Conjecture In Theory Research Articles

Exceptional cosmetic surgeries on homology spheres

The cosmetic surgery conjecture is a longstanding conjecture in 3-manifold theory. We present a theorem about exceptional cosmetic surgery for homology spheres. Along the way we prove that if the surgery is not a small Seifert Z/2Z-homology sphere or a toroidal irreducible non-Seifert surgery then there is at most one pair of exceptional truly cosmetic slope. We also prove that toroidal truly cosmetic surgeries on integer homology spheres must be integer homology spheres.

On self-dual negacirculant codes of index two and four

In this paper, we study a special kind of factorization of $x^n+1$ over $\mathbb{F}_q, $ with $q$ a prime power $\equiv 3~({\rm mod}~4)$ when $n=2p,$ with $p\equiv 3~({\rm mod}~4)$ and $p$ is a prime. Given such a $q$ infinitely many such $p$'s exist that admit $q$ as a primitive root by the Artin conjecture in arithmetic progressions. This number theory conjecture is known to hold under GRH. We study the double (resp. four)-negacirculant codes over finite fields $\mathbb{F}_q, $ of co-index such $n$'s, including the exact enumeration of the self-dual subclass, and a modified Varshamov-Gilbert bound on the relative distance of the codes it contains.

Leaf-Reconstructibility of Phylogenetic Networks

An important problem in evolutionary biology is to reconstruct the evolutionary history of a set $X$ of species. This history is often represented as a phylogenetic network, that is, a connected graph with leaves labelled by elements in $X$ (for example, an evolutionary tree), which is usually also binary, i.e., all vertices have degree 1 or 3. A common approach used in phylogenetics to build a phylogenetic network on $X$ involves constructing it from networks on subsets of $X$. Here we consider the question of which (unrooted) phylogenetic networks are leaf-reconstructible, i.e., which networks can be uniquely reconstructed from the set of networks obtained from it by deleting a single leaf (its $X$-deck). This problem is closely related to the (in)famous reconstruction conjecture in graph theory but, as we shall show, presents distinct challenges. We show that some large classes of phylogenetic networks are reconstructible from their $X$-deck. This includes phylogenetic trees, binary networks containing at least one nontrivial cut-edge, and binary level-4 networks. (The level of a network measures how far it is from being a tree.) We also show that for fixed $k$, almost all binary level-$k$ phylogenetic networks are leaf-reconstructible. As an application of our results, we show that a level-3 network $N$ can be reconstructed from its quarnets, that is, 4-leaved networks that are induced by $N$ in a certain recursive fashion. Our results lead to several interesting open problems which we discuss, including the conjecture that all phylogenetic networks with at least five leaves are leaf-reconstructible.

On the proof of the thin sandwich conjecture in arbitrary dimensions

In this paper we show the validity, under certain geometric conditions, of Wheeler’s thin sandwich conjecture for higher dimensional theories of gravity. We extend the results for the 3-dimensional case in Phys. Rev. D 48, 3596–3599 (1993) in two ways. On the one hand, we show that the results presented in Phys. Rev. D 48, 3596–3599 (1993) are valid in arbitrary dimensions, and on the other hand, we show that the geometric hypotheses needed for the proofs can always be satisfied, which constitutes in itself a new result for the 3-dimensional case. In this way, we show that on any compact n-dimensional manifold, n≥3, there is an open set in the space of all possible initial data where the thin sandwich problem is well-posed.

On orbifold theory

Let V be a simple vertex operator algebra and G a finite automorphism group of V such that VG is regular. It is proved that every irreducible VG-module occurs in an irreducible g-twisted V-module for some g∈G. Moreover, the quantum dimensions of irreducible VG-modules are determined and a global dimension formula for V in terms of twisted modules is obtained. In particular, the orbifold theory conjecture is completely solved if G is solvable.

A p-adic analytic family of the D-th Shintani lifting for a Coleman family and congruences between the central L-values

We will construct a p-adic analytic family of D-th Shintani lifting generalized by Kojima and Tokuno for a Coleman family. Consequently, we will have a p-adic L-function which interpolates the central L-values attached to a Coleman family and obtain a congruence between the central L-values. Focusing on the case of p-ordinary, we will obtain two applications. One of them states that a congruence between Hecke eigenforms of different weights sufficiently close, p-adically, derives a congruence between their central L-values. The other one is about the Goldfeld conjecture in analytic number theory. We will show that there exists a primitive form satisfying the conjecture for each even weight sufficiently close to 2, 3-adically, thanks to a result of Vatsal.

Combining SAT Solvers with Computer Algebra Systems to Verify Combinatorial Conjectures

We present a method and an associated system, called MathCheck, that embeds the functionality of a computer algebra system (CAS) within the inner loop of a conflict-driven clause-learning SAT solver. SAT+CAS systems, a la MathCheck, can be used as an assistant by mathematicians to either find counterexamples or finitely verify open universal conjectures on any mathematical topic (e.g., graph and number theory, algebra, geometry, etc.) supported by the underlying CAS. Such a SAT+CAS system combines the efficient search routines of modern SAT solvers, with the expressive power of CAS, thus complementing both. The key insight behind the power of the SAT+CAS combination is that the CAS system can help cut down the search-space of the SAT solver, by providing learned clauses that encode theory-specific lemmas, as it searches for a counterexample to the input conjecture (just like the T in DPLL (T)). In addition, the combination enables a more efficient encoding of problems than a pure Boolean representation. In this paper, we leverage the capabilities of several different CAS, namely the SAGE, Maple, and Magma systems. As case studies, we study three long-standing open mathematical conjectures, two from graph theory regarding properties of hypercubes, and one from combinatorics about Hadamard matrices. The first conjecture states that any matching of any d-dimensional hypercube can be extended to a Hamiltonian cycle; the second states that given an edge-antipodal coloring of a hypercube there always exists a monochromatic path between two antipodal vertices; the third states that Hadamard matrices exist for all orders divisible by 4. Previous results on the graph theory conjectures have shown the conjectures true up to certain low-dimensional hypercubes, and attempts to extend them have failed until now. Using our SAT+CAS system, MathCheck, we extend these two conjectures to higher-dimensional hypercubes. Regarding Hadamard matrices, we demonstrate the advantages of SAT+CAS by constructing Williamson matrices up to order 42 (equivalently, Hadamard up to order $$4\times 42=168$$4×42=168), improving the bounds up to which Williamson matrices of even order have been constructed. Prior state-of-the-art construction was only feasibly performed for odd numbers, where possible.

On the isospectral orbifold–manifold problem for nonpositively curved locally symmetric spaces

An old problem asks whether a Riemannian manifold can be isospectral to a Riemannian orbifold with nontrivial singular set. In this short note we show that under the assumption of Schanuel’s conjecture in transcendental number theory, this is impossible whenever the orbifold and manifold in question are length-commensurable compact locally symmetric spaces of nonpositive curvature associated to simple Lie groups.

Theory’s Doubt: History, Theory and Image in Robin Evans’s Physiognomy of Morals

This paper begins with a simple proposition: theory is an image animated by history. By exploring the notion of image in the work of architect and historian Robin Evans, the nature of this animation is elucidated. The paper discusses Evans’s infatuation with the “powers of the image” from a threefold perspective: what is a theory-image; how is it constructed in and through Evans’s writings; and why is it relevant for the discipline of architecture. Evans’s project is therefore the recovery of the image and its space of projection as a site of theoretical inquiry, one that is crucially distinct from language. Through the analysis of Evans’s work, I would argue, it is possible to sustain a radical conceptualization of theory not merely as specular history – a Doppelgänger of sorts – but as a productive field of inquiry fundamentally decoupled from, yet forever linked to, history. It is necessary in this radical conceptualization to embrace the conjecture of productive theory in lieu of the disciplinarity of history.

The importance of being atomic: Ecological invasions as random walks instead of waves

Invasions are one of the most easily identified spatial phenomena in ecology, and have inspired a rich variety of theories for ecologists’ and naturalists’ consideration. However, a number of arguments over the sensitivities of invasion rates to stochasticity, density-dependence, dimension, and discreteness persist in the literature.The standard mathematical approach to invasions is based on Fisher’s analysis of traveling waves solutions for the spread of an advantageous allele. In this paper, we exploit an alternative theory based on Ellner’s premise that species invasions are best interpreted not as waves, but as random walks, and that the discreteness of living organisms is fundamentally important. Using a density-dependent invasion model in a stationary environment with indivisible (atomic) individuals where reproduction and dispersal are stochastic and independent, we show 4 key properties of Ellner’s invasions previously suggested by simulation analysis: (1) greater spatial dispersal stochasticity quickens invasions, (2) greater demographic stochasticity slows invasions, (3) negative density-dependence slows invasions, and (4) greater temporal dispersal stochasticity quickens invasions. We prove the first three results by using generating functions and stochastic-dominance methods to rank furthest-forward dispersal distributions. The fourth result is proven in the special case of atomless theory, but remains an open conjecture in atomic theory. In addition, we explain why, unlike atomless invasions, an infinitely wide atomic invasion in two-dimensions can travel faster than a finite-width invasion and a one-dimensional invasion. The paper concludes with a classification of invasion dynamics based on dispersal kernel tails.

A TOPOLOGICAL VARIATION OF THE RECONSTRUCTION CONJECTURE

AbstractThis paper investigates topological reconstruction, related to the reconstruction conjecture in graph theory. We ask whether the homeomorphism types of subspaces of a space X which are obtained by deleting singletons determine X uniquely up to homeomorphism. If the question can be answered affirmatively, such a space is called reconstructible. We prove that in various cases topological properties can be reconstructed. As main result we find that familiar spaces such as the reals ℝ, the rationals ℚ and the irrationals ℙ are reconstructible, as well as spaces occurring as Stone–Čech compactifications. Moreover, some non-reconstructible spaces are discovered, amongst them the Cantor set C.

Note on the index conjecture in zero-sum theory and its connection to a Dedekind-type sum

Let S=(a1)⋯(ak) be a minimal zero-sum sequence over a finite cyclic group G. The index conjecture states that if k=4 and gcd⁡(|G|,6)=1, then S has index 1. In this note we study the index conjecture and connect it to a Dedekind-type sum. In particular we reprove a special case of the conjecture when |G| is prime.

A Conditional Construction of Restricted Isometries

We study the restricted isometry property of a matrix that is built from the discrete Fourier transform matrix by collecting rows indexed by quadratic residues. We find an $\epsilon>0$ such that, conditioned on a folklore conjecture in number theory, this matrix satisfies the restricted isometry property with sparsity parameter $K=\Omega(M^{1/2+\epsilon})$, where $M$ is the number of rows.

Counterexamples in multimarginal optimal transport with Coulomb cost and spherically symmetric data

We disprove a conjecture in Density Functional Theory, relative to multimarginal optimal transport maps with Coulomb cost. In the case of spherically symmetric data, which model for instance Lithium and Beryllium atoms, we show that some special maps, introduced by Seidl, Gori-Giorgi and Savin are not always optimal in the corresponding transport problem. We also provide examples of maps satisfying optimality conditions for special classes of data.

David Rees, FRS 1918–2013

David Rees, FRS 1918–2013

A Brush with Research: Teaching Grounded Theory in the Art and Design Classroom

Grounded Theory is a systematic approach to social research that allows for new concepts and theories to emerge from gathered data, as opposed to relying on either established theory or personal conjecture to interpret social processes. Although Grounded Theory is a well-known method within social science literature, it is relatively unknown in art and design literature, even though it is useful for developing social context and human empathy. In recent years, the model has gained relative credibility in many other disciplines, offering potential advantages for interdisciplinary scenarios. This paper provides a detailed review of Grounded Theory from an educational perspective, and then concludes with some proposed pedagogical methods for art and design based practices.

A universality theorem for projectively unique polytopes and a conjecture of Shephard

We prove that every polytope described by algebraic coordinates is the face of a projectively unique polytope. This provides a universality property for projectively unique polytopes. Using a closely related result of Below, we construct a combinatorial type of 5-dimensional polytope that is not realizable as a subpolytope of any stacked polytope. This disproves a classical conjecture in polytope theory, first formulated by Shephard in the seventies.

Anick's space and Hopf invariants

A major conjecture in homotopy theory is that the fibre of the double suspension is homotopy equivalent to the double loops on a certain Anick space. We approach the problem by studying lifts of looped Hopf invariants, and give a first approximation to the existence of such a lift.

Regular Languages Are Church-Rosser Congruential

This article shows a general result about finite monoids and weight reducing string rewriting systems. As a consequence it proves a long standing conjecture in formal language theory: All regular languages are Church-Rosser congruential. The class of Church-Rosser congruential languages was introduced by McNaughton, Narendran, and Otto in 1988. A language L is Church-Rosser congruential if there exists a finite, confluent, and length-reducing semi-Thue system S such that L is a finite union of congruence classes modulo S . It was known that there are deterministic linear context-free languages which are not Church-Rosser congruential, but the conjecture was that all regular languages are of this form. The article offers a stronger statement: A language is regular if and only if it is strongly Church-Rosser congruential. It is the journal version of the conference abstract which was presented at ICALP 2012.

Tracy–Widom asymptotics for $q$-TASEP

We consider the q-TASEP that is a q-deformation of the totally asymmetric simple exclusion process (TASEP) on Z for q in [0,1) where the jump rates depend on the gap to the next particle. For step initial condition, we prove that the current fluctuation of q-TASEP at time t are of order t^{1/3} and asymptotically distributed as the GUE Tracy-Widom distribution, which confirms the KPZ scaling theory conjecture.