On specifications,subset types and interpretation of proposition in type theory

In this paper we discuss some practical aspects of using type theory as a programming and specification language, where the viewpoint is to use it not only as a basis for program synthesis but also as a programming language with a programming logic allowing us to do ordinary verification.The subset type has been added to type theory in order to avoid irrelevant information in programs. We give an example of a proof which illustrates the problems that may occur if the subset type is used in specifications when we have the standard interpretation of propositions as types. Harrop-formulas and Squash are then discussed as solutions to these problems. It is argued that they are not acceptable from a practical point of view.An extension of the theory to include the two new judgment forms:A is a proposition, andA is true, is then given and explained in terms of the old theory. The logical constants are no longer identified with the corresponding type theoretical constants, but propositions are interpreted as Gödel formulas, which allow us to introduce and justify logical rules similar to rules for classical logic. The interpretation is extended to include predicates defined by using reflections of the ordinary definition of Gödel formulas in a type of small propositions.The programming example is then revisited and stronger elimination rules are discussed.     

Remarks on Fourier series

We prove the following propositions. An even integrable function whose Fourier coefficients form a convex sequence is absolutely continuous if and only if its Fourier series converges absolutely. If the function f(t)is convex on [0, ],f(t)=f(—t), then for odd n while for even n, b₀=0.Translated from Matematicheskie Zametki, Vol. 3, No. 5, pp. 597–604, May, 1968.     

Une réciproque générique du théorème Sunada

Let G a compact group of isometries of a closed riemannian manifold(X,m). Sunada proved that if are twofinite almost-conjugated subgroups of G, then and are isospectral. We prove that if G is finite, there exists an open dense set in the set of G-invariant metrics for which the converse ofthis resukt is true. If G is infinite, the situations is more complicated and we obtain some partial results.     

16-Dimensional Smooth Projective Planes with Large Collineation Groups

Smooth projective planes are projective planes defined on smooth manifolds (i.e. the set of points and the set of lines are smooth manifolds) such that the geometric operations of join and intersection are smooth. A systematic study of such planes and of their collineation groups can be found in previous works of the author. We prove in this paper that a 16-dimensional smooth projective plane which admits a collineation group of dimension d 39 is isomorphic to the octonion projective plane P₂ O. For topological compact projective planes this is true if d 41. Note that there are nonclassical topological planes with a collineation group of dimension 40.     

Every Matrix is a Product of Toeplitz Matrices

We show that every \(n\,\times \,n\) matrix is generically a product of \(\lfloor n/2 \rfloor + 1\) Toeplitz matrices and always a product of at most \(2n+5\) Toeplitz matrices. The same result holds true if the word ‘Toeplitz’ is replaced by ‘Hankel,’ and the generic bound \(\lfloor n/2 \rfloor + 1\) is sharp. We will see that these decompositions into Toeplitz or Hankel factors are unusual: We may not, in general, replace the subspace of Toeplitz or Hankel matrices by an arbitrary \((2n-1)\)-dimensional subspace of \({n\,\times \,n}\) matrices. Furthermore, such decompositions do not exist if we require the factors to be symmetric Toeplitz or persymmetric Hankel, even if we allow an infinite number of factors.     

Dynamic uniqueness for stochastic chains with unbounded memory

We say that a probability kernel exhibits dynamic uniqueness (DU) if all the stochastic chains starting from a fixed past coincide on the future tail $\sigma$-algebra. Our first theorem is a set of properties that are pairwise equivalent to DU which allow us to understand how it compares to other more classical concepts. In particular, we prove that DU is equivalent to a weak-${?}^2$ summability condition on the kernel. As a corollary to this theorem, we prove that the Bramson–Kalikow and the long-range Ising models both exhibit DU if and only if their kernels are ${?}^2$ summable. Finally, if we weaken the condition for DU, asking for coincidence on the future $\sigma$-algebra for almost every pair of pasts, we obtain a condition that is equivalent to $\beta$-mixing (weak-Bernoullicity) of the compatible stationary chain. As a consequence, we show that a modification of the weak-${?}^2$ summability condition on the kernel is equivalent to the $\beta$-mixing of the compatible stationary chain.     

Kollineationen und Schliessungssätze für Ebene Faserungen

Every affine central collineation of a translation plane induces a special collineation of the projective space spanned by the spreadF belonging to . Here the relations between these special collineations of and certain incidence propositions inF are investigated; so new proofs are given for some characterisations of (A,B)-regular spreads included in [7].     

For which reproducing kernel Hilbert spaces is Pick's theorem true?

Pick's theorem tells us that there exists a function inH , which is bounded by 1 and takes given values at given points, if and only if a certain matrix is positive.H is the space of multipliers ofH ², and this theorem has a natural generalisation whenH is replaced by the space of multipliers of a general reproducing kernel Hilbert spaceH(K) (whereK is the reproducing kernel). J. Agler has shown that this generalised theorem is true whenH(K) is a certain Sobolev space or the Dirichlet space, so it is natural to ask for which reproducing kernel Hilbert spaces this generalised theorem is true. This paper widens Agler's approach to cover reproducing kernel Hilbert spaces in general, replacing Agler's use of the deep theory of co-analytic models by a relatively elementary, and more general, matrix argument. The resulting theorem gives sufficient (and usable) conditions on the kernelK, for the generalised Pick's theorem to be true forH(K), and these are then used to prove Pick's theorem for certain weighted Hardy and Sobolev spaces and for a functional Hilbert space introduced by Saitoh.     

Baer and Baer *-ring characterizations of Leavitt path algebras

We characterize Leavitt path algebras which are Rickart, Baer, and Baer ?-rings in terms of the properties of the underlying graph. In order to treat non-unital Leavitt path algebras as well, we generalize these annihilator-related properties to locally unital rings and provide a more general characterizations of Leavitt path algebras which are locally Rickart, locally Baer, and locally Baer ?-rings. Leavitt path algebras are also graded rings and we formulate the graded versions of these annihilator-related properties and characterize Leavitt path algebras having those properties as well.Our characterizations provide a quick way to generate a wide variety of examples of rings. For example, creating a Baer and not a Baer ?-ring, a Rickart ?-ring which is not Baer, or a Baer and not a Rickart ?-ring, is straightforward using the graph-theoretic properties from our results. In addition, our characterizations showcase more properties which distinguish behavior of Leavitt path algebras from their $C^{?}$-algebra counterparts. For example, while a graph $C^{?}$-algebra is Baer (and a Baer ?-ring) if and only if the underlying graph is finite and acyclic, a Leavitt path algebra is Baer if and only if the graph is finite and no cycle has an exit, and it is a Baer ?-ring if and only if the graph is a finite disjoint union of graphs which are finite and acyclic or loops.     

Analytic hypoellipticity for sums of squares and the Treves conjecture

We are concerned with the problem of real analytic regularity of the solutions of sums of squares with real analytic coefficients. Treves conjecture states that an operator of this type is analytic hypoelliptic if and only if all the strata in the Poisson–Treves stratification are symplectic.We produce a model operator, $P_1$, having a single symplectic stratum and prove that it is Gevrey $s_0$ hypoelliptic and not better. See Theorem 2.1 for a definition of $s_0$. We also show that this phenomenon has a microlocal character.We point out explicitly that this is a counterexample to the sufficient part of Treves conjecture and not to the necessary part, which is still an open problem.     

Roots of Independence Polynomials of Well Covered Graphs

Let G be a well covered graph, that is, all maximal independent sets of G have the same cardinality, and let i_k denote the number of independent sets of cardinality k in G. We investigate the roots of the independence polynomial i(G, x) = i_kx^k. In particular, we show that if G is a well covered graph with independence number , then all the roots of i(G, x) lie in in the disk |z| (this is far from true if the condition of being well covered is omitted). Moreover, there is a family of well covered graphs (for each ) for which the independence polynomials have a root arbitrarily close to –.     

Nilpotent elements and Armendariz rings

We study the structure of the set of nilpotent elements in Armendariz rings and introduce nil-Armendariz as a generalization. We also provide some new examples by proving that if D is a K-algebra and $n?2$, the coproduct $D{1}_{K}K〈x|x^n=0〉$ is Armendariz if and only if D is a domain with $K?\{0\}$ as its group of units. Finally we study the conditions under which the polynomial ring over a nil-Armendariz ring is nil-Armendariz, which is related to a question of Amitsur.     

An improved bound for the de Bruijn–Newman constant

The Riemann hypothesis is equivalent to the conjecture that the de Bruijn–Newman constant satisfies 0. However, so far all the bounds that have been proved for go in the other direction, and provide support for the conjecture of Newman that 0. This paper shows how to improve previous lower bounds and prove that –2.710^–9<. This can be done using a pair of zeros of the Riemann zeta function near zero number 10²⁰ that are unusually close together. The new bound provides yet more evidence that the Riemann hypothesis, if true, is just barely true.     

Spreads of right quadratic skew field extensions

LetL/K be a right quadratic (skew) field extension and let be a 3-dimensional projective space overK which is embedded in a 3-dimensional projective space overL. Moreover, let be a line of which carries no point of. The main result is that — even whenL orK is a skew field — the following holds true: A Desarguesian spread of is given by the set of all lines of which are indicated by the points of . A spread of arises in this way if, and only if, there exists an isomorphism ofL onto the kernel of the spread such thatK is elementwise invariant. Furthermore, a geometric characterization of right quadratic extensions with a left degree other than 2 and of quadratic Galois extensions is given.Dedicated to O. Giering on the occasion of this 60th birthdayThe term field is to mean a not necessarily commutative field.     

Representation of artinian partially ordered sets over semiartinian von Neuman regular algebras

If R is a semiartinian von Neumann regular ring, then the set ${\mathbf{Prim}}_R$ of primitive ideals of R, ordered by inclusion, is an artinian poset in which all maximal chains have a greatest element. Moreover, if ${\mathbf{Prim}}_R$ has no infinite antichains, then the lattice ${\mathbb{L}}_2(R)$ of all ideals of R is anti-isomorphic to the lattice of all upper subsets of ${\mathbf{Prim}}_R$. Since the assignment $U?r_R(U)$ defines a bijection from any set ${\mathbf{Simp}}_R$ of representatives of simple right R-modules to ${\mathbf{Prim}}_R$, a natural partial order is induced in ${\mathbf{Simp}}_R$, under which the maximal elements are precisely those simple right R-modules which are finite dimensional over the respective endomorphism division rings; these are always R-injective. Given any artinian poset I with at least two elements and having a finite cofinal subset, a lower subset $I^{\prime }?I$ and a field D, we present a construction which produces a semiartinian and unit-regular D-algebra $D_I$ having the following features: (a) ${\mathbf{Simp}}_{D_I}$ is order isomorphic to I; (b) the assignment $H?{\mathbf{Simp}}_{D_I/H}$ realizes an anti-isomorphism from the lattice ${\mathbb{L}}_2(D_I)$ to the lattice of all upper subsets of ${\mathbf{Simp}}_{D_I}$; (c) a non-maximal element of ${\mathbf{Simp}}_{D_I}$ is injective if and only if it corresponds to an element of $I^{\prime }$, thus $D_I$ is a right V-ring if and only if $I^{\prime }=I$; (d) $D_I$ is a right and left V-ring if and only if I is an antichain; (e) if I has finite dual Krull length, then $D_I$ is (right and left) hereditary; (f) if I is at most countable and $I^{\prime }=?$, then $D_I$ is a countably dimensional D-algebra.