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This paper considers comparison problems of predictor and estimator in the context of seemingly unrelated regression models ( $$\mathrm{SURM}$$ s). $$\mathrm{SURM}$$ s are a class of multiple regression equations with correlated errors among the equations from linear regression models. Our aim is to establish a variety of equalities and inequalities for comparing covariance matrices of the best linear unbiased predictors ( $$\mathrm{BLUP}$$ s) and the ordinary least squares predictors ( $$\mathrm{OLSP}$$ s) of unknown vectors under $$\mathrm{SURM}$$ s by using various rank and inertia formulas of block matrices. The results for comparisons of the best linear unbiased estimators ( $$\mathrm{BLUE}$$ s) and the ordinary least squares estimators ( $$\mathrm{OLSE}$$ s) in the models are also considered.

Let $G$ be a finite group, and let $\Delta(G)$ be the prime graph built on the set of conjugacy class sizes of $G$: this is the simple undirected graph whose vertices are the prime numbers dividing some conjugacy class size of $G$, two vertices $p$ and $q$ being adjacent if and only if $pq$ divides some conjugacy class size of $G$. In the present paper, we classify the finite groups $G$ for which $\Delta(G)$ has a cut vertex.

In a recent pair of papers, Gorin and Shkolnikov (Ann Probab 46: 2287–2344, 2018) and Hariya (Electron Commun Probab 21: 6, 2016) have shown that the area under normalized Brownian excursion minus one half the integral of the square of its total local time is a centered normal random variable with variance $$\frac{1}{12}$$ . Lamarre and Shkolnikov generalized this to Brownian bridges (Lamarre and Shkolnikov in Ann Inst Henri Poincare Probab Stat 55: 1402–1438, 2019) and ask for a combinatorial interpretation. We provide a combinatorial interpretation using random forests on n vertices. In particular, we show that there is a process level generalization for a certain infinite forest model. We also show analogous results for a variety of other related models using stochastic calculus.

This paper considers the problem of solving a special quartic–quadratic optimization problem with a single sphere constraint, namely, finding a global and local minimizer of $$\frac{1}{2}\mathbf {z}^{*}A\mathbf {z}+\frac{\beta }{2}\sum _{k=1}^{n}|z_{k}|^{4}$$ such that $$\Vert \mathbf {z}\Vert _{2}=1$$ . This problem spans multiple domains including quantum mechanics and chemistry sciences and we investigate the geometric properties of this optimization problem. Fourth-order optimality conditions are derived for characterizing local and global minima. When the matrix in the quadratic term is diagonal, the problem has no spurious local minima and global solutions can be represented explicitly and calculated in $$O(n\log {n})$$ operations. When A is a rank one matrix, the global minima of the problem are unique under certain phase shift schemes. The strict-saddle property, which can imply polynomial time convergence of second-order-type algorithms, is established when the coefficient $$\beta $$ of the quartic term is either at least $$O(n^{3/2})$$ or not larger than O(1). Finally, the Kurdyka–Łojasiewicz exponent of quartic–quadratic problem is estimated and it is shown that the largest exponent is at least 1/4 for a broad class of stationary points.

This paper deals with lattice congruences of the weak order on the symmetric group, and initiates the investigation of the cover graphs of the corresponding lattice quotients. These graphs also arise as the skeleta of the so-called quotientopes, a family of polytopes recently introduced by Pilaud and Santos [Bull. Lond. Math. Soc., 51:406–420, 2019], which generalize permutahedra, associahedra, hypercubes and several other polytopes. We prove that all of these graphs have a Hamilton path, which can be computed by a simple greedy algorithm. This is an application of our framework for exhaustively generating various classes of combinatorial objects by encoding them as permutations. We also characterize which of these graphs are vertex-transitive or regular via their arc diagrams, give corresponding precise and asymptotic counting results, and we determine their minimum and maximum degrees.\ud \ud

Consider the generalized absolute value function defined by \[ a(t) = \vert t \vert t^{n-1}, \qquad t \in \mathbb{R}, n \in \mathbb{N}_{\geq 1}. \] Further, consider the $n$-th order divided difference function $a^{[n]}: \mathbb{R}^{n+1} \rightarrow \mathbb{C}$ and let $1 < p_1, \ldots, p_n < \infty$ be such that $\sum_{l=1}^n p_l^{-1} = 1$. Let $\mathcal{S}_{p_l}$ denote the Schatten-von Neumann ideals and let $\mathcal{S}_{1,\infty}$ denote the weak trace class ideal. We show that for any $(n+1)$-tuple ${\bf A}$ of bounded self-adjoint operators the multiple operator integral $T_{a^{[n]}}^{\bf A}$ maps $\mathcal{S}_{p_1} \times \ldots \times \mathcal{S}_{p_n}$ to $\mathcal{S}_{1, \infty}$ boundedly with uniform bound in ${\bf A}$. The same is true for the class of $C^{n+1}$-functions that outside the interval $[-1, 1]$ equal $a$. In [CLPST16] it was proved that for a function $f$ in this class such boundedness of $T^{ {\bf A} }_{f^{[n]}}$ from $\mathcal{S}_{p_1} \times \ldots \times \mathcal{S}_{p_n}$ to $\mathcal{S}_{1}$ may fail, resolving a problem by V. Peller. This shows that the estimates in the current paper are optimal. The proof is based on a new reduction method for arbitrary multiple operator integrals of divided differences., to appear in Israel Journal of Mathematics

For any group $G$ and any set $A$, consider the shift action of $G$ on the full shift $A^G$. A configuration $x \in A^G$ has \emph{least period} $H \leq G$ if the stabiliser of $x$ is precisely $H$. Among other things, the number of such configurations is interesting as it provides an upper bound for the size of the corresponding $\text{Aut}(A^G)$-orbit. In this paper we show that if $G$ is finitely generated and $H$ is of finite index, then the number of configurations in $A^G$ with least period $H$ may be computed using the M\"obius function of the lattice of subgroups of finite index in $G$. Moreover, when $H$ is a normal subgroup, we classify all situations such that the number of $G$-orbits with least period $H$ is at most $10$., Comment: 8 pages

The goal of this paper is to describe the relationship between generalized B-opers, generalized $$\mathrm {SO}(2n,{\mathbb {C}})$$ -opers and (G, P)-opers. In particular, we show that to each generalized B-oper there is a naturally associated (G, P)-oper, but there are some (G, P)-opers that do not arise as generalized B-opers or $$\mathrm {SO}(2n,{\mathbb {C}})$$ -opers.

In this paper, we study the relationship between wide subcategories and torsion classes of an abelian length category $\mathcal {A}$ from the point of view of lattice theory. Motivated by τ-tilting reduction of Jasso, we mainly focus on intervals $[\mathcal {U},\mathcal {T}]$ in the lattice $\operatorname {\mathsf {tors}} \mathcal {A}$ of torsion classes in $\mathcal {A}$ such that $\mathcal {W}:=\mathcal {U}^{\perp } \cap \mathcal {T}$ is a wide subcategory of $\mathcal {A}$ ; we call these intervals wide intervals. We prove that a wide interval $[\mathcal {U},\mathcal {T}]$ is isomorphic to the lattice $\operatorname {\mathsf {tors}} \mathcal {W}$ of torsion classes in the abelian category $\mathcal {W}$ . We also characterize wide intervals in two ways: First, in purely lattice theoretic terms based on the brick labeling established by Demonet–Iyama–Reading–Reiten–Thomas; and second, in terms of the Ingalls–Thomas correspondences between torsion classes and wide subcategories, which were further developed by Marks–Sťovicek.

Let $$G_n:=U(n)\ltimes {\mathbb {H}}_n$$ be the semidirect product of the unitary group acting by automorphisms on the Heisenberg group $${\mathbb {H}}_n.$$ It was pointed out in [5], that the unitary dual $$\widehat{G_n}$$ of $$G_n$$ is homeomorphic to the space of admissible coadjoint orbits $${\mathfrak {g}}_n^\ddagger /G_n$$ of $$G_n.$$ One of the important subset of $$\widehat{G_n}$$ is what is called the cortex of $$G_n,$$ cor $$(G_n),$$ which is the set of all $$\pi \in \widehat{G_n}$$ that cannot be Hausdorff separated from the identity representation $$1_{G_n}$$ of $$G_n.$$ In the present paper, we use the orbit method of Lipsman to determine the cortex of $$G_n.$$

The equitable tree-coloring can formulate a structure decomposition problem on the communication network with some security considerations. Namely, an equitable tree-$k$-coloring of a graph is a vertex coloring using $k$ distinct colors such that every color class induces a forest and the sizes of any two color classes differ by at most one. In this paper, we show some theoretical results on the equitable tree-coloring of graphs by proving that every $d$-degenerate graph with maximum degree at most $\Delta$ is equitably tree-$k$-colorable for every integer $k\geq (\Delta+1)/2$ provided that $\Delta\geq 9.818d$, confirming the equitable vertex arboricity conjecture for graphs with low degeneracy., Comment: This is the final version acceptted for publication in Acta Mathematica Sinica, English Series, with an improvement of the lower bound for the maximum degree and with the title changed

Many practical integer programming problems involve variables with one or two-sided bounds. Dunkel and Schulz (A refined Gomory–Chvatal closure for polytopes in the unit cube, http://www.optimization-online.org/DB_FILE/2012/03/3404.pdf , 2012) considered a strengthened version of Chvatal–Gomory (CG) inequalities that use 0–1 bounds on variables, and showed that the set of points in a rational polytope that satisfy all these strengthened inequalities is a polytope. Recently, we generalized this result by considering strengthened CG inequalities that use all variable bounds. In this paper, we generalize further by considering not just variable bounds, but general linear constraints on variables. We show that all points in a rational polyhedron that satisfy such strengthened CG inequalities form a rational polyhedron. We also extend this polyhedrality result to mixed-integer sets defined by linear constraints.

In this paper we investigate the spectra and the ergodic properties of the multiplication operators and the convolution operators acting on the Schwartz space $${\mathcal S}({\mathbb R})$$ S ( R ) of rapidly decreasing functions, i.e., operators of the form $$M_h: {\mathcal S}({\mathbb R})\rightarrow {\mathcal S}({\mathbb R})$$ M h : S ( R ) → S ( R ) , $$f \mapsto h f $$ f ↦ h f , and $$C_T:{\mathcal S}({\mathbb R})\rightarrow {\mathcal S}({\mathbb R})$$ C T : S ( R ) → S ( R ) , $$f\mapsto T\star f$$ f ↦ T ⋆ f . Precisely, we determine their spectra and characterize when those operators are power bounded and mean ergodic.

Linear projections from $$\mathbb {P}^k$$ P k to $$\mathbb {P}^h$$ P h appear in computer vision as models of images of dynamic or segmented scenes. Given multiple projections of the same scene, the identification of sufficiently many correspondences between the images allows, in principle, to reconstruct the position of the projected objects. A critical locus for the reconstruction problem is a variety in $$\mathbb {P}^k$$ P k containing the set of points for which the reconstruction fails. Critical loci turn out to be determinantal varieties. In this paper we determine and classify all the smooth critical loci, showing that they are classical projective varieties.

Let G be a simple, connected graph and let A(G) be the adjacency matrix of G. If D(G) is the diagonal matrix of the vertex degrees of G, then for every real $$\alpha \in [0,1]$$ , the matrix $$A_{\alpha }(G)$$ is defined as $$A_{\alpha }(G) = \alpha D(G) + (1- \alpha ) A(G).$$ The eigenvalues of the matrix $$A_{\alpha }(G)$$ form the $$A_{\alpha }$$ -spectrum of G. Let $$G_1 {\dot{\vee }} G_2$$ , $$G_1 {\underline{\vee }} G_2$$ , $$G_1 \langle \text {v} \rangle G_2$$ and $$G_1 \langle \text {e} \rangle G_2$$ denote the subdivision-vertex join, subdivision-edge join, R-vertex join and R-edge join of two graphs $$G_1$$ and $$G_2$$ , respectively. In this paper, we compute the $$A_{\alpha }$$ -spectra of $$G_1 {\dot{\vee }} G_2$$ , $$G_1 {\underline{\vee }} G_2$$ , $$G_1 \langle \text {v} \rangle G_2$$ and $$G_1 \langle \text {e} \rangle G_2$$ for a regular graph $$G_1$$ and an arbitrary graph $$G_2$$ in terms of their $$A_{\alpha }$$ -eigenvalues. As an application of these results, we construct infinitely many pairs of $$A_{\alpha }$$ -cospectral graphs.

In this paper, criteria for strict convexity of Orlicz sequence spaces equipped with the p-Amemiya norms $$(1\le p\le \infty )$$ are presented. The obtained results widen the characterization of strict convexity of Orlicz sequence spaces. From our results it follows that there are Orlicz sequence spaces which are strictly convex for p-Amemiya norms with $$1

Slater introduced the point-addition operation on graphs to characterize 4-connected graphs. The $$\varGamma $$ -extension operation on binary matroids is a generalization of the point-addition operation. In this paper, for a given binary matroid M we characterize the closure operator and the flats of the $$\varGamma $$ -extension binary matroid $$M^X$$ in terms of the closure operator of original binary matroid M.

Schmieder and Shiba investigated conformal embeddings of a marked open Riemann surface R of finite genus g into closed Riemann surfaces of the same genus, and they showed that for each given j $$(1\le j \le g)$$ , the set $${\mathfrak {M}}_j$$ of j-th diagonal elements of the period matrices of all closings of R is a closed disk. In this paper, for a marked open Riemann surface R of genus g $$(1 \le g < \infty )$$ and a real g-vector $$\mathbf{a}=(a_1, \ldots , a_g)\ne \mathbf{0}$$ , we introduce the $$\mathbf{a}$$ -span $$\rho _\mathbf{a}$$ of R, defined in terms of $$L_1$$ - and $$L_0$$ -canonical semi-exact differentials normalized by $$\mathbf{a}$$ , and establish a new relation between $$\rho _\mathbf{a}$$ and $${\mathfrak {M}}_j$$ . From the viewpoint of several complex variables, a variational formula of $$\rho _\mathbf{a}(t)$$ is obtained for a smooth family $${\mathcal {R}}$$ of open Riemann surfaces R(t) with a complex parameter t in a disk $$\varDelta $$ . As an application, we prove that if $${\mathcal {R}}$$ is a two-dimensional pseudoconvex domain fibered by open Riemann surfaces of the same topological type, then $$\rho _\mathbf{a}(t)$$ is subharmonic on $$\varDelta $$ . This means the subharmonicity of the diameter of the $$\mathbf{a}$$ -directional moduli disk for higher genera when $${\mathcal {R}}$$ is pseudoconvex.

When $$A=3$$ , the positive integral solutions of the so-called Markoff equation $$\begin{aligned} M_A:x^2 + y^2 + z^2 = Axyz \end{aligned}$$ can be generated from the single solution (1, 1, 1) by the action of certain automorphisms of the hypersurface. Since Markoff’s proof of this fact, several authors have showed that the structure of $$M_A(R)$$ , when R is $${\mathbb Z}[i]$$ or certain orders in number fields, behave in a similar fashion. Moreover, for $$R={\mathbb Z}$$ and $$R={\mathbb Z}[i]$$ , Zagier and Silverman, respectively, have found asymptotic formulae for the number of integral points of bounded height. In this paper, we investigate these problems when R is a polynomial ring over a field K of odd characteristic. We characterize the set $$M_A(K[t])$$ in a similar fashion as Markoff and previous authors. We also give an asymptotic formula that is similar to Zagier’s and Silverman’s formula.

In this paper, the Harnack inequalities for $G$-SDEs with degenerate noise are derived by method of coupling by change of measure. Moreover, the gradient estimate for the associated nonlinear semigroup $\bar{P}_t$ $$|\nabla \bar{P}_t f|\leq c(p,t)(\bar{P}_t |f|^p)^{\frac{1}{p}}, \ \ p>1, t>0$$ is also obtained for bounded and continuous function $f$. As an application of Harnack inequality, we prove the weak existence of degenerate $G$-SDEs under some integrable conditions. Finally, an example is presented. All of the above results extends the existed results in the linear expectation setting., 17 pages

The edge-connectivity of a connected hypergraph H is the minimum number of edges (named as edge-cut) whose removal makes H disconnected. It is known that the edge-connectivity of a hypergraph is bounded above by its minimum degree. H is super edge-connected, if every edge-cut consists of edges incident with a vertex of minimum degree. A hypergraph H is linear if any two edges of H share at most one vertex. We call H uniform if all edges of H have the same cardinality. Sufficient conditions for equality of edge-connectivity and minimum degree of graphs and super edge-connected graphs are known. In this paper, we present a generalization of some of these sufficient conditions to linear and/or uniform hypergraphs.

Let $$K = \mathbb {Q} (\alpha )$$ be a pure number field generated by a complex root $$\alpha$$ of a monic irreducible polynomial $$F(x) = x^{20}-m$$ , with $$m \ne \mp 1$$ a square free rational integer. In this paper, we study the monogenity of K. We prove that if $$m\not \equiv 1\ \text{(mod } {4})$$ and $$\overline{m}\not \in \{\overline{1}, \overline{7}, \overline{18}, \overline{24}\} \ \text{(mod } {25})$$ , then K is monogenic. But if $$m\equiv 1\ \text{(mod } {16})$$ or $$m\equiv 1\ \text{(mod } {25})$$ , then K is not monogenic. Some illustrating examples are given too.

Let $$z_0$$ and $$w_0$$ be given points in the open unit disk $$\mathbb {D}$$ with $$|w_0| < |z_0|$$ , and $$\mathcal {H}_0$$ be the class of all analytic self-maps f of $$\mathbb {D}$$ normalized by $$f(0)=0$$ . In this paper, we establish the third-order Dieudonne’s Lemma and apply it to explicitly determine the variability region $$\{f'''(z_0): f\in \mathcal {H}_0,f(z_0) =w_0, f'(z_0)=w_1\}$$ for given $$z_0,w_0,w_1$$ and give the form of all the extremal functions.

Let $$K = \mathbb {Q} (\alpha )$$ be a pure number field generated by a root $$\alpha$$ of a monic irreducible polynomial $$F(x) = x^{3^r} -m$$ , with $$m \ne \pm 1$$ is a square-free rational integer and r is a positive integer. In this paper, we study the monogenity of K. We prove that if $$m \not \equiv \pm 1 \ \text { (mod }{9})$$ , then K is monogenic. We give also sufficient conditions on r and m for K to be not monogenic. Some illustrating examples are given too.

Let $${{\mathfrak {A}}}\, $$ and $${{\mathfrak {A}}}\, '$$ be two $$C^*$$ -algebras with identities $$I_{{{\mathfrak {A}}}\, }$$ and $$I_{{{\mathfrak {A}}}\, '}$$ , respectively, and $$P_1$$ and $$P_2 = I_{{{\mathfrak {A}}}\, } - P_1$$ nontrivial projections in $${{\mathfrak {A}}}\, $$ . In this paper, we study the characterization of multiplicative $$*$$ -Lie–Jordan-type maps, where the notion of these maps arise here. In particular, if $${\mathcal {M}}_{{{\mathfrak {A}}}\, }$$ is a von Neumann algebra relative $$C^{*}$$ -algebra $${{\mathfrak {A}}}\, $$ without central summands of type $$I_1$$ then every bijective unital multiplicative $$*$$ -Lie–Jordan-type maps are $$*$$ -ring isomorphisms.

A finite group $${{{\mathbb {G}}}}$$ is said to have perfect order subsets if for every d, the number of elements of $$ {{{\mathbb {G}}}}$$ of order d (if there are any) divides $$|{{{\mathbb {G}}}}|.$$ In this paper, we make a modest improvement on the result of Ford, Konyagin and Luca [3] about perfect order subsets.

Over the past two decades several different approaches to defining a geometry over$${{\mathbb F}_1}$$F1have been proposed. In this paper, relying on Toën and Vaquié’s formalism (J.K-Theory 3: 437–500, 2009), we investigate a new category$${\mathsf {Sch}}_{\widetilde{{\mathsf B}}}$$SchB~of schemes admitting a Zariski cover by affine schemes relative to the category of blueprints introduced by Lorscheid (Adv. Math. 229: 1804–1846, 2012). A blueprint, which may be thought of as a pair consisting of a monoidMand a relation on the semiring$$M\otimes _{{{\mathbb F}_1}} {\mathbb N}$$M⊗F1N, is a monoid object in a certain symmetric monoidal category$${\mathsf B}$$B, which is shown to be complete, cocomplete, and closed. We prove that every$${\widetilde{{\mathsf B}}}$$B~-scheme$$\Sigma $$Σcan be associated, through adjunctions, with both a classical scheme$$\Sigma _{\mathbb Z}$$ΣZand a scheme$$\underline{\Sigma }$$Σ̲over$${{\mathbb F}_1}$$F1in the sense of Deitmar (in van der Geer G., Moonen B., Schoof R. (eds.) Progress in mathematics 239, Birkhäuser, Boston, 87–100, 2005), together with a natural transformation$$\Lambda :\Sigma _{\mathbb Z}\rightarrow \underline{\Sigma }\otimes _{{{\mathbb F}_1}}{\mathbb Z}$$Λ:ΣZ→Σ̲⊗F1Z. Furthermore, as an application, we show that the category of “$${{\mathbb F}_1}$$F1-schemes” defined by Connes and Consani in (Compos. Math. 146: 1383–1415, 2010) can be naturally merged with that of$${\widetilde{{\mathsf B}}}$$B~-schemes to obtain a larger category, whose objects we call “$${{\mathbb F}_1}$$F1-schemes with relations”.

In this paper, we address the additivity of $n$-multiplicative isomorphisms and $n$-multiplicative derivations on Gamma rings. We proved that, if $\M$ is a $\Gamma$-ring satisfying the some conditions, then any $n$-multiplicative isomorphism $\left(\varphi, \phi\right)$ of $\M$ onto an arbitrary gamma ring is additive., Comment: 10 pages

In this paper, we consider compositions of integers when only parts of size at most 3 are allowed in both composition and its conjugate composition. First we obtain a relation between the number of such compositions and the Fibonacci numbers. Then we provide combinatorial identities between these compositions and compositions into 1’s and 2’s, compositions into odd parts, and compositions into parts greater than 1. Further, several generalized results for these compositions are obtained.

Let $p\equiv -q \equiv 5\pmod 8$ be two prime integers. In this paper, we investigate the unit groups of the fields $ L_1 =\mathbb{Q}(\sqrt 2, \sqrt{p}, \sqrt{q}, \sqrt{-1} )$ and $ L_1^+=\mathbb{Q}(\sqrt 2, \sqrt{p}, \sqrt{q} )$. Furthermore , we give the second $ 2$-class groups of the subextensions of $L_1$ as well the $2$-class groups of the fields $ L_n =\mathbb{Q}( \sqrt{p}, \sqrt{q}, \zeta_{2^{n+2}} )$ and their maximal real subfelds., Comment: 18 pages

In 2005, the second author and Todorov introduced an upper bound on the finitistic dimension of an Artin algebra, now known as the {\phi}-dimension. The {\phi}-dimension conjecture states that this upper bound is always finite, a fact that would imply the finitistic dimension conjecture. In this paper, we present a counterexample to the {\phi}-dimension conjecture and explain where it comes from. We also discuss implications for further research and the finitistic dimension conjecture., Comment: final version

We deal with an alienation problem for an Euler–Lagrange type functional equation $$\begin{aligned} f(\alpha x + \beta y) + f(\alpha x - \beta y) = 2\alpha ^2f(x) + 2\beta ^2f(y) \end{aligned}$$ f ( α x + β y ) + f ( α x - β y ) = 2 α 2 f ( x ) + 2 β 2 f ( y ) assumed for fixed nonzero real numbers $$\alpha ,\beta ,\, 1 \ne \alpha ^2 \ne \beta ^2$$ α , β , 1 ≠ α 2 ≠ β 2 , and the classic quadratic functional equation $$\begin{aligned} g(x+y) + g(x-y) = 2g(x) + 2g(y). \end{aligned}$$ g ( x + y ) + g ( x - y ) = 2 g ( x ) + 2 g ( y ) . We were inspired by papers of Kim et al. (Abstract and applied analysis, vol. 2013, Hindawi Publishing Corporation, 2013) and Gordji and Khodaei (Abstract and applied analysis, vol. 2009, Hindawi Publishing Corporation, 2009), where the special case $$g = \gamma f$$ g = γ f was examined.

It is well-known that the notion of limit in the sharp topology of sequences of Colombeau generalized numbers$$\widetilde{\mathbb {R}}$$R~does not generalize classical results. E.g. the sequence$$\frac{1}{n}\not \rightarrow 0$$1n↛0and a sequence$$(x_{n})_{n\in \mathbb {N}}$$(xn)n∈Nconvergesifand only if$$x_{n+1}-x_{n}\rightarrow 0$$xn+1-xn→0. This has several deep consequences, e.g. in the study of series, analytic generalized functions, or sigma-additivity and classical limit theorems in integration of generalized functions. The lacking of these results is also connected to the fact that$$\widetilde{\mathbb {R}}$$R~is necessarily not a complete ordered set, e.g. the set of all the infinitesimals has neither supremum nor infimum. We present a solution of these problems with the introduction of the notions of hypernatural number, hypersequence, close supremum and infimum. In this way, we can generalize all the classical theorems for the hyperlimit of a hypersequence. The paper explores ideas that can be applied to other non-Archimedean settings.

Subset selection for matrices is the task of extracting a column sub-matrix from a given matrix $B\in\mathbb{R}^{n\times m}$ with $m>n$ such that the pseudoinverse of the sampled matrix has as small Frobenius or spectral norm as possible. In this paper, we consider a more general problem of subset selection for matrices that allows a block to be fixed at the beginning. Under this setting, we provide a deterministic method for selecting a column sub-matrix from $B$. We also present a bound for both the Frobenius and spectral norms of the pseudoinverse of the sampled matrix, showing that the bound is asymptotically optimal. The main technology for proving this result is the interlacing families of polynomials developed by Marcus, Spielman, and Srivastava. This idea also results in a deterministic greedy selection algorithm that produces the sub-matrix promised by our result.

Let X be a complex quasi-projective algebraic variety. In this paper we study the mixed Hodge structures of the symmetric products $$\mathrm {Sym}^{n}X$$ when the cohomology of X is given by exterior products of cohomology classes with odd degree. We obtain an expression for the equivariant mixed Hodge polynomials $$\mu _{X^{n}}^{S_{n}}\left( t,u,v\right) $$ , codifying the permutation action of $$S_{n}$$ as well as its subgroups. This allows us to deduce formulas for the mixed Hodge polynomials of its symmetric products $$\mu _{\mathrm {Sym}^{n}X}\left( t,u,v\right) $$ . These formulas are then applied to the case of linear algebraic groups.

Let p be an odd prime, and let m be a positive integer satisfying $$p^m \equiv 3~(\text {mod }4).$$ Let $$\mathbb {F}_{p^m}$$ be the finite field with $$p^m$$ elements, and let $$R=\mathbb {F}_{p^m}[u]/\left\langle u^2\right\rangle$$ be the finite commutative chain ring with unity. In this paper, we determine all constacyclic codes of length $$4p^s$$ over R and their dual codes, where s is a positive integer. We also determine their sizes and list some isodual constacyclic codes of length $$4p^s$$ over R.

In this paper we characterize the algebraic structure of two-dimensional $$(\alpha ,\beta )$$ -constacyclic codes of arbitrary length $$s\ell $$ and of their duals over a finite field $$\mathbb{F}_q $$ , where $$\alpha ,\beta$$ are non zero elements of $$\mathbb{F}_q $$ . For $$\alpha ,\beta \in \{1,-1\}$$ , we give necessary and sufficient conditions for a two-dimensional $$(\alpha ,\beta )$$ -constacyclic code to be self-dual. We also show that a two-dimensional $$(\alpha ,1 )$$ -constacyclic code $${\mathcal {C}}$$ of length $$n=s\ell $$ cannot be self-dual if $$\gcd (s,q)= 1$$ . Finally, we give some examples of self-dual, isodual, MDS and quasi-twisted codes corresponding to two-dimensional $$(\alpha ,\beta )$$ -constacyclic codes.

Let \(S\) be a transformation semigroup of degree \(n\). To each element \(s\in S\) we associate a permutation group \(G_R(s)\) acting on the image of \(s\), and we find a natural generating set for this group. It turns out that the \(\mathcal{R}\)-class of \(s\) is a disjoint union of certain sets, each having size equal to the size of \(G_R(s)\). As a consequence, we show that two \(\mathcal{R}\)-classes containing elements with equal images have the same size, even if they do not belong to the same \(\mathcal{D}\)-class. By a certain duality process we associate to \(s\) another permutation group \(G_L(s)\) on the image of \(s\), and prove analogous results for the \(\mathcal{L}\)-class of \(S\). Finally we prove that the Schutzenberger group of the \(\mathcal{H}\)-class of \(s\) is isomorphic to the intersection of \(G_R(s)\) and \(G_L(s)\). The results of this paper can also be applied in new algorithms for investigating transformation semigroups, which will be described in a forthcoming paper.

This paper is a sequel to [5]. The starting point of both papers is a conjecture of the author and W. Dwyer [2]. Suppose that F is a number field, with ring of integers OF . Let ` denote an odd prime and let R = OF [1/`]. Let J∗ a denote the mod ` cohomology of the homotopy-fibre of the ` -th power mapSU→SU (all homology and cohomology groups in this paper have Z/` coefficients). Thus as an algebra, J∗ a is the tensor product of the polynomial algebra P ∗ a dual toH∗BU /(` -th powers) and a companion exterior algebra E∗ a . Let P denote the algebra of Steenrod `-th power operations. Conjecture A. Suppose F contains the `-th roots of unityμ`. Then as P-Hopf algebras H∗(GLR;Z/`) ∼= (H∗BU)s ⊗ (H∗U)r2+s ⊗ J∗ a1 ⊗ ...⊗ J∗ am

The non-commutative analytic Toeplitz algebra is the wot– closed algebra generated by the left regular representation of the free semigroup on n generators. We develop a detailed picture of the algebraic structure of this algebra. In particular, we show that there is a canonical homomorphism of the automorphism group onto the group of conformal automorphisms of the complex n-ball. The k-dimensional representations form a generalized maximal ideal space with a canonical surjection onto the ball of k × kn matrices which is a homeomorphism over the open ball analogous to the fibration of the maximal ideal space of H∞ over the unit disk. In [6, 17, 18, 20], a good case is made that the appropriate analogue for the analytic Toeplitz algebra in n non-commuting variables is the wotclosed algebra generated by the left regular representation of the free semigroup on n generators. The papers cited obtain a compelling analogue of Beurling’s theorem and inner–outer factorization. In this paper, we add further evidence. The main result is a short exact sequence determined by a canonical homomorphism of the automorphism group onto this algebra onto the group of conformal automorphisms of the unit ball of Cn. The kernel is the subgroup of quasi-inner automorphisms, which are trivial modulo the wot-closed commutator ideal. Additional evidence of analytic properties comes from the structure of k-dimensional (completely contractive) representations, which have a structure very similar to the fibration of the maximal ideal space of H∞ over the unit disk. An important tool in our analysis is a detailed structure theory for wot-closed right ideals. Curiously, left ideals remain more obscure. The non-commutative analytic Toeplitz algebra Ln is determined by the left regular representation of the free semigroup Fn on n generators z1, . . . , zn which acts on `2(Fn) by λ(w)ξv = ξwv for v, w in Fn. In particular, the algebra Ln is the unital, wot-closed algebra generated by the isometries Li = λ(zi) for 1 ≤ i ≤ n. This algebra and its norm-closed version (the noncommutative disk algebra) were introduced by Popescu [19] in an abstract sense in connection with a non-commutative von Neumann inequality and 1991 Mathematics Subject Classification. 47D25. March 9, 1997; October 9, 1997 final draft. First author partially supported by an NSERC grant and a Killam Research Fellowship. Second author partially supported by an NSF grant.

Almost perfect sequences with $ \theta = 2 $ are studied in this paper. Recently Arasu, Ma and Voss [1] studied such sequences and they could only obtain sequences having periods 8, 12 and 28. In this paper, we prove that no other almost perfect sequences exist for the case $ \theta = 2 $ .

In a previous paper the authors introduced seven complete congruences on the lattice ℒev(ℛI of e-varieties of regular semigroups of the form ρP:UρPV⇔P∘U=P∘V, whereP is drawn from a small set of e-varieties: left zero, right zero, rectangular bands, groups, left groups, right groups and completely simple semigroups. Four new complete congruences are introduced here of the form αP:UαPV⇔P∩U=P∩V, whereP is one of the following classes of regular semigroups: left monoids, right monoids, monoids, idempotent generated semigroups. For each complete congruence ρ on ℒev(ℛI) and eachU∈ℒev(ℛI), the ρ-class ofU is an interval [Uρ,Uρ] so that there is associated with each such congruence an idempotent operatorU→Uρ on ℒev(ℛI). This paper establishes numerous results concerning the commutativity of operators of this form.

Given the Cayley graph of a finitely generated group $G$, with respect to a presentation $G^{\alpha}$ with $n$ generators, the quotient of the rank of the fundamental group of subgraphs of the Cayley graph by the cardinality of the set of vertices of the subgraphs gives rise to the definition of the normalized cyclomatic quotient $\Xi (G^{\alpha})$. The asymptotic behavior of this quotient is similar to the asymptotic behavior of the quotient of the cardinality of the boundary of the subgraph by the cardinality of the subgraph. Using Følner's criterion for amenability one gets that $\Xi (G^{\alpha})$ vanishes for infinite groups if and only if they are amenable. When $G$ is finite then $\Xi (G^{\alpha})=1/|G|$, where $|G|$'> is the cardinality of $G$, and when $G$ is non-amenable then $1-n\leq\Xi (G^{\alpha})\le 0$, with $\Xi (G^{\alpha})=1-n$ if and only if $G$ is free of rank $n$. Thus we see that on special cases $\Xi (G^{\alpha})$ takes the values of the Euler characteristic of $G$. Most of the paper is concerned with formulae for the value of $\Xi (G^{\alpha})$ with respect to that of subgroups and factor groups, and with respect to the decomposition of the group into direct product and free product. Some of the formulae and bounds we get for $\Xi (G^{\alpha})$ are similar to those given for the spectral radius of symmetric random walks on the graph of $G^{\alpha}$, but this is not always the case. In the last section of the paper we define and touch very briefly the balanced cyclomatic quotient, which is defined on concentric balls in the graph and is related to the growth of $G$., Comment: LaTex, 23 pages, no figures

In this paper, we give examples of two new phenomena in Kleinian groups. We rst exhibit a sequence of homeomorphic marked hyperbolic 3-manifolds whose algebraic limit is not homeomorphic to any element in the sequence. We then use this construction to exhibit situations where the space of convex co-compact representations of a given 3-manifold group has many components but its closure is connected. Let M be a compact, irreducible, oriented 3-manifold and let D( 1(M)) denote the space of all discrete, faithful representations of 1(M) into PSL2(C). A sequence of representations { n} ⊂ D( 1(M)) converging to ∈ D( 1(M)) gives rise to a sequence {N n = H= n( 1(M))} of hyperbolic 3-manifolds, each of which is homotopy equivalent to M . The sequence {N n} is said to converge algebraically to N = H= ( 1(M)). (See [7, 13, 14] for more information about algebraic convergence of Kleinian groups.) In many situations (see [1, 6, 15, 24, 25, 27]), it has been shown that N n must be homeomorphic to N for all large enough n, and we had suspected that this would always be the case. In this paper, we give a collection of examples where N n is not homeomorphic to N for any n. Our sequences are quite well-behaved: the n( 1(M)) are convex co-compact and mutually quasiconformally conjugate, and the algebraic limit ( 1(M)) is geometrically nite. In our examples, M is obtained by gluing a collection of I -bundles to a solid torus along a family of parallel annuli. These manifolds are particularly simple examples of books of I -bundles (see [9]) where, to explain the terminology, one should think of the solid torus as the binding and the I -bundles

We address the structure identification and the uniform approximation of two fully nonlinear layer neural networks of the type$$f(x)=1^T h(B^T g(A^T x))$$f(x)=1Th(BTg(ATx))on$$\mathbb R^d$$Rd, where$$g=(g_1,\dots , g_{m_0})$$g=(g1,⋯,gm0),$$h=(h_1,\dots , h_{m_1})$$h=(h1,⋯,hm1),$$A=(a_1|\dots |a_{m_0}) \in \mathbb R^{d \times m_0}$$A=(a1|⋯|am0)∈Rd×m0and$$B=(b_1|\dots |b_{m_1}) \in \mathbb R^{m_0 \times m_1}$$B=(b1|⋯|bm1)∈Rm0×m1, from a small number of query samples. The solution of the case of two hidden layers presented in this paper is crucial as it can be further generalized to deeper neural networks. We approach the problem by sampling actively finite difference approximations to Hessians of the network. Gathering several approximate Hessians allows reliably to approximate the matrix subspace$$\mathcal W$$Wspanned by symmetric tensors$$a_1 \otimes a_1,\dots ,a_{m_0}\otimes a_{m_0}$$a1⊗a1,⋯,am0⊗am0formed by weights of the first layer together with the entangled symmetric tensors$$v_1 \otimes v_1 ,\dots ,v_{m_1}\otimes v_{m_1}$$v1⊗v1,⋯,vm1⊗vm1, formed by suitable combinations of the weights of the first and second layer as$$v_\ell =A G_0 b_\ell /\Vert A G_0 b_\ell \Vert _2$$vℓ=AG0bℓ/‖AG0bℓ‖2,$$\ell \in [m_1]$$ℓ∈[m1], for a diagonal matrix$$G_0$$G0depending on the activation functions of the first layer. The identification of the 1-rank symmetric tensors within$$\mathcal W$$Wis then performed by the solution of a robust nonlinear program, maximizing the spectral norm of the competitors constrained over the unit Frobenius sphere. We provide guarantees of stable recovery under a posteriori verifiable conditions. Once the 1-rank symmetric tensors$$\{a_i \otimes a_i, i\in [m_0]\}\cup \{v_\ell \otimes v_\ell , \ell \in [m_1] \}$${ai⊗ai,i∈[m0]}∪{vℓ⊗vℓ,ℓ∈[m1]}are computed, we address their correct attribution to the first or second layer ($$a_i$$ai’s are attributed to the first layer). The attribution to the layers is currently based on a semi-heuristic reasoning, but it shows clear potential of reliable execution. Having the correct attribution of the$$a_i,v_\ell $$ai,vℓto the respective layers and the consequent de-parametrization of the network, by using a suitably adapted gradient descent iteration, it is possible to estimate, up to intrinsic symmetries, the shifts of the activations functions of the first layer and compute exactly the matrix$$G_0$$G0. Eventually, from the vectors$$v_\ell =A G_0 b_\ell /\Vert A G_0 b_\ell \Vert _2$$vℓ=AG0bℓ/‖AG0bℓ‖2’s and$$a_i$$ai’s one can disentangle the weights$$b_\ell $$bℓ’s, by simple algebraic manipulations. Our method of identification of the weights of the network is fully constructive, with quantifiable sample complexity and therefore contributes to dwindle the black box nature of the network training phase. We corroborate our theoretical results by extensive numerical experiments, which confirm the effectiveness and feasibility of the proposed algorithmic pipeline.

Let P be a poset and $$\alpha :P\rightarrow P$$ be a function. The aim of this paper is to introduce and study the notion of skew derivations on P. We prove some fundamental properties of posets involving skew derivations. In particular, apart from proving the other results, we prove that if d and g are two skew derivations of P associated with an automorphism $$\alpha $$ such that $$d\alpha =\alpha d$$ and $$g\alpha =\alpha g,$$ then $$d \le g $$ if and only if $$g d =\alpha d$$ . Also, we prove that $$ Fix_{\alpha ,d}(P)\cap l(\alpha (x)) = l(d(x))$$ for all $$x\in P.$$ Furthermore, we give some examples to demonstrate that various restrictions imposed in the hypotheses of our results are not superfluous.

In this paper, we investigate the large deviations of sums of weighted random variables that are approximately independent, generalizing and improving some results of Montgomery and Odlyzko. We are motivated by examples arising from number theory, including the sequences pit, χ(p), χd(p), λf (p), and Klq(a − n, b), where p ranges over the primes, t varies in a large interval, χ varies among all characters modulo q, χd varies over quadratic characters attached to fundamental discriminants |d| ≤ x, λf (n) are the Fourier coefficients of holomorphic cusp forms f of (a large) weight k for the full modular group, and Klq(a, b) are the normalized Kloosterman sums modulo a large prime q, where a, b vary in (𝔽q)×.

Let $G$ be a simply connected Lie group with Lie algebra $\mathfrak{g}$. We show that the following categories are naturally equivalent. The category $\mathsf{Mod}(C(G))$, of sufficiently smooth modules over the DG-algebra of singular chains on $G$. The category $\mathsf{Rep}(Tg)$ of representations of the DG-Lie algebra $T\mathfrak{g}$, which is universal for the Cartan relations. This equivalence extends the correspondence between representations of $G$ and representations of $\mathfrak{g}$. In a companion paper we show that in the compact case, the equivalence can be extended to an $\mathsf{A}_\infty$ equivalence of DG-categories., Changes in notation. Added references. All comments are welcome

In this paper, we establish a non-trivial upper bound for the exponential sum $$\begin{aligned} \sum _{y

In this paper, we study ICE-closed (= Image-Cokernel-Extension-closed) subcategories of an abelian length category using torsion classes. To each interval $[\mathcal{U},\mathcal{T}]$ in the lattice of torsion classes, we associate a subcategory $\mathcal{T} \cap \mathcal{U}^\perp$ called the heart. We show that every ICE-closed subcategory can be realized as a heart of some interval of torsion classes, and give a lattice-theoretic characterization of intervals whose hearts are ICE-closed. In particular, we prove that ICE-closed subcategories are precisely torsion classes in some wide subcategories. For an artin algebra, we introduce the notion of wide $\tau$-tilting modules as a generalization of support $\tau$-tilting modules. Then we establish a bijection between wide $\tau$-tilting modules and doubly functorially finite ICE-closed subcategories, which extends Adachi--Iyama--Reiten's bijection on torsion classes. For the hereditary case, we discuss the Hasse quiver of the poset of ICE-closed subcategories by introducing a mutation of rigid modules., Comment: 31 pages, final version, to appear in Math. Z. Section 4.4 rewritten using sincere intervals