Special values of anticyclotomic L-functions modulo λ

The purpose of this article is to generalize some results of Vatsal on the special values of Rankin–Selberg L-functions in an anticyclotomic \({\mathbb{Z}_{p}}\) -extension. Let g be a cuspidal Hilbert modular newform of parallel weight \({(2,\ldots,2)}\) and level \({\mathcal{N}}\) over a totally real field F, and let K/F be a totally imaginary quadratic extension of relative discriminant \({\mathcal{D}}\) . We study the l-adic valuation of the special values \({L(g,\chi,\frac{1}{2})}\) as \({\chi}\) varies over the ring class characters of K of \({\mathcal{P}}\) -power conductor, for some fixed prime ideal \({\mathcal{P}}\) . We prove our results under the only assumption that the prime to \({\mathcal{P}}\) part of \({\mathcal{N}}\) is relatively prime to \({\mathcal{D}}\) .     

Categorical group invariants of 3-manifolds

For a given class \({\mathcal{G}}\) of groups, a 3-manifold M is of \({\mathcal{G}}\) -category \({\leq k}\) if it can be covered by k open subsets such that for each path-component W of the subsets the image of its fundamental group \({ \pi_1(W) \rightarrow \pi(M )}\) belongs to \({\mathcal{G}}\) . The smallest number k such that M admits such a covering is the \({\mathcal{G}}\) -category, \({cat_{\mathcal{G}}(M)}\) . If M is closed, it has \({\mathcal{G}}\) -category between 1 and 4. We characterize all closed 3-manifolds of \({\mathcal{G}}\) -category 1, 2, and 3 for various classes \({\mathcal{G}}\) .     

Forbidden Triples Containing a Complete Graph and a Complete Bipartite Graph of Small Order

For a graph G and a set \({\mathcal{F}}\) of connected graphs, G is said be \({\mathcal{F}}\) -free if G does not contain any member of \({\mathcal{F}}\) as an induced subgraph. We let \({\mathcal{G} _{3}(\mathcal{F})}\) denote the set of all 3-connected \({\mathcal{F}}\) -free graphs. This paper is concerned with sets \({\mathcal{F}}\) of connected graphs such that \({\mathcal{F}}\) contains no star, \({|\mathcal{F}|=3}\) and \({\mathcal{G}_{3}(\mathcal{F})}\) is finite. Among other results, we show that for a connected graph T( ≠ K ₁) which is not a star, \({\mathcal{G}_{3}(\{K_{4},K_{2,2},T\})}\) is finite if and only if T is a path of order at most 6.     

Procrustes Problems and Parseval Quasi-Dual Frames

Parseval frames have particularly useful properties, and in some cases, they can be used to reconstruct signals which were analyzed by a non-Parseval frame. In this paper, we completely describe the degree to which such reconstruction is feasible. Indeed, notice that for fixed frames \({\mathcal{F}}\) and \({\mathcal{X}}\) with synthesis operators F and X, the operator norm of FX ^??I measures the (normalized) worst-case error in the reconstruction of vectors when analyzed with \({\mathcal{X}}\) and synthesized with \({\mathcal{F}}\) . Hence, for any given frame \({\mathcal{F}}\) , we compute explicitly the infimum of the operator norm of FX ^??I, where \({\mathcal{X}}\) is any Parseval frame. The \({\mathcal{X}}\) ’s that minimize this quantity are called Parseval quasi-dual frames of \({\mathcal{F}}\) . Our treatment considers both finite and infinite Parseval quasi-dual frames.     

Comparison of fine structural mice via coarse iteration

Let \({\mathcal{M}}\) be a fine structural mouse. Let \({\mathbb{D}}\) be a fully backgrounded \({L[\mathbb{E}]}\) -construction computed inside an iterable coarse premouse S. We describe a process comparing \({\mathcal{M}}\) with \({\mathbb{D}}\) , through forming iteration trees on \({\mathcal{M}}\) and on S. We then prove that this process succeeds.     

Weak theories of concatenation and minimal essentially undecidable theories

We consider weak theories of concatenation, that is, theories for strings or texts. We prove that the theory of concatenation \({\mathsf{WTC}^{-\varepsilon}}\) , which is a weak subtheory of Grzegorczyk’s theory \({\mathsf{TC}^{-\varepsilon}}\) , is a minimal essentially undecidable theory, that is, the theory \({\mathsf{WTC}^{-\varepsilon}}\) is essentially undecidable and if one omits an axiom scheme from \({\mathsf{WTC}^{-\varepsilon}}\) , then the resulting theory is no longer essentially undecidable. Moreover, we give a positive answer to Grzegorczyk and Zdanowski’s conjecture that ‘The theory \({\mathsf{TC}^{-\varepsilon}}\) is a minimal essentially undecidable theory’. For the alternative theories \({\mathsf{WTC}}\) and \({\mathsf{TC}}\) which have the empty string, we also prove that the each theory without the neutrality of \({\varepsilon}\) is to be such a theory too.     

A Remark on The Geometry of Spaces of Functions with Prime Frequencies

For any positive integer r, denote by \({\mathcal{P}_{r}}\) the set of all integers \({\gamma \in \mathbb{Z}}\) having at most r prime divisors. We show that \({C_{\mathcal{P}_{r}}(\mathbb{T})}\) , the space of all continuous functions on the circle \({\mathbb{T}}\) whose Fourier spectrum lies in \({\mathcal{P}_{r}}\) , contains a complemented copy of \({\ell^{1}}\) . In particular, \({C_{\mathcal{P}_{r}}(\mathbb{T})}\) is not isomorphic to \({C(\mathbb{T})}\) , nor to the disc algebra \({A(\mathbb{D})}\) . A similar result holds in the L ¹ setting.     

Finite frames,P-frames and basically disconnected frames

Let \({\mathcal{P}}\) be an ideal of closed quotients of a completely regular frame L and \({\mathcal{R}_{\mathcal{P}}(L)}\) the collection of all functions in the ring \({\mathcal{R}(L)}\) whose support belong to \({\mathcal{P}}\) . We show that \({\mathcal{R}(L)}\) is a Noetherian ring if and only if \({\mathcal{R}(L)}\) is an Artinian ring if and only if L is a finite frame. Using this result, we next show that if \({\mathcal{P}}\) is the ideal of all compact closed quotients of L and L is \({\mathcal{P}}\) -continuous, then \({\mathcal{R}_{\mathcal{P}}(L)}\) is a Noetherian ring if and only if L is finite. Moreover, we show that L is a P-frame if and only if each ideal of \({\mathcal{R}(L)}\) is of the form \({\mathcal{R}_{\mathcal{P}}(L)}\) for some choice of \({\mathcal{P}}\) . We furnish equivalent conditions for \({\mathcal{R}_{\mathcal{P}}(L)}\) to be a prime ideal, a free ideal, and an essential ideal of \({\mathcal{R}(L)}\) separately in terms of the cozero elements of L. Finally, we show that L is basically disconnected if and only if \({\mathcal{R}(L)}\) is a coherent ring.     

Anderson’s Orthogonality Catastrophe for One-Dimensional Systems

The overlap, \({\mathcal{D}_N}\) , between the ground state of N free fermions and the ground state of N fermions in an external potential in one spatial dimension is given by a generalized Gram determinant. An upper bound is \({\mathcal{D}_N\leq\exp(-\mathcal{I}_N)}\) with the so-called Anderson integral \({\mathcal{I}_N}\) . We prove, provided the external potential satisfies some conditions, that in the thermodynamic limit \({\mathcal{I}_N = \gamma\ln N + O(1)}\) as \({N\to\infty}\) . The coefficient γ > 0 is given in terms of the transmission coefficient of the one-particle scattering matrix. We obtain a similar lower bound on \({\mathcal{D}_N}\) concluding that \({\tilde{C} N^{-\tilde{\gamma}} \leq \mathcal{D}_N \leq CN^{-\gamma}}\) with constants C, \({\tilde{C}}\) , and \({\tilde{\gamma}}\) . In particular, \({\mathcal{D}_N\to 0}\) as \({N\to\infty}\) which is known as Anderson’s orthogonality catastrophe.     

On the Relaxed Colouring Game and the Unilateral Colouring Game

In the paper we introduce the new game—the unilateral \({\mathcal{P}}\) -colouring game which can be used as a tool to study the r-colouring game and the (r, d)-relaxed colouring game. Let be given a graph G, an additive hereditary property \({\mathcal {P}}\) and a set C of r colours. In the unilateral \({\mathcal {P}}\) -colouring game similarly as in the r-colouring game, two players, Alice and Bob, colour the uncoloured vertices of the graph G, but in the unilateral \({\mathcal {P}}\) -colouring game Bob is more powerful than Alice. Alice starts the game, the players play alternately, but Bob can miss his move. Bob can colour the vertex with an arbitrary colour from C, while Alice must colour the vertex with a colour from C in such a way that she cannot create a monochromatic minimal forbidden subgraph for the property \({\mathcal {P}}\) . If after |V(G)| moves the graph G is coloured, then Alice wins the game, otherwise Bob wins. The \({\mathcal {P}}\) -unilateral game chromatic number, denoted by \({\chi_{ug}^\mathcal {P}(G)}\) , is the least number r for which Alice has a winning strategy for the unilateral \({\mathcal {P}}\) -colouring game with r colours on G. We prove that the \({\mathcal {P}}\) -unilateral game chromatic number is monotone and is the upper bound for the game chromatic number and the relaxed game chromatic number. We give the winning strategy for Alice to play the unilateral \({\mathcal {P}}\) -colouring game. Moreover, for k ≥  2 we define a class of graphs \({\mathcal {H}_k =\{G|{\rm every \;block \;of\;}G \; {\rm has \;at \;most}\; k \;{\rm vertices}\}}\) . The class \({\mathcal {H}_k }\) contains, e.g., forests, Husimi trees, line graphs of forests, cactus graphs. Let \({\mathcal {S}_d}\) be the class of graphs with maximum degree at most d. We find the upper bound for the \({\mathcal {S}_2}\) -unilateral game chromatic number for graphs from \({\mathcal {H}_3}\) and we study the \({\mathcal {S}_d}\) -unilateral game chromatic number for graphs from \({\mathcal {H}_4}\) for \({d \in \{2,3\}}\) . As the conclusion from these results we obtain the result for the d-relaxed game chromatic number: if \({G \in \mathcal {H}_k}\) , then \({\chi_g^{(d)}(G) \leq k + 2-d}\) , for \({k \in \{3, 4\}}\) and \({d \in \{0, \ldots, k-1\}}\) . This generalizes a known result for trees.     

Representations of algebras in varieties generated by infinite primal algebras

If \({\mathcal{A}}\) is an infinite primal algebra, then we shall represent any algebra in the variety \({V\,(\mathcal{A}}\) ) generated by \({\mathcal{A}}\) as a limit reduced power of \({\mathcal{A}}\) . Furthermore, we show that any homomorphism between algebras in \({V\,(\mathcal{A}}\) ) can be induced by mappings between underlying sets of the limit reduced powers. With this representation of the morphisms between algebras in \({V\,(\mathcal{A}}\) ) at hand, we will construct a category equivalent to the category \({V\,(\mathcal{A}}\) ).     

Reeb orbits trapped by Denjoy minimal sets

Let φ be any flow on T ⁿ obtained as the suspension of a smooth diffeomorphism of \({T^{n-1}}\) , and let \({\mathcal {A}}\) be any compact invariant set of φ. We realize \({(\mathcal{A}, \varphi|_{\mathcal{A}})}\) up to reparametrization as an invariant set of the Reeb flow of a contact form on \({\mathbb{R}^{2n+1}}\) equal to the standard contact form outside a compact set and defining the standard contact structure on all of \({\mathbb{R}^{2n+1}}\) . This uses the method from Geiges, Röttgen, and Zehmisch.     

Newton Polytopes and Witness Sets

We present two algorithms that compute the Newton polytope of a polynomial f defining a hypersurface \({\mathcal{H}}\) in \({\mathbb{C}^n}\) using numerical computation. The first algorithm assumes that we may only compute values of f—this may occur if f is given as a straight-line program, as a determinant, or as an oracle. The second algorithm assumes that \({\mathcal{H}}\) is represented numerically via a witness set. That is, it computes the Newton polytope of \({\mathcal{H}}\) using only the ability to compute numerical representatives of its intersections with lines. Such witness set representations are readily obtained when \({\mathcal{H}}\) is the image of a map or is a discriminant. We use the second algorithm to compute a face of the Newton polytope of the Lüroth invariant, as well as its restriction to that face.     

The model companion of the class of pseudocomplemented semilattices is finitely axiomatizable

It is shown that the class \({\mathcal{PCSL}^{ec}}\) of existentially closed pseudocomplemented semilattices is finitely axiomatizable by appropriately extending a finite axiomatization of the class \({\mathcal{PCSL}^{ac}}\) of algebraically closed pseudocomplemented semilattices. Because \({\mathcal{PCSL}^{ec}}\) coincides with the model companion of the class \({\mathcal{PCSL}}\) of pseudocomplemented semilattices, this answers the question asked by Albert and Burris in a paper in 1986: “Does the class of pseudocomplemented semilattices have a finitely axiomatizable model companion?"