Related papers: The Reduction Property Revisited
Steinberg's tensor product theorem shows that for semisimple algebraic groups the study of irreducible representations of higher Frobenius kernels reduces to the study of irreducible representations of the first Frobenius kernel. In the…
Let $A$ be an $AH$ algebra, that is, $A$ is the inductive limit $C^{*}$-algebra of $$A_{1}\xrightarrow{\phi_{1,2}}A_{2}\xrightarrow{\phi_{2,3}}A_{3}\longrightarrow\cdots\longrightarrow A_{n}\longrightarrow\cdots$$ with…
The induction principle for natural numbers expresses that when a property holds for some natural number a and is hereditary, then it holds for all numbers greater than or equal to a. We present a similar principle for real numbers.
We introduce a fragment of continuous first-order logic, analogue of Palyutin formulas (or h-formulas) in classical model theory, which is preserved under reduced products in both directions. We use it to extend classical results on…
The enterprise of comparing mathematical theorems according to their logical strength is an active area in mathematical logic. In this setting, called reverse mathematics, one investigates which theorems provably imply which others in a…
A new large-cardinal property is introduced which enables one to give a relative consistency proof of restricted versions of the reflection principles discussed by Tait in his essay "Constructing Cardinals from Below".
Residuation theory concerns the study of partially ordered algebraic structures, most often monoids, equipped with a weak inverse for the monoidal operator. One of its area of application has been constraint programming, whose key…
In conjugate gradient method, it is well known that the recursively computed residual differs from true one as the iteration proceeds in finite arithmetic. Some work have been devoted to analyze this be-havior and to evaluate the lower and…
In this paper, we introduce a family of topological spaces that captures the existence of preservation theorems. The structure of those spaces allows us to study the relativisation of preservation theorems under suitable definitions of…
We show that under certain conditions, well-studied algebraic properties transfer from the class $\mathcal{Q}_{_\text{RFSI}}$ of the relatively finitely subdirectly irreducible members of a quasivariety $\mathcal{Q}$ to the whole…
We generalize the notion of proof term to the realm of transfinite reduction. Proof terms represent reductions in the first-order term format, thereby facilitating their formal analysis. We show that any transfinite reduction can be…
Pattern avoidance classes of permutations that cannot be expressed as unions of proper subclasses can be described as the set of subpermutations of a single bijection. In the case that this bijection is a permutation of the natural numbers…
Why do language models trained on contradictory data prefer correct answers? In controlled experiments with small transformers (3.5M--86M parameters), we show that this preference tracks the compressibility structure of errors rather than…
We generalize the derivation of the Wallis formula for $\pi$ from a variational computation of the spectrum of the Hydrogen atom. We obtain infinite product formulas for certain combinations of gamma functions, which include irrational…
Various problems on integers lead to the class of congruence preserving functions on rings, i.e. functions verifying $a-b$ divides $f(a)-f(b)$ for all $a,b$. We characterized these classes of functions in terms of sums of rational…
Three philosophical principles are often quoted in connection with Leibniz: "objects sharing the same properties are the same object", "everything can possibly exist, unless it yields contradiction", "the ideal elements correctly determine…
In this paper we discuss the notion of reducibility for matrix weights and introduce a real vector space $\mathcal C_\mathbb{R}$ which encodes all information about the reducibility of $W$. In particular a weight $W$ reduces if and only if…
Rado's Conjecture is a compactness/reflection principle that says any nonspecial tree of height $\omega_1$ has a nonspecial subtree of size $\leq \aleph_1$. Though incompatible with Martin's Axiom, Rado's Conjecture turns out to have many…
Deduction modulo is a way to express a theory using computation rules instead of axioms. We present in this paper an extension of deduction modulo, called Polarized deduction modulo, where some rules can only be used at positive…
Polynomial reduction is one of the main tools in computational algebra with innumerable applications in many areas, both pure and applied. Since many years both the theory and an efficient design of the related algorithm have been solidly…