Two-step estimation of ergodic Lévy driven SDE

We consider high frequency samples from ergodic Lévy driven stochastic differential equation with drift coefficient \(a(x,\alpha )\) and scale coefficient \(c(x,\gamma )\) involving unknown parameters \(\alpha \) and \(\gamma \). We suppose that the Lévy measure \(\nu _{0}\), has all order moments but is not fully specified. We will prove the joint asymptotic normality of some estimators of \(\alpha \), \(\gamma \) and a class of functional parameter \(\int \varphi (z)\nu _0(dz)\), which are constructed in a two-step manner: first, we use the Gaussian quasi-likelihood for estimation of \((\alpha ,\gamma )\); and then, for estimating \(\int \varphi (z)\nu _0(dz)\) we make use of the method of moments based on the Euler-type residual with the the previously obtained quasi-likelihood estimator.     

Functional equations for the Stieltjes constants

The Stieltjes constants \(\gamma _k(a)\) appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function \(\zeta (s,a)\) about \(s=1\). We present the evaluation of \(\gamma _1(a)\) and \(\gamma _2(a)\) at rational arguments, this being of interest to theoretical and computational analytic number theory and elsewhere. We give multiplication formulas for \(\gamma _0(a)\), \(\gamma _1(a)\), and \(\gamma _2(a)\), and point out that these formulas are cases of an addition formula previously presented. We present certain integral evaluations generalizing Gauss’ formula for the digamma function at rational argument. In addition, we give the asymptotic form of \(\gamma _k(a)\) as \(a \rightarrow 0\) as well as a novel technique for evaluating integrals with integrands with \(\ln (-\ln x)\) and rational factors.     

Coalescece ad Meetig Times o $$$$-Block Markov Chais

We consider finite-state, discrete-time, mixing Markov chains \((V,P)\), where \(V\) is the state space and \(P\) is the transition matrix. To each such chain \((V,P)\), we associate a sequence of chains \((V_n,P_n)\) by coding trajectories of \((V,P)\) according to their overlapping \(n\)-blocks. The chain \((V_n,P_n)\), called the \(n\)-block Markov chain associated with \((V,P)\), may be considered an alternate version of \((V,P)\) having memory of length \(n\). Along such a sequence of chains, we characterize the asymptotic behavior of coalescence times and meeting times as \(n\) tends to infinity. In particular, we define an algebraic quantity \(L(V,P)\) depending only on \((V,P)\), and we show that if the coalescence time on \((V_n,P_n)\) is denoted by \(C_n\), then the quantity \(\frac{1}{n} \log C_n\) converges in probability to \(L(V,P)\) with exponential rate. Furthermore, we fully characterize the relationship between \(L(V,P)\) and the entropy of \((V,P)\).     

On some results for a class of meromorphic functions having quasiconformal extension

We consider the class \(\Sigma (p)\) of univalent meromorphic functions f on \({\mathbb D}\) having a simple pole at \(z=p\in [0,1)\) with residue 1. Let \(\Sigma _k(p)\) be the class of functions in \(\Sigma (p)\) which have k-quasiconformal extension to the extended complex plane \({\hat{\mathbb C}}\), where \(0\le k < 1\). We first give a representation formula for functions in this class and using this formula, we derive an asymptotic estimate of the Laurent coefficients for the functions in the class \(\Sigma _k(p)\). Thereafter, we give a sufficient condition for functions in \(\Sigma (p)\) to belong to the class \(\Sigma _k(p).\) Finally, we obtain a sharp distortion result for functions in \(\Sigma (p)\) and as a consequence, we obtain a distortion estimate for functions in \(\Sigma _k(p).\)     

Solvability of Minimal Graph Equation Under Pointwise Pinching Condition for Sectional Curvatures

We study the asymptotic Dirichlet problem for the minimal graph equation on a Cartan–Hadamard manifold M whose radial sectional curvatures outside a compact set satisfy an upper bound

$$\begin{aligned} K(P)\le - \frac{\phi (\phi -1)}{r(x)^2} \end{aligned}$$

and a pointwise pinching condition

$$\begin{aligned} |K(P) |\le C_K|K(P') | \end{aligned}$$

for some constants \(\phi >1\) and \(C_K\ge 1\), where P and \(P'\) are any 2-dimensional subspaces of \(T_xM\) containing the (radial) vector \(\nabla r(x)\) and \(r(x)=d(o,x)\) is the distance to a fixed point \(o\in M\). We solve the asymptotic Dirichlet problem with any continuous boundary data for dimensions \(n=\dim M>4/\phi +1\).

     

Sampling and Cubature on Sparse Grids Based on a B-spline Quasi-Interpolation

Let \(X_n = \{x^j\}_{j=1}^n\) be a set of n points in the d-cube \({\mathbb {I}}^d:=[0,1]^d\), and \(\Phi _n = \{\varphi _j\}_{j =1}^n\) a family of n functions on \({\mathbb {I}}^d\). We consider the approximate recovery of functions f on \({{\mathbb {I}}}^d\) from the sampled values \(f(x^1), \ldots , f(x^n)\), by the linear sampling algorithm \( L_n(X_n,\Phi _n,f) := \sum _{j=1}^n f(x^j)\varphi _j. \) The error of sampling recovery is measured in the norm of the space \(L_q({\mathbb {I}}^d)\)-norm or the energy quasi-norm of the isotropic Sobolev space \(W^\gamma _q({\mathbb {I}}^d)\) for \(1 < q < \infty \) and \(\gamma > 0\). Functions f to be recovered are from the unit ball in Besov-type spaces of an anisotropic smoothness, in particular, spaces \(B^{\alpha ,\beta }_{p,\theta }\) of a “hybrid” of mixed smoothness \(\alpha > 0\) and isotropic smoothness \(\beta \in {\mathbb {R}}\), and spaces \(B^a_{p,\theta }\) of a nonuniform mixed smoothness \(a \in {\mathbb {R}}^d_+\). We constructed asymptotically optimal linear sampling algorithms \(L_n(X_n^*,\Phi _n^*,\cdot )\) on special sparse grids \(X_n^*\) and a family \(\Phi _n^*\) of linear combinations of integer or half integer translated dilations of tensor products of B-splines. We computed the asymptotic order of the error of the optimal recovery. This construction is based on B-spline quasi-interpolation representations of functions in \(B^{\alpha ,\beta }_{p,\theta }\) and \(B^a_{p,\theta }\). As consequences, we obtained the asymptotic order of optimal cubature formulas for numerical integration of functions from the unit ball of these Besov-type spaces.     

Geometric compactification of moduli spaces of half-translation structures on surfaces

In this paper, we give an equivariant compactification of the space \({\mathbb {P}}{\text {Flat}}(\Sigma )\) of homothety classes of half-translation structures on a compact, connected, orientable surface \(\Sigma \). We introduce the space \({\mathbb {P}}{\text {Mix}}(\Sigma )\) of homothety classes of mixed structures on \(\Sigma \), that are \({\text {CAT}}(0)\) tree-graded spaces in the sense of Drutu and Sapir, with pieces which are \({\mathbb {R}}\)-trees and completions of surfaces endowed with half-translation structures. Endowing \({\text {Mix}}(\Sigma )\) with the equivariant Gromov topology, and using asymptotic cone techniques, we prove that \({\mathbb {P}}{\text {Mix}}(\Sigma )\) is an equivariant compactification of \({\mathbb {P}}{\text {Flat}}(\Sigma )\), thus allowing us to understand in a geometric way the degenerations of half-translation structures on \(\Sigma \). We finally compare our compactification to the one of Duchin–Leininger–Rafi, based on geodesic currents on \(\Sigma \), by the mean of the translation distances of the elements of the covering group of \(\Sigma \).     

The crank moments weighted by the parity of cranks

In this note, we introduce the 2kth crank moment \(\mu _{2k}(-1,n)\) weighted by the parity of cranks and show that \((-1)^n \mu _{2k}(-1,n)>0\) for \(n\ge k \ge 0\). When \(k=0\), the inequality \((-1)^n \mu _{2k}(-1,n)>0\) reduces to Andrews and Lewis’s inequality \((-1)^n(M_e(n)-M_o(n))>0\) for \(n\ge 0\), where \(M_e(n)\) (resp. \(M_o(n)\)) denotes the number of partitions of n with even (resp. odd) crank. Several generating functions of \(\mu _{2k}(-1,n)\) are also studied in order to show the positivity of \((-1)^n\mu _{2k}(-1,n)\).     

On spt-crank-type functions

In a recent paper, Andrews, Dixit, and Yee introduced a new spt-type function \({\mathrm {spt}}_{\omega }(n)\), which is closely related to Ramanujan’s third-order mock theta function \(\omega (q)\). Garvan and Jennings-Shaffer introduced a crank function which explains congruences for \({\mathrm {spt}}_{\omega }(n)\). In this article, we study the asymptotic behavior of this crank function and confirm a positivity conjecture of the crank asymptotically. We also study a sign pattern of the crank and congruences for \({\mathrm {spt}}_{\omega }(n)\).     

Plethysm and Lattice Point Counting

We apply lattice point counting methods to compute the multiplicities in the plethysm of \(\textit{GL}(n)\). Our approach gives insight into the asymptotic growth of the plethysm and makes the problem amenable to computer algebra. We prove an old conjecture of Howe on the leading term of plethysm. For any partition \(\mu \) of 3, 4, or 5, we obtain an explicit formula in \(\lambda \) and k for the multiplicity of \(S^\lambda \) in \(S^\mu (S^k)\).     

On generalized Ramanujan primes

Let \(\Delta = \sum _{m=0}^\infty q^{(2m+1)^2} \in \mathbf {F}_2[[q]]\) be the reduction mod 2 of the \(\Delta \) series. A modular form of level 1, \(f=\sum _{n\geqslant 0} c(n) \,q^n\), with integer coefficients, is congruent modulo \(2\) to a polynomial in \(\Delta \). Let us set \(W_f(x)=\sum _{n\leqslant x,\ c(n)\text { odd }} 1\), the number of odd Fourier coefficients of \(f\) of index \(\leqslant x\). The order of magnitude of \(W_f(x)\) (for \(x\rightarrow \infty \)) has been determined by Serre in the seventies. Here, we give an asymptotic equivalent for \(W_f(x)\). Let \(p(n)\) be the partition function and \(A_0(x)\) (resp. \(A_1(x)\)) be the number of \(n\leqslant x\) such that \(p(n)\) is even (resp. odd). In the preceding papers, the second-named author has shown that \(A_0(x)\geqslant 0.28 \sqrt{x\;\log \log x}\) for \(x\geqslant 3\) and \(A_1(x)>\frac{4.57 \sqrt{x}}{\log x}\) for \(x\geqslant 7\). Here, it is proved that \(A_0(x)\geqslant 0.069 \sqrt{x}\;\log \log x\) holds for \(x>1\) and that \(A_1(x) \geqslant \frac{0.037 \sqrt{x}}{(\log x)^{7/8}}\) holds for \(x\geqslant 2\). The main tools used to prove these results are the determination of the order of nilpotence of a modular form of level-\(1\) modulo \(2\), and of the structure of the space of those modular forms as a module over the Hecke algebra, which have been given in a recent work of Serre and the second-named author.     

On $$L^p$$-Boundedness of Pseudo-Differential Operators of Sjöstrand’s Class

We extended the known result that symbols from modulation spaces \(M^{\infty ,1}(\mathbb {R}^{2n})\), also known as the Sjöstrand’s class, produce bounded operators in \(L^2(\mathbb {R}^n)\), to general \(L^p\) boundedness at the cost of loss of derivatives. Indeed, we showed that pseudo-differential operators acting from \(L^p\)-Sobolev spaces \(L^p_s(\mathbb {R}^n)\) to \(L^p(\mathbb {R}^n)\) spaces with symbols from the modulation space \(M^{\infty ,1}(\mathbb {R}^{2n})\) are bounded, whenever \(s\ge n|1/p-1/2|.\) This estimate is sharp for all \(1< p<\infty \).     

On the elliptic curve endomorphism generator

For an elliptic curve \({E}\) over a finite field we define the point sequence \((P_n)\) recursively by \(P_n=\vartheta (P_{n-1})=\vartheta ^n(P_0)\) with an endomorphism \(\vartheta \in {{\mathrm{End}}}({E})\) and with some initial point \(P_0\) on \({E}\). We study the distribution and the linear complexity of sequences obtained from \((P_n)\).     

Zeta functions and asymptotic additive bases with some unusual sets of primes

Fix \(\delta \in (0,1]\), \(\sigma _0\in [0,1)\) and a real-valued function \(\varepsilon (x)\) for which \(\varlimsup _{x\rightarrow \infty }\varepsilon (x)\leqslant 0\). For every set of primes \(\mathcal {P}\) whose counting function \(\pi _\mathcal {P}(x)\) satisfies an estimate of the form

$$\begin{aligned} \pi _\mathcal {P}(x)=\delta \,\pi (x)+O\bigl (x^{\sigma _0+\varepsilon (x)}\bigr ), \end{aligned}$$

we define a zeta function \(\zeta _\mathcal {P}(s)\) that is closely related to the Riemann zeta function \(\zeta (s)\). For \(\sigma _0\leqslant \frac{1}{2}\), we show that the Riemann hypothesis is equivalent to the non-vanishing of \(\zeta _\mathcal {P}(s)\) in the region \(\{\sigma >\frac{1}{2}\}\).

For every set of primes \(\mathcal {P}\) that contains the prime 2 and whose counting function satisfies an estimate of the form

$$\begin{aligned} \pi _\mathcal {P}(x)=\delta \,\pi (x)+O\bigl ((\log \log x)^{\varepsilon (x)}\bigr ), \end{aligned}$$

we show that \(\mathcal {P}\) is an exact asymptotic additive basis for \(\mathbb {N}\), i.e. for some integer \(h=h(\mathcal {P})>0\) the sumset \(h\mathcal {P}\) contains all but finitely many natural numbers. For example, an exact asymptotic additive basis for \(\mathbb {N}\) is provided by the set

$$\begin{aligned} \{2,547,1229,1993,2749,3581,4421,5281\ldots \}, \end{aligned}$$

which consists of 2 and every hundredth prime thereafter.

     

Some results on generalized strong external difference families

A generalized strong external difference family (briefly \((v, m; k_1,\dots ,k_m; \lambda _1,\dots ,\lambda _m)\)-GSEDF) was introduced by Paterson and Stinson in 2016. In this paper, we give some nonexistence results for GSEDFs. In particular, we prove that a \((v, 3;k_1,k_2,k_3; \lambda _1,\lambda _2,\lambda _3)\)-GSEDF does not exist when \(k_1+k_2+k_3< v\). We also give a first recursive construction for GSEDFs and prove that if there is a \((v,2;2\lambda ,\frac{v-1}{2};\lambda ,\lambda )\)-GSEDF, then there is a \((vt,2;4\lambda ,\frac{vt-1}{2};2\lambda ,2\lambda )\)-GSEDF with \(v>1\), \(t>1\) and \(v\equiv t\equiv 1\pmod 2\). Then we use it to obtain some new GSEDFs for \(m=2\). In particular, for any prime power q with \(q\equiv 1\pmod 4\), we show that there exists a \((qt, 2;(q-1)2^{n-1},\frac{qt-1}{2};(q-1)2^{n-2},(q-1)2^{n-2})\)-GSEDF, where \(t=p_1p_2\dots p_n\), \(p_i>1\), \(1\le i\le n\), \(p_1, p_2,\dots ,p_n\) are odd integers.     

Chains,antichains, and complements in infinite partition lattices

We consider the partition lattice \(\Pi (\lambda )\) on any set of transfinite cardinality \(\lambda \) and properties of \(\Pi (\lambda )\) whose analogues do not hold for finite cardinalities. Assuming AC, we prove: (I) the cardinality of any maximal well-ordered chain is always exactly \(\lambda \); (II) there are maximal chains in \(\Pi (\lambda )\) of cardinality \(> \lambda \); (III) a regular cardinal \(\lambda \) is strongly inaccessible if and only if every maximal chain in \(\Pi (\lambda )\) has size at least \(\lambda \); if \(\lambda \) is a singular cardinal and \(\mu ^{< \kappa } < \lambda \le \mu ^\kappa \) for some cardinals \(\kappa \) and (possibly finite) \(\mu \), then there is a maximal chain of size \(< \lambda \) in \(\Pi (\lambda )\); (IV) every non-trivial maximal antichain in \(\Pi (\lambda )\) has cardinality between \(\lambda \) and \(2^{\lambda }\), and these bounds are realised. Moreover, there are maximal antichains of cardinality \(\max (\lambda , 2^{\kappa })\) for any \(\kappa \le \lambda \); (V) all cardinals of the form \(\lambda ^\kappa \) with \(0 \le \kappa \le \lambda \) occur as the cardinalities of sets of complements to some partition \(\mathcal {P} \in \Pi (\lambda )\), and only these cardinalities appear. Moreover, we give a direct formula for the number of complements to a given partition. Under the GCH, the cardinalities of maximal chains, maximal antichains, and numbers of complements are fully determined, and we provide a complete characterisation.     

Gradient Estimates for Commutators of Square Roots of Elliptic Operators with Complex Bounded Measurable Coefficients

Let \(L=-\mathrm{div}(A\nabla )\) be a second order divergence form elliptic operator and A an accretive \(n\times n\) matrix with bounded measurable complex coefficients in \({\mathbb R}^n\). Let \(\nabla b\in L^n({\mathbb R}^n)\,(n>2)\). In this paper, we prove that the commutator generated by b and the square root of L, which is defined by \([b,\sqrt{L}]f(x)=b(x)\sqrt{L}f(x)-\sqrt{L}(bf)(x)\), is bounded from the homogenous Sobolev space \({\dot{L}}_1^2({\mathbb R}^n)\) to \(L^2({\mathbb R}^n)\).     

rthogonality Property $$\mathcal {}$$ and Compact Perturbations

A bounded linear operator T acting on a Hilbert space is said to have orthogonality property \(\mathcal {O}\) if the subspaces \(\ker (T-\alpha )\) and \(\ker (T-\beta )\) are orthogonal for all \(\alpha , \beta \in \sigma _p(T)\) with \(\alpha \ne \beta \). In this paper, the authors investigate the compact perturbations of operators with orthogonality property \(\mathcal {O}\). We give a sufficient and necessary condition to determine when an operator T has the following property: for each \(\varepsilon >0\), there exists \(K\in \mathcal {K(H)}\) with \(\Vert K\Vert <\varepsilon \) such that \(T+K\) has orthogonality property \(\mathcal {O}\). Also, we study the stability of orthogonality property \(\mathcal {O}\) under small compact perturbations and analytic functional calculus.     

Endotrivial modules for the general linear group in a nondefining characteristic

Suppose that \(G\) is a finite group such that \(\mathrm{SL }(n,q)\subseteq G \subseteq \mathrm{GL }(n,q)\), and that \(Z\) is a central subgroup of \(G\). Let \(T(G/Z)\) be the abelian group of equivalence classes of endotrivial \(k(G/Z)\)-modules, where \(k\) is an algebraically closed field of characteristic \(p\) not dividing \(q\). We show that the torsion free rank of \(T(G/Z)\) is at most one, and we determine \(T(G/Z)\) in the case that the Sylow \(p\)-subgroup of \(G\) is abelian and nontrivial. The proofs for the torsion subgroup of \(T(G/Z)\) use the theory of Young modules for \(\mathrm{GL }(n,q)\) and a new method due to Balmer for computing the kernel of restrictions in the group of endotrivial modules.     

On the asymptotics of supremum distribution for some iterated processes

In this paper, we study the asymptotic behavior of supremum distribution of some classes of iterated stochastic processes \(\{X(Y(t)) : t \in [0, \infty )\}\), where \(\{X(t) : t \in \mathbb {R} \}\) is a centered Gaussian process and \(\{Y(t): t \in [0, \infty )\}\) is an independent of {X(t)} stochastic process with a.s. continuous sample paths. In particular, the asymptotic behavior of \(\mathbb {P}(\sup _{s\in [0,T]} X(Y (s)) > u)\) as \(u \to \infty \), where T>0, as well as \(\lim _{u\to \infty } \mathbb {P}(\sup _{s\in [0,h(u)]} X(Y (s)) > u)\), for some suitably chosen function h(u) are analyzed. As an illustration, we study the asymptotic behavior of the supremum distribution of iterated fractional Brownian motion process.