The conservation relation for cuspidal representations

Let \({\pi \in Cusp ({\rm U}(V_n))}\) be a smooth cuspidal irreducible representation of a unitary group U(V _n ) of dimension n over a non-Archimedean locally compact field. Let \({W^\pm_m}\) be the two isomorphism classes of Hermitian spaces of dimension m, and the denote by \({\tau^+ \in Cusp ({\rm U}(W_{m^+}^+))}\) and \({\tau^- \in Cusp ({\rm U}(W_{m^-}^-))}\) the first non-zero theta lifts of π. In this article we prove that m ⁺ + m ^? = 2n + 2, which was conjectured in Harris et al. (J AMS 9:941–1004, 1996, Speculations 7.5 and 7.6). We prove similar equalities for the other dual pairs of type I: the symplectic-orthogonal dual pairs and the quaternionic dual pairs.     

Stability of volume comparison for complex convex bodies

We prove the stability of the affirmative part of the solution to the complex Busemann–Petty problem. Namely, if K and L are origin-symmetric convex bodies in \({{\mathbb C}^n}\), n = 2 or n = 3, \({\varepsilon >0 }\) and \({{\rm Vol}_{2n-2}(K\cap H) \le {\rm Vol}_{2n-2}(L \cap H) + \varepsilon}\) for any complex hyperplane H in \({{\mathbb C}^n}\) , then \({({\rm Vol}_{2n}(K))^{\frac{n-1}n}\le({\rm Vol}_{2n}(L))^{\frac{n-1}n} + \varepsilon}\) , where Vol_2n is the volume in \({{\mathbb C}^n}\) , which is identified with \({{\mathbb R}^{2n}}\) in the natural way.     

More,or less,trivial matrix functional equations

Various functional equations satisfied by one or two (N × N)-matrices \({\mathbf{F}(z) }\) and \({\mathbf{G}(z) }\) depending on the scalar variable z are investigated, with N an arbitrary positive integer. Some of these functional equations are generalizations to the matrix case (N > 1) of well-known functional equations valid in the scalar (N = 1) case, such as \({\mathbf{F}(x) \, \mathbf{F}(y) = \, \mathbf{F}(x y) \, \rm and \, \mathbf{G}({\it x}) \, \mathbf{G}({\it y}) = \mathbf{G}({\it x+y}) }\); others—such as \({\mathbf{G}(y) \, \mathbf{F}(x) = \mathbf{F}(x) \, \mathbf{G}(xy) }\)—possess nontrivial solutions only in the matrix case (N > 1), namely their scalar (N = 1) counterparts only feature quite trivial solutions. It is also pointed out that if two (N × N)-matrices \({\mathbf{F}(x) \, \rm and \, \mathbf{G}({\it y})}\) satisfy the triplet of functional equations written above—and nontrivial examples of such matrices are exhibited— then they also satisfy an endless hierarchy of matrix functional relations involving an increasing number of scalar independent variables, the first items of which read \({\mathbf{F}(x_{1}) \, \mathbf{G}(y_{1}) \, \mathbf{F} (x_{2}) = \mathbf{F}(x_{1} x_{2}) \, \mathbf{G } (x_{2} y_{1}) \, \rm and \, \mathbf{G}({\it y}_{1}) \, \mathbf{F} ({\it x}_{1}) \, \mathbf{G} ({\it y}_{2}) = \mathbf{F} ({\it x}_{1}) \, \mathbf{G} ({\it x}_{1} {\it y}_{1}+{\it y}_{2}) }\).     

Interassociates of the bicyclic semigroup

An interassociate of a semigroup \((S,\cdot )\) is a semigroup \((S, *)\) such that for all \(a, b, c \in S\), \(a\cdot (b*c)=(a\cdot b) *c\) and \(a*(b\cdot c)=(a*b) \cdot c\). We investigate the bicyclic semigroup C and its interassociates. In particular, if p and q are the generators of the bicyclic semigroup and m and n are fixed nonnegative integers, the operation \(a*_{m,n} b= aq^mp^n b\) is known to be an interassociate. We show that for distinct pairs (m, n) and (s, t), the interassociates \((C, *_{m,n})\) and \((C, *_{s,t})\) are not isomorphic. We also generalize a result regarding homomorphisms on C to homomorphisms on its interassociates.     

Infinite dimensional multiplicity free spaces III: matrix coefficients and regular functions

In earlier papers we studied direct limits \({(G,\,K) = \varinjlim\, (G_n,K_n)}\) of two types of Gelfand pairs. The first type was that in which the G _n /K _n are compact Riemannian symmetric spaces. The second type was that in which \({G_n = N_n\rtimes K_n}\) with N _n nilpotent, in other words pairs (G _n , K _n ) for which G _n /K _n is a commutative nilmanifold. In each we worked out a method inspired by the Frobenius–Schur Orthogonality Relations to define isometric injections \({\zeta_{m,n}: L^2(G_n/K_n) \hookrightarrow L^2(G_m/K_m)}\) for m ≧ n and prove that the left regular representation of G on the Hilbert space direct limit \({L^2(G/K) := \varinjlim L^2(G_n/K_n)}\) is multiplicity-free. This left open questions concerning the nature of the elements of L ²(G/K). Here we define spaces \({\mathcal{A}(G_n/K_n)}\) of regular functions on G _n /K _n and injections \({\nu_{m,n} : \mathcal{A}(G_n/K_n) \to \mathcal{A}(G_m/K_m)}\) for m ≧ n related to restriction by \({\nu_{m,n}(f)|_{G_n/K_n} = f}\). Thus the direct limit \({\mathcal{A}(G/K) := \varinjlim \{\mathcal{A}(G_n/K_n), \nu_{m,n}\}}\) sits as a particular G-submodule of the much larger inverse limit \({\varprojlim \{\mathcal{A}(G_n/K_n), {\rm restriction}\}}\). Further, we define a pre Hilbert space structure on \({\mathcal{A}(G/K)}\) derived from that of L ²(G/K). This allows an interpretation of L ²(G/K) as the Hilbert space completion of the concretely defined function space \({\mathcal{A}(G/K)}\), and also defines a G-invariant inner product on \({\mathcal{A}(G/K)}\) for which the left regular representation of G is multiplicity-free.     

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\({(S/I)\leq \nu_{o}(G)}\).     

On the value distribution of positive definite quadratic forms

Denote by 0 = λ ₀ < λ ₁ ≤ λ ₂ ≤ . . . the infinite sequence given by the values of a positive definite irrational quadratic form in k variables at integer points. For l ≥ 2 and an (l ?1)-dimensional interval I = I ₂×. . .×I _l we consider the l-level correlation function \({K^{(l)}_I(R)}\) which counts the number of tuples (i ₁, . . . , i _l ) such that \({\lambda_{i_1},\ldots,\lambda_{i_l}\leq R^2}\) and \({\lambda_{i_{j}}-\lambda_{i_{1}}\in I_j}\) for 2 ≤ j ≤ l. We study the asymptotic behavior of \({K^{(l)}_I(R)}\) as R tends to infinity. If k ≥ 4 we prove \({K^{(l)}_I(R)\sim c_l(Q)\,{\rm vol}(I)R^{lk-2(l-1)}}\) for arbitrary l, where c _l (Q) is an explicitly determined constant. This remains true for k = 3 under the restriction l ≤ 3.     

The range of approximate unitary equivalence classes of homomorphisms from AH-algebras

Let C be a unital AH-algebra and A be a unital simple C*-algebras with tracial rank zero. It has been shown that two unital monomorphisms \({\phi, \psi: C\to A}\) are approximately unitarily equivalent if and only if

$ [\phi]=[\psi]\quad {\rm in}\quad KL(C,A)\quad {\rm and}\quad \tau\circ \phi=\tau\circ \psi \quad{\rm for\, all}\tau\in T(A),$

where T(A) is the tracial state space of A. In this paper we prove the following: Given \({\kappa\in KL(C,A)}\) with \({\kappa(K_0(C)_+\setminus\{0\})\subset K_0(A)_+\setminus\{0\}}\) and with κ([1 _C ]) = [1 _A ] and a continuous affine map \({\lambda: T(A)\to T_{\mathfrak f}(C)}\) which is compatible with κ, where \({T_{\mathfrak f}(C)}\) is the convex set of all faithful tracial states, there exists a unital monomorphism \({\phi: C\to A}\) such that

$[\phi]=\kappa\quad{\rm and}\quad \tau\circ \phi(c)=\lambda(\tau)(c)$

for all \({c\in C_{s.a.}}\) and \({\tau\in T(A).}\) Denote by \({{\rm Mon}_{au}^e(C,A)}\) the set of approximate unitary equivalence classes of unital monomorphisms. We provide a bijective map

$\Lambda: {\rm Mon}_{au}^e (C,A)\to KLT(C,A)^{++},$

where KLT(C, A)⁺⁺ is the set of compatible pairs of elements in KL(C, A)⁺⁺ and continuous affine maps from T(A) to \({T_{\mathfrak f}(C).}\) Moreover, we found that there are compact metric spaces X, unital simple AF-algebras A and \({\kappa\in KL(C(X), A)}\) with \({\kappa(K_0(C(X))_+\setminus\{0\})\subset K_0(A)_+\setminus\{0\}}\) for which there is no homomorphism h: C(X) → A so that [h] = κ.

     

Some infinite families of Ramsey ($${\varvec{P}}_\mathbf{3},{\varvec{P}}_{{\varvec{n}}}$$)-minimal trees

For any given two graphs G and H, the notation \(F\rightarrow \) (G, H) means that for any red–blue coloring of all the edges of F will create either a red subgraph isomorphic to G or a blue subgraph isomorphic to H. A graph F is a Ramsey (G, H)-minimal graph if \(F\rightarrow \) (G, H) but \(F-e\nrightarrow (G,H)\), for every \(e \in E(F)\). The class of all Ramsey (G, H)-minimal graphs is denoted by \(\mathcal {R}(G,H)\). In this paper, we construct some infinite families of trees belonging to \(\mathcal {R}(P_3,P_n)\), for \(n=8\) and 9. In particular, we give an algorithm to obtain an infinite family of trees belonging to \(\mathcal {R}(P_3,P_n)\), for \(n\ge 10\).     

$${\mathcal{L}_1}$$-spaces with the Radon-Nikodým property containing reflexive subspaces

We show that for every sequence \({(p_n)_{n\in\mathbb{N}}}\) with 1 ≤ p _n  ≤ 2 there exists an \({\mathcal{L}_1}\) -space with the Radon-Nikodým containing an isomorphic copy of \({\ell_1(\ell_{p_n})}\) .     

Generic irreducibilty of Laplace eigenspaces on certain compact Lie groups

If G is a compact Lie group endowed with a left invariant metric g, then G acts via pullback by isometries on each eigenspace of the associated Laplace operator \(\Delta _g\). We establish algebraic criteria for the existence of left invariant metrics g on G such that each eigenspace of \(\Delta _g\), regarded as the real vector space of the corresponding real eigenfunctions, is irreducible under the action of G. We prove that generic left invariant metrics on the Lie groups \(G={ SU}(2)\times \cdots \times { SU}(2)\times T\), where T is a (possibly trivial) torus, have the property just described. The same holds for quotients of such groups G by discrete central subgroups. In particular, it also holds for \({ SO}(3)\), \({ U}(2)\), \({ SO}(4)\).     

Normal coverings of solvable groups

For a finite non cyclic group G, let γ(G) be the smallest integer k such that G contains k proper subgroups H ₁, . . . , H _k with the property that every element of G is contained in \({H_i^g}\) for some \({i \in \{1,\dots,k\}}\) and \({g \in G.}\) We prove that for every n ≥ 2, there exists a finite solvable group G with γ(G) = n.     

An inequality between Willmore functional and Weyl functional for submanifolds in space forms

Let \({\phi : M \to R^{n+p}(c)}\) be an n-dimensional submanifold in an (n + p)-dimensional space form R ^n+p(c) with the induced metric g. Willmore functional of \({\phi}\) is \({W(\phi) = \int_{M}(S - nH^{2})^{n/2}dv}\) , where \({S = \sum_{\alpha,i, j}(h^{\alpha}_{ij} )^2}\) is the square of the length of the second fundamental form, H is the mean curvature of M. The Weyl functional of (M, g) is \({\nu(g) = \int_{M}|W_{g}|^{n/2}dv}\) , where \({|W_{g}|^{2} = \sum_{i, j,k,l} W^{2}_{ijkl}}\) and W _ijkl are the components of the Weyl curvature tensor W _g of (M, g). In this paper, we discover an inequality relation between Willmore functional \({W(\phi)}\) and Weyl funtional ν(g).     

The “strange term” in the periodic homogenization for multivalued Leray–Lions operators in perforated domains

Using the periodic unfolding method of Cioranescu, Damlamian and Griso, we study the homogenization for equations of the form

$-{\rm div}\,\,d_\varepsilon=f,\,\,{\rm with}\,\,\left(\nabla u_{\varepsilon , \delta }(x),d_{\varepsilon , \delta }(x)\right) \in A_\varepsilon(x)$

in a perforated domain with holes of size \({\varepsilon \delta }\) periodically distributed in the domain, where \({A_\varepsilon }\) is a function whose values are maximal monotone graphs (on R ^N ). Two different unfolding operators are involved in such a geometric situation. Under appropriate growth and coercivity assumptions, if the corresponding two sequences of unfolded maximal monotone graphs converge in the graph sense to the maximal monotone graphs A(x, y) and A ₀(x, z) for almost every \({(x,y,z)\in \Omega \times Y \times {\rm {\bf R}}^N}\), as \({\varepsilon \to 0}\), then every cluster point (u ₀, d ₀) of the sequence \({(u_{\varepsilon , \delta }, d_{\varepsilon , \delta } )}\) for the weak topology in the naturally associated Sobolev space is a solution of the homogenized problem which is expressed in terms of u ₀ alone. This result applies to the case where \({A_{\varepsilon}(x)}\) is of the form \({B(x/\varepsilon)}\) where B(y) is periodic and continuous at y = 0, and, in particular, to the oscillating p-Laplacian.

     

O n ( m n,n, m n,m) relative differe nce sets with $$\text {gcd}( m,n)=1$$gcd( m,n)=1

There has been much research on \((p^{a},p^{b},p^{a},p^{a-b})\) relative difference sets with p a prime, while there are only a few results on (mn, n, mn, m) relative difference sets with \(\text {gcd}(m,n)=1\). The non-existence results on (mn, n, mn, m) relative difference sets with \(\text {gcd}(m,n)=1\) have only been obtained for the following five cases: (1) \(m=p,\ n=q,\ p>q\); (2) \(m=pq,\ n=3,\ p,q>3\); (3) \(m=4,\ n=p\); (4) \(m=2\) and (5) \(n=p\), where p, q are distinct odd primes. For the existence results, there are only four constructions of semi-regular relative difference sets in groups of size not a prime power with the forbidden subgroup having size larger than 2. In this paper, we present some more non-existence results on (mn, n, mn, m) relative difference sets with \(\text {gcd}(m,n)=1\). In particular, our result is a generalization of the main result of Hiramine’s work (J Comb Theory Ser A 117(7):996–1003, 2010). Meanwhile, we give a construction of non-abelian (16q, q, 16q, 16) relative difference sets, where q is a prime power with \(q\equiv 1\pmod {4}\) and \(q>4.2\times 10^{8}\). This is the third known infinite classes of non-abelian semi-regular relative difference sets.     

Topological entropy of sets of generic points for actions of amenable groups

Let G be a countable discrete infinite amenable group which acts continuously on a compact metric space X and let μ be an ergodic G-invariant Borel probability measure on X. For a fixed tempered F?lner sequence {Fn} in G with limn→+∞|Fn|/log n= ∞, we prove the following result:h_top~B(G_μ, {F_n}) = h_μ(X, G),where G_μ is the set of generic points for μ with respect to {F_n} and h_top~B(G_μ, {F_n}) is the Bowen topological entropy(along {F_n}) on G_μ. This generalizes the classical result of Bowen(1973).     

Representing an isotone map between two bounded ordered sets by principal lattice congruences

For bounded lattices L₁ and L₂, let \({f\colon L_1 \to L_2}\) be a lattice homomorphism. Then the map \({{\rm Princ}(f)\colon \rm {Princ}(\it L_1) \to {\rm Princ}(\it L_2)}\), defined by \({{\rm con}(x,y) \mapsto {\rm con}(f(x),f(y))}\), is a 0-preserving isotone map from the bounded ordered set Princ(L₁) of principal congruences of L₁ to that of L₂. We prove that every 0-preserving isotone map between two bounded ordered sets can be represented in this way. Our result generalizes a 2016 result of G. Grätzer from \({\{0,1}\}\)-preserving isotone maps to 0-preserving isotone maps.     

A note on Jeśmanowicz’ conjecture

Je?manowicz [9] conjectured that, for positive integers m and n with m > n, gcd(m,n) = 1 and \({m\not\equiv n\pmod{2}}\), the exponential Diophantine equation \({(m^2-n^2)^x+(2mn)^y=(m^2+n^2)^z}\) has only the positive integer solution (x, y, z) = (2, 2, 2). We prove the conjecture for \({2 \| mn}\) and m + n has a prime factor p with \({p\not\equiv1\pmod{16}}\).     

Commutators of Riesz transforms of magnetic Schrödinger operators

Let \({A=-(\nabla-i{\vec a})\cdot (\nabla-i{\vec a}) +V}\) be a magnetic Schrödinger operator acting on \({L^2({\mathbb R}^n)}\), n ≥  1, where \({{\vec a}=(a_1, \ldots, a_n)\in L^2_{\rm loc}({\mathbb R}^n, {\mathbb R}^n)}\) and \({0\leq V\in L^1_{\rm loc}({\mathbb R}^n)}\). In this paper, we show that when a function \({b\in {\rm BMO}({\mathbb R}^n)}\), the commutators [b, T _k ]f = T _k (b f) ? b T _k f, k = 1, . . . , n, are bounded on \({L^p({\mathbb R}^n)}\) for all 1 < p < 2, where the operators T _k are Riesz transforms (?/?x _k  ? i a _k )A ^?1/2 associated with A.     

Approximation orders of the unit in the <Emphasis Type="Italic">β</Emphasis>-dynamical systems

For any real number β > 1, let S _n (β) be the partial sum of the first n items of the β-expansion of 1. It was known that the approximation order of 1 by S _n (β) is β ^?n for Lebesgue almost all β > 1. We consider the size of the set of β > 1 for which 1 can be approximated with the other orders \({\beta^{-\varphi(n)}}\) , where \({\varphi}\) is a positive function defined on \({\mathbb N}\) . More precisely, the size of the sets

$$\left\{\beta\in \mathfrak{B}:\limsup_{n\rightarrow\infty}\frac{\log_{\beta}(1-S_n(\beta))}{\varphi(n)}=-1\right\}$$

and

$$\left\{\beta\in \mathfrak{B}:\liminf_{n\rightarrow\infty}\frac{\log_{\beta}(1-S_n(\beta))}{\varphi(n)}=-1\right\}$$

are determined, where \({\mathfrak{B}=\{ \beta>1:\beta \text{ is not a simple Parry number}\}}\) .