Marcel Maria Wolfgang Wild

Verified

Marcel verified their affiliation via an institutional email.

Learn more

Verified

Marcel verified their affiliation via an institutional email.

Learn more

• PHD 1987, University of Zurich
• Professor (Emeritus) at Stellenbosch University

We present a novel technique for converting a Boolean CNF into an orthogonal DNF, aka exclusive sum of products. Our method (which will be pitted against a hardwired command from Mathematica) zooms in on the models of the CNF by imposing its clauses one by one. Clausal Imposition invites parallelization, and wildcards beyond the common don't-care s...

Achieving the goals in the title (and others) relies on a cardinality-wise scanning of the ideals of the poset. Specifically, the relevant numbers attached to the k+1 element ideals are inferred from the corresponding numbers of the k-element (order) ideals. Crucial in all of this is a compressed representation (using wildcards) of the ideal lattic...

Representing lattices L by equivalence relations amounts to embed them into the lattice Part(V) of all partitions of a set V, and has a long history. Here we are concerned with MODULAR lattices L and aim for sets V as small as possible, i.e. |V| = d(L)+1 where d(L) is the length of L. In other words, we strive for a tight (=cover-preserving) lattic...

Various algorithms have been proposed to enumerate all connected induced subgraphs of a graph $G=(V,E)$. As a variation we enumerate all "conn-partitions", i.e. partitions $\Pi$ of $V$ with the property that each part of $\Pi$ induces a connected subgraph. In another vein, we enumerate all $X\subseteq V$ which induce a subgraph that is (respectivel...

By "geodesic" we mean any sequence of vertices $(v_1,v_2,...,v_k)$ of a graph $G$ that constitute a shortest path from $v_1$ to $v_k$. We propose a novel, output-polynomial algorithm to enumerate all geodesics of $G$. The graph can be directed or not, and weighted or not.

Driven by applications in the natural, social and computer sciences several algorithms have been proposed to enumerate all sets $X\s V$ of vertices of a graph $G=(V,E)$ that induce a {\it connected} subgraph. We offer two algorithms for enumerating all $X$'s that induce (more exquisite) {\it metric} subgraphs. Specifically, the first algorithm, cal...

Every mathematician is familiar with the beautiful structure of finite commutative groups. What is less well known is that finite commutative semigroups also have a neat and well-described structure. We prove this in an efficient fashion. We unravel the structural details of many concrete finite commutative semigroups. Here "concrete" comes in two...

Our objective is the compressed enumeration (based on wildcards) of all minimal hitting sets of general hypergraphs. To the author’s best knowledge, the only previous attempt towards compression, due to Toda, is based on binary decision diagrams and much different from our techniques. Traditional one-by-one enumeration schemes cannot compete when t...

We show that the R minimal hitting sets of a hypergraph H ⊆ P(W) can be enumerated in time O(Rh^2 w^2) where w := |W | and h := |H|.

By processing all minimal cutsets of a graph G, and by using novel wildcards, all spanning trees of G can be compactly encoded. Thus, different from all previous enumeration schemes, the spanning trees are not generated one-by-one. The Mathematica implementation of one of our algorithms generated for a random (11,50)-graph its 819'603'181 spanning...

Let $W$ be a finite set which simultaneously serves as the universe of any poset $(W,\preceq)$ and as the vertex set of any graph $G$. Our algorithm, abbreviated A-I-I, enumerates (in a compressed format using don't-care symbols) all $G$-independent order ideals of $(W,\preceq)$. For many instances the high-end Mathematica implementation of A-I-I c...

Part B (of a project involving four Parts) is about "bases of lines", a concept introduced by C. Herrmann and the author in the late 80's. Bases of lines attempt to describe a given modular lattice in a geometric way akin to how projective geometries describe complemented modular lattices. This e.g. yields the result that each modular lattice L of...

This is Part A of four Parts dedicated to modular lattices of finite length. It builds on 1992 notes of the author (available on ResearchGate), and in so doing heeds a wish of the late Gian-Carlo Rota. Part A is in fairly final form and mainly features known material, exceptions being a short proof of the distributivity of congruence lattices of la...

Our main objective is the COMPRESSED enumeration (based on wildcards) of all minimal hitting sets of general hypergraphs. To the author's best knowledge the only previous attempt towards compression, due to Toda, is based on BDD's and much different from our techniques. Numerical experiments show that traditional one-by-one enumeration schemes cann...

Despite the more handy terminology of abstract simplicial complexes SC, in its core this article is about antitone Boolean functions. Given the maximal faces (=facets) of SC, our main algorithm, called Facets-To-Faces, outputs SC in a compressed format. The degree of compression of Facets-To-Faces, which is programmed in high-level Mathematica code...

It is well known that the number of models x of a Boolean function f: {0,1}^n ---> {0,1} can be calculated fast if f is given by a BDD. Let H(x) be the Hamming weight, i.e. the number of 1's in x. Knuth gives a nifty way to calculate the number of k-models, i.e. having H(x)=k. Our main topic however is ENUMERATION (=generation) as opposed to counti...

Frequent Set Mining (FSM) is a popular data mining technique. Given a binary table (thus a 'database' in FSM parlance) we suggest an apparently novel way to calculate the maximal frequent sets. The more important innovation however is this. Assume the maximal frequent sets are available (obtained in whatever way). Then the family of ALL frequent se...

An odd cycle cover is a vertex set whose removal makes a graph bipartite. We show that if a k-element odd cycle cover of a graph with w vertices is known then all N maximum anticliques (= independent sets) can be generated in time O(2^k w^3 + N w^2)). An important ingredient is the efficient generation of the maximum anticliques in bipartite graphs...

The model set of a general Boolean function in CNF is calculated in a compressed format, using wildcards. This novel method can be explained in very visual ways. Preliminary comparison with existing methods (BDD's and ESOPs) looks promising but our algorithm begs for a C encoding which would render it comparable in more systematic ways. Addendum 26...

We present a systematic approach to decide shellability, which goes beyond extending partial shellings (that can confirm but hardly disprove shellability). Furthermore, our method (unlike Moriyama's algorithm) does not need the face-numbers. Instead of the n! permutations of the n facets we deal with certain admissible chains in certain posets of c...

When evaluating the lengthy inclusion-exclusion expansion $N({ }) - N({1}) - N({2}) - .... + N({1,2}) + N({1,3}) + ....$ many of the terms $N(X)$ may turn out to be zero, and hence should be discarded BEFOREHAND. Often this can be done. The main idea is that the index sets X of nonzero terms N(X) constitute a set ideal (called the NERVE) which can...

A newer version of this has been published 2021 in Quaestiones Mathematicae (also on RG). Nevertheless, not all worthy bits of the version in front of you could be accomodated in the QM version.

Representing lattices L by equivalence relations amounts to embed them into the lattice Part(V) of all partitions of a set V, and has a long history. Here we are concerned with MODULAR lattices L and aim for sets V as small as possible, i.e. |V| = d(L)+1 where d(L) is the length of L. In other words, we strive for a tight (=cover-preserving) lattic...

Apart from a brief look at applications (Relational Databases, Formal Concept Analysis et al.) this article is devoted to the mathematical t h e o r y of implications (=pure Horn formulas). It is mainly a survey of results obtained in the last thirty years, but features a few novelties as well. Some keywords: The Duquenne-Guiges (implicational) bas...

This is my Inaugural Lecture from 10 October 2011. Comment from August 19, 2014: I find myself browsing these notes on a regular base. I'm glad I managed to take stock of my disparate research by capturing fading memories. I cherish the 46 footnotes, not all of them purely mathematical. (My exceedingly thick booklet was greeted with a 10000 Rand bi...

This article has become o b s o l e t e. It is covered and much expanded in the article 'Tight embedding of modular lattices into partition lattices: progress and program' (also on RG).

Analysis of Algorithms International audience The classic Coupon-Collector Problem (CCP) is generalized. Only basic probability theory is used. Centerpiece rather is an algorithm that efficiently counts all k-element transversals of a set system.

Many nonlinear filters used in practise are stack filters. An algorithm is presented which calculates the output distribution of an arbitrary stack filter S from the disjunctive normal form (DNF) of its underlying positive Boolean function. The so called selection probabilities can be computed along the way. [Newest version in the arXiv is from Aug...

An algorithm is presented that compactly (thus not one by one) generates all models of a Horn formula. This was my first publication that embraced 'clause-wise processing' with wildcards.

The N cardinality k ideals of any w-element poset (w, k variable) can be enumerated in time O(Nw^4). With hindsight (and Google) the other result about k-element subtrees is inferior to much faster algorithms. Interestingly though, both results fit a common hat, i.e the output-polynomial enumerability of all fixed-cardinality closed subsets of cert...

For any positive integers M and N we define a selfdually ordered band ℬ(M,N) of cardinality ((M+N+2) || (N+1))-2{M+N+2\choose N+1}-2 and ask whether or not it is lattice-ordered. The origin of ℬ(M,N) in nonlinear signal processing is outlined.

We show that the edge set of the n-dimensional hypercube Qn is the disjoint union of the edge sets of n isomorphic trees.

A catalogue of all non-isomorphic simple connected regular matroids ${\cal M}$ of cardinality $n \leq 15$ is provided on the net. These matroids are given as binary matrix matroids and are sieved from the large pool of all non-isomorphic binary matrix matroids of cardinality $\leq 15$. For each ${\cal M}$ its Tutte polynomial is determined by an al...

An algorithm to count, or alternatively generate, all k-element transversals of a set system is presented and compared with three known methods. For special cases it works in output-linear time. [Newest version in the arXiv is from Nov 2012]

This article is the second part of an essay dedicated to lattices freely generated by posets within a variety. The first part dealt with four easy varieties while this part is concerned with finitely generated varieties. Here we present a method of constructing a subdirect product L of a finite family F of finite lattices, exploiting a set of speci...

We introduce an algorithm for computing closure systems derived from a family of implications on a set. Semilattices presentations are explored and used in conjunction with the algorithm to compute various types of lattices freely generated by partially ordered sets within four easy varieties. Comment: 18 pages, 6 figures

Two procedures to compute the output distribution phi_S of certain stack filters S (so called erosion-dilation cascades) are given. One rests on the disjunctive normal form of S and also yields the rank selection probabilities. The other is based on inclusion-exclusion and e.g. yields phi_S for some important LULU-operators S. Properties of phi_S c...

The Edelman–Jamison problem is to characterize those abstract convex geometries that are representable by a set of points in the plane. We show that some natural modification of the Edelman–Jamison problem is equivalent to the well known NP-hard order type problem. The relation to the realizability of oriented matroids is clarified.

For practical purposes median smoothers have often been considered the basic components for the construction of simple non-linear smoothers for removing impulsive noise before any subsequent linear signal extraction from a sequence. Alternatives exist that are computationally convenient, conceptually simpler and provide significant insight into the...

To what extent is the isomorphism type of an incidence algebra determined by the zero–nonzero pattern of a matrix representation? We settle the question in a natural framework where the matrices are subdivided into four blocks: The lower left is zero, the diagonal blocks are fixed, and the upper right is variable We also reprove a related Theorem o...

Distributive supermatroids generalize matroids to partially ordered sets. Completing earlier work of Barnabei, Nicoletti and Pezzoli we characterize the lattice of flats of a distributive supermatroid. For the prominent special case of a polymatroid the description of the flat lattice is particularly simple. Large portions of the proofs reduce to p...

An efficient method to generate all edge sets X⊆E of a graph G=(V,E), which are vertex-disjoint unions of cycles, is presented. It can be tweaked to generate (i) all cycles, (ii) all cycles of cardinality ⩽5, (iii) all chordless cycles, (iv) all Hamiltonian cycles.

In Part II of this work I survey in coherent form, with polished proofs when given, my theoretical endeavours (I'm no "practitioner") in nonlinear signal processing from their beginnings in 1998. It is not merely a survey paper; some new material features as well. That includes, e.g., a more elegant derivation of the conditions characterizing the...

It is shown how a combinatorial analysis of the positive Boolean functions underlying stack filters, can reveal many of their properties, including idempotency.

An R-module is said to be "linearly induced" if every lattice automorphism of its submodule lattice is induced by a suitable R-module automorphism. It is shown that up to trivial cases EACH length two R-module with at most 6 submodules is linearly induced, but NONE with at least 7 submodules is.

Let $$\mathcal{D}$$ be a lattice of finite height. The correspondence between closure operators $$cl:\mathcal{D} \to \mathcal{D}$$ and ∧-subsemilattices $$\mathcal{L} \subseteq \mathcal{D}$$ is well known. Here we investigate what type of number-valued function $$\mathcal{D} \to \mathbb{N}$$ is induces a ∧-subsemilattice $$\mathcal{L}$$; and if so,...

The asyptotic number of nonequivalent binary n-codes is determined. This is also the asymptotic number of nonisomorphic binary n-matroids. The connection to a result of Lefmann, Roedl, Phelps is explored. The latter states that almost all binary n-codes have a trivial automorphism group.

Stack filters are nonlinear filters used for image processing (examples: median filters, order statistics). In the translation-invariant case a stack filter is determined by a positive Boolean function b. Many important properties of stack filters (idempotency, co-idempotency, order relations) can be tested in polynomial time if the DNF and/or CNF...

An R-module M is "hom-proj" if each homogeneous (not necessarily linear) bijection M --> M induces an automorphism L(M) --> L(M) of the submodule lattice. The focus is on finite length modules here. For instance, some concrete finite modular lattices L are pointed out such that every module M with L(M) isomorphic to L must be hom-proj.

An R-module V is a ray if for every R-module W, the R-homogeneous functions from V to W are additive. We use properties of the lattice L(V) of submodules of V to determine conditions for V to be a ray. We also use the lattice structure of L(V) to further study those rings such that every R-module V is a ray. As a result, we can characterize all se...

A structural matrix algebra R of n × n matrices over a field F has a distributive lattice Lat(R) of invariant subspaces ⊆F^n. This and related known results are reproven here in a fresh way. Further we investigate what happens when R still operates on F^n but is isomorphic to a structural matrix algebra of m × m matrices (m ≠ n). Then m < n and Lat...

There are fourteen groups of order 16, and they do of course make their appearance in higher level texts. However, the classification of the groups of order 16 is always obtained as a special case of a sophisticated theory of p-groups, and many details are left to the reader to verify. Here we give a complete classification of the groups of order s...

Viewing the elements of as images f with pixels i∈S of grey-scale value f(i) motivates the study of certain nonlinear operators . For translation invariant Φ (called stack filters in the signal processing literature) we derive the first necessary and sufficient condition for idempotency which can be tested in polynomial time. Various related proper...

It is e.g. shown that every strong (in the sense of JP Serra) stack filter S must be co-idempotent, i.e. if I is the identity then I-S composed with itself equals I-S. For the special case of stack filters which are openings (and thus strong) this was known by a completely different argument of Christian Ronse.

Whereas computing an optimal implicational base of an arbitrary finite closure system is NP-hard (David Maier 1980), there is a quadratic time algorithm for modular closure systems.

The asymptotic value as n→∞ of the number b(n) of inequivalent binary n-codes is determined. It was long known that b(n) also gives the number of nonisomorphic binary n-matroids.

Given two subspaces F and G of a quadratic space E, when is there an isometry on E which induces an isometry between F and G? For finite-dimensional E the question is settled by a famous theorem of Witt. Here we push to infinite dim(E), which requires to cope with a variety of new phenomena.

The first note settles questions of Metropolis-Rota-Stein and Kamara respectively. Both are concerned with polarities on finite posets and distributive lattices. For this purpose an old result (1960) of Monteiro is reproven in a fresh way. The second note gives, in the distributive case, an elementary proof of Joseph Kung's famous matching theorem...

The principal theme of the present paper is to consider isomorphism classes of binary matroids as orbits of a suitable group action. This interpretation is based on a theorem of Brylawski–Lucas. A refinement of the Burnside Lemma is used in order to enumerate these orbits. Ternary matroids are dealt with in much the same way (Section 2). Counting r...

Whether or not a finite universal algebra has a modular congruence lattice, can be tested in polynomial time. This settles a question raised by Ralph Mc Kenzie.

New base exchange properties of binary and graphic matroids are derived. The graphic matroids within the class of 4-connected binary matroids are characterized by base exchange properties. Some progress with the characterization of arbitrary graphic matroids is made. Characterizing various types of matroids by base exchange properties is e.g. impor...

It is shown that the number j(L) of join irreducibles of a finite modular lattice is at least 2d(L)-s(L), where d(L) and s(L) are the length and number of subdirectly irreducible factors of L respectively. The bound is sharp for distributive lattices, but also for some non-distributive ones.

Closure systems C on a finite set M arise in many areas of discrete mathematics. They are conveniently encoded by either the family Irr(C) of meet irreducible closed sets, or by implicational bases . We significantly improve six (partly little known) algorithms in order to settle the problems (a),(b),... (g) listed below. In particular, the algori...

Although many notions familiar from topology and matroid theory make sense for arbitrary closure spaces, we claim that "implicational bases" are most worthwhile studying. They are applied in the theory of relational data bases and in formal concept analysis. Further applications in combinatorics and algebra are foreseeable. The main theorem describ...

This article is a follow up (in French) of the article [Cover preserving order embeddings into Boolean lattices, Order 9 (1992) 209-232]. It provides a leisurely introduction into that topic and contains some new examples.

As is well known, every matroid is uniquely determined by its family of circuits. It is shown that each simple binary matroid is already uniquely determined by its family of c l o s e d circuits.

When is a finite modular lattice cover preserving embeddable into a partition lattice? We give some necessary, and slightly sharper sufficient conditions. For example, the class of cover preserving embeddable modular lattices strictly contains the class of acyclic modular lattices.

It is not known which finite graphs occur as induced subgraphs of a hypercube. This is relevant in the theory of parallel computing. The ordered version of the problem is: Which finite posets P occur as cover-preserving subposets of a Boolean lattice? Our main Theorem gives (for 0,1-posets) a necessary and sufficient condition, which involves the c...

Surveys, in a leisurely way, the structure theory of modular lattices, including results obtained in my PHD and in collaboration with Christian Herrmann. Comment from march 2022: Recently these notes have been thoroughly revised and extended. So rather have a look at "Modular lattices of finite length (Part A)", and Part B!

Finite length 2-distributive modular lattices of finite representation type are characterized in various ways and all their representations are determined. Also, the second Brauer-Thrall Conjecture is verified for this setting.

If two subspaces V and V of a sesquilinear space E are congruent (i.e., there is an isometry : E E with (V)=V) then their corresponding quadratic lattices V(V, E) and V(V, E) are isomorphic. It is shown that the converse holds for important types of sesquilinear spaces E, provided that dim(E) 3. However, the converse generally fails if dim(E) 3.

MARCEL WILD It will be shown that a complete join epimorphism F between two complete lattices L~ and L2 which maps principal ideals upon principal ideals preserves so-called meet weak lattice identities that hold in L1 (e.g. modularity, distributivity). Each onto map between two closure spaces (A, -) and (X, -) which is continuous and closed, induc...

This aricle, which was published in [Informatik-Bericht 89/3, Institut fuer Informatik, Clausthal-Zellerfeld], is a precursor but not a subset of my article "A theory of finite closure spaces based on implications" in [Advances in. Math. 108 (1994) 118-139].

The classic Theorem of Witt about isometries between quadratic spaces was lifted from finite dimension to uncountable dimension Aleph_2 by Herbert Gross. The crucial technical tool was a finite distributive lattice of subspaces. I settled matters for dimension Aleph_3 albeit the auxiliary lattice ceases to be distributive and has 957 elements. For...

Two procedures to compute the output distribution φ S of certain stack filters S (so called erosion-dilation cascades) are given. One rests on the disjunctive normal form of S and also yields the rank selection probabilities. The other is based on inclusion-exclusion and e.g. yields φ S for some important LU LU -operators S. Properties of φ S can b...