Constructing Ramsey graphs from Boolean function representations

Explicit construction of Ramsey graphs or graphs with no large clique or independent set, has remained a challenging open problem for a long time. While Erdös’ probabilistic argument shows the existence of graphs on 2n vertices with no clique or independent set of size 2 ⁿ , the best explicit constructions achieve a far weaker bound. There is a connection between Ramsey graph constructions and polynomial representations of Boolean functions due to Grolmusz; a low degree representation for the OR function can be used to explicitly construct Ramsey graphs [17,18]. We generalize the above relation by proposing a new framework. We propose a new definition of OR representations: a pair of polynomials represent the OR function if the union of their zero sets contains all points in {0, 1} ⁿ except the origin. We give a simple construction of a Ramsey graph using such polynomials. Furthermore, we show that all the known algebraic constructions, ones to due to Frankl-Wilson [12], Grolmusz [18] and Alon [2] are captured by this framework; they can all be derived from various OR representations of degree O(√n) based on symmetric polynomials. Thus the barrier to better Ramsey constructions through such algebraic methods appears to be the construction of lower degree representations. Using new algebraic techniques, we show that better bounds cannot be obtained using symmetric polynomials.     

On minimal representations of Rational Arrival Processes

Rational Arrival Processes (RAPs) form a general class of stochastic processes which include Markovian Arrival Processes (MAPs) as a subclass. In this paper we study RAPs and their representations of different sizes. We show some transformation methods between different representations and present conditions to evaluate the size of the minimal representation. By using some analogous results from linear systems theory, a minimization approach is defined which allows one to transform a RAP (from a redundant high dimension) into one of its minimal representations. An algorithm for computing a minimal representation is also given. Furthermore, we extend the approach to RAPs with batch arrivals (BRAPs) and to RAPs with arrivals of different customer types (MRAPs).     

Canonical monotone decompositions of fractional stable matchings

This paper continues recent work that introduced algebraic methods for studying the stable marriage problem of Gale and Shapley [1962]. Vande Vate [1989] and Rothblum [1992] identified a set of linear inequalities which define a polytope whose extreme points correspond to the stable matchings. Points in this polytope are called fractional stable matchings. Here we identify a unique representation of fractional stable matchings as a convex combination of stable matchings that are arrangeable in a man-decreasing order. We refer to this representation and to a dual one, in terms of woman-decreasing order, as the canonical monotone representations. These representations can be interpreted as time-sharing stable matchings where particular stable matchings are used at each time-instance but the scheduled stable matchings are (occasionally) switched over time. The new representations allow us to extend, in a natural way, the lattice structure of the set of stable matchings to the set of all fractional stable matchings.     

Multivariate stochastic delay differential equations and CAR representations of CARMA processes

In this study we show how to represent a continuous time autoregressive moving average (CARMA) as a higher order stochastic delay differential equation, which may be thought of as a CAR($\infty$) representation. Furthermore, we show how the CAR($\infty$) representation gives rise to a prediction formula for CARMA processes. To be used in the above mentioned results we develop a general theory for multivariate stochastic delay differential equations, which will be of independent interest, and which will have particular focus on existence, uniqueness and representations.     

Homotopy equivalent group representations and Picard groups op the burnside ring and the character ring

We are concerned with the homotopy theory of group representations and its relation to character theory and the theory of the Burnside ring. We combine the methods of tom Dieck — Petrie [4] and torn Dieck [3] to show that the canonical map from the J-group jO(G), a subquotient of the representation ring RO(G), into the Picard group of the rational representation ring is injective for p-groups G. Moreover we compute the order of the cokernel of this map. We show that the Picard group of the rational representation ring is a direct summand in the Picard group of the Burnside ring. Finally we compute the Picard groups if G is abelian and indicate a computation for general G.     

Martingale solutions and Markov selections for stochastic partial differential equations

We present a general framework for solving stochastic porous medium equations and stochastic Navier–Stokes equations in the sense of martingale solutions. Following Krylov [N.V. Krylov, The selection of a Markov process from a Markov system of processes, and the construction of quasidiffusion processes, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973) 691–708] and Flandoli–Romito [F. Flandoli, N. Romito, Markov selections for the 3D stochastic Navier–Stokes equations, Probab. Theory Related Fields 140 (2008) 407–458], we also study the existence of Markov selections for stochastic evolution equations in the absence of uniqueness.     

On the representations of projective geometries in algebraic combinatorial geometries

In this paper, we show that the full algebraic combinatorial geometry is not a projective geometry, it is only semimodular, but the p-polynomial points give a projective subgeometry. Also, we show that the subgeometry can be coordinatized by a skew field, which is quotient ring of an Ore domain. As a corollary, we prove the existence of algebraic representations over fields of prime characteristic of the non-Pappus matroid and its dual matroid. Regarding the existence of algebraic representations of the non-Pappus matroid, this result was earlier proved by Lindström [7] for finite fields.     

Worst portfolios for dynamic monetary utility processes

We study the worst portfolios for a class of law invariant dynamic monetary utility functions with domain in a class of stochastic processes. The concept of comonotonicity is introduced for these processes in order to prove the existence of worst portfolios. Using robust representations of monetary utility function processes in discrete time, a relation between the worst portfolios at different periods of time is presented. Finally, we study conditions to achieve the maximum in the representation theorems for concave monetary utility functions that are continuous for bounded decreasing sequences.     

Generalising Group Algebras

We generalise group algebras to other algebraic objects withbounded Hilbert space representation theory; the generalisedgroup algebras are called ‘host’ algebras. The mainproperty of a host algebra is that its representation theoryshould be isomorphic (in the sense of the Gelfand–Raikovtheorem) to a specified subset of representations of the algebraicobject. Here we obtain both existence and uniqueness theoremsfor host algebras as well as general structure theorems forhost algebras. Abstractly, this solves the question of whena set of Hilbert space representations is isomorphic to therepresentation theory of a C*-algebra. To make contact withharmonic analysis, we consider general convolution algebrasassociated to representation sets, and consider conditions fora convolution algebra to be a host algebra.     

Representations of homogeneous quantum Lévy fields

We study homogeneous quantum Lévy processes and fields with independent additive increments over a noncommutative *-monoid. These are described by infinitely divisible generating state functionals, invariant with respect to an endomorphic injective action of a symmetry semigroup. A strongly covariant GNS representation for the conditionally positive logarithmic functionals of these states is constructed in the complex Minkowski space in terms of canonical quadruples and isometric representations on the underlying pre-Hilbert field space. This is of much use in constructing quantum stochastic representations of homogeneous quantum Lévy fields on Itô monoids, which is a natural algebraic way of defining dimension free, covariant quantum stochastic integration over a space-time indexing set.     

Second order backward stochastic differential equations with quadratic growth

We extend the well posedness results for second order backward stochastic differential equations introduced by Soner, Touzi and Zhang (2012)  [31] to the case of a bounded terminal condition and a generator with quadratic growth in the z$z$ variable. More precisely, we obtain uniqueness through a representation of the solution inspired by stochastic control theory, and we obtain two existence results using two different methods. In particular, we obtain the existence of the simplest purely quadratic 2BSDEs through the classical exponential change, which allows us to introduce a quasi-sure version of the entropic risk measure. As an application, we also study robust risk-sensitive control problems. Finally, we prove a Feynman–Kac formula and a probabilistic representation for fully non-linear PDEs in this setting.     

A Note on Integral Structures in Some Locally Algebraic Representations of GL2

In the p-adic local Langlands correspondence for GL_2(Q_p), the following theorem of Berger and Breuil has played an important role: the locally algebraic representations of GL_2(Q_p) associated to crystabelline Galois representations admit a unique unitary completion. In this note, we give a new proof of the weaker statement that the locally algebraic representations admit at most one unitary completion and such a completion is automatically admissible. Our proof is purely representation theoretic, involving neither(ψ, Γ)-module techniques nor global methods. When F is a finite extension of Q_p, we also get a simpler proof of a theorem of Vignéras for the existence of integral structures for(locally algebraic) special series and for(smooth) tamely ramified principal series.     

Ulrich and aCM bundles from invariant theory

We use certain special prehomogeneous representations of algebraic groups in order to construct aCM vector bundles, possibly Ulrich, on certain families of hypersurfaces. Among other results, we show that a general cubic hypersurface of dimension seven admits an indecomposable Ulrich bundle of rank nine, and that a general quartic fourfold admits an unsplit aCM bundle of rank six.     

Quaternionic structures

Any oriented 4-dimensional real vector bundle is naturally a line bundle over a bundle of quaternion algebras. In this paper we give an account of modules over bundles of quaternion algebras, discussing Morita equivalence, characteristic classes and K-theory. The results have been used to describe obstructions for the existence of almost quaternionic structures on 8-dimensional Spin^c manifolds in ?adek et al. (2008) [5] and may be of some interest, also, in quaternionic and algebraic geometry.     

Mathematical problems of stochastic quantum mechanics: Harmonic analysis on phase space and quantum geometry

In this paper we review the mathematical methods and problems that are specific to the programme of stochastic quantum mechanics and quantum spacetime. The physical origin of these problems is explained, and then the mathematical models are developed. Three notions emerge as central to the programme: positive operator-valued (POV) measures on a Hilbert space, reproducing kernel Hilbert spaces, and fibre bundle formulations of quantum geometries. A close connection between the first two notions is shown to exist, which provides a natural setting for introducing a fibration on the associated overcomplete family of vectors. The introduction of group covariance leads to an extended version of harmonic analysis on phase space. It also yields a theory of induced group representations, which extends the results of Mackey on imprimitivity systems for locally compact groups to the more general case of systems of covariance. Quantum geometries emerge as fibre bundles whose base spaces are manifolds of mean stochastic locations for quantum test particles (i.e., spacetime excitons) that display a phase space structure, and whose fibres and structure groups contain, respectively, the aforementioned overcomplete families of vectors and unitary group representations of phase space systems of covariance.Work supported in part by the Natural Science and Engineering Research Council of Canada (NSERC) grants.     

Multivariate fractionally integrated CARMA processes

A multivariate analogue of the fractionally integrated continuous time autoregressive moving average (FICARMA) process defined by Brockwell [Representations of continuous-time ARMA processes, J. Appl. Probab. 41 (A) (2004) 375-382] is introduced. We show that the multivariate FICARMA process has two kernel representations: as an integral over the fractionally integrated CARMA kernel with respect to a Lévy process and as an integral over the original (not fractionally integrated) CARMA kernel with respect to the corresponding fractional Lévy process (FLP). In order to obtain the latter representation we extend FLPs to the multivariate setting. In particular we give a spectral representation of FLPs and consequently, derive a spectral representation for FICARMA processes. Moreover, various probabilistic properties of the multivariate FICARMA process are discussed. As an example we consider multivariate fractionally integrated Ornstein-Uhlenbeck processes.     

Tensor Approximation of Generalized Correlated Diffusions and Functional Copula Operators

In this paper we develop a class of applied probabilistic continuous time but discretized state space decompositions of the characterization of a multivariate generalized diffusion process. This decomposition is novel and, in particular, it allows one to construct families of mimicking classes of processes for such continuous state and continuous time diffusions in the form of a discrete state space but continuous time Markov chain representation. Furthermore, we present this novel decomposition and study its discretization properties from several perspectives. This class of decomposition both brings insight into understanding locally in the state space the induced dependence structures from the generalized diffusion process as well as admitting computationally efficient representations in order to evaluate functionals of generalized multivariate diffusion processes, which is based on a simple rank one tensor approximation of the exact representation. In particular, we investigate aspects of semimartingale decompositions, approximation and the martingale representation for multidimensional correlated Markov processes. A new interpretation of the dependence among processes is given using the martingale approach. We show that it is possible to represent, in both continuous and discrete space, that a multidimensional correlated generalized diffusion is a linear combination of processes originated from the decomposition of the starting multidimensional semimartingale. This result not only reconciles with the existing theory of diffusion approximations and decompositions, but defines the general representation of infinitesimal generators for both multidimensional generalized diffusions and, as we will demonstrate, also for the specification of copula density dependence structures. This new result provides immediate representation of the approximate weak solution for correlated stochastic differential equations. Finally, we demonstrate desirable convergence results for the proposed multidimensional semimartingales decomposition approximations.     

On the regular representation of a nonunimodular locally compact group

We study the discrete part of the regular representation of a locally compact group and also its Type I part if the group is separable. Our results extend to nonunimodular groups' known results for unimodular groups about formal degrees of square integrable representations, and the Plancherel formula. We establish orthogonality relations for matrix coefficients of square integrable representations and we show that the formal degree in general is not a positive number, but a positive self-adjoint unbounded operator, semi-invariant under the representation. Integrable representations are also studied in this context. Finally we show that when the group is nonunimodular, “Plancherel measure” is not a true measure, but a measure multiplied by a section of a certain real oriented line bundle on the dual space of the group.     

Some time change representations of stable integrals, via predictable transformations of local martingales

From the predictable reduction of a marked point process to Poisson, we derive a similar reduction theorem for purely discontinuous martingales to processes with independent increments. Both results are then used to examine the existence of stochastic integrals with respect to stable Lévy processes, and to prove a variety of time change representations for such integrals. The Knight phenomenon, where possibly dependent but orthogonal processes become independent after individual time changes, emerges as a general principle.     

The Fermion Model of Representations of Affine Krichever–Novikov Algebras

To a generic holomorphic vector bundle on an algebraic curve and an irreducible finite-dimensional representation of a semisimple Lie algebra, we assign a representation of the corresponding affine Krichever–Novikov algebra in the space of semi-infinite exterior forms. It is shown that equivalent pairs of data give rise to equivalent representations and vice versa.