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 \((V,P)\).     

Left annihilator of generalized derivations on Lie ideals in prime rings

Let \(R\) be a prime ring, \(L\) a noncentral Lie ideal of \(R\), \(F\) a generalized derivation with associated nonzero derivation \(d\) of \(R\). If \(a\in R\) such that \(a(d(u)^{l_1} F(u)^{l_2} d(u)^{l_3} F(u)^{l_4} \ldots F(u)^{l_k})^{n}=0\) for all \(u\in L\), where \(l_1,l_2,\ldots ,l_k\) are fixed non negative integers not all are zero and \(n\) is a fixed integer, then either \(a=0\) or \(R\) satisfies \(s_4\), the standard identity in four variables.     

A Polynomial-Type Jost Solution and Spectral Properties of a Self-Adjoint Quantum-Difference Operator

In this paper, we find a polynomial-type Jost solution of a self-adjoint \(q\)-difference equation of second order. Then we investigate the analytical properties and asymptotic behavior of the Jost solution. We prove that the self-adjoint operator \(L\) generated by the \(q\)-difference expression of second order has essential spectrum filling the segment \([-2\sqrt{q},2\sqrt{q}]\), \(q>1\). Finally, we examine the properties of the eigenvalues of \(L\).     

Calabi-Yau manifolds with isolated conical singularities

Let \(X\) be a complex projective variety with only canonical singularities and with trivial canonical bundle. Let \(L\) be an ample line bundle on \(X\). Assume that the pair \((X,L)\) is the flat limit of a family of smooth polarized Calabi-Yau manifolds. Assume that for each singular point \(x \in X\) there exist a Kähler-Einstein Fano manifold \(Z\) and a positive integer \(q\) dividing \(K_{Z}\) such that \(-\frac{1}{q}K_{Z}\) is very ample and such that the germ \((X,x)\) is locally analytically isomorphic to a neighborhood of the vertex of the blow-down of the zero section of \(\frac{1}{q}K_{Z}\). We prove that up to biholomorphism, the unique weak Ricci-flat Kähler metric representing \(2\pi c_{1}(L)\) on \(X\) is asymptotic at a polynomial rate near \(x\) to the natural Ricci-flat Kähler cone metric on \(\frac{1}{q}K_{Z}\) constructed using the Calabi ansatz. In particular, our result applies if \((X, \mathcal{O}(1))\) is a nodal quintic threefold in \(\mathbf {P}^{4}\). This provides the first known examples of compact Ricci-flat manifolds with non-orbifold isolated conical singularities.     

Elimination of extremal index zeroes from generic paths of closed $$$$-forms

Let \( \alpha \) be a Morse closed \( 1 \)-form of a smooth \( n \)-dimensional manifold \( M \). The zeroes of \( \alpha \) of index \( 0 \) or \( n \) are called centers. It is known that every non-vanishing de Rham cohomology class \( u \) contains a Morse representative without centers. The result of this paper is the one-parameter analogue of the last statement: every generic path \( (\alpha _t)_{ t\in [0,1] }\) of closed \( 1 \)-forms in a fixed class \( u\ne 0 \) such that \( \alpha _0,\alpha _1 \) have no centers, can be modified relatively to its extremities to another such path \( (\beta _t)_{t \in [0,1]} \) having no center at all.     

Diagonalization of the Finite Hilbert Transform on Two Adjacent Intervals

We continue the study of stability of solving the interior problem of tomography. The starting point is the Gelfand–Graev formula, which converts the tomographic data into the finite Hilbert transform (FHT) of an unknown function f along a collection of lines. Pick one such line, call it the x-axis, and assume that the function to be reconstructed depends on a one-dimensional argument by restricting f to the x-axis. Let \(I_1\) be the interval where f is supported, and \(I_2\) be the interval where the Hilbert transform of f can be computed using the Gelfand–Graev formula. The equation to be solved is \(\left. {\mathcal {H}}_1 f=g\right| _{I_2}\), where \({\mathcal {H}}_1\) is the FHT that integrates over \(I_1\) and gives the result on \(I_2\), i.e. \({\mathcal {H}}_1: L^2(I_1)\rightarrow L^2(I_2)\). In the case of complete data, \(I_1\subset I_2\), and the classical FHT inversion formula reconstructs f in a stable fashion. In the case of interior problem (i.e., when the tomographic data are truncated), \(I_1\) is no longer a subset of \(I_2\), and the inversion problems becomes severely unstable. By using a differential operator L that commutes with \({\mathcal {H}}_1\), one can obtain the singular value decomposition of \({\mathcal {H}}_1\). Then the rate of decay of singular values of \({\mathcal {H}}_1\) is the measure of instability of finding f. Depending on the available tomographic data, different relative positions of the intervals \(I_{1,2}\) are possible. The cases when \(I_1\) and \(I_2\) are at a positive distance from each other or when they overlap have been investigated already. It was shown that in both cases the spectrum of the operator \({\mathcal {H}}_1^*{\mathcal {H}}_1\) is discrete, and the asymptotics of its eigenvalues \(\sigma _n\) as \(n\rightarrow \infty \) has been obtained. In this paper we consider the case when the intervals \(I_1=(a_1,0)\) and \(I_2=(0,a_2)\) are adjacent. Here \(a_1 < 0 < a_2\). Using recent developments in the Titchmarsh–Weyl theory, we show that the operator L corresponding to two touching intervals has only continuous spectrum and obtain two isometric transformations \(U_1\), \(U_2\), such that \(U_2{\mathcal {H}}_1 U_1^*\) is the multiplication operator with the function \(\sigma (\lambda )\), \(\lambda \ge (a_1^2+a_2^2)/8\). Here \(\lambda \) is the spectral parameter. Then we show that \(\sigma (\lambda )\rightarrow 0\) as \(\lambda \rightarrow \infty \) exponentially fast. This implies that the problem of finding f is severely ill-posed. We also obtain the leading asymptotic behavior of the kernels involved in the integral operators \(U_1\), \(U_2\) as \(\lambda \rightarrow \infty \). When the intervals are symmetric, i.e. \(-a_1=a_2\), the operators \(U_1\), \(U_2\) are obtained explicitly in terms of hypergeometric functions.     

On Locally Gabriel Geometric Graphs

Let \(P\) be a set of \(n\) points in the plane. A geometric graph \(G\) on \(P\) is said to be locally Gabriel if for every edge \((u,v)\) in \(G\), the Euclidean disk with the segment joining \(u\) and \(v\) as diameter does not contain any points of \(P\) that are neighbors of \(u\) or \(v\) in \(G\). A locally Gabriel graph(LGG) is a generalization of Gabriel graph and is motivated by applications in wireless networks. Unlike a Gabriel graph, there is no unique LGG on a given point set since no edge in a LGG is necessarily included or excluded. Thus the edge set of the graph can be customized to optimize certain network parameters depending on the application. The unit distance graph(UDG), introduced by Erdos, is also a LGG. In this paper, we show the following combinatorial bounds on edge complexity and independent sets of LGG: (i) For any \(n\), there exists LGG with \(\Omega (n^{5/4})\) edges. This improves upon the previous best bound of \(\Omega (n^{1+\frac{1}{\log \log n}})\). (ii) For various subclasses of convex point sets, we show tight linear bounds on the maximum edge complexity of LGG. (iii) For any LGG on any \(n\) point set, there exists an independent set of size \(\Omega (\sqrt{n}\log n)\).     

A potential well for an integral functional with singular Lagrangian

This paper concerns a functional of the form

$$\begin{aligned} \Phi (u)=\int _\Omega L(x,u(x),\nabla u(x))\, dx \end{aligned}$$

on the Sobolev space \(H_0^1(\Omega )\) where \(\Omega \) is a bounded open subset of \({\mathbb {R}}^N\) with \(N\ge 3\) and \(0\in \Omega \). The hypotheses on L ensure that \(u\equiv 0\) is a critical point of \(\Phi \), but allow the Lagrangian to be singular at \(x=0\). It is shown that, under these assumptions, the usual conditions associated with Jacobi (positive definiteness of the second variation of \(\Phi \) at \(u\equiv 0\)), Legendre (ellipticity at \(u\equiv 0\)) and Weierstrass [strict convexity of \(L(x,s,\xi )\) with respect to \(\xi \)] from the calculus of variations are not sufficient ensure that \(u\equiv 0\) is a local minimum of \(\Phi \). Using recent criteria for the existence of a potential well of a \(C^1\)-functional on a real Hilbert space, conditions implying that \(u\equiv 0\) lies in a potential well of \(\Phi \) are established. They are shown to be sharp in some cases.

     

A new theorem on the prime-counting function

For \(x>0\), let \(\pi (x)\) denote the number of primes not exceeding x. For integers a and \(m>0\), we determine when there is an integer \(n>1\) with \(\pi (n)=(n+a)/m\). In particular, we show that, for any integers \(m>2\) and \(a\leqslant \lceil e^{m-1}/(m-1)\rceil \), there is an integer \(n>1\) with \(\pi (n)=(n+a)/m\). Consequently, for any integer \(m>4\), there is a positive integer n with \(\pi (mn)=m+n\). We also pose several conjectures for further research; for example, we conjecture that, for each \(m=1,2,3,\ldots \), there is a positive integer n such that \(m+n\) divides \(p_m+p_n\), where \(p_k\) denotes the k-th prime.     

Meromorphic tensor equivalence for Yangians and quantum loop algebras

Let \(\mathfrak{g}\) be a complex semisimple Lie algebra, and \(Y_{\hbar }(\mathfrak{g})\), \(U_{q}(L\mathfrak{g})\) the corresponding Yangian and quantum loop algebra, with deformation parameters related by \(q=e^{\pi \iota \hbar }\). When \(\hbar \) is not a rational number, we constructed in Gautam and Toledano Laredo (J. Am. Math. Soc. 29:775, 2016) a faithful functor \(\Gamma \) from the category of finite-dimensional representations of \(Y_{\hbar }(\mathfrak{g})\) to those of \(U_{q}(L \mathfrak{g})\). The functor \(\Gamma \) is governed by the additive difference equations defined by the commuting fields of the Yangian, and restricts to an equivalence on a subcategory of \(\operatorname{Rep}_{\operatorname{fd}}(Y_{\hbar }(\mathfrak{g}))\) defined by choosing a branch of the logarithm. In this paper, we construct a tensor structure on \(\Gamma \) and show that, if \(|q|\neq 1\), it yields an equivalence of meromorphic braided tensor categories, when \(Y_{\hbar }(\mathfrak{g})\) and \(U_{q}(L\mathfrak{g})\) are endowed with the deformed Drinfeld coproducts and the commutative part of their universal \(R\)-matrices. This proves in particular the Kohno–Drinfeld theorem for the abelian \(q\)KZ equations defined by \(Y_{\hbar }(\mathfrak{g})\). The tensor structure arises from the abelian \(q\)KZ equations defined by an appropriate regularisation of the commutative part of the \(R\)-matrix of \(Y_{\hbar }(\mathfrak{g})\).     

New nonbinary code bounds based on divisibility arguments

For \(q,n,d \in \mathbb {N}\), let \(A_q(n,d)\) be the maximum size of a code \(C \subseteq [q]^n\) with minimum distance at least d. We give a divisibility argument resulting in the new upper bounds \(A_5(8,6) \le 65\), \(A_4(11,8)\le 60\) and \(A_3(16,11) \le 29\). These in turn imply the new upper bounds \(A_5(9,6) \le 325\), \(A_5(10,6) \le 1625\), \(A_5(11,6) \le 8125\) and \(A_4(12,8) \le 240\). Furthermore, we prove that for \(\mu ,q \in \mathbb {N}\), there is a 1–1-correspondence between symmetric \((\mu ,q)\)-nets (which are certain designs) and codes \(C \subseteq [q]^{\mu q}\) of size \(\mu q^2\) with minimum distance at least \(\mu q - \mu \). We derive the new upper bounds \(A_4(9,6) \le 120\) and \(A_4(10,6) \le 480\) from these ‘symmetric net’ codes.     

Examples of Ricci-Mean Curvature Flows

Let \(\pi :{\mathbb {P}}({\mathcal {O}}(0)\oplus {\mathcal {O}}(k))\rightarrow {\mathbb {P}}^{n-1}\) be a projective bundle over \({\mathbb {P}}^{n-1}\) with \(1\le k \le n-1\). We denote \({\mathbb {P}}({\mathcal {O}}(0)\oplus {\mathcal {O}}(k))\) by \(N_{k}^{n}\) and endow it with the U(n)-invariant gradient shrinking Kähler Ricci soliton structure constructed by Cao (Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994), A K Peters, Wellesley, 1996) and Koiso (Recent topics in differential and analytic geometry. Advanced studies in pure mathematics, Boston, 1990). In this paper, we show that lens space \(L(k\, ;1)(r)\) with radius r embedded in \(N_{k}^{n}\) is a self-similar solution. We also prove that there exists a pair of critical radii \(r_{1}<r_{2}\), which satisfies the following. The lens space \(L(k\, ;1)(r)\) is a self-shrinker if \(r<r_{2}\) and self-expander if \(r_{2}<r\), and the Ricci-mean curvature flow emanating from \(L(k\, ;1)(r)\) collapses to the 0-section of \(\pi \) if \(r<r_{1}\) and to the \(\infty \)-section of \(\pi \) if \(r_{1}<r\). This paper gives explicit examples of Ricci-mean curvature flows.     

A note on generalized skew derivations on Lie ideals

Let \(\mathcal {R}\) be a prime ring, \(\mathcal {Z(R)}\) its center, \(\mathcal {C}\) its extended centroid, \(\mathcal {L}\) a Lie ideal of \(\mathcal {R}, \mathcal {F}\) a generalized skew derivation associated with a skew derivation d and automorphism \(\alpha \). Assume that there exist \(t\ge 1\) and \(m,n\ge 0\) fixed integers such that \( vu = u^m\mathcal {F}(uv)^tu^n\) for all \(u,v \in \mathcal {L}\). Then it is shown that either \(\mathcal {L}\) is central or \(\mathrm{char}(\mathcal {R})=2, \mathcal {R}\subseteq \mathcal {M}_2(\mathcal {C})\), the ring of \(2\times 2\) matrices over \(\mathcal {C}, \mathcal {L}\) is commutative and \(u^2\in \mathcal {Z(R)}\), for all \(u\in \mathcal {L}\). In particular, if \(\mathcal {L}=[\mathcal {R,R}]\), then \(\mathcal {R}\) is commutative.     

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.     

Standard Jordan partitions with three parameters

We extend previous work on standard two-parameter Jordan partitions by Barry (Commun Algebra 43:4231–4246, 2015) to three parameters. Let \(J_r\) denote an \(r \times r\) matrix with minimal polynomial \((t-1)^r\) over a field F of characteristic p. For positive integers \(n_1\), \(n_2\), and \(n_3\) satisfying \(n_1 \le n_2 \le n_3\), the Jordan canonical form of the \(n_1 n_2 n_3 \times n_1 n_2 n_3\) matrix \(J_{n_1} \otimes J_{n_2} \otimes J_{n_3}\) has the form \(J_{\lambda _1} \oplus J_{\lambda _2} \oplus \cdots \oplus J_{\lambda _m}\) where \(\lambda _1 \ge \lambda _2 \ge \cdots \ge \lambda _m>0\) and \(\sum _{i=1}^m \lambda _i=n_1 n_2 n_3\). The partition \(\lambda (n_1,n_2,n_3:p)=(\lambda _1, \lambda _2,\ldots , \lambda _m)\) of \(n_1 n_2 n_3\), which depends on \(n_1\), \(n_2\), \(n_3\), and p, will be called a Jordan partition. We will define what we mean by a standard Jordan partition and give necessary and sufficient conditions for its existence.     

Why there is no an existence theorem for a convex polytope with prescribed directions and perimeters of the faces?

We choose some special unit vectors \({\mathbf {n}}_1,\ldots ,{\mathbf {n}}_5\) in \({\mathbb {R}}^3\) and denote by \({\mathscr {L}}\subset {\mathbb {R}}^5\) the set of all points \((L_1,\ldots ,L_5)\in {\mathbb {R}}^5\) with the following property: there exists a compact convex polytope \(P\subset {\mathbb {R}}^3\) such that the vectors \({\mathbf {n}}_1,\ldots ,{\mathbf {n}}_5\) (and no other vector) are unit outward normals to the faces of P and the perimeter of the face with the outward normal \({\mathbf {n}}_k\) is equal to \(L_k\) for all \(k=1,\ldots ,5\). Our main result reads that \({\mathscr {L}}\) is not a locally-analytic set, i.e., we prove that, for some point \((L_1,\ldots ,L_5)\in {\mathscr {L}}\), it is not possible to find a neighborhood \(U\subset {\mathbb {R}}^5\) and an analytic set \(A\subset {\mathbb {R}}^5\) such that \({\mathscr {L}}\cap U=A\cap U\). We interpret this result as an obstacle for finding an existence theorem for a compact convex polytope with prescribed directions and perimeters of the faces.     

Sharp boundary behaviour of solutions to semilinear nonlocal elliptic equations

We investigate quantitative properties of nonnegative solutions \(u(x)\ge 0\) to the semilinear diffusion equation \(\mathcal {L}u= f(u)\), posed in a bounded domain \(\Omega \subset \mathbb {R}^N\) with appropriate homogeneous Dirichlet or outer boundary conditions. The operator \(\mathcal {L}\) may belong to a quite general class of linear operators that include the standard Laplacian, the two most common definitions of the fractional Laplacian \((-\Delta )^s\) (\(0<s<1\)) in a bounded domain with zero Dirichlet conditions, and a number of other nonlocal versions. The nonlinearity f is increasing and looks like a power function \(f(u)\sim u^p\), with \(p\le 1\). The aim of this paper is to show sharp quantitative boundary estimates based on a new iteration process. We also prove that, in the interior, solutions are Hölder continuous and even classical (when the operator allows for it). In addition, we get Hölder continuity up to the boundary. Particularly interesting is the behaviour of solution when the number \(\frac{2s}{1-p}\) goes below the exponent \(\gamma \in (0,1]\) corresponding to the Hölder regularity of the first eigenfunction \(\mathcal {L}\Phi _1=\lambda _1 \Phi _1\). Indeed a change of boundary regularity happens in the different regimes \(\frac{2s}{1-p} \gtreqqless \gamma \), and in particular a logarithmic correction appears in the “critical” case \(\frac{2s}{1-p} = \gamma \). For instance, in the case of the spectral fractional Laplacian, this surprising boundary behaviour appears in the range \(0<s\le (1-p)/2\).     

The Fischer–Marsden conjecture and contact geometry

In this paper, we consider the Fischer–Marsden conjecture within the frame-work of K-contact manifolds and \((\kappa ,\mu )\)-contact manifolds. First, we prove that a complete K-contact metric satisfying \(\mathcal {L}^{*}_g(\lambda )=0\) is Einstein and is isometric to a unit sphere \(S^{2n+1}\). Next, we prove that if a non-Sasakian \((\kappa ,\mu )\)-contact metric satisfies \(\mathcal {L}^{*}_g(\lambda )=0\), then \( M^{3} \) is flat, and for \(n > 1\), \(M^{2n+1}\) is locally isometric to the product of a Euclidean space \(E^{n+1}\) and a sphere \(S^n(4)\) of constant curvature \(+\,4\).