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1201  Free Multivariate w*Semicrossed Products: Reflexivity and the Bicommutant Property Bickerton, Robert T.; Kakariadis, Evgenios T.A.
We study w*semicrossed products over actions of the free semigroup
and the free abelian semigroup on (possibly nonselfadjoint)
w*closed algebras.
We show that they are reflexive when the dynamics are implemented
by uniformly bounded families of invertible row operators.
Combining with results of Helmer we derive that w*semicrossed
products of factors (on a separable Hilbert space) are reflexive.
Furthermore we show that w*semicrossed products of automorphic
actions on maximal abelian selfadjoint algebras are reflexive.
In all cases we prove that the w*semicrossed products have the
bicommutant property if and only if the ambient algebra of the
dynamics does also.


1236  Unperforated Pairs of Operator Spaces and Hyperrigidity of Operator Systems Clouâtre, Raphaël
We study restriction and extension properties for states on C$^*$algebras
with an eye towards hyperrigidity of operator systems. We use
these ideas to provide supporting evidence for Arveson's hyperrigidity
conjecture. Prompted by various characterizations of hyperrigidity
in terms of states, we examine unperforated pairs of selfadjoint
subspaces in a C$^*$algebra. The configuration of the subspaces
forming an unperforated pair is in some sense compatible with
the order structure of the ambient C$^*$algebra. We prove
that commuting pairs are unperforated, and obtain consequences
for hyperrigidity. Finally, by exploiting recent advances in
the tensor theory of operator systems, we show how the weak expectation
property can serve as a flexible relaxation of the notion of
unperforated pairs.


1261  Range Spaces of Coanalytic Toeplitz Operators Fricain, Emmanuel; Hartmann, Andreas; Ross, William T.
In this paper we discuss the range of a coanalytic Toeplitz
operator. These range spaces are closely related to de BrangesRovnyak
spaces (in some cases they are equal as sets). In order to understand
its structure, we explore when
the range space decomposes into the range of an associated analytic
Toeplitz operator and an identifiable orthogonal complement.
For certain cases, we compute this orthogonal complement in terms
of the kernel of a certain Toeplitz operator on the Hardy space
where we focus on when this kernel is a model space (backward
shift invariant subspace).
In the spirit of AhernClark, we also discuss the nontangential
boundary behavior in these range spaces. These results give us
further insight into the description of the range of a coanalytic
Toeplitz operator as well as its orthogonal decomposition. Our
AhernClark type results, which are stated in a general abstract
setting, will also have applications to related subHardy Hilbert
spaces of analytic functions such as the de BrangesRovnyak spaces
and the harmonically weighted Dirichlet spaces.


1284  Long Sets of Lengths with Maximal Elasticity Geroldinger, Alfred; Zhong, Qinghai
We introduce a new invariant describing the structure of sets of lengths in atomic monoids and domains. For an atomic monoid $H$, let $\Delta_{\rho} (H)$ be the set of all positive integers $d$ which occur as differences of arbitrarily long arithmetical progressions contained in sets of lengths having maximal elasticity $\rho (H)$. We study $\Delta_{\rho} (H)$ for transfer Krull monoids of finite type (including commutative Krull domains with finite class group) with methods from additive combinatorics, and also for a class of weakly Krull domains (including orders in algebraic number fields) for which we use ideal theoretic methods.


1319  Multiplicative Energy of Shifted Subgroups and Bounds On Exponential Sums with Trinomials in Finite Fields Macourt, Simon; Shkredov, Ilya D.; Shparlinski, Igor E.
We give a new bound on collinear triples in
subgroups of prime finite
fields and use it to give some new bounds on exponential sums
with trinomials.


1339  Relative Discrete Series Representations for Two Quotients of $p$adic $\mathbf{GL}_n$ Smith, Jerrod Manford
We provide an explicit construction of representations in the
discrete spectrum of two $p$adic symmetric spaces. We consider
$\mathbf{GL}_n(F) \times \mathbf{GL}_n(F) \backslash \mathbf{GL}_{2n}(F)$
and $\mathbf{GL}_n(F) \backslash \mathbf{GL}_n(E)$, where $E$
is a quadratic Galois extension of a nonarchimedean local field
$F$ of characteristic zero and odd residual characteristic. The
proof of the main result involves an application of a symmetric
space version of Casselman's Criterion for square integrability
due to Kato and Takano.


1373  A New Proof of the HansenMullen Irreducibility Conjecture Tuxanidy, Aleksandr; Wang, Qiang
We give a new proof of the HansenMullen irreducibility conjecture.
The proof relies on an application of a (seemingly new) sufficient
condition for the existence of
elements of degree $n$ in the support of functions on finite
fields.
This connection to irreducible polynomials is made via the least
period of the discrete Fourier transform (DFT) of functions with
values in finite fields.
We exploit this relation and prove, in an elementary fashion,
that a relevant function related to the DFT of characteristic
elementary symmetric functions (which produce the coefficients
of characteristic polynomials)
satisfies a simple requirement on the least period.
This bears a sharp contrast to previous techniques in literature
employed to tackle existence
of irreducible polynomials with prescribed coefficients.


1390  Squarefree Values of Decomposable Forms Xiao, Stanley Yao
In this paper we prove that decomposable forms,
or homogeneous polynomials $F(x_1, \cdots, x_n)$ with integer
coefficients which split completely into linear factors over
$\mathbb{C}$, take on infinitely many squarefree values subject
to simple necessary conditions and $\deg f \leq 2n + 2$ for all
irreducible factors $f$ of $F$. This work generalizes a theorem
of Greaves.


1416  A Special Case of Completion Invariance for the $c_2$ Invariant of a Graph Yeats, Karen
The $c_2$ invariant is an arithmetic graph invariant defined
by Schnetz. It is useful for understanding Feynman periods.
Brown and Schnetz conjectured that the $c_2$ invariant has a
particular symmetry known as completion invariance.
This paper will prove completion invariance of the $c_2$ invariant
in the case that we are over the field with 2 elements and the
completed graph has an odd number of vertices.
The methods involve enumerating certain edge bipartitions of
graphs; two different constructions are needed.

