Conway’s Wizards

I present and discuss a puzzle about wizards invented by John H. Conway.     

The Alperin and dade conjectures for the simple Conway’s third group

This paper is part of a program to study Alperin’s weight conjecture and Dade’s conjecture on counting ordinary characters in blocks for several finite groups. The classifications of radical subgroups and certain radical chains and their local structures of the simple Conway’s third group have been obtained by using the computer algebra system CAYLEY. The Alperin weight conjecture and the Dade final conjecture have been confirmed for the group.     

On Conway’s potential function for colored links

The Conway potential function(CPF) for colored links is a convenient version of the multivariable Alexander–Conway polynomial. We give a skein characterization of CPF, much simpler than the one by Murakami. In particular, Conway's "smoothing of crossings" is not in the axioms. The proof uses a reduction scheme in a twisted group-algebra P_nB_n, where B_n is a braid group and P_n is a domain of multi-variable rational fractions. The proof does not use computer algebra tools. An interesting by-product is a characterization of the Alexander–Conway polynomial of knots.     

Chase’s lemma and its context

Chase’s lemma provides a powerful tool for translating properties of (co)products in abelian categories into chain conditions. This note discusses the context in which the lemma is used, making explicit what is often neglected in the literature because of its technical nature.     

The SOC in cells’ living expectations of Conway’s Game of Life and its extended version

In self-organized systems such as Conway’s Game of Life (CGL). Wikipedia, Conway’s game of the life, https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life., though whether the single cell will survive or die seems unpredictable, the log–log distribution of all cells living frequency satisfies the 1/f linear law, thus meets the Self-organized Criticality(SOC) rule, which not only proves that CGL is a self-organized system, but more significantly, that the chance of living for each cell is spatial heterogeneous, and is statistical fractal.After carried out CGL, the specified iterative period which begins with a random initial condition and ends when it reaches the homeostasis, add up all the states which the living cells are marked by 1s, and the dead are marked by 0s. The resulted sum picture consisting of cells having its gray level representing the living times during the iterative process. By plotting the gray level distribution of the sum picture on log–log scale, the graph indicates the spatial living expectations distributions. Then we find the curve of the graph satisfies the Self-organized Criticality(SOC) rule, showing its linear feature in the intermediate zone, which also has name of 1/f feature.To examine its universality, we designed a more complicated self-organized cellular automata with each cell having five possible states thus the rule table becomes more complicated. As expected, the consequence shows the similar feature, and the linear feature is even more obvious when the similar experiments are carried out.To conclude, it is a new discovery of SOC from a new perspective. And with the self-organized systems expanding to other different rule tables, this feature may still be satisfied.More further, considering the natural self-organized systems of living creatures, the spatial living expectations of different phenotypes may satisfy the 1/f law, too. Though we regard this as an inspirational orientation, the supposition needs more designed experiments to prove.     

Feynman’s operational calculi for noncommuting operators: The monogenic calculus

In recent papers the authors presented their approach to Feynman’s operational calculi for a system of not necessarily commuting bounded linear operators acting on a Banach space. The central objects of the theory are the disentangling algebra, a commutative Banach algebra, and the disentangling map which carries this commutative structure into the noncommutative algebra of operators. Under assumptions concerning the growth of disentangled exponential expressions, the associated functional calculus for the system of operators is a distribution with compact support which we view as the joint spectrum of the operators with respect to the disentangling map. In this paper, the functional calculus is represented in terms of a higher-dimensional analogue of the Riesz-Dunford calculus using Clifford analysis.     

Completeness for μ-calculi: A coalgebraic approach

We set up a generic framework for proving completeness results for variants of the modal mu-calculus, using tools from coalgebraic modal logic. We illustrate the method by proving two new completeness results: for the graded mu-calculus (which is equivalent to monadic second-order logic on the class of unranked tree models), and for the monotone modal mu-calculus.Besides these main applications, our result covers the Kozen–Walukiewicz completeness theorem for the standard modal mu-calculus, as well as the linear-time mu-calculus and modal fixpoint logics on ranked trees. Completeness of the linear-time mu-calculus is known, but the proof we obtain here is different and places the result under a common roof with Walukiewicz' result.Our approach combines insights from the theory of automata operating on potentially infinite objects, with methods from the categorical framework of coalgebra as a general theory of state-based evolving systems. At the interface of these theories lies the notion of a coalgebraic modal one-step language. One of our main contributions here is the introduction of the novel concept of a disjunctive basis for a modal one-step language. Generalizing earlier work, our main general result states that in case a coalgebraic modal logic admits such a disjunctive basis, then soundness and completeness at the one-step level transfer to the level of the full coalgebraic modal mu-calculus.     

Läuchli’s Completeness Theorem from a Topos-Theoretic Perspective

We prove a variant of Läuchli’s completeness theorem for intuitionistic predicate calculus. The formulation of the result relies on the observation (due to Lawvere) that Läuchli’s theorem is related to the logic of the canonical indexing of the atomic topos of \(\mathbb{Z}{\text{ - sets}}\). We show that the process that transforms Kripke-counter-models into Läuchli-counter-models is (essentially) the inverse image of a geometric morphism. Completeness follows because this geometric morphism is an open surjection.     

The ‘corrected Durfee’s inequality’ for homogeneous complete intersections

We address the conjecture of Durfee (Math Ann 232:85–98, 1978), bounding the singularity genus $p_g$ by a multiple of the Milnor number $\mu $ for an $n$ -dimensional isolated complete intersection singularity. We show that the original conjecture of Durfee, namely $(n +1)!\cdot p_g \le \mu $ , fails whenever the codimension $r$ is greater than one. Moreover, we propose a new inequality $C_{n,r}\cdot p_g \le \mu $ , and we verify it for homogeneous complete intersections. In the homogeneous case the inequality is guided by a ‘combinatorial inequality’, that might have an independent interest.     

The application of multi-dimensional Jensen’s inequality for G-martingale

Jensen’s inequalities play a key role in probability theory. In this paper, we aim to construct G-martingales by Jensen’s inequalities in the form of G-expectation, and extend the above results to the n-dimensional case.     

Weights in Serre’s conjecture for Hilbert modular forms: The ramified case

Let F be a totally real field and p ≥ 3 a prime. If ρ : is continuous, semisimple, totally odd, and tamely ramified at all places of F dividing p, then we formulate a conjecture specifying the weights for which ρ is modular. This extends the conjecture of Diamond, Buzzard, and Jarvis, which required p to be unramified in F. We also prove a theorem that verifies one half of the conjecture in many cases and use Dembélé’s computations of Hilbert modular forms over to provide evidence in support of the conjecture. The author thanks the NSF for a Graduate Research Fellowship that supported him during part of this work.     

The reason of Hopf’s and Oleinik’s proofs for countability of shocks being wrong

For the number of complete shock curves of a conservation law with one space variable,Hopf in 1950 for the Burger equation,and Oleinik in 1956 for the general,stated that it is at most countable.In 1979,the present author published an example to show that the statement of Hopf and Oleinik is wrong.But after so long time,the wrong statement for countability still appeared in some publications,which is at least partly due to that some ones felt difficult to understand Hopf and Oleinik’s proofs being wrong.So,pointing out where they went wrong becomes very necessary.     

The Gaussian approximation for multi-color generalized Friedman’s urn model

The generalized Friedman’s urn model is a popular urn model which is widely used in many disciplines.In particular,it is extensively used in treatment allocation schemes in clinical trials.In this paper,we show that both the urn composition process and the allocation proportion process can be approximated by a multi-dimensional Gaussian process almost surely for a multi-color generalized Friedman’s urn model with both homogeneous and non-homogeneous generating matrices.The Gaussian process is a solution of ...     

The relevance of Freiman’s theorem for combinatorial commutative algebra

Mathematische Zeitschrift - Freiman’s theorem gives a lower bound for the cardinality of the doubling of a finite set in $${mathbb R}^n$$ . In this paper we give an interpretation of his...     

The small index property for homogeneous models in AEC’s

We prove a version of a small index property theorem for strong amalgamation classes. Our result builds on an earlier theorem by Lascar and Shelah (in their case, for saturated models of uncountable first-order theories). We then study versions of the small index property for various non-elementary classes. In particular, we obtain the small index property for quasiminimal pregeometry structures.