On the Taylor expansion of $\lambda$-terms and the groupoid structure of their rigid approximants

We show that the normal form of the Taylor expansion of a $\lambda$-term is isomorphic to its B\"ohm tree, improving Ehrhard and Regnier's original proof along three independent directions. First, we simplify the final step of the proof by following the left reduction strategy directly in the resource calculus, avoiding to introduce an abstract machine ad hoc. We also introduce a groupoid of permutations of copies of arguments in a rigid variant of the resource calculus, and relate the coefficients of Taylor expansion with this structure, while Ehrhard and Regnier worked with groups of permutations of occurrences of variables. Finally, we extend all the results to a nondeterministic setting: by contrast with previous attempts, we show that the uniformity property that was crucial in Ehrhard and Regnier's approach can be preserved in this setting.

The D-CTC Condition is Generically Fulfilled in Classical (Non-quantum) Statistical Systems

The D-CTC condition, introduced by David Deutsch as a condition to be fulfilled by analogues for processes of quantum systems in the presence of closed timelike curves, is investigated for classical statistical (non-quantum) bi-partite systems. It is shown that the D-CTC condition can generically be fulfilled in classical statistical systems, under very general, model-independent conditions. The central property used is the convexity and completeness of the state space that allows it to generalize Deutsch’s original proof for q-bit systems to more general classes of statistically described systems. The results demonstrate that the D-CTC condition, or the conditions under which it can be fulfilled, is not characteristic of, or dependent on, the quantum nature of a bi-partite system.

The consistency of arithmetic

This chapter opens the part of the book that deals with ordinal proof theory. Here the systems of interest are not purely logical ones, but rather formalized versions of mathematical theories, and in particular the first-order version of classical arithmetic built on top of the sequent calculus. Classical arithmetic goes beyond pure logic in that it contains a number of specific axioms for, among other symbols, 0 and the successor function. In particular, it contains the rule of induction, which is the essential rule characterizing the natural numbers. Proving a cut-elimination theorem for this system is hopeless, but something analogous to the cut-elimination theorem can be obtained. Indeed, one can show that every proof of a sequent containing only atomic formulas can be transformed into a proof that only applies the cut rule to atomic formulas. Such proofs, which do not make use of the induction rule and which only concern sequents consisting of atomic formulas, are called simple. It is shown that simple proofs cannot be proofs of the empty sequent, i.e., of a contradiction. The process of transforming the original proof into a simple proof is quite involved and requires the successive elimination, among other things, of “complex” cuts and applications of the rules of induction. The chapter describes in some detail how this transformation works, working through a number of illustrative examples. However, the transformation on its own does not guarantee that the process will eventually terminate in a simple proof.

Nonnegative and Strictly Positive Linearization of Jacobi and Generalized Chebyshev Polynomials

In the theory of orthogonal polynomials, as well as in its intersection with harmonic analysis, it is an important problem to decide whether a given orthogonal polynomial sequence $$(P_n(x))_{n\in \mathbb {N}_0}$$ ( P n ( x ) ) n ∈ N 0 satisfies nonnegative linearization of products, i.e., the product of any two $$P_m(x),P_n(x)$$ P m ( x ) , P n ( x ) is a conical combination of the polynomials $$P_{|m-n|}(x),\ldots ,P_{m+n}(x)$$ P | m - n | ( x ) , … , P m + n ( x ) . Since the coefficients in the arising expansions are often of cumbersome structure or not explicitly available, such considerations are generally very nontrivial. Gasper (Can J Math 22:582–593, 1970) was able to determine the set V of all pairs $$(\alpha ,\beta )\in (-1,\infty )^2$$ ( α , β ) ∈ ( - 1 , ∞ ) 2 for which the corresponding Jacobi polynomials $$(R_n^{(\alpha ,\beta )}(x))_{n\in \mathbb {N}_0}$$ ( R n ( α , β ) ( x ) ) n ∈ N 0 , normalized by $$R_n^{(\alpha ,\beta )}(1)\equiv 1$$ R n ( α , β ) ( 1 ) ≡ 1 , satisfy nonnegative linearization of products. Szwarc (Inzell Lectures on Orthogonal Polynomials, Adv. Theory Spec. Funct. Orthogonal Polynomials, vol 2, Nova Sci. Publ., Hauppauge, NY pp 103–139, 2005) asked to solve the analogous problem for the generalized Chebyshev polynomials $$(T_n^{(\alpha ,\beta )}(x))_{n\in \mathbb {N}_0}$$ ( T n ( α , β ) ( x ) ) n ∈ N 0 , which are the quadratic transformations of the Jacobi polynomials and orthogonal w.r.t. the measure $$(1-x^2)^{\alpha }|x|^{2\beta +1}\chi _{(-1,1)}(x)\,\mathrm {d}x$$ ( 1 - x 2 ) α | x | 2 β + 1 χ ( - 1 , 1 ) ( x ) d x . In this paper, we give the solution and show that $$(T_n^{(\alpha ,\beta )}(x))_{n\in \mathbb {N}_0}$$ ( T n ( α , β ) ( x ) ) n ∈ N 0 satisfies nonnegative linearization of products if and only if $$(\alpha ,\beta )\in V$$ ( α , β ) ∈ V , so the generalized Chebyshev polynomials share this property with the Jacobi polynomials. Moreover, we reconsider the Jacobi polynomials themselves, simplify Gasper’s original proof and characterize strict positivity of the linearization coefficients. Our results can also be regarded as sharpenings of Gasper’s one.

Relative vertex asphericity

Diagrammatic reducibility DR and its generalization, vertex asphericity VA, are combinatorial tools developed for detecting asphericity of a 2-complex. Here we present tests for a relative version of VA that apply to pairs of 2-complexes $(L,K)$ , where K is a subcomplex of L. We show that a relative weight test holds for injective labeled oriented trees, implying that they are VA and hence aspherical. This strengthens a result obtained by the authors in 2017 and simplifies the original proof.

Uniform convergence in von Neumann’s ergodic theorem in the absence of a spectral gap

Von Neumann’s original proof of the ergodic theorem is revisited. A uniform convergence rate is established under the assumption that one can control the density of the spectrum of the underlying self-adjoint operator when restricted to suitable subspaces. Explicit rates are obtained when the bound is polynomial, with applications to the linear Schrödinger and wave equations. In particular, decay estimates for time averages of solutions are shown.

An L-function free proof of Hua's Theorem on sums of five prime squares

We provide a new proof of Hua's result that every sufficiently large integer N ≡ 5 (mod 24) can be written as the sum of the five prime squares. Hua's original proof relies on the circle method and uses results from the theory of L-functions. Here, we present a proof based on the transference principle first introduced in[5]. Using a sieve theoretic approach similar to ([10]), we do not require any results related to the distributions of zeros of L- functions. The main technical difficulty of our approach lies in proving the pseudo-randomness of the majorant of the characteristic function of the W-tricked primes which requires a precise evaluation of the occurring Gaussian sums and Jacobi symbols.

IDENTITIES IN BRANDT SEMIGROUPS, REVISITED

We present a new proof for the main claim made in the author's paper "On the identity bases of Brandt semigroups" (Ural. Gos. Univ. Mat. Zap., 14, no.1 (1985), 38–42); this claim provides an identity basis for an arbitrary Brandt semigroup over a group of finite exponent. We also show how to fill a gap in the original proof of the claim in loc. cit.