, L) functions. We show that, under the appropriate Coxeter group action, the resulting set of three-term relations (mixed and otherwise) among J and L functions partitions into eighteen orbits. We provide an explicit example of a relation from each orbit. We further classify the eighteen orbits into five types, with each type uniquely determined by the distances (under a certain natural metric) between the J and L functions in the relation. We show that the type of a relation dictates the complexity (in terms of both number of summands and number of factors in each summand) of the coefficients of the J and L functions therein.     

Congruences for the coefficients of the Gordon and McIntosh mock theta function $$\xi (q)$$ ξ ( q )

The Ramanujan Journal - Recently Gordon and McIntosh introduced the third order mock theta function $$\xi (q)$$ defined by $$\begin{aligned} \xi (q)=1+2\sum _{n=1}^{\infty...     

On a Poincaré type formula for solutions of singular and degenerate elliptic equations

We provide a geometric Poincaré type formula for stable solutions of ?Δ _p (u) = f(u). From this, we derive a symmetry result in the plane. This work is a refinement of previous results obtained by the authors under further integrability and regularity assumptions.     

On the Riemann–Hilbert problem of the matrix Lakshmanan–Porsezian–Daniel system with a $$4\times4$$ AKNS-type matrix Lax pair

Theoretical and Mathematical Physics - The initial-boundary value problems for the matrix Lakshmanan–Porsezian–Daniel system are studied by utilizing the Fokas unified transform...     

Itô’s formula for the L p -norm of stochastic {W^{1}_{p}} -valued processes

We prove Itô’s formula for the L _p -norm of a stochastic ${W^{1}_{p}}$ -valued processes appearing in the theory of SPDEs in divergence form.     

A Gelfand–Tsetlin-type basis for the algebra $$mathfrak{sp}_4$$ and hypergeometric functions

Theoretical and Mathematical Physics - We consider a realization of a representation of the $$ mathfrak{sp} _4$$ Lie algebra in the space of functions on a Lie group $$Sp_4$$ . We find a function...     

The Aronszajn–Donoghue Theory for Rank One Perturbations of the $$\mathcal{H}_{-2} {\text{-Class}}$$

A singular rank one perturbation of a self-adjoint operator A in a Hilbert space is considered, where and but with the usual A–scale of Hilbert spaces. A modified version of the Aronszajn-Krein formula is given. It has the form where F denotes the regularized Borel transform of the scalar spectral measure of A associated with . Using this formula we develop a variant of the well known Aronszajn–Donoghue spectral theory for a general rank one perturbation of the class.Submitted: March 14, 2002 Revised: December 15, 2002     

Weak $$A_\infty $$ Weights and Weak Reverse Hölder Property in a Space of Homogeneous Type

In the Euclidean setting, the Fujii–Wilson-type \(A_\infty \) weights satisfy a reverse Hölder inequality (RHI), but in spaces of homogeneous type the best-known result has been that \(A_\infty \) weights satisfy only a weak reverse Hölder inequality. In this paper, we complement the results of Hytönen, Pérez and Rela and show that there exist both \(A_\infty \) weights that do not satisfy an RHI and a genuinely weaker weight class that still satisfies a weak RHI. We also show that all the weights that satisfy a weak RHI have a self-improving property, but the self-improving property of the strong reverse Hölder weights fails in a general space of homogeneous type. We prove most of these purely non-dyadic results using convenient dyadic systems and techniques.