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.     

On the Almost Sure Location of the Singular Values of Certain Gaussian Block-Hankel Large Random Matrices

This paper studies the almost sure location of the eigenvalues of matrices \({\mathbf{W}}_N {\mathbf{W}}_N^{*}\), where \({\mathbf{W}}_N = ({\mathbf{W}}_N^{(1)T}, \ldots , {\mathbf{W}}_N^{(M)T})^{T}\) is a \({\textit{ML}} \times N\) block-line matrix whose block-lines \(({\mathbf{W}}_N^{(m)})_{m=1, \ldots , M}\) are independent identically distributed \(L \times N\) Hankel matrices built from i.i.d. standard complex Gaussian sequences. It is shown that if \(M \rightarrow +\infty \) and \(\frac{{\textit{ML}}}{N} \rightarrow c_* (c_* \in (0, \infty ))\), then the empirical eigenvalue distribution of \({\mathbf{W}}_N {\mathbf{W}}_N^{*}\) converges almost surely towards the Marcenko–Pastur distribution. More importantly, it is established using the Haagerup–Schultz–Thorbjornsen ideas that if \(L = O(N^{\alpha })\) with \(\alpha < 2/3\), then, almost surely, for \(N\) large enough, the eigenvalues of \({\mathbf{W}}_N {\mathbf{W}}_N^{*}\) are located in the neighbourhood of the Marcenko–Pastur distribution. It is conjectured that the condition \(\alpha < 2/3\) is optimal.     

Central Limit Theorem for Linear Eigenvalue Statistics for a Tensor Product Version of Sample Covariance Matrices

For \(k,m,n\in {\mathbb {N}}\), we consider \(n^k\times n^k\) random matrices of the form

$$\begin{aligned} {\mathcal {M}}_{n,m,k}({\mathbf {y}})=\sum _{\alpha =1}^m\tau _\alpha {Y_\alpha }Y_\alpha ^T,\quad {Y}_\alpha ={\mathbf {y}}_\alpha ^{(1)}\otimes \cdots \otimes {\mathbf {y}}_\alpha ^{(k)}, \end{aligned}$$

where \(\tau _{\alpha }\), \(\alpha \in [m]\), are real numbers and \({\mathbf {y}}_\alpha ^{(j)}\), \(\alpha \in [m]\), \(j\in [k]\), are i.i.d. copies of a normalized isotropic random vector \({\mathbf {y}}\in {\mathbb {R}}^n\). For every fixed \(k\ge 1\), if the Normalized Counting Measures of \(\{\tau _{\alpha }\}_{\alpha }\) converge weakly as \(m,n\rightarrow \infty \), \(m/n^k\rightarrow c\in [0,\infty )\) and \({\mathbf {y}}\) is a good vector in the sense of Definition 1.1, then the Normalized Counting Measures of eigenvalues of \({\mathcal {M}}_{n,m,k}({\mathbf {y}})\) converge weakly in probability to a nonrandom limit found in Marchenko and Pastur (Math USSR Sb 1:457–483, 1967). For \(k=2\), we define a subclass of good vectors \({\mathbf {y}}\) for which the centered linear eigenvalue statistics \(n^{-1/2}{{\mathrm{Tr}}}\varphi ({\mathcal {M}}_{n,m,2}({\mathbf {y}}))^\circ \) converge in distribution to a Gaussian random variable, i.e., the Central Limit Theorem is valid.

     

Decorated marked surfaces: spherical twists versus braid twists

We are interested in the 3-Calabi-Yau categories \({\mathcal {D}}\) arising from quivers with potential associated to a triangulated marked surface \(\mathbf {S}\) (without punctures). We prove that the spherical twist group \(\mathrm{ST}\) of \({\mathcal {D}}\) is isomorphic to a subgroup (generated by braid twists) of the mapping class group of the decorated marked surface \({\mathbf {S}}_\bigtriangleup \). Here \({\mathbf {S}}_\bigtriangleup \) is the surface obtained from \(\mathbf {S}\) by decorating with a set of points, where the number of points equals the number of triangles in any triangulations of \(\mathbf {S}\). For instance, when \(\mathbf {S}\) is an annulus, the result implies that the corresponding spaces of stability conditions on \({\mathcal {D}}\) are contractible.     

Semigroups of rectangular matrices under a sandwich operation

Let \({\mathcal {M}}_{mn}={\mathcal {M}}_{mn}({\mathbb {F}})\) denote the set of all \(m\times n\) matrices over a field \({\mathbb {F}}\), and fix some \(n\times m\) matrix \(A\in {\mathcal {M}}_{nm}\). An associative operation \(\star \) may be defined on \({\mathcal {M}}_{mn}\) by \(X\star Y=XAY\) for all \(X,Y\in {\mathcal {M}}_{mn}\), and the resulting sandwich semigroup is denoted \({\mathcal {M}}_{mn}^A={\mathcal {M}}_{mn}^A({\mathbb {F}})\). These semigroups are closely related to Munn rings, which are fundamental tools in the representation theory of finite semigroups. We study \({\mathcal {M}}_{mn}^A\) as well as its subsemigroups \(\hbox {Reg}({\mathcal {M}}_{mn}^A)\) and \({\mathcal {E}}_{mn}^A\) (consisting of all regular elements and products of idempotents, respectively), and the ideals of \(\hbox {Reg}({\mathcal {M}}_{mn}^A)\). Among other results, we characterise the regular elements; determine Green’s relations and preorders; calculate the minimal number of matrices (or idempotent matrices, if applicable) required to generate each semigroup we consider; and classify the isomorphisms between finite sandwich semigroups \({\mathcal {M}}_{mn}^A({\mathbb {F}}_1)\) and \({\mathcal {M}}_{kl}^B({\mathbb {F}}_2)\). Along the way, we develop a general theory of sandwich semigroups in a suitably defined class of partial semigroups related to Ehresmann-style “arrows only” categories; we hope this framework will be useful in studies of sandwich semigroups in other categories. We note that all our results have applications to the variants \({\mathcal {M}}_n^A\) of the full linear monoid \({\mathcal {M}}_n\) (in the case \(m=n\)), and to certain semigroups of linear transformations of restricted range or kernel (in the case that \(\hbox {rank}(A)\) is equal to one of m, n).     

Sharp MSE Bounds for Proximal Denoising

Denoising has to do with estimating a signal \(\mathbf {x}_0\) from its noisy observations \(\mathbf {y}=\mathbf {x}_0+\mathbf {z}\). In this paper, we focus on the “structured denoising problem,” where the signal \(\mathbf {x}_0\) possesses a certain structure and \(\mathbf {z}\) has independent normally distributed entries with mean zero and variance \(\sigma ^2\). We employ a structure-inducing convex function \(f(\cdot )\) and solve \(\min _\mathbf {x}\{\frac{1}{2}\Vert \mathbf {y}-\mathbf {x}\Vert _2^2+\sigma {\lambda }f(\mathbf {x})\}\) to estimate \(\mathbf {x}_0\), for some \(\lambda >0\). Common choices for \(f(\cdot )\) include the \(\ell _1\) norm for sparse vectors, the \(\ell _1-\ell _2\) norm for block-sparse signals and the nuclear norm for low-rank matrices. The metric we use to evaluate the performance of an estimate \(\mathbf {x}^*\) is the normalized mean-squared error \(\text {NMSE}(\sigma )=\frac{{\mathbb {E}}\Vert \mathbf {x}^*-\mathbf {x}_0\Vert _2^2}{\sigma ^2}\). We show that NMSE is maximized as \(\sigma \rightarrow 0\) and we find the exact worst-case NMSE, which has a simple geometric interpretation: the mean-squared distance of a standard normal vector to the \({\lambda }\)-scaled subdifferential \({\lambda }\partial f(\mathbf {x}_0)\). When \({\lambda }\) is optimally tuned to minimize the worst-case NMSE, our results can be related to the constrained denoising problem \(\min _{f(\mathbf {x})\le f(\mathbf {x}_0)}\{\Vert \mathbf {y}-\mathbf {x}\Vert _2\}\). The paper also connects these results to the generalized LASSO problem, in which one solves \(\min _{f(\mathbf {x})\le f(\mathbf {x}_0)}\{\Vert \mathbf {y}-{\mathbf {A}}\mathbf {x}\Vert _2\}\) to estimate \(\mathbf {x}_0\) from noisy linear observations \(\mathbf {y}={\mathbf {A}}\mathbf {x}_0+\mathbf {z}\). We show that certain properties of the LASSO problem are closely related to the denoising problem. In particular, we characterize the normalized LASSO cost and show that it exhibits a “phase transition” as a function of number of observations. We also provide an order-optimal bound for the LASSO error in terms of the mean-squared distance. Our results are significant in two ways. First, we find a simple formula for the performance of a general convex estimator. Secondly, we establish a connection between the denoising and linear inverse problems.     

On height isotopy classes of embeddings in the plane of a Morse function of a circle

Let \(\mathrm{SM}_{2n}(S^1,\mathbb {R})\) be a set of stable Morse functions of an oriented circle such that the number of singular points is \(2n\in \mathbb {N}\) and the order of singular values satisfies the particular condition. For an orthogonal projection \(\pi :\mathbb {R}^2\rightarrow \mathbb {R}\), let \({\tilde{f}}_0\) and \({\tilde{f}}_1:S^1\rightarrow \mathbb {R}^2\) be embedding lifts of f. If there is an ambient isotopy \(\tilde{\varphi }_t:\mathbb {R}^2\rightarrow \mathbb {R}^2\) \((t\in [0,1])\) such that \({\pi \circ \tilde{\varphi }}_t(y_1,y_2)=y_1\) and \(\tilde{\varphi }_1\circ {\tilde{f}}_0={\tilde{f}}_1\), we say that \({\tilde{f}}_0\) and \({\tilde{f}}_1\) are height isotopic. We define a function \(I:\mathrm{SM}_{2n}(S^1,\mathbb {R})\rightarrow \mathbb {N}\) as follows: I(f) is the number of height isotopy classes of embeddings such that each rotation number is one. In this paper, we determine the maximal value of the function I equals the n-th Baxter number and the minimal value equals \(2^{n-1}\).     

Behaviour at the non-positive integers of Dirichlet series associated to polynomials of several variables

In this paper, we relate the special values at a non-positive integer \({\underline{\mathbf{s}}=(s_{1},\ldots, s_{r})= -\underline{\mathbf{N}}= (-N_{1},\ldots, -N_{r})}\) obtained by meromorphic continuation of the multiple Dirichlet series \({{Z(\underline{\mathbf{P}}, \underline{\mathbf{s}})=\sum_{\underline{m}\in {\mathbb{N}}^{*n}}{\frac{1}{\prod_{i=1}^{r}{P_{i}^{ s_{i}}(\underline{m})}}}}}\) to special values of the function \({Y(\underline{\mathbf{P}}, \underline{\mathbf{s}})=\int_{[1, +\infty[^{n}} {\prod_{i=1}^{r}{P_{i}^{- s_{i}}(\underline{\mathbf{x}})}\; d{\underline{\mathbf{x}}}}}\) where \({\underline{\mathbf{P}}=(P_{1},..., P_{r}),\; (r\geq 1)}\) are elliptic polynomials in “\({n}\) ” variables. We prove a simple relation between \({Z(\underline{\mathbf{P}}_{\underline{\mathbf{a}}}, -\underline{\mathbf{N}})}\) and \({Y(\underline{\mathbf{P}}_{\underline{\mathbf{a}}}, -\underline{\mathbf{N}})}\), such that for all \({\underline{\mathbf{a}} \in {\mathbb{R}}^{n}_{+}}\), we denote \({\underline{\mathbf{P}}_{\underline{\mathbf{a}}}:=(P_{1 \underline{\mathbf{a}}},\ldots, P_{r \underline{\mathbf{a}}})}\), where \({P_{i\;\underline{\mathbf{a}}}(\underline{\mathbf{x}}):= P_i(\underline{\mathbf{x}}+ \underline{\mathbf{a}})\; (1\leq i\leq r)}\) is the shifted polynomial.     

A System Coupling and Donoghue Classes of Herglotz–Nevanlinna Functions

We study the impedance functions of conservative L-systems with the unbounded main operators. In addition to the generalized Donoghue class \({\mathfrak {M}}_\kappa \) of Herglotz–Nevanlinna functions considered by the authors earlier, we introduce “inverse” generalized Donoghue classes \({\mathfrak {M}}_\kappa ^{-1}\) of functions satisfying a different normalization condition on the generating measure, with a criterion for the impedance function \(V_\Theta (z)\) of an L-system \(\Theta \) to belong to the class \({\mathfrak {M}}_\kappa ^{-1}\) presented. In addition, we establish a connection between “geometrical” properties of two L-systems whose impedance functions belong to the classes \({\mathfrak {M}}_\kappa \) and \({\mathfrak {M}}_\kappa ^{-1}\), respectively. In the second part of the paper we introduce a coupling of two L-system and show that if the impedance functions of two L-systems belong to the generalized Donoghue classes \({\mathfrak {M}}_{\kappa _1}\)(\({\mathfrak {M}}_{\kappa _1}^{-1}\)) and \({\mathfrak {M}}_{\kappa _2}\)(\({\mathfrak {M}}_{\kappa _2}^{-1}\)), then the impedance function of the coupling falls into the class \({\mathfrak {M}}_{\kappa _1\kappa _2}\). Consequently, we obtain that if an L-system whose impedance function belongs to the standard Donoghue class \({\mathfrak {M}}={\mathfrak {M}}_0\) is coupled with any other L-system, the impedance function of the coupling belongs to \({\mathfrak {M}}\) (the absorbtion property). Observing the result of coupling of n L-systems as n goes to infinity, we put forward the concept of a limit coupling which leads to the notion of the system attractor, two models of which (in the position and momentum representations) are presented. All major results are illustrated by various examples.     

Chebyshev polynomials of the second kind via raising operator preserving the orthogonality

It is well known that monic orthogonal polynomial sequences \(\{T_n\}_{n\ge 0}\) and \(\{U_n\}_{n\ge 0}\), the Chebyshev polynomials of the first and second kind, satisfy the relation \(DT_{n+1}=(n+1)U_n\) (\(n\ge 0\)). One can also easily check that the following “inverse” of the mentioned formula holds: \({\mathcal {U}}_{-1}(U_n)=(n+1)T_{n+1}\) (\(n\ge 0\)), where \({\mathcal {U}}_\xi =x(xD+{\mathbb {I}})+\xi D\) with \(\xi \) being an arbitrary nonzero parameter and \({\mathbb {I}}\) representing the identity operator. Note that whereas the first expression involves the operator D which lowers the degree by one, the second one involves \({\mathcal {U}}_\xi \) which raises the degree by one (i.e. it is a “raising operator”). In this paper it is shown that the scaled Chebyshev polynomial sequence \(\{a^{-n}U_n(ax)\}_{n\ge 0}\) where \(a^2=-\xi ^{-1}\), is actually the only monic orthogonal polynomial sequence which is \({\mathcal {U}}_\xi \)-classical (i.e. for which the application of the raising operator \({\mathcal {U}}_\xi \) turns the original sequence into another orthogonal one).     

A new generalization of Hermite’s reciprocity law

Given a partition \(\lambda \) of n, the Schur functor \({\mathbb {S}}_\lambda \) associates to any complex vector space V, a subspace \({\mathbb {S}}_\lambda (V)\) of \(V^{\otimes n}\). Hermite’s reciprocity law, in terms of the Schur functor, states that \({\mathbb {S}}_{(p)}\left( {\mathbb {S}}_{(q)}({\mathbb {C}}^2)\right) \simeq {\mathbb {S}}_{(q)}\left( {\mathbb {S}}_{(p)}({\mathbb {C}}^2)\right) . \) We extend this identity to many other identities of the type \({\mathbb {S}}_{\lambda }\left( {\mathbb {S}}_{\delta }({\mathbb {C}}^2)\right) \simeq {\mathbb {S}}_{\mu }\left( {\mathbb {S}}_{\epsilon }({\mathbb {C}}^2)\right) \).     

Normes cyclotomiques naïves et unités logarithmiques

We compute the \({\mathbb {Z}}\)-rank of the subgroup \(\widetilde{E}_K =\bigcap _{n\in {\mathbb {N}}} N_{K_n/K}(K_n^\times )\) of elements of the multiplicative group of a number field K that are norms from every finite level of the cyclotomic \({\mathbb {Z}}_\ell \)-extension \(K^c\) of K. Thus we compare its \(\ell \)-adification \({\mathbb {Z}}_\ell \otimes _{\mathbb {Z}}\widetilde{E}_K\) with the group of logarithmic units \(\widetilde{\varepsilon }_K\). By the way we point out an easy proof of the Gross–Kuz’min conjecture for \(\ell \)-undecomposed extensions of abelian fields.     

Uniqueness of self-shrinkers to the degree-one curvature flow with a tangent cone at infinity

Given a smooth, symmetric and homogeneous of degree one function \(f\left( \lambda _{1},\ldots ,\lambda _{n}\right) \) satisfying \(\partial _{i}f>0\quad \forall \,i=1,\ldots , n\), and a properly embedded smooth cone \({\mathcal {C}}\) in \({\mathbb {R}}^{n+1}\), we show that under suitable conditions on f, there is at most one f self-shrinker (i.e. a hypersurface \(\Sigma \) in \({\mathbb {R}}^{n+1}\) satisfying \(f\left( \kappa _{1},\ldots ,\kappa _{n}\right) +\frac{1}{2}X\cdot N=0\), where \(\kappa _{1},\ldots ,\kappa _{n}\) are principal curvatures of \(\Sigma \)) that is asymptotic to the given cone \({\mathcal {C}}\) at infinity.     

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 \(\Gamma _m\backslash {\mathcal {N}}_m\).     

A Note on the Deformations of Almost Complex Structures on Closed Four-Manifolds

In this paper, we calculate the dimension of the J-anti-invariant cohomology subgroup \(H_J^-\) on \({\mathbb {T}}^4\). Inspired by the concrete example, \({\mathbb {T}}^4\), we get that: on a closed symplectic 4-dimensional manifold \((M, \omega )\), \(h^-_J=0\) for generic \(\omega \)-compatible almost complex structures.     

There is exactly one $${\mathbb {Z}}_2{\mathbb {Z}}_4$$-cyclic 1-perfect code

Let \(\mathcal{C}\) be a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-additive code of length \(n > 3\). We prove that if the binary Gray image of \(\mathcal{C}\) is a 1-perfect nonlinear code, then \(\mathcal{C}\) cannot be a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-cyclic code except for one case of length \(n=15\). Moreover, we give a parity check matrix for this cyclic code. Adding an even parity check coordinate to a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-additive 1-perfect code gives a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-additive extended 1-perfect code. We also prove that such a code cannot be \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-cyclic.     

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.