Dimension Reduction for Finite Trees in \varvec{\ell _1}

We show that every $n$ -point tree metric admits a $(1+\varepsilon )$ -embedding into $\ell _1^{C(\varepsilon ) \log n}$ , for every $\varepsilon > 0$ , where $C(\varepsilon ) \le O\big ((\frac{1}{\varepsilon })^4 \log \frac{1}{\varepsilon })\big )$ . This matches the natural volume lower bound up to a factor depending only on $\varepsilon $ . Previously, it was unknown whether even complete binary trees on $n$ nodes could be embedded in $\ell _1^{O(\log n)}$ with $O(1)$ distortion. For complete $d$ -ary trees, our construction achieves $C(\varepsilon ) \le O\big (\frac{1}{\varepsilon ^2}\big )$ .     

Boundaries of Disk-Like Self-affine Tiles

Let $T:= T(A, \mathcal{D })$ T : = T ( A , D ) be a disk-like self-affine tile generated by an integral expanding matrix $A$ A and a consecutive collinear digit set $\mathcal{D }$ D , and let $f(x)=x^{2}+px+q$ f ( x ) = x 2 + px + q be the characteristic polynomial of $A$ A . In the paper, we identify the boundary $\partial T$ ? T with a sofic system by constructing a neighbor graph and derive equivalent conditions for the pair $(A,\mathcal{D })$ ( A , D ) to be a number system. Moreover, by using the graph-directed construction and a device of pseudo-norm $\omega $ ω , we find the generalized Hausdorff dimension $\dim _H^{\omega } (\partial T)=2\log \rho (M)/\log |q|$ dim H ω ( ? T ) = 2 log ρ ( M ) / log | q | where $\rho (M)$ ρ ( M ) is the spectral radius of certain contact matrix $M$ M . Especially, when $A$ A is a similarity, we obtain the standard Hausdorff dimension $\dim _H (\partial T)=2\log \rho /\log |q|$ dim H ( ? T ) = 2 log ρ / log | q | where $\rho $ ρ is the largest positive zero of the cubic polynomial $x^{3}-(|p|-1)x^{2}-(|q|-|p|)x-|q|$ x 3 ? ( | p | ? 1 ) x 2 ? ( | q | ? | p | ) x ? | q | , which is simpler than the known result.     

A metrical lower bound on the star discrepancy of digital sequences

In this paper we study uniform distribution properties of digital sequences over a finite field of prime order. In 1998 it was shown by Larcher that for almost all $s$ -dimensional digital sequences $\mathcal{S }$ the star discrepancy $D_N^*$ satisfies an upper bound of the form $D_N^*(\mathcal{S })=O((\log N)^s (\log \log N)^{2+\varepsilon })$ for any $\varepsilon >0$ . Generally speaking it is much more difficult to obtain good lower bounds for specific sequences than upper bounds. Here we show that Larchers result is best possible up to some $\log \log N$ term. More detailed, we prove that for almost all $s$ -dimensional digital sequences $\mathcal{S }$ the star discrepancy satisfies $D_N^*(\mathcal{S }) \ge c(q,s) (\log N)^s \log \log N$ for infinitely many $N \in \mathbb{N }$ , where $c(q,s)>0$ only depends on $q$ and $s$ but not on $N$ .     

A Reduction Theorem for the Kripke–Joyal Semantics: Forcing Over an Arbitrary Category can Always be Replaced by Forcing Over a Complete Heyting Algebra

It is assumed that a Kripke–Joyal semantics ${\mathcal{A} = \left\langle \mathbb{C},{\rm Cov}, {\it F},\Vdash \right\rangle}$ A = C , Cov , F , ? has been defined for a first-order language ${\mathcal{L}}$ L . To transform ${\mathbb{C}}$ C into a Heyting algebra ${\overline{\mathbb{C}}}$ C ¯ on which the forcing relation is preserved, a standard construction is used to obtain a complete Heyting algebra made up of cribles of ${\mathbb{C}}$ C . A pretopology ${\overline{{\rm Cov}}}$ Cov ¯ is defined on ${\overline{\mathbb{C}}}$ C ¯ using the pretopology on ${\mathbb{C}}$ C . A sheaf ${\overline{{\it F}}}$ F ¯ is made up of sections of F that obey functoriality. A forcing relation ${\overline{\Vdash}}$ ? ¯ is defined and it is shown that ${\overline{\mathcal{A}} = \left\langle \overline{\mathbb{C}},\overline{\rm{Cov}},\overline{{\it F}}, \overline{\Vdash} \right\rangle }$ A ¯ = C ¯ , Cov ¯ , F ¯ , ? ¯ is a Kripke–Joyal semantics that faithfully preserves the notion of forcing of ${\mathcal{A}}$ A . That is to say, an object a of ${\mathbb{C}Ob}$ C O b forces a sentence with respect to ${\mathcal{A}}$ A if and only if the maximal a-crible forces it with respect to ${\overline{\mathcal{A}}}$ A ¯ . This reduces a Kripke–Joyal semantics defined over an arbitrary site to a Kripke–Joyal semantics defined over a site which is based on a complete Heyting algebra.     

Real Clifford Algebras as Tensor Products over Centers

Denote by ${\mathcal{C}\ell_{p,q}}$ the Clifford algebra on the real vector space ${\mathbb{R}^{p,q}}$ . This paper gives a unified tensor product expression of ${\mathcal{C}\ell_{p,q}}$ by using the center of ${\mathcal{C}\ell_{p,q}}$ . The main result states that for nonnegative integers p, q, ${\mathcal{C}\ell_{p,q} \simeq \otimes^{\kappa-\delta}\mathcal{C}_{1,1} \otimes Cen(\mathcal{C}\ell_{p,q}) \otimes^{\delta} \mathcal{C}\ell_{0,2},}$ where ${p + q \equiv \varepsilon}$ mod 2, ${\kappa = ((p + q) - \varepsilon)/2, p - |q - \varepsilon| \equiv i}$ mod 8 and ${\delta = \lfloor i / 4 \rfloor}$ .     

1D Schrödinger Operators with Short Range Interactions: Two-Scale Regularization of Distributional Potentials

For real ${L_\infty(\mathbb{R})}$ -functions ${\Phi}$ and ${\Psi}$ of compact support, we prove the norm resolvent convergence, as ${\varepsilon}$ and ${\nu}$ tend to 0, of a family ${S_{\varepsilon \nu}}$ of one-dimensional Schrödinger operators on the line of the form $$S_{\varepsilon \nu} = -\frac{d^2}{dx^2} + \frac{\alpha}{\varepsilon^2} \Phi \left( \frac{x}{\varepsilon} \right) + \frac{\beta}{\nu} \Psi \left(\frac{x}{\nu} \right),$$ provided the ratio ${\nu/\varepsilon}$ has a finite or infinite limit. The limit operator S ₀ depends on the shape of ${\Phi}$ and ${\Psi}$ as well as on the limit of ratio ${\nu/\varepsilon}$ . If the potential ${\alpha\Phi}$ possesses a zero-energy resonance, then S ₀ describes a non trivial point interaction at the origin. Otherwise S ₀ is the direct sum of the Dirichlet half-line Schrödinger operators.     

Interface foliation for an inhomogeneous Allen–Cahn equation in Riemannian manifolds

Let ${(\mathcal{M}, \tilde{g})}$ be an N-dimensional smooth compact Riemannian manifold. We consider the problem ${\varepsilon^2 \triangle_{\tilde{g}} \tilde{u} + V(\tilde{z})\tilde{u}(1-\tilde{u}^2)=0\; {\rm in}\; \mathcal{M}}$ , where ${\varepsilon > 0}$ is a small parameter and V is a positive, smooth function in ${\mathcal{M}}$ . Let ${\kappa \subset \mathcal{M}}$ be an (N ? 1)-dimensional smooth submanifold that divides ${\mathcal{M}}$ into two disjoint components ${\mathcal{M}_{\pm}}$ . We assume κ is stationary and non-degenerate relative to the weighted area functional ${\int_{\kappa}V^{\frac{1}{2}}}$ . For each integer m ≥ 2, we prove the existence of a sequence ${\varepsilon = \varepsilon_\ell \rightarrow 0}$ , and two opposite directional solutions with m-transition layers near κ, whose mutual distance is ${{\rm O}(\varepsilon | \log \varepsilon | )}$ . Moreover, the interaction between neighboring layers is governed by a type of Jacobi–Toda system.     

Ranks of backward shift invariant subspaces of the Hardy space over the bidisk

Let $\{\varphi _n(z)\}_{n\ge 0}$ be a sequence of inner functions satisfying that $\zeta _n(z):=\varphi _n(z)/\varphi _{n+1}(z)\in H^\infty (z)$ for every $n\ge 0$ and $\{\varphi _n(z)\}_{n\ge 0}$ has no nonconstant common inner divisors. Associated with it, we have a Rudin type invariant subspace $\mathcal{M }$ of $H^2(\mathbb{D }^2)$ . The ranks of $\mathcal{M }\ominus w\mathcal{M }$ for $\mathcal{F }_z$ and $\mathcal{F }^*_z$ respectively are determined, where $\mathcal{F }_z$ is the fringe operator on $\mathcal{M }\ominus w\mathcal{M }$ . Let $\mathcal{N }= H^2(\mathbb{D }^2)\ominus \mathcal{M }$ . It is also proved that the rank of $\mathcal{M }\ominus w\mathcal{M }$ for $\mathcal{F }^*_z$ equals to the rank of $\mathcal{N }$ for $T^*_z$ and $T^*_w$ .     

The universal Glivenko–Cantelli property

Let ${\mathcal{F}}$ be a separable uniformly bounded family of measurable functions on a standard measurable space ${(X, \mathcal{X})}$ , and let ${N_{[]}(\mathcal{F}, \varepsilon, \mu)}$ be the smallest number of ${\varepsilon}$ -brackets in L ¹(μ) needed to cover ${\mathcal{F}}$ . The following are equivalent:

1. ${\mathcal{F}}$ is a universal Glivenko–Cantelli class.
2. ${N_{[]}(\mathcal{F},\varepsilon,\mu) < \infty}$ for every ${\varepsilon > 0}$ and every probability measure μ.
3. ${\mathcal{F}}$ is totally bounded in L ¹(μ) for every probability measure μ.
4. ${\mathcal{F}}$ does not contain a Boolean σ-independent sequence.

It follows that universal Glivenko–Cantelli classes are uniformity classes for general sequences of almost surely convergent random measures.     

Hypercentralizing generalized skew derivations on left ideals in prime rings

Let $\mathcal{R }$ be a prime ring of characteristic different from $2, \mathcal{Q }_r$ the right Martindale quotient ring of $\mathcal{R }, \mathcal{C }$ the extended centroid of $\mathcal{R }, \mathcal{I }$ a nonzero left ideal of $\mathcal{R }, F$ a nonzero generalized skew derivation of $\mathcal{R }$ with associated automorphism $\alpha $ , and $n,k \ge 1$ be fixed integers. If $[F(r^n),r^n]_k=0$ for all $r \in \mathcal{I }$ , then there exists $\lambda \in \mathcal{C }$ such that $F(x)=\lambda x$ , for all $x\in \mathcal{I }$ . More precisely one of the following holds: (1) $\alpha $ is an $X$ -inner automorphism of $\mathcal{R }$ and there exist $b,c \in \mathcal{Q }_r$ and $q$ invertible element of $\mathcal{Q }_r$ , such that $F(x)=bx-qxq^{-1}c$ , for all $x\in \mathcal{Q }_r$ . Moreover there exists $\gamma \in \mathcal{C }$ such that $\mathcal{I }(q^{-1}c-\gamma )=(0)$ and $b-\gamma q \in \mathcal{C }$ ; (2) $\alpha $ is an $X$ -outer automorphism of $\mathcal{R }$ and there exist $c \in \mathcal{Q }_r, \lambda \in \mathcal{C }$ , such that $F(x)=\lambda x-\alpha (x)c$ , for all $x\in \mathcal{Q }_r$ , with $\alpha (\mathcal{I })c=0$ .     

A unified approach for minimizing composite norms

We propose a first-order augmented Lagrangian algorithm (FALC) to solve the composite norm minimization problem $$\begin{aligned} \begin{array}{ll} \min \limits _{X\in \mathbb{R }^{m\times n}}&\mu _1\Vert \sigma (\mathcal{F }(X)-G)\Vert _\alpha +\mu _2\Vert \mathcal{C }(X)-d\Vert _\beta ,\\ \text{ subject} \text{ to}&\mathcal{A }(X)-b\in \mathcal{Q }, \end{array} \end{aligned}$$ where $\sigma (X)$ denotes the vector of singular values of $X \in \mathbb{R }^{m\times n}$ , the matrix norm $\Vert \sigma (X)\Vert _{\alpha }$ denotes either the Frobenius, the nuclear, or the $\ell _2$ -operator norm of $X$ , the vector norm $\Vert .\Vert _{\beta }$ denotes either the $\ell _1$ -norm, $\ell _2$ -norm or the $\ell _{\infty }$ -norm; $\mathcal{Q }$ is a closed convex set and $\mathcal{A }(.)$ , $\mathcal{C }(.)$ , $\mathcal{F }(.)$ are linear operators from $\mathbb{R }^{m\times n}$ to vector spaces of appropriate dimensions. Basis pursuit, matrix completion, robust principal component pursuit (PCP), and stable PCP problems are all special cases of the composite norm minimization problem. Thus, FALC is able to solve all these problems in a unified manner. We show that any limit point of FALC iterate sequence is an optimal solution of the composite norm minimization problem. We also show that for all $\epsilon >0$ , the FALC iterates are $\epsilon $ -feasible and $\epsilon $ -optimal after $\mathcal{O }(\log (\epsilon ^{-1}))$ iterations, which require $\mathcal{O }(\epsilon ^{-1})$ constrained shrinkage operations and Euclidean projection onto the set $\mathcal{Q }$ . Surprisingly, on the problem sets we tested, FALC required only $\mathcal{O }(\log (\epsilon ^{-1}))$ constrained shrinkage, instead of the $\mathcal{O }(\epsilon ^{-1})$ worst case bound, to compute an $\epsilon $ -feasible and $\epsilon $ -optimal solution. To best of our knowledge, FALC is the first algorithm with a known complexity bound that solves the stable PCP problem.     

Characterization of Lie higher derivations on triangular algebras

Let $\mathcal{A}$ and $\mathcal{B}$ be unital rings, and $\mathcal{M}$ be an $\left( {\mathcal{A},\mathcal{B}} \right)$ -bimodule, which is faithful as a left $\mathcal{A}$ -module and also as a right $\mathcal{B}$ -module. Let $\mathcal{U} = Tri\left( {\mathcal{A},\mathcal{M},\mathcal{B}} \right)$ be the triangular algebra. In this paper, we give some different characterizations of Lie higher derivations on $\mathcal{U}$ .     

Some results on metric n-Lie algebras

We study the structure of a metric n-Lie algebra G over the complex field C. Let G = SR be the Levi decomposition, where R is the radical of G and S is a strong semisimple subalgebra of G. Denote by m(G) the number of all minimal ideals of an indecomposable metric n-Lie algebra and R ⊥ the orthogonal complement of R. We obtain the following results. As S-modules, R ⊥ is isomorphic to the dual module of G/R. The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on G is equal to that of the vector space of certain linear transformations on G; this dimension is greater than or equal to m(G) + 1. The centralizer of R in G is equal to the sum of all minimal ideals; it is the direct sum of R ⊥ and the center of G. Finally, G has no strong semisimple ideals if and only if R⊥■R.     

Existence of semiclassical states for a quasilinear Schrödinger equation with critical exponent in {\mathbb{R}^N}

We consider the following perturbed version of quasilinear Schrödinger equation $$\begin{array}{lll}-\varepsilon^2\Delta u +V(x)u-\varepsilon^2\Delta (u^2)u=h(x,u)u+K(x)|u|^{22^*-2}u\end{array}$$ in ${\mathbb{R}^N}$ , where N ≥ 3, 22* = 4N/(N ? 2), V(x) is a nonnegative potential, and K(x) is a bounded positive function. Using minimax methods, we show that this equation has at least one positive solution provided that ${\varepsilon \leq \mathcal{E}}$ ; for any ${m\in\mathbb{N}}$ , it has m pairs of solutions if ${\varepsilon \leq \mathcal{E}_m}$ , where ${\mathcal{E}}$ and ${\mathcal{E}_m}$ are sufficiently small positive numbers. Moreover, these solutions ${u_\varepsilon \to 0}$ in ${H^1(\mathbb{R}^N)}$ as ${\varepsilon \to 0}$ .     

Tilings of Parallelograms with Similar Right Triangles

We say that a triangle $T$ T tiles the polygon $\mathcal A $ A if $\mathcal A $ A can be decomposed into finitely many non-overlapping triangles similar to $T$ T . A tiling is called regular if there are two angles of the triangles, say $\alpha $ α and $\beta $ β , such that at each vertex $V$ V of the tiling the number of triangles having $V$ V as a vertex and having angle $\alpha $ α at $V$ V is the same as the number of triangles having angle $\beta $ β at $V$ V . Otherwise the tiling is called irregular. Let $\mathcal P (\delta )$ P ( δ ) be a parallelogram with acute angle $\delta $ δ . In this paper we prove that if the parallelogram $\mathcal P (\delta )$ P ( δ ) is tiled with similar triangles of angles $(\alpha , \beta , \pi /2)$ ( α , β , π / 2 ) , then $(\alpha , \beta )=(\delta , \pi /2-\delta )$ ( α , β ) = ( δ , π / 2 - δ ) or $(\alpha , \beta )=(\delta /2, \pi /2-\delta /2)$ ( α , β ) = ( δ / 2 , π / 2 - δ / 2 ) , and if the tiling is regular, then only the first case can occur.     

Planar Beurling Transform and Bergman Type Spaces

In this paper we describe the actions of the operator $S_\mathbb{D }$ or its adjoint $S_\mathbb{D }^*$ on the poly-Bergman spaces of the unit disk $\mathbb{D }.$ Let $k$ and $j$ be positive integers. We prove that $(S_\mathbb{D })^{j}$ is an isometric isomorphism between the true poly-Bergman subspace $\mathcal{A }_{(k)}^2(\mathbb{D })\ominus N_{(k),j}$ onto the true poly-Bergman space $\mathcal{A }_{(j+k)}^2(\mathbb{D }),$ where the linear space $N_{(k),j}$ have finite dimension $j.$ The action of $(S_\mathbb{D })^{j-1}$ on the canonical Hilbert base for the Bergman subspace $\mathcal{A }^2(\mathbb{D })\ominus \mathcal{P }_{j-1},$ gives a Hilbert base $\{ \phi _{ j , k } \}_{ k }$ for $\mathcal{A }_{(j)}^2(\mathbb{D }).$ It is shown that $\{ \phi _{ j , k } \}_{ j, k }$ is a Hilbert base for $L^2(\mathbb{D },d A)$ such that whenever $j$ and $k$ remain constant we obtain a Hilbert base for the true poly-Bergman space $\mathcal{A }_{(j)}^2(\mathbb{D })$ and $\mathcal{A }_{(-k)}^2(\mathbb{D }),$ respectively. The functions $\phi _{ j , k }$ are polynomials in $z$ and $\overline{z}$ and are explicitly given in terms of the $(2,1)$ -hypergeometric polynomials. We prove explicit representations for the true poly-Bergman kernels and the Koshelev representation for the poly-Bergman kernels of $\mathbb{D }.$ The action of $S_\Pi $ on the true poly-Bergman spaces of the upper half-plane $\Pi $ allows one to introduce Hilbert bases for the true poly-Bergman spaces, and to give explicit representations of the true poly-Bergman and poly-Bergman kernels.     

Interior-point algorithms for P_{*}(\kappa )-LCP based on a new class of kernel functions

In this paper, we propose interior-point algorithms for $P_* (\kappa )$ -linear complementarity problem based on a new class of kernel functions. New search directions and proximity measures are defined based on these functions. We show that if a strictly feasible starting point is available, then the new algorithm has $\mathcal{O }\bigl ((1+2\kappa )\sqrt{n}\log n \log \frac{n\mu ^0}{\epsilon }\bigr )$ and $\mathcal{O }\bigl ((1+2\kappa )\sqrt{n} \log \frac{n\mu ^0}{\epsilon }\bigr )$ iteration complexity for large- and small-update methods, respectively. These are the best known complexity results for such methods.     

Necessary and Sufficient Conditions for Associated Differential Operators in Quaternionic Analysis and Applications to Initial Value Problems

This paper deals with the initial value problem of type $$\begin{array}{ll} \qquad \frac{\partial u}{\partial t} = \mathcal{L} u := \sum \limits^3_{i=0} A^{(i)} (t, x) \frac{\partial u}{\partial x_{i}} + B(t, x)u + C(t, x)\\ u (0, x) = u_{0}(x)\end{array}$$ in the space of generalized regular functions in the sense of Quaternionic Analysis satisfying the differential equation $$\mathcal{D}_{\lambda}u := \mathcal{D} u + \lambda u = 0,$$ where ${t \in [0, T]}$ is the time variable, x runs in a bounded and simply connected domain in ${\mathbb{R}^{4}, \lambda}$ is a real number, and ${\mathcal{D}}$ is the Cauchy-Fueter operator. We prove necessary and sufficient conditions on the coefficients of the operator ${\mathcal{L}}$ under which ${\mathcal{L}}$ is associated with the operator ${\mathcal{D}_{\lambda}}$ , i.e. ${\mathcal{L}}$ transforms the set of all solutions of the differential equation ${\mathcal{D}_{\lambda}u = 0}$ into solutions of the same equation for fixedly chosen t. This criterion makes it possible to construct operators ${\mathcal{L}}$ for which the initial value problem is uniquely soluble for an arbitrary initial generalized regular function u ₀ by the method of associated spaces constructed by W. Tutschke (Teubner Leipzig and Springer Verlag, 1989) and the solution is also generalized regular for each t.     

An extension of one direction in Marty’s normality criterion

We prove the following extension of one direction in Marty’s theorem: If $k$ is a natural number, $\alpha >1$ and $\mathcal{F }$ is a family of functions meromorphic on a domain $D$ all of whose poles have multiplicity at least $\frac{k}{\alpha -1}$ , then the normality of $\mathcal{F }$ implies that the family $$\begin{aligned} \left\{ \frac{|f^{(k)}|}{1+|f|^\alpha }\,:\, f\in \mathcal{F }\right\} \end{aligned}$$ is locally uniformly bounded.     

Congruence Properties of Binary Partition Functions

Let ${\mathcal{A}}$ be a finite subset of ${\mathbb{N}}$ containing 0, and let f (n) denote the number of ways to write n in the form ${\sum \varepsilon _{j}2^{j}}$ , where ${\varepsilon _{j} \epsilon \mathcal{A}}$ . We show that there exists a computable ${T = T (\mathcal{A})}$ so that the sequence (f (n) mod 2) is periodic with period T. Variations and generalizations of this problem are also discussed.