Selection problems of $${{\mathbb {Z}}}^2$$-periodic entropy solutions and viscosity solutions

\({{\mathbb {Z}}}^2\)-periodic entropy solutions of hyperbolic scalar conservation laws and \({{\mathbb {Z}}}^2\)-periodic viscosity solutions of Hamilton–Jacobi equations are not unique in general. However, uniqueness holds for viscous scalar conservation laws and viscous Hamilton–Jacobi equations. Bessi (Commun Math Phys 235:495–511, 2003) investigated the convergence of approximate \({{\mathbb {Z}}}^2\)-periodic solutions to an exact one in the process of the vanishing viscosity method, and characterized this physically natural \({{\mathbb {Z}}}^2\)-periodic solution with the aid of Aubry–Mather theory. In this paper, a similar problem is considered in the process of the finite difference approximation under hyperbolic scaling. We present a selection criterion different from the one in the vanishing viscosity method, which may depend on the approximation parameter.     

Magnetic Trajectories in an Almost Contact Metric Manifold $${\mathbb{R}^{2N+1}}$$

In this paper we classify magnetic trajectories γ in \({{\mathbb{R}}^{2N+1}}\) endowed with a canonical quasi-Sasakian structure, corresponding to a magnetic field proportional to the fundamental 2-form. We prove that they are helices of order 5 and we show that there exists a totally geodesic \({{\mathbb{R}}^5}\) in \({\mathbb{R}^{2N+1}}\) such that γ lies in \({{\mathbb{R}}^5}\). Moreover, the quasi-Sasakian structure of \({{\mathbb{R}}^5}\) is that induced from the ambient manifold.     

A partition relation for pairs on $$\omega ^{\omega ^\omega }$$

We consider colorings of the pairs of a family \(\mathcal {F}\subseteq {{\mathrm{FIN}}}\) of topological type \(\omega ^{\omega ^k}\), for \(k>1\); and we find a homogeneous family \(\mathcal {G}\subseteq \mathcal {F}\) for each coloring. As a consequence, we complete our study of the partition relation \({\forall l>1,\, \alpha \rightarrow ({{\mathrm{top}}}\;\omega ^2+1)^2_{l,m}}\) identifying \(\omega ^{\omega ^\omega }\) as the smallest ordinal space \(\alpha <\omega _1\) satisfying \({\forall l>1,\, \alpha \rightarrow ({{\mathrm{top}}}\;\omega ^2+1)^2_{l,4}}\).     

Short-time behavior of the heat kernel and Weyl’s law on $${{\mathrm{RCD}}}^*(K,N)$$ spaces

In this paper, we prove pointwise convergence of heat kernels for mGH-convergent sequences of \({{\mathrm{RCD}}}^{*}(K,N)\)-spaces. We obtain as a corollary results on the short-time behavior of the heat kernel in \({{\mathrm{RCD}}}^*(K,N)\)-spaces. We use then these results to initiate the study of Weyl’s law in the \({{\mathrm{RCD}}}\) setting.     

Lagrangian isotopy of tori in $${S^2\times S^2}$$ and $${{\mathbb{C}}P^2}$$CP2

We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic four-manifolds: the symplectic vector space \({{\mathbb{R}}^4}\), the projective plane \({{\mathbb{C}}P^2}\), and the monotone \({S^2 \times S^2}\). The result is proven by studying pseudoholomorphic foliations while performing the splitting construction from symplectic field theory along the Lagrangian torus. A number of other related results are also shown. Notably, the nearby Lagrangian conjecture is established for \({T^*{\mathbb{T}}^2}\), i.e. it is shown that every closed exact Lagrangian submanifold in this cotangent bundle is Hamiltonian isotopic to the zero-section.     

Extremal Problems for Mappings with Generalized Parametric Representation in $${{\mathbb {C}}}^{n}$$

In this paper we are concerned with the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) (\(t\ge 0\)) of normalized biholomorphic mappings on the Euclidean unit ball \(\mathbb {B}^n\) in \({\mathbb {C}}^n\) that can be embedded in normal Loewner chains whose normalizations are given by time-dependent operators \(A\in \widetilde{\mathcal {A}}\), where \(\widetilde{\mathcal {A}}\) is a family of measurable mappings from \([0,\infty )\) into \(L({\mathbb {C}}^n)\) which satisfy certain natural assumptions. In particular, we consider extreme points and support points associated with the compact family \(\widetilde{S}^t_A(\mathbb {B}^n)\), where \(A\in \widetilde{\mathcal {A}}\). We prove that if \(f(z,t)=V(t)^{-1}z+\cdots \) is a normal Loewner chain such that \(V(s)f(\cdot ,s)\in \mathrm{ex}\,\widetilde{S}^s_A(\mathbb {B}^n)\) (resp. \(V(s)f(\cdot ,s)\in \mathrm{supp}\,\widetilde{S}^s_A(\mathbb {B}^n)\)), then \(V(t)f(\cdot ,t)\in \mathrm{ex}\, \widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\) (resp. \(V(t)f(\cdot ,t)\in \mathrm{supp}\,\widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\)), where V(t) is the unique solution on \([0,\infty )\) of the initial value problem: \(\frac{d V}{d t}(t)=-A(t)V(t)\), a.e. \(t\ge 0\), \(V(0)=I_n\). Also, we obtain an example of a bounded support point for the family \(\widetilde{S}_A^t(\mathbb {B}^2)\), where \(A\in \widetilde{\mathcal {A}}\) is a certain time-dependent operator. We also consider the notion of a reachable family with respect to time-dependent linear operators \(A\in \widetilde{\mathcal {A}}\), and obtain characterizations of extreme/support points associated with these families of bounded biholomorphic mappings on \(\mathbb {B}^n\). Useful examples and applications yield that the study of the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) for time-dependent operators \(A\in \widetilde{\mathcal {A}}\) is basically different from that in the case of constant time-dependent linear operators.     

On the classification of 4-dimensional $$(m,\rho )$$-quasi-Einstein manifolds with harmonic Weyl curvature

In this paper we study four-dimensional \((m,\rho )\)-quasi-Einstein manifolds with harmonic Weyl curvature when \(m\notin \{0,\pm 1,-2,\pm \infty \}\) and \(\rho \notin \{\frac{1}{4},\frac{1}{6}\}\). We prove that a non-trivial \((m,\rho )\)-quasi-Einstein metric g (not necessarily complete) is locally isometric to one of the following: (i) \({\mathcal {B}}^2_\frac{R}{2(m+2)}\times {\mathbb {N}}^2_\frac{R(m+1)}{2(m+2)}\), where \({\mathcal {B}}^2_\frac{R}{2(m+2)}\) is the northern hemisphere in the two-dimensional (2D) sphere \({\mathbb {S}}^2_\frac{R}{2(m+2)}\), \({\mathbb {N}}_\delta \) is a 2D Riemannian manifold with constant curvature \(\delta \), and R is the constant scalar curvature of g. (ii) \({\mathcal {D}}^2_\frac{R}{2(m+2)}\times {\mathbb {N}}^2_\frac{R(m+1)}{2(m+2)}\), where \({\mathcal {D}}^2_\frac{R}{2(m+2)}\) is half (cut by a hyperbolic line) of hyperbolic plane \({\mathbb {H}}^2_\frac{R}{2(m+2)}\). (iii) \({\mathbb {H}}^2_\frac{R}{2(m+2)}\times {\mathbb {N}}^2_\frac{R(m+1)}{2(m+2)}\). (iv) A certain singular metric with \(\rho =0\). (v) A locally conformal flat metric. By applying this local classification, we obtain a classification of the complete \((m,\rho )\)-quasi-Einstein manifolds given the condition of a harmonic Weyl curvature. Our result can be viewed as a local classification of gradient Einstein-type manifolds. A corollary of our result is the classification of \((\lambda ,4+m)\)-Einstein manifolds, which can be viewed as (m, 0)-quasi-Einstein manifolds.     

The algebra of generating functions for multiple divisor sums and applications to multiple zeta values

We study the algebra \({{\mathrm{{\mathcal {MD}}}}}\) of generating functions for multiple divisor sums and its connections to multiple zeta values. The generating functions for multiple divisor sums are formal power series in q with coefficients in \({\mathbb {Q}}\) arising from the calculation of the Fourier expansion of multiple Eisenstein series. We show that the algebra \({{\mathrm{{\mathcal {MD}}}}}\) is a filtered algebra equipped with a derivation and use this derivation to prove linear relations in \({{\mathrm{{\mathcal {MD}}}}}\). The (quasi-)modular forms for the full modular group \({{\mathrm{SL}}}_2({\mathbb {Z}})\) constitute a subalgebra of \({{\mathrm{{\mathcal {MD}}}}}\), and this also yields linear relations in \({{\mathrm{{\mathcal {MD}}}}}\). Generating functions of multiple divisor sums can be seen as a q-analogue of multiple zeta values. Studying a certain map from this algebra into the real numbers we will derive a new explanation for relations between multiple zeta values, including those of length 2, coming from modular forms.     

Algebraic and abelian solutions to the projective translation equation

Let \({{\mathrm x}=(x,y)}\). A projective two-dimensional flow is a solution to a 2-dimensional projective translation equation (PrTE) \({(1-z)\phi({\mathrm x})=\phi(\phi({\mathrm x}z)(1-z)/z)}\), \({\phi:\mathbb{C}^{2}\mapsto\mathbb{C}^{2}}\). Previously we have found all solutions of the PrTE which are rational functions. The rational flow gives rise to a vector field \({\varpi(x,y)\bullet \varrho(x,y)}\) which is a pair of 2-homogenic rational functions. On the other hand, only very special pairs of 2-homogenic rational functions, such as vector fields, give rise to rational flows. The main ingredient in the proof of the classifying theorem is a reduction algorithm for a pair of 2-homogenic rational functions. This reduction method in fact allows us to derive more results. Namely, in this work we find all projective flows with rational vector fields whose orbits are algebraic curves. We call these flows abelian projective flows, since either these flows are described in terms of abelian functions and with the help of 1-homogenic birational plane transformations (1-BIR), and the orbits of these flows can be transformed into algebraic curves \({x^{A}(x-y)^{B}y^{C}\equiv{\mathrm{const.}}}\) (abelian flows of type I), or there exists a 1-BIR which transforms the orbits into the lines \({y\equiv{\mathrm{const.}}}\) (abelian flows of type II), and generally the latter flows are described in terms of non-arithmetic functions. Our second result classifies all abelian flows which are given by two variable algebraic functions. We call these flows algebraic projective flows, and these are abelian flows of type I. We also provide many examples of algebraic, abelian and non-abelian flows.     

Banach Random Walk in the Unit Ball $$S\subset l^{2}$$ and Chaotic Decomposition of $$l^{2}\left( S,{{\mathbb {P}}}\right) $$l2S,P

A Banach random walk in the unit ball S in \(l^{2}\) is defined, and we show that the integral introduced by Banach (Theory of the integral. Warszawa-Lwów, 1937) can be expressed as the expectation with respect to the measure \({{\mathbb {P}}}\) induced by this walk. A decomposition \(l^{2}\left( S,{{\mathbb {P}}}\right) =\bigoplus _{i=0}^{\infty } {{\mathfrak {B}}}_{i}\) in terms of what we call Banach chaoses is given.     

$${{\mathbb {Z}}}_2$$-double cyclic codes

A binary linear code C is a \({\mathbb {Z}}_2\)-double cyclic code if the set of coordinates can be partitioned into two subsets such that any cyclic shift of the coordinates of both subsets leaves invariant the code. These codes can be identified as submodules of the \({\mathbb {Z}}_2[x]\)-module \({\mathbb {Z}}_2[x]/(x^r-1)\times {\mathbb {Z}}_2[x]/(x^s-1).\) We determine the structure of \({\mathbb {Z}}_2\)-double cyclic codes giving the generator polynomials of these codes. We give the polynomial representation of \({\mathbb {Z}}_2\)-double cyclic codes and its duals, and the relations between the generator polynomials of these codes. Finally, we study the relations between \({{\mathbb {Z}}}_2\)-double cyclic and other families of cyclic codes, and show some examples of distance optimal \({\mathbb {Z}}_2\)-double cyclic codes.     

Fourth-order Schrödinger type operator with singular potentials

We prove that a deformation of a hypersurface in an (n + 1)-dimensional real space form \({{\mathbb S}^{n+1}_{p,1}}\) induces a Hamiltonian variation of the normal congruence in the space \({{\mathbb L}({\mathbb S}^{n+1}_{p,1})}\) of oriented geodesics. As an application, we show that every Hamiltonian minimal submanifold in \({{\mathbb L}({\mathbb S}^{n+1})}\) (resp. \({{\mathbb L}({\mathbb H}^{n+1})}\)) with respect to the (para-)Kähler Einstein structure is locally the normal congruence of a hypersurface \({\Sigma}\) in \({{\mathbb S}^{n+1}}\) (resp. \({{\mathbb H}^{n+1}}\)) that is a critical point of the functional \({{\mathcal W}(\Sigma) = \int_\Sigma\left(\Pi_{i=1}^n|\epsilon+k_i^2|\right)^{1/2}}\), where k_i denote the principal curvatures of \({\Sigma}\) and \({\epsilon \in \{-1, 1\}}\). In addition, for \({n = 2}\), we prove that every Hamiltonian minimal surface in \({{\mathbb L}({\mathbb S}^{3})}\) (resp. \({{\mathbb L}({\mathbb H}^{3})}\)), with respect to the (para-)Kähler conformally flat structure, is the normal congruence of a surface in \({{\mathbb S}^{3}}\) (resp. \({{\mathbb H}^{3}}\)) that is a critical point of the functional \({{\mathcal W}\prime(\Sigma) = \int_\Sigma\sqrt{H^2-K+1}}\) (resp. \({{\mathcal W}\prime(\Sigma) = \int_\Sigma\sqrt{H^2-K-1}}\)), where H and K denote, respectively, the mean and Gaussian curvature of \({\Sigma}\).     

On the 2-adic valuation of the cardinality of elliptic curves over finite extensions of $$\mathbb {F}_{\varvec{q}} $$

In this paper we study the difference between the 2-adic valuations of the cardinalities \( \# E( \mathbb {F}_{q^k} ) \) and \( \# E( \mathbb {F}_q ) \) of an elliptic curve E over \( \mathbb {F}_q \). We also deduce information about the structure of the 2-Sylow subgroup \( E[ 2^\infty ]( \mathbb {F}_{q^k} ) \) from the exponents of \( E[ 2^\infty ]( \mathbb {F}_q ) \).     

Depth,Stanley depth,and regularity of ideals associated to graphs

Let \({\mathbb{K}}\) be a field and \({S=\mathbb{K}[x_1,\dots,x_n]}\) be the polynomial ring in n variables over \({\mathbb{K}}\). Let G be a graph with n vertices. Assume that \({I=I(G)}\) is the edge ideal of G and \({J=J(G)}\) is its cover ideal. We prove that \({{\rm sdepth}(J)\geq n-\nu_{o}(G)}\) and \({{\rm sdepth}(S/J)\geq n-\nu_{o}(G)-1}\), where \({\nu_{o}(G)}\) is the ordered matching number of G. We also prove the inequalities \({{\rmsdepth}(J^k)\geq {\rm depth}(J^k)}\) and \({{\rm sdepth}(S/J^k)\geq {\rmdepth}(S/J^k)}\), for every integer \({k\gg 0}\), when G is a bipartite graph. Moreover, we provide an elementary proof for the known inequality reg\({(S/I)\leq \nu_{o}(G)}\).     

Disintegration of positive isometric group representations on $$\varvec{\mathrm {L}^{p}}$$-spaces

Let G be a Polish locally compact group acting on a Polish space \({{X}}\) with a G-invariant probability measure \(\mu \). We factorize the integral with respect to \(\mu \) in terms of the integrals with respect to the ergodic measures on X, and show that \(\mathrm {L}^{p}({{X}},\mu )\) (\(1\le p<\infty \)) is G-equivariantly isometrically lattice isomorphic to an \({\mathrm {L}^p}\)-direct integral of the spaces \(\mathrm {L}^{p}({{X}},\lambda )\), where \(\lambda \) ranges over the ergodic measures on X. This yields a disintegration of the canonical representation of G as isometric lattice automorphisms of \(\mathrm {L}^{p}({{X}},\mu )\) as an \({\mathrm {L}^p}\)-direct integral of order indecomposable representations. If \(({{X}}^\prime ,\mu ^\prime )\) is a probability space, and, for some \(1\le q<\infty \), G acts in a strongly continuous manner on \(\mathrm {L}^{q}({{X}}^\prime ,\mu ^\prime )\) as isometric lattice automorphisms that leave the constants fixed, then G acts on \(\mathrm {L}^{p}({{X}}^{\prime },\mu ^{\prime })\) in a similar fashion for all \(1\le p<\infty \). Moreover, there exists an alternative model in which these representations originate from a continuous action of G on a compact Hausdorff space. If \(({{X}}^\prime ,\mu ^\prime )\) is separable, the representation of G on \(\mathrm {L}^p(X^\prime ,\mu ^\prime )\) can then be disintegrated into order indecomposable representations. The notions of \({\mathrm {L}^p}\)-direct integrals of Banach spaces and representations that are developed extend those in the literature.     

The finite Littlewood problem in $${{\mathbb {F}}}_p$$

Let p be a large prime number and f(x) be an integer-valued function defined in \({\mathbb F}_p\). The Littlewood problem in \({{\mathbb {F}}}_p\) is to establish non-trivial lower bounds for the \(\ell _1\) norm of exponential sums involving f(x). In the present paper, we establish new lower bounds for exponential sums including polynomials, powers of any primitive root and subgroups of \(\mathbb {F}_p^*.\)     

Counting signed swallowtails of polynomial selfmaps of {\mathbb {R}^3}

For a generic \({f \in C^\infty({\mathbb {R}}^3,{\mathbb {R}}^3)}\), there is a discrete set of swallowtail critical points. At any swallowtail point p there exists a well-oriented coordinate system centred at p, and a coordinate system centred at f(p), such that locally f has the form \({f_\pm(x, y, z) = (\pm xy+x^2 z+x^4, y, z)}\), so one may associate with p a sign \({I(f, p) \in \{\pm 1\}}\). We shall show how to compute the number of swallowtail points having the positive/negative sign, in the case where \({f : {\mathbb {R}}^3 \rightarrow {\mathbb {R}}^3}\) is a polynomial mapping, in terms of signatures of quadratic forms.     

Helicoidal surfaces satisfying {\Delta ^{II}\mathbf{G}=f(\mathbf{G}+C)}

In this paper, we study helicoidal surfaces without parabolic points in Euclidean 3-space \({\mathbb{R} ^{3}}\), satisfying the condition \({\Delta ^{II}\mathbf{G}=f(\mathbf{G}+C)}\), where \({\Delta ^{II}}\) is the Laplace operator with respect to the second fundamental form, f is a smooth function on the surface and C is a constant vector. Our main results state that helicoidal surfaces without parabolic points in \({ \mathbb{R} ^{3}}\) which satisfy the condition \({\Delta ^{II} \mathbf{G}=f(\mathbf{G}+C)}\), coincide with helicoidal surfaces with non-zero constant Gaussian curvature.     

Infra-nilmanifolds modeled on the group of uni-triangular matrices

Let \({\mathcal {N}}_m\) be the group of \(m\times m\) upper triangular real matrices with all the diagonal entries 1. Then it is an \((m-1)\)-step nilpotent Lie group, diffeomorphic to \({\mathbb {R}}^{\frac{1}{2} m(m-1)}\). It contains all the integer matrices as a lattice \(\Gamma _m\). The automorphism group of \({\mathcal {N}}_m \ (m\ge 4)\) turns out to be extremely small. In fact, \(\mathrm {Aut}({\mathcal {N}})=\mathcal {I} \rtimes \mathrm {Out}({\mathcal {N}})\), where \(\mathcal {I}\) is a connected, simply connected nilpotent Lie group, and \(\mathrm {Out}({\mathcal {N}})={{\tilde{K}}}={(\mathbb {R}^*)^{m-1}\rtimes \mathbb {Z}_2}\). With a nice left-invariant Riemannian metric on \({\mathcal {N}}\), the isometry group is \(\mathrm {Isom}({\mathcal {N}})= {\mathcal {N}} \rtimes K\), where \(K={(\mathbb {Z}_2)^{m-1}\rtimes \mathbb {Z}_2}\subset {{\tilde{K}}}\) is a maximal compact subgroup of \(\mathrm {Aut}({\mathcal {N}})\). We prove that, for odd \(m\ge 4\), there is no infra-nilmanifold which is essentially covered by the nilmanifold \(\Gamma _m\backslash {\mathcal {N}}_m\). For \(m=2n\ge 4\) (even), there is a unique infra-nilmanifold which is essentially (and doubly) covered by the nilmanifold \(\Gamma _m\backslash {\mathcal {N}}_m\).