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Invariant manifolds for random and stochastic partial differential equations. (arXiv:0901.0382v1 [math.DS]) - Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic influences. Under a nonuniform hyperbolicity or a nonuniform exponential dichotomy condition, the existence of random pseudo-stable and pseudo-unstable manifolds for a class of \emph{random} partial differential equations and \emph{stochastic} partial differential equations is shown. Unlike the invariant manifold theory for stochastic \emph{ordinary} differential equations, random norms are not used. The result is then applied to a nonlinear stochastic partial differential equation with linear multiplicative noise. ...Feed Source: arxiv.org
On Cox rings of K3-surfaces. (arXiv:0901.0369v1 [math.AG]) - We study Cox rings of K3-surfaces. A first result is that a K3-surface has a
finitely generated Cox ring if and only if its effective cone is polyhedral.
Moreover, we investigate degrees of generators and relations for Cox rings of
K3-surfaces of Picard number two, and explicitly compute the Cox rings of
generic K3-surfaces admitting a non-symplectic involution and have Picard
number 2 to 5 or occur as double covers of del Pezzo surfaces.
...
Feed Source: arxiv.org
New inductive constructions of complete caps in $PG(N,q)$, $q$ even. (arXiv:0901.0367v1 [math.CO]) - Some new families of small complete caps in $PG(N,q)$, $q$ even, are
described. By using inductive arguments, the problem of the construction of
small complete caps in projective spaces of arbitrary dimensions is reduced to
the same problem in the plane. The caps constructed in this paper provide an
improvement on the currently known upper bounds on the size of the smallest
complete cap in $PG(N,q),$ $N\geq 4,$ for all $q\geq 2^{3}.$ In particular,
substantial improvements are obtained for infinite values of $q$ square,
including $ q=2^{2Cm},$ $C\geq 5,$ $m\geq 3;$ for $q=2^{Cm},$ $C\geq 5,$ $m\geq
9,$ with $C,m$ odd; and for all $q\leq 2^{18}.$
...
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Riemann-Stieltjes operators and multipliers on $Q_p$ spaces in the unit ball of $C^n$. (arXiv:0901.0366v1 [math.CV]) - This paper is devoted to characterizing the Riemann-Stieltjes operators and
pointwise multipliers acting on M${\rm \ddot{o}}$bius invariant spaces $Q_p$,
which unify BMOA and Bloch space in the scale of $p$. The boundedness and
compactness of these operators on $Q_p$ spaces are determined by means of an
embedding theorem, i.e. $Q_p$ spaces boundedly embedded in the non-isotropic
tent type spaces $T_q^\infty$.
...
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Convexity properties of generalized moment maps. (arXiv:0901.0361v1 [math.DG]) - In this paper, we consider generalized moment maps for Hamiltonian actions on
$H$-twisted generalized complex manifolds introduced by Lin and Tolman
\cite{Lin}. The main purpose of this paper is to show convexity and
connectedness properties for generalized moment maps. We study Hamiltonian
torus actions on compact $H$-twisted generalized complex manifolds and prove
that all components of the generalized moment map are Bott-Morse functions.
Based on this, we shall show that the generalized moment maps have a convex
image and connected fibers. Furthermore, by applying the arguments of Lerman,
Meinrenken, Tolman, and Woodward \cite{Ler2} we extend our results to the case
of Hamiltonian actions of general compact Lie groups on $H$-twisted generalized
complex orbifolds.
...
Feed Source: arxiv.org
Information, Divergence and Risk for Binary Experiments. (arXiv:0901.0356v1 [stat.ML]) - We unify f-divergences, Bregman divergences, surrogate loss bounds (regret
bounds), proper scoring rules, matching losses, cost curves, ROC-curves and
information. We do this by systematically studying integral and variational
representations of these objects and in so doing identify their primitives
which all are related to cost-sensitive binary classification. As well as
clarifying relationships between generative and discriminative views of
learning, the new machinery leads to tight and more general surrogate loss
bounds and generalised Pinsker inequalities relating f-divergences to
variational divergence. The new viewpoint illuminates existing algorithms: it
provides a new derivation of Support Vector Machines in terms of divergences
and relates Maximum Mean Discrepancy to Fisher Linear Discriminants. It also
suggests new techniques for estimating f-divergences.
...
Feed Source: arxiv.org
Generic Rigidity of Laurent polynomials. (arXiv:0901.0354v1 [math.NT]) - Generic Newton polygon of L-functions of all $p^m$-power order exponential
sums are determined.
...
Feed Source: arxiv.org
Implications of Energy Conditions on Standard Static Space-times. (arXiv:0901.0370v1 [math.DG]) - In the framework of standard static space times, we state a family of
sufficient or necessary conditions for a set of physically reasonable energy
and convergence conditions in relativity and related theories. We concentrate
our study on questions about the sub-harmonicity of the warping function, the
scalar curvature map, conformal hyperbolicity, conjugate points and the
time-like diameter of this class of space-times.
...
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Global Existence Proof for Relativistic Boltzmann Equation with Hard Interactions. (arXiv:0901.0372v1 [math-ph]) - By combining the DiPerna and Lions techniques for the nonrelativistic
Boltzmann equation and the Dudy\'{n}ski and Ekiel-Je\.{z}ewska device of the
causality of the relativistic Boltzmann equation, it is shown that there exists
a global mild solution to the Cauchy problem for the relativistic Boltzmann
equation with the assumptions of the relativistic scattering cross section
including some relativistic hard interactions and the initial data satisfying
finite mass, energy and entropy. This is in fact an extension of the result of
Dudy\'{n}ski and Ekiel-Je\.{z}ewska to the case of the relativistic Boltzmann
equation with hard interactions.
...
Feed Source: arxiv.org
The Atiyah-Patodi-Singer index theorem for Dirac operators over C*-algebras. (arXiv:0901.0381v1 [math.DG]) - We prove an Atiyah-Patodi-Singer index theorem for Dirac operators twisted by
C*-vector bundles. We use it to derive a general product formula for Eta-forms
and to define and study new Rho-invariants generalizing Lott's higher Rho-form.
The higher Atiyah-Patodi-Singer index theorem of Leichtnam-Piazza can be
recovered by applying the theorem to Dirac operators twisted by the
Mishenko-Fomenko bundle associated to the reduced C*-algebra of the fundamental
group.
...
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Rational linking and contact geometry. (arXiv:0901.0380v1 [math.SG]) - In the note we study Legendrian and transverse knots in rationally
null-homologous knot types. In particular we generalize the standard
definitions of self-linking number, Thurston-Bennequin invariant and rotation
number. We then prove a version of Bennequin's inequality for these knots and
classify precisely when the Bennequin bound is sharp for fibered knot types.
Finally we study rational unknots and show they are weakly Legendrian and
transversely simple.
...
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On certain categories of modules for twisted affine Lie algebras. (arXiv:0901.0377v1 [math.RA]) - We classify integrable irreducible $\hat{g}[\sigma]$-modules in categories E
and C, where E is proved to contain the well known evaluation modules and C to
unify highest weight modules, evaluation modules and their tensor product
modules.
...
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Global Solution to the Relativistic Enskog Equation With Near-Vacuum Data. (arXiv:0901.0375v1 [math-ph]) - We give two hypotheses of the relativistic collision kernal and show the
existence and uniqueness of the global mild solution to the relativistic Enskog
equation with the initial data near the vacuum for a hard sphere gas.
...
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A $C^0$-estimate for the parabolic Monge-Amp\`{e}re equation on complete non-compact K\"ahler manifolds. (arXiv:0901.0374v1 [math.DG]) - In this article we study the K\"ahler Ricci flow, the corresponding parabolic
Monge Amp\`{e}re equation and complete non-compact K\"ahler Ricci flat
manifolds. In our main result Theorem \ref{mainthm} we prove that if $(M, g)$
is sufficiently close to being K\"ahler Ricci flat in a suitable sense, then
the K\"ahler Ricci flow \eqref{KRF} has a long time smooth solution $g(t)$
converging smoothly uniformly on compact sets to a complete K\"ahler Ricci flat
metric on $M$. The main step is to obtain a uniform $C^0$-estimates for the
corresponding parabolic Monge Amp\`{e}re equation. Our results on this can be
viewed as a parabolic version of the main results in \cite{TY3} on the elliptic
Monge Amp\`{e}re equation.
...
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New approach to q-Genocch, Euler numbers and polynomials and their interpolation functions. (arXiv:0901.0353v1 [math.NT]) - We give a new construction of q-Genocchi numbers, Euler numbers of higher
order, which are different than the q-Genocchi numbers of Cangul-Ozden-Simsek.
By using our q-Genoucchi, Euler nimbers of higher order, we can investigate the
interesting relationship between w-q-Euler polynomials and w-q-Genocchi
polynomials.
...
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Compressible flows with a density-dependent viscosity coefficient. (arXiv:0901.0352v1 [math.AP]) - We prove the global existence of weak solutions for the 2-D compressible
Navier-Stokes equations with a density-dependent viscosity coefficient
($\lambda=\lambda(\rho)$). Initial data and solutions are small in energy-norm
with nonnegative densities having arbitrarily large sup-norm. Then, we show
that if there is a vacuum domain at the initial time, then the vacuum domain
will retain for all time, and vanishes as time goes to infinity. At last, we
show that the condition of $\mu=$constant will induce a singularity of the
system at vacuum. Thus, the viscosity coefficient $\mu$ plays a key role in the
Navier-Stokes equations.
...
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New estimates for the Beurling-Ahlfors operator on differential forms. (arXiv:0901.0345v1 [math.CA]) - We establish new $p$-estimates for the norm of the generalized
Beurling--Ahlfors transform $\mathcal{S}$ acting on form-valued functions.
Namely, we prove that $\norm{\mathcal{S}}_{L^p(\R^n;\Lambda)\to
L^p(\R^n;\Lambda)}\leq n(p^{*}-1)$ where $p^*=\max\{p, p/(p-1)\},$ thus
extending the recent Nazarov--Volberg estimates to higher dimensions. The
even-dimensional case has important implications for quasiconformal mappings.
Some promising prospects for further improvement are discussed at the end.
...
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Beta Jacobi processes. (arXiv:0901.0324v1 [math.PR]) - We define and study a multidimensional process that generalizes the
eigenvalues of matrix Jacobi processes on the one hand and whose stationary
distribution is given by the beta Jacobi ensemble on the other hand.
...
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Convolution symmetries of integrable hierarchies, matrix models and $\tau$-functions. (arXiv:0901.0323v1 [math-ph]) - Generalized convolution symmetries of integrable hierarchies of KP-Toda and
2KP-Toda type have the effect of multiplying the Fourier coefficients of the
Baker-Akhiezer function by a specified sequence of constants. The induced
action on the associated fermionic Fock space is diagonal in the standard
orthonormal base determined by occupation sites and labeled by partitions. The
coefficients in the single and double Schur function expansions of the
associated $\tau$-functions, which are the Pl\"ucker coordinates of a
decomposable element, are multiplied by the corresponding diagonal factors.
Applying such transformations to matrix integrals, we obtain new matrix models
of externally coupled type which are also KP-Toda or 2KP-Toda $\tau$-functions.
More general multiple integral representations of tau functions are similarly
obtained, as well as finite determinantal expressions for them.
...
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The Weil algebra and the Van Est isomorphism. (arXiv:0901.0322v1 [math.DG]) - This paper belongs to a series of papers devoted to the study of the
cohomology of classifying spaces. Generalizing the Weil algebra of a Lie
algebra and Kalkman's BRST model, here we introduce the Weil algebra $W(A)$
associated to any Lie algebroid $A$. We then show that this Weil algebra is
related to the Bott-Shulman complex (computing the cohomology of the
classifying space) via a Van Est map and we prove a Van Est isomorphism
theorem. As an application, we generalize and give a simpler more conceptual
proof of the main result of Bursztyn et.al. on the integration of Poisson and
Dirac structures and of the reconstruction of connection 1-forms on
prequantizations (Weinstein-Xu, Crainic).
...
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Approximate Parametrization of Plane Algebraic Curves by Linear Systems of Curves. (arXiv:0901.0320v1 [math.AG]) - It is well known that an irreducible algebraic curve is rational (i.e.
parametric) if and only if its genus is zero. In this paper, given a tolerance
$\epsilon>0$ and an $\epsilon$-irreducible algebraic affine plane curve
$\mathcal C$ of proper degree $d$, we introduce the notion of
$\epsilon$-rationality, and we provide an algorithm to parametrize
approximately affine $\epsilon$-rational plane curves, without exact
singularities at infinity, by means of linear systems of $(d-2)$-degree curves.
The algorithm outputs a rational parametrization of a rational curve
$\bar{\mathcal C}$ of degree at most $d$ which has the same points at infinity
as $\mathcal C$. Moreover, although we do not provide a theoretical analysis,
our empirical analysis shows that $\bar{\mathcal C}$ and $\mathcal C$ are close
in practice.
...
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Representations up to homotopy of Lie algebroids. (arXiv:0901.0319v1 [math.DG]) - This is the first in a series of papers devoted to the study of the
cohomology of classifying spaces. The aim of this paper is to introduce and
study the notion of representation up to homotopy and to make sense of the
adjoint representation of a Lie algebroid. Our construction is inspired by
Quillen's notion of superconnection and fits into the general theory of
structures up to homotopy. The advantage of considering such representations is
that they are flexible and general enough to contain interesting examples which
are the correct generalization of the corresponding notions for Lie algebras.
They also allow one to identify seemingly ad-hoc constructions and cohomology
theories as instances of the cohomology with coefficients in representations
(up to homotopy). In particular, we show that the adjoint representation of a
Lie algebroid makes sense as a representation up to homotopy and that, similar
to the case of Lie algebras, the resulting cohomology controls the deformations
o...
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Thoughts on an Unified Framework for Artificial Chemistries. (arXiv:0901.0318v1 [cs.AI]) - Artificial Chemistries (ACs) are symbolic chemical metaphors for the
exploration of Artificial Life, with specific focus on the problem of
biogenesis or the origin of life. This paper presents authors thoughts towards
defining a unified framework to characterize and classify symbolic artificial
chemistries by devising appropriate formalism to capture semantic and
organizational information. We identify three basic high level abstractions in
initial proposal for this framework viz., information, computation, and
communication. We present an analysis of two important notions of information,
namely, Shannon's Entropy and Algorithmic Information, and discuss inductive
and deductive approaches for defining the framework.
...
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Invisible Parts of Attractors. (arXiv:0901.0316v1 [math.DS]) - This paper deals with the attractors of generic dynamical systems. We
introduce the notion of epsilon-invisible set, which is an open set in which
almost all orbits spend on average a fraction of time no greater than epsilon.
For extraordinarily small values of epsilon (say, smaller than 2^{-100}), these
are areas of the phase space which an observer virtually never sees when
following a generic orbit.
We construct an open set in the space of all dynamical systems which have an
epsilon-invisible set that includes parts of attractors of size comparable to
the entire attractor of the system, for extraordinarily small values of
epsilon. The open set consists of C^1 perturbations of a particular skew
product over the Smale-Williams solenoid. Thus for all such perturbations, a
sizable portion of the attractor is almost never visited by generic orbits and
practically never seen by the observer.
...
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Levantamiento de Wagner de una Metrica de Riemann al Haz de Marcos Ortonormales. (arXiv:0901.0326v1 [math.DG]) - In the present work we construct a lift of a metric $g$ on a 2-dimensional
oriented Riemannian manifold $M$ to a metric $\hat{g}$ on the total space $P$
of the orthonormal frame bundle of $M$. We call this lift the \textit {Wagner
lift}. Viktor Vladimirovich Wagner (1908 -1981) proposed a technique to extend
a metric defined on a non-holonomic distribution to its prolongation via the
Lie brackets. We apply the Wagner construction to the specific case when the
distribution is the infinitesimal connection in the orthonormal frame bundle
which corresponds to a Levi-Civita connection. We find relations between the
geometry of the Riemannian manifold $(M,g)$ and of the total space $(P,G)$ of
the orthonormal frame bundle endowed with the lifted metric.
...
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Azumaya structure on D-branes and resolution of ADE orbifold singularities revisited: Douglas-Moore vs. Polchinski-Grothendieck. (arXiv:0901.0342v1 [math.AG]) - In this continuation of [L-Y1] and [L-L-S-Y], we explain how the Azumaya
structure on D-branes together with a netted categorical quotient construction
produces the same resolution of ADE orbifold singularities as that arises as
the vacuum manifold/variety of the supersymmetric quantum field theory on the
D-brane probe world-volume, given by Douglas and Moore [D-M] under the
string-theory contents and constructed earlier through hyper-K\"{a}hler
quotients by Kronheimer and Nakajima. This is consistent with the moral behind
this project that Azumaya-type structure on D-branes themselves -- stated as
the Polchinski-Grothendieck Ansatz in [L-Y1] -- gives a mathematical reason for
many originally-open-string-induced properties of D-branes.
...
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The Froissart-Gribov representation of Jost function of Dirac operators in arbitrary-dimension space. (arXiv:0901.0341v1 [math-ph]) - A dynamic scheme basing on equation for T-matrix momentum transfer spectral
density and integral representation for Jost function is proposed for local
Dirac Hamiltonians in arbitrary N- dimension spaces and for Schrodinger one
with singular or nonlocal generalized Yukawa-type potentials. A generalization
of the off-shell-Jost function method for that Hamiltonians and universal
renormalization procedure of Jost function calculation for singular and
nonlocal potentials is proposed.
...
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The Causal Perturbation Expansion Revisited: Rescaling the Interacting Dirac Sea. (arXiv:0901.0334v1 [math-ph]) - The causal perturbation expansion defines the Dirac sea in the presence of a
time-dependent external field. It yields an operator whose image generalizes
the vacuum solutions of negative energy and thus gives a canonical splitting of
the solution space into two subspaces. After giving a self-contained
introduction to the ideas and techniques, we show that this operator is in
general not idempotent. We modify the standard construction by a rescaling
procedure giving a projector on the generalized negative-energy subspace. The
resulting rescaled causal perturbation expansion uniquely defines the fermionic
projector in terms of a series of distributional solutions of the Dirac
equation. The technical core of the paper is to work out the combinatorics of
the expansion in detail. It is also shown that the fermionic projector with
interaction can be obtained from the free projector by a unitary
transformation. We finally analyze the consequences of the rescaling procedure
on the light-con...
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Biextensions of 1-motives in Voevodsky's category of motives. (arXiv:0901.0331v1 [math.KT]) - Let k be a perfect field. In this paper we prove that biextensions of
1-motives define multilinear morphisms between 1-motives in Voevodsky's
triangulated category of effective geometrical motives over k with rational
coefficients.
...
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The phase transition of the quantum Ising model is sharp. (arXiv:0901.0328v1 [math-ph]) - An analysis is presented of the phase transition of the quantum Ising model
with transverse field on the d-dimensional hypercubic lattice. It is shown that
there is a unique sharp transition. The value of the critical point is
calculated rigorously in one dimension. The first step is to express the
quantum Ising model in terms of a (continuous) classical Ising model in d+1
dimensions. A so-called `random-parity' representation is developed for the
latter model, similar to the random-current representation for the classical
Ising model on a discrete lattice. Certain differential inequalities are
proved. Integration of these inequalities yields the sharpness of the phase
transition, and also a number of other facts concerning the critical and
near-critical behaviour of the model under study.
...
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On a new method for controlling exponential processes. (arXiv:0901.0327v1 [math.NA]) - Unlike the classical polynomial case there has not been invented up to very
recently a tool similar to the Bernstein-Bezier representation which would
allow us to control the behavior of the exponential polynomials. The
exponential analog to the classical Bernstein polynomials has been introduced
in a recent authors' paper which appeared in Constructive Approximations, and
this analog retains all basic properties of the classical Bernstein
polynomials. The main purpose of the present paper is to contribute in this
direction, by proving some important properties of the "Bernstein exponential
operator" which has been introduced. We also fix our attention upon some
special type of exponential polynomials which are particularly important for
the further development of theory of representation of Multivariate data.
...
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On the Second Boundary Value Problem for a Class of Modified-Hessian Equations. (arXiv:0901.0312v1 [math.AP]) - In this paper a new class of modified-Hessian equations, closely related to
the Optimal Transportation Equation, will be introduced and studied. In
particular, the existence of globally smooth, classical solutions of these
equations satisfying the second boundary value problem will be proven. This
proof follows a standard method of continuity argument, which subsequently
requires various a priori estimates to be made on classical solutions. These
estimates are modifications of and generalise the corresponding estimates of
Trudinger and Wang for the Optimal Transportation Equation. Of particular note
is the fact that the global C^2 estimate contained in this paper makes no use
of duality in regards to the original equation.
...
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Linear Transformations and Restricted Isometry Property. (arXiv:0901.0541v1 [cs.IT]) - The Restricted Isometry Property (RIP) introduced by Cand\'es and Tao is a
fundamental property in compressed sensing theory. It says that if a sampling
matrix satisfies the RIP of certain order proportional to the sparsity of the
signal, then the original signal can be reconstructed even if the sampling
matrix provides a sample vector which is much smaller in size than the original
signal. This short note addresses the problem of how a linear transformation
will affect the RIP. This problem arises from the consideration of extending
the sensing matrix and the use of compressed sensing in different bases. As an
application, the result is applied to the redundant dictionary setting in
compressed sensing.
...
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Critical mass phenomenon for a chemotaxis kinetic model with spherically symmetric initial data. (arXiv:0901.0503v1 [math.AP]) - The goal of this paper is to exhibit a critical mass phenomenon occuring in a
model for cell self-organization via chemotaxis. The very well known dichotomy
arising in the behavior of the macroscopic Keller-Segel system is derived at
the kinetic level, being closer to microscopic features. Indeed, under the
assumption of spherical symmetry, we prove that solutions with initial data of
large mass blow-up in finite time, whereas solutions with initial data of small
mass do not. Blow-up is the consequence of a virial identity and the existence
part is derived from a comparison argument. Spherical symmetry is crucial
within the two approaches. We also briefly investigate the drift-diffusion
limit of such a kinetic model. We recover partially at the limit the
Keller-Segel criterion for blow-up, thus arguing in favour of a global link
between the two models.
...
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Transmission Capacities for Overlaid Wireless Ad Hoc Networks with Outage Constraints. (arXiv:0901.0492v1 [cs.IT]) - We study the transmission capacities of two coexisting wireless networks (a
primary network vs. a secondary network) that operate in the same geographic
region and share the same spectrum. We define transmission capacity as the
product among the density of transmissions, the transmission rate, and the
successful transmission probability (1 minus the outage probability). The
primary (PR) network has a higher priority to access the spectrum without
particular considerations for the secondary (SR) network, where the SR network
limits its interference to the PR network by carefully controlling the density
of its transmitters. Assuming that the nodes are distributed according to
Poisson point processes and the two networks use different transmission ranges,
we quantify the transmission capacities for both of these two networks and
discuss their tradeoff based on asymptotic analyses. Our results show that if
the PR network permits a small increase of its outage probability, the sum
transm...
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Product Structures for Legendrian Contact Homology. (arXiv:0901.0490v1 [math.SG]) - Legendrian contact homology (LCH) and its associated differential graded
algebra are powerful non-classical invariants of Legendrian knots.
Linearization makes the LCH computationally tractable at the expense of
discarding nonlinear (and noncommutative) information. To recover some of the
nonlinear information while preserving computability, we introduce invariant
cup and Massey products - and, more generally, an A_\infty structure - on the
linearized LCH. We apply the products and A_\infty structure in three ways: to
find infinite families of Legendrian knots that are not isotopic to their
Legendrian mirrors, to reinterpret the duality theorem of the fourth author in
terms of the cup product, and to recover higher-order linearizations of the
LCH.
...
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On the ranks and border ranks of symmetric tensors. (arXiv:0901.0487v1 [math.AG]) - Motivated by questions arising in signal processing, computational
complexity, and other areas, we study the ranks and border ranks of symmetric
tensors using geometric methods. We provide improved lower bounds for the rank
of a symmetric tensor (i.e., a homogeneous polynomial) obtained by considering
the singularities of the hypersurface defined by the polynomial. We obtain
normal forms for polynomials of border rank up to five, and compute or bound
the ranks of several classes of polynomials, including monomials, the
determinant, and the permanent.
...
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General discrete random walk with variable absorbing probabilities. (arXiv:0901.0469v1 [math.PR]) - We obtain expected number of arrivals, probability of arrival, absorption
probabilities and expected time before absorption for a general discrete random
walk with variable absorbing probabilities on a finite interval using Fibonacci
numbers
...
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Fundamental solutions for a class of three dimensional elliptic equations with singular coefficients. (arXiv:0901.0468v1 [math-ph]) - In this article we consider an equation $$ L_{\alpha ,\beta ,\gamma} (u)
\equiv u_{xx} + u_{yy} + u_{zz} + \displaystyle \frac{{2\alpha}}{x}u_x +
\displaystyle \frac{{2\beta}}{y}u_y + \displaystyle \frac{{2\gamma}}{z}u_z = 0
$$ in a domain ${\bf R}_3^ + \equiv {{({x,y,z}): x > 0, y > 0, z > 0}}$. Here
$\alpha ,\beta ,\gamma$ are constants, moreover $0 < 2\alpha, 2\beta, 2\gamma <
1$. Main result of this paper is a construction of eight fundamental solutions
for above-given equation which are expressed by Lauricella's hypergeometric
functions with three variables. Using expansion of Lauricella's hypergeometric
function by products of Gauss's hypergeometric function, it is proved that the
found solutions have a singularity of the order $1/r$ when $r \to 0$.
...
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A Law of Likelihood for Composite Hypotheses. (arXiv:0901.0463v1 [math.ST]) - The law of likelihood underlies a general framework, known as the likelihood
paradigm, for representing and interpreting statistical evidence. As stated,
the law applies only to simple hypotheses, and there have been reservations
about extending the law to composite hypotheses, despite their tremendous
relevance in statistical applications. This paper proposes a generalization of
the law of likelihood for composite hypotheses. The generalized law is
developed in an axiomatic fashion, illustrated with real examples, and examined
in an asymptotic analysis. Previous concerns about including composite
hypotheses in the likelihood paradigm are discussed in light of the new
developments. The generalized law of likelihood is compared with other
likelihood-based methods and its practical implications are noted. Lastly, a
discussion is given on how to use the generalized law to interpret published
results of hypothesis tests as reduced data when the full data are not
available.
...
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Likelihood Inference in Exponential Families and Directions of Recession. (arXiv:0901.0455v1 [math.ST]) - When in a full exponential family the maximum likelihood estimate (MLE) does
not exist, the MLE may exist in the Barndorff-Nielsen completion of the family.
We propose a practical algorithm for finding the MLE in the completion based on
repeated linear programming using the R contributed package rcdd and illustrate
it with two generalized linear model examples. When the MLE for the null
hypothesis lies in the completion, likelihood ratio tests of model comparison
are almost unchanged from the usual case. Only the degrees of freedom need to
be adjusted. When the MLE lies in the completion, confidence intervals are
changed much more from the usual case. The MLE of the natural parameter can be
thought of as having gone to infinity in a certain direction, which we call a
generic direction of recession. We propose a new one-sided confidence interval
which says how close to infinity the natural parameter may be. This maps to
one-sided confidence intervals for mean values showing how close...
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Derived categories of coherent sheaves on Calabi-Yau fibrations. (arXiv:0901.0509v1 [math.AG]) - In this note we prove that the derived categories of an abelian fibration and
its dual are derived equivalent if and only if there is fiberwise equivalence.
We also study the problem for $K3$ fibrations.
...
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Platonic polyhedra tune the 3-sphere: II. Harmonic analysis on cubic spherical 3-manifolds. (arXiv:0901.0511v1 [math.DG]) - From the homotopy groups of two cubic spherical 3-manifolds we construct the
isomorphic groups of deck transformations acting on the 3-sphere. These groups
become the cyclic group of order eight and the quaternion group respectively.
By reduction of representations from the orthogonal group to the identity
representation of these subgroups we provide two subgroup-periodic bases for
the harmonic analysis on the 3-manifolds. This harmonic analysis has
applications to cosmic topology.
...
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A Family of Nonlinear Fourth Order Equations of Gradient Flow Type. (arXiv:0901.0540v1 [math.AP]) - Global existence and long-time behavior of solutions to a family of nonlinear
fourth order evolution equations on $R^d$ are studied. These equations
constitute gradient flows for the perturbed information functionals $F[u] =
1/(2\alpha) \int | D (u^\alpha) |^2 dx + \lambda/2 \int |x|^2 u dx$ with
respect to the $L^2$-Wasserstein metric. The value of $\alpha$ ranges from
$\alpha=1/2$, corresponding to a simplified quantum drift diffusion model, to
$\alpha=1$, corresponding to a thin film type equation.
...
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Born-Oppenheimer-type Approximations for Degenerate Potentials: Recent Results and a Survey on the area. (arXiv:0901.0539v1 [math-ph]) - This paper is devoted to the asymptotics of eigenvalues for a Schr\"o-dinger
operator in the case when the potential V does not tend to infinity at
infinity. Such a potential is called degenerate. The point is that the set in
the phase space where the associated hamiltonian is smaller than a fixed energy
E may have an infinite volume, so that the Weyl formula which gives the
behaviour of the counting function has to be revisited. We recall various
results in this area, in the classical context as well as in the semi-classical
one and comment the different methods. In sections 3, 4 we present our joint
works with A Morame, (Universit\'e de Nantes),concerning a degenerate potential
V(x) =f(y) g(z), where g is assumed to be a homogeneous positive function of m
variables, and f is a smooth and strictly positive function of n variables,
with a minimum in 0. In the case where f tends to infinity at infinity, we give
the semi-classical asymptotic behaviour of the number of eigenvalues less...
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Polarization Codes: Characterization of Exponent, Bounds, and Constructions. (arXiv:0901.0536v1 [cs.IT]) - Polarization codes were recently introduced by Ar\i kan. They achieve the
capacity of arbitrary symmetric binary-input discrete memoryless channels (and
even extensions thereof) under a low complexity successive decoding strategy.
The original polar code construction is closely related to the recursive
construction of Reed-Muller codes and is based on the $2 \times 2$ matrix
$\bigl[ 1 &0 1& 1 \bigr]$. It was shown by Ar\i kan and Telatar that this
construction achieves an error exponent of $\frac12$, i.e., that for
sufficiently large blocklengths the error probability decays exponentially in
the square root of the length. It was already mentioned by Ar\i kan that in
principle larger matrices can be used to construct polar codes. A fundamental
question then is to see whether there exist matrices with exponent exceeding
$\frac12$. We first show that any $\ell \times \ell$ matrix none of whose
column permutations is upper triangular polarizes symmetric channels. We then
charact...
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On a geometric black hole of a compact manifold. (arXiv:0901.0528v1 [math.GT]) - Using a smooth triangulation and a Riemannian metric on a compact, connected,
closed manifold M of dimension n we have got that every such M can be
represented as a union of a n-dimensional cell and a connected union K of some
subsimplexes of the triangulation. A sufficiently small closed neighborhood of
K is called a geometric black hole. Any smooth tensor field T (or other
structure) can be deformed into a continuous and sectionally smooth tensor
field T1 where T1 has a very simple construction out of the black hole.
...
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On Multipath Fading Channels at High SNR. (arXiv:0901.0521v1 [cs.IT]) - This work studies the capacity of multipath fading channels. A noncoherent
channel model is considered, where neither the transmitter nor the receiver is
cognizant of the realization of the path gains, but both are cognizant of their
statistics. It is shown that if the delay spread is large in the sense that the
variances of the path gains decay exponentially or slower, then capacity is
bounded in the signal-to-noise ratio (SNR). For such channels, capacity does
not tend to infinity as the SNR tends to infinity. In contrast, if the
variances of the path gains decay faster than exponentially, then capacity is
unbounded in the SNR. It is further demonstrated that if the number of paths is
finite, then at high SNR capacity grows double-logarithmically with the SNR,
and the capacity pre-loglog, defined as the limiting ratio of capacity to
log(log(SNR)) as SNR tends to infinity, is 1 irrespective of the number of
paths.
...
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Abstract Hardy-Sobolev spaces and interpolation. (arXiv:0901.0518v1 [math.CA]) - The purpose of this work is to describe an abstract theory of Hardy-Sobolev
spaces on doubling Riemannian manifolds via an atomic decomposition. We study
the real interpolation of these spaces with Sobolev spaces and finally give
applications to Riesz transforms.
...
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Submanifolds associated to Toda theories. (arXiv:0901.0516v1 [math-ph]) - A set of two-dimensional semi-riemannian submanifolds of flat semi-riemannian
manifolds is associated to each Toda theory. The method and an example are
given to Toda theories associated to real finite dimensional Lie algebras.
...
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An Unusual Proof that the Reals are Uncountable. (arXiv:0901.0446v1 [math.HO]) - This somewhat unusual proof for the fact that the reals are uncountable,
which is adapted from one of Bourbaki's proofs in "Fonctions d'une variable
reelle", may be of some interest.
...
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Kashiwara and Zelevinsky involutions in affine type A. (arXiv:0901.0443v1 [math.RT]) - We first describe how the Kashiwara involution on crystals of affine type $A$
is encoded by the combinatorics of aperiodic multisegments. This yields a
simple relation between this involution and the Zelevinsky involution on the
set of simple modules for the affine Hecke algebras. We then give efficient
procedures for computing these involutions. Remarkably, these procedures do not
use the underlying crystal structure. They also permit to match explicitly the
Ginzburg and Ariki parametrizations of the simple modules associated to affine
and cyclotomic Hecke algebras, respectively .
...
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Non-vanishing theorem for log canonical pairs. (arXiv:0901.0409v1 [math.AG]) - We obtain a correct generalization of Shokurov's non-vanishing theorem for
log canonical pairs. It implies the base point free theorem for log canonical
pairs. We also prove the rationality theorem for log canonical pairs. As a
corollary, we obtain the cone theorem for log canonical pairs. We do not need
Ambro's theory of quasi-log varieties. Our proof is very similar to the
original one for kawamata log terminal pairs.
...
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The Tau Constant of A Metrized Graph and Its Behavior Under Graph Operations. (arXiv:0901.0407v1 [math.CO]) - This paper concerns the tau constant, which is an important invariant of a
metrized graph, and which has applications in arithmetic properties of curves.
We give several formulas for the tau constant, and show how it changes under
graph operations including deletion of an edge, contraction of an edge, union
of graphs along one or two points. We show how the tau constant changes when
edges of a graph are replaced by arbitrary graphs. We prove Rumely and Baker's
lower bound conjecture on the tau constant for some large classes of metrized
graphs.
...
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Fusion Rings of Loop Group Representations. (arXiv:0901.0391v1 [math.RT]) - We compute the fusion rings of positive energy representations of the loop
groups of the simple, simply connected Lie groups.
...
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Deformations of canonical pairs and Fano varieties. (arXiv:0901.0389v1 [math.AG]) - This paper is devoted to the study of various aspects of deformations of log
pairs, especially in connection to questions related to the invariance of
singularities and log plurigenera. In particular, using recent results from the
minimal model program, we obtain an extension theorem for adjoint divisors in
the spirit of Siu and Kawamata and more recent works of Hacon and McKernan. Our
main motivation however comes from the study of deformations of Fano varieties.
Our first application regards the behavior of Mori chamber decompositions in
families of Fano varieties. We expect that, in the case of mild singularities,
such decomposition is rigid under deformation; we prove that this is the case
for low dimensional varieties and, in all dimensions, for toric varieties. We
then turn to analyze deformation properties of toric Fano varieties, and prove
that in fact every simplicial toric Fano variety with at most terminal
singularities is rigid under deformations (and in particular is no...
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Confirming Two Conjectures of Su and Wang. (arXiv:0901.0385v1 [math.CO]) - Two conjectures of Su and Wang (2008) concerning binomial coefficients are
proved. For $n\geq k\geq 0$ and $b>a>0$, we show that the finite sequence
$C_j=\binom{n+ja}{k+jb}$ is a P\'{o}lya frequency sequence. For $n\geq k\geq 0$
and $a>b>0$, we show that there exists an integer $m\geq 0$ such that the
infinite sequence $\binom{n+ja}{k+jb}, j=0, 1, $, is log-concave for $0\leq
j\leq m$ and log-convex for $j\geq m$. The proof of the first result exploits
the connection between total positivity and planar networks, while that of the
second uses a variation-diminishing property of the Laplace transform.
...
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Stein's lemma, Malliavin calculus, and tail bounds, with application to polymer fluctuation exponent. (arXiv:0901.0383v1 [math.PR]) - We consider a random variable X satisfying almost-sure conditions involving
G:=<DX,-DL^{-1}X> where DX is X's Malliavin derivative and L^{-1} is the
inverse Ornstein-Uhlenbeck operator. A lower- (resp. upper-) bound condition on
G is proved to imply a Gaussian-type lower (resp. upper) bound on the tail
P[X>z]. Bounds of other natures are also given. A key ingredient is the use of
Stein's lemma, including the explicit form of the solution of Stein's equation
relative to the function 1_{x>z}, and its relation to G. Another set of
comparable results is established, without the use of Stein's lemma, using
instead a formula for the density of a random variable based on G, recently
devised by the author and Ivan Nourdin. As an application, via a Mehler-type
formula for G, we show that the Brownian polymer in a Gaussian environment
which is white-noise in time and positively correlated in space has deviations
of Gaussian type and a fluctuation exponent \chi=1/2. We also show th...
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Anti-Pluricanonical Systems On Q-Fano Threefolds. (arXiv:0901.0413v1 [math.AG]) - We investigate birationality of the anti-pluricanonical map $\phi_{-m}$, the
rational map defined by the anti-pluricanonical system $|-mK|$, on
$\mathbb{Q}$-Fano threefolds.
...
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Braids and Open Book Decompositions II. (arXiv:0901.0414v1 [math.GT]) - We construct an immersed surface for a braid in an annulus open book
decomposition, which is a generalization of the Bennequin surface for a braid
in R^3. By resolving the singularities of the immersed surface, we obtain an
embedded Seifert surface for the braid. We find a self-linking number formula
associated to the surface and prove that it is a generalization of the
Bennequin's self-linking formula for a braid in R^3. We also prove that our
self-linking formula is invariant up to mod k under transversal isotopy of the
contact structure compatible with the open book decomposition.
...
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The Borel Conjecture for hyperbolic and CAT(0)-groups. (arXiv:0901.0442v1 [math.GT]) - We prove the Borel Conjecture for a class of groups containing
word-hyperbolic groups and groups acting properly, isometrically and
cocompactly on a finite dimensional CAT(0)-space.
...
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Back to balls in billiards. (arXiv:0901.0441v1 [math.DS]) - We consider a billiard in the plane with periodic configuration of convex
scatterers. This system is recurrent, in the sense that almost every orbit
comes back arbitrarily close to the initial point. In this paper we study the
time needed to get back in an r-ball about the initial point, in the phase
space and also for the position, in the limit when r->0. We establish the
existence of an almost sure convergence rate, and prove a convergence in
distribution for the rescaled return times.
...
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Convergence of ray sequences of Pade approximants to 2F1(a,1;c;z), c>a>0. (arXiv:0901.0435v1 [math.CA]) - The Pad\'e table of $\phantom{}_2F_1(a,1;c;z)$ is normal for $c>a>0$ (cf.
\cite{3}). For $m \geq n-1$ and $c \notin {\zz}^{\phantom{}^-}$, the
denominator polynomial $Q_{mn}(z)$ in the $[m/n]$ Pad\'e approximant
$P_{mn}(z)/Q_{mn}(z)$ for $\phantom{}_2F_1(a,1;c;z)$ and the remainder term
$Q_{mn}(z)\phantom{}_2F_1(a,1;c;z)-P_{mn}(z)$ were explicitly evaluated by
Pad\'e (cf. \cite{2}, \cite{5} or \cite{7}). We show that for $c>a>0$ and
$m\geq n-1$, the poles of $P_{mn}(z)/Q_{mn}(z)$ lie on the cut $(1,\infty)$. We
deduce that the sequence of approximants $P_{mn}(z)/Q_{mn}(z)$ converges to
$\phantom{}_2F_1(a,1;c;z)$ as $m \to \infty$, $ n/m \to \rho$ with $0<\rho \leq
1$, uniformly on compact subsets of the unit disc $|z|<1$ for $c>a>0$
...
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Einstein and conformally flat critical metrics of the volume functional. (arXiv:0901.0422v1 [math.DG]) - Let $R$ be a constant. Let $\mathcal{M}^R_\gamma$ be the space of smooth
metrics $g$ on a given compact manifold $\Omega^n$ ($n\ge 3$) with smooth
boundary $\Sigma $ such that $g$ has constant scalar curvature $R$ and
$g|_{\Sigma}$ is a fixed metric $\gamma$ on $\Sigma$. Let $V(g)$ be the volume
of $g\in\mathcal{M}^R_\gamma$. In this work, we classify all Einstein or
conformally flat metrics which are critical points of $V(\cdot)$ in
$\mathcal{M}^R_\gamma$.
...
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Optimal regularity for the Signorini problem. (arXiv:0901.0421v1 [math.AP]) - We prove under general assumptions that solutions of the thin obstacle or
Signorini problem in any space dimension achieve the optimal regularity
$C^{1,1/2}$. This improves the known optimal regularity results by allowing the
thin obstacle to be defined in an arbitrary $C^{1,\beta}$ hypersurface,
$\beta>1/2$, additionally, our proof covers any linear elliptic operator in
divergence form with smooth coefficients. The main ingredients of the proof are
a version of Almgren's monotonicity formula and the optimal regularity of
global solutions.
...
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A finiteness theorem for hyperbolic 3-manifolds. (arXiv:0901.0300v1 [math.GT]) - We prove that there are only finitely many closed hyperbolic 3-manifolds with
injectivity radius and first eigenvalue of the Laplacian bounded below whose
fundamental groups can be generated by a given number of elements. An
application to arithmetic manifolds is also given.
...
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A rarefaction-tracking method for hyperbolic conservation law. (arXiv:0901.0298v1 [math.NA]) - We present a numerical method for scalar conservation laws in one space
dimension. The solution is approximated by local similarity solutions. While
traditional approaches use shocks, the presented method uses rarefaction and
compression waves. The solution is represented by particles that carry function
values and move according to the method of characteristics. Between two
neighboring particles, an interpolation is defined by an analytical similarity
solution of the conservation law. An interaction of particles represents a
collision of characteristics. The resulting shock is resolved by merging
particles so that the total area under the function is conserved. The method is
variation diminishing, nevertheless, it has no numerical dissipation away from
shocks. Although shocks are not explicitly tracked, they can be located
accurately. We present numerical examples, and outline specific applications
and extensions of the approach.
...
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Parabolic Subgroups of Real Direct Limit Groups. (arXiv:0901.0295v1 [math.RT]) - Let $G_R$ be a classical real direct limit Lie group and $g_R$ its Lie
algebra. The parabolic subalgebras of the complexification $g_C$ were described
by the first two authors. In the present paper we extend these results to
$g_R$. This also gives a description of the parabolic subgroups of $G_R$.
Furthermore, we give a geometric criterion for a parabolic subgroup $P_C$ of
$G_C$ to intersect $G_R$ in a parabolic subgroup. This criterion involves the
$G_R$-orbit structure of the flag ind-manifold $G_C/P_C$.
...
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Tensor products of irreducible representations of the group G = GL(3,q). (arXiv:0901.0292v1 [math.RT]) - We describe the tensor products of two irreducible linear complex
representations of the finite general linear group G = GL(3,q) in terms of
induced representations by linear characters of maximal torii and also in terms
of Gelfand-Graev representations. Our results include MacDonald's conjectures
for G and at the same time they are extensions to G of finite counterparts to
classical results on tensor products of holomorphic and anti-holomorphic
representations of the group SL(2, R). Moreover they provide an easy way to
decompose these tensor products, with the help of Frobenius reciprocity. We
also state some conjectures for the general case of GL(n,q).
...
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Matrices of unitary moments. (arXiv:0901.0288v1 [math.OA]) - We investigate certain matrices composed of mixed, second-order moments of
unitaries. The unitaries are taken from C*-algebras with moments taken with
respect to traces, or, alternatively, from matrix algebras with the usual
trace. These sets are of interest in light of a theorem of E. Kirchberg about
Connes' embedding problem.
...
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The Problem Of Gauge Theory. (arXiv:0812.4512v3 [math.DG] UPDATED) - I sketch what it is supposed to mean to quantize gauge theory, and how this
can be made more concrete in perturbation theory and also by starting with a
finite-dimensional lattice approximation. Based on real experiments and
computer simulations, quantum gauge theory in four dimensions is believed to
have a mass gap. This is one of the most fundamental facts that makes the
Universe the way it is. This article is the written form of a lecture presented
at the conference "Geometric Analysis: Past and Future" (Harvard University,
August 27-September 1, 2008), in honor of the 60th birthday of S.-T. Yau.
...
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Q-Fano threefolds of large Fano index, I. (arXiv:0812.1695v2 [math.AG] UPDATED) - We study Q-Fano threefolds of large Fano index. In particular, we prove that
the maximum of Fano index is attained for the weighted projective space
P(3,4,5,7).
...
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Liouville type of theorems with weights for the Navier-Stokes equations and the Euler equations. (arXiv:0811.4647v2 [math.AP] UPDATED) - We study Liouville type of theorems for the Navier-Stokes and the Euler
equations on $\Bbb R^N$, $N\geq 2$. Specifically, we prove that if a weak
solution $(v,p)$ satisfies $|v|^2 +|p| \in L^1 (0,T; L^1(\Bbb R^N, w_1(x)dx))$
and $\int_{\Bbb R^N} p(x,t)w_2 (x)dx \geq0$ for some weight functions $w_1(x)$
and $w_2 (x)$, then the solution is trivial, namely $v=0$ almost everywhere on
$\Bbb R^N \times (0, T)$. Similar results hold for the MHD Equations on $\Bbb
R^N$, $N\geq3$.
...
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Groupoids and the Brauer group. (arXiv:0811.3882v7 [math.KT] UPDATED) - Using a groupoid of matrix subalgebras in a fixed matrix algebra we define
some nonabelian bundle gerbe which probably gives rise to some new twistings in
K-theory.
...
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A cofinite universal space for proper actions for mapping class groups. (arXiv:0811.3871v2 [math.GT] UPDATED) - We prove that the mapping class group $\Gamma_{g,n}$ for surfaces of negative
Euler characteristic has a cofinite universal space $\E$ for proper actions
(the resulting quotient is a finite $CW$-complex). The approach is to construct
a truncated Teichmueller space $\T_{g,n}(\epsilon)$ by introducing a lower
bound for the length of shortest closed geodesics and showing that
$\T_{g,n}(\epsilon)$ is a $\Gamma_{g,n}$ equivariant deformation retract of the
Teichmueller space $\T_{g, n}$. The existence of such a cofinite universal
space is important in the study of the cohomology of the group $\gag$. As an
application, we note that there are only finitely many conjugacy classes of
finite subgroups of $\Gamma_{g,n}$. Another application is that the rational
Novikov conjecture in K-theory holds for $\Gamma_{g,n}$.
...
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A Thermodynamic Classification of Real Numbers. (arXiv:0811.1369v2 [math.NT] UPDATED) - A new classification scheme for real numbers is given, motivated by ideas
from statistical mechanics in general and work of Knauf and of Fiala and Kleban
in particular. Critical for this classification of a real number will be the
Diophantine properties of its continued fraction expansion.
...
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A self-regulating and patch subdivided population. (arXiv:0811.1279v2 [math.PR] UPDATED) - We consider an interacting particle process on a graph which, from a
macroscopic point of view, looks like $\Z^d$ and, at a microscopic level, is a
complete graph of degree $N$ (called a patch). There are two birth rates: an
inter-patch one $\lambda$ and an intra-patch one $\phi$. Once a site is
occupied, there is no breeding from outside the patch and the probability
$c(i)$ of success of an intra-patch breeding decreases with the size $i$ of the
population in the site. We prove the existence of a critical value
$\lambda_{cr}(\phi, c, N)$ and a critical value $\phi_{cr}(\lambda, c, N)$. We
consider a sequence of processes generated by the families of control functions
$\{c_i\}_{i \in \N}$ and degrees $\{N_i\}_{i \in \N}$; we prove, under mild
assumptions, the existence of a critical value $i_{cr}$. Roughly speaking we
show that, in the limit, these processes behave as the branching random walk on
$\Z^d$ with external birth rate $\lambda$ and internal birth rate $\phi$. Some
examples...
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On the Banach-Mazur Type for Normed Spaces. (arXiv:0812.2216v2 [math.FA] UPDATED) - In order to measure qualitative properties we introduce a notion of a type
for arbitrary normed spaces which measures the worst possible growth of partial
sums of sequences weakly converging to zero. The ideas can be traced back to
Banach and Mazur who used this type to compare the so-called linear dimension
of classical Banach spaces. As an application we compare the linear dimension
and investigate isomorphy of some classical Banach spaces.
...
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Parametric Bing and Krasinkiewicz maps: revisited. (arXiv:0812.2899v3 [math.GN] UPDATED) - Let $M$ be a complete metric $ANR$-space such that for any metric compactum
$K$ the function space $C(K,M)$ contains a dense set of Bing (resp.,
Krasinkiewicz) maps. It is shown that $M$ has the following property: If
$f\colon X\to Y$ is a perfect surjection between metric spaces, then $C(X,M)$
with the source limitation topology contains a dense $G_\delta$-subset of maps
$g$ such that all restrictions $g|f^{-1}(y)$, $y\in Y$, are Bing (resp.,
Krasinkiewicz) maps. We apply the above result to establish some mapping
theorems for extensional dimension.
...
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Classifying finite 2-nilpotent p-groups, Lie algebras and graphs: equivalent wild problems. (arXiv:0812.4158v2 [math.GR] UPDATED) - We reduce the graph isomorphism problem to 2-nilpotent p-groups isomorphism
problem and to finite 2-nilpotent Lie algebras over the ring Z/p^3 Z.
Furthermore, we show that classifying problems in categories graphs, finite
2-nilpotent p-groups, and 2-nilpotent Lie algebras over are polynomially
equivalent and wild.
...
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Closed Spaces in Cosmology. (arXiv:0812.4103v3 [gr-qc] UPDATED) - This paper deals with two aspects of relativistic cosmologies with closed
(compact and boundless) spatial sections. These spacetimes are based on the
theory of General Relativity, and admit a foliation into space sections S(t),
which are spacelike hypersurfaces satisfying the postulate of the closure of
space: each S(t) is a 3-dimensional, closed Riemannian manifold. The discussed
topics are: (1) A comparison, previously obtained, between Thurston's
geometries and Bianchi-Kantowski-Sachs metrics for such 3-manifolds is here
clarified and developed. (2) Some implications of global inhomogeneity for
locally homogeneous 3-spaces of constant curvature are analyzed from an
observational viewpoint.
...
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Universal convex coverings. (arXiv:0812.3525v2 [math.NT] UPDATED) - In every dimension $d\ge1$, we establish the existence of a positive finite
constant $v_d$ and of a subset $\mathcal U_d$ of $\mathbb R^d$ such that the
following holds: $\mathcal C+\mathcal U_d=\mathbb R^d$ for every convex set
$\mathcal C\subset \mathbb R^d$ of volume at least $v_d$ and $\mathcal U_d$
contains at most $\log(r)^{d-1}r^d$ points at distance at most $r$ from the
origin, for every large $r$.
...
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Exceptional Point and Gauge Invariance in Particle Models and Related Field Theories. (arXiv:0812.3512v2 [math-ph] UPDATED) - We propose appearance of Exceptional Point (EP) in a real parameter space, in
a novel type of Hermitian model. This is possible because the {\it{constraint
structure}} changes discontinuously at the EP leading to a coalescence of a
full tower of quantum (Harmonic Oscillator) states. We also find interesting
consequences of complexifying the parameter space. We show that this model is a
descendant of a well known relativistic field theory in 1+1-dimension- the
bosonized Chiral Schwinger Model - and in the latter the features of EP are
retained. We also show that the Cranking Model, recently studied in the context
of EP, is a descendant of another well studied relativistic field theory in
2+1-dimension- the Maxwell-Chern-Simons-Proca Model.
...
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Convergence and Monotonicity Problems in an Information-Theoretic Law of Small Numbers. (arXiv:0810.5203v2 [cs.IT] UPDATED) - A version of the law of small numbers is analyzed in information-theoretic
terms. Specifically, let $f=\{f_i, i=0, 1, \}$ be a probability mass function
(pmf) on nonnegative integers with mean $\lambda<\infty$. Denote the $n$th
convolution of $f$ by $f^{*n}$ and denote the $\alpha$-thinning of $f$ by
$T_\alpha(f)$. Then, as $n\to\infty$, the entropy $H(T_{1/n}(f^{*n}))$ tends to
$H(po(\lambda))$, where $po(\lambda)$ denotes the pmf of the Poisson
distribution with mean $\lambda$, and the relative entropy $D(T_{1/n
(f^{*n})|po(\lambda))$ tends to zero, if it ever becomes finite. Moreover,
$\alpha^{-1} D(T_\alpha(f)|po(\alpha\lambda))$ increases in $\alpha\in (0,1)$,
and $n^{-1} D(f^{*n}|po(n\lambda))$ decreases in $n=1,2, $. It follows that
$D(T_{1/n}(f^{*n})|po(\lambda))$ decreases monotonically in $n$. Furthermore,
assuming that $f$ is ultra-log-concave (i.e., log-concave relative to the
Poisson pmf), we show that $H(T_{1/n}(f^{*n}))$ increases monotonically in $n$.
This is...
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Finiteness and super-rigidity of J-holomorphic curves in symplectic three-folds. (arXiv:0810.1640v2 [math.SG] UPDATED) - This paper has been withdrawn by the author, due a crucial mistake in proof
of lemma 3.2.
...
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Dynamical deformations of 3d Lie algebras in Bianchi classification over harmonic oscillator. (arXiv:0807.0428v2 [math.RT] UPDATED) - Operadic Lax representations for the harmonic oscillator are used to
construct the dynamical deformations of 3d real Lie algebras in Bianchi
classification. It is shown that the energy conservation of the harmonic
oscillator is related to the Jacobi identities of the dynamically deformed
algebras. Based on this observation, it is proved that the dynamical
deformations of 3d real Lie algebras in Bianchi classification over the
harmonic oscillator are Lie algebras.
...
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Bounds on the Sum Capacity of Synchronous Binary CDMA Channels. (arXiv:0806.1659v2 [cs.IT] UPDATED) - In this paper, we obtain a family of lower bounds for the sum capacity of
Code Division Multiple Access (CDMA) channels assuming binary inputs and binary
signature codes in the presence of additive i.i.d. noise with an arbitrary
distribution. The envelope of this family gives a relatively tight lower bound
in terms of the number of users, spreading gain and the noise distribution. The
derivation methods for the noiseless and the noisy channels are different but
when the noise variance goes to zero, the noisy channel bound approaches the
noiseless case. The behavior of the lower bound shows that for small noise
power, the number of users can be much more than the spreading gain without any
significant loss of information (overloaded CDMA). An upper bound is also
derived under the usual assumption that the users send out equally likely
binary bits in the presence of additive i.i.d. noise with an arbitrary
distribution. As the noise level increases, and/or, the ratio of the number of
u...
Feed Source: arxiv.org
Robust Cognitive Beamforming With Partial Channel State Information. (arXiv:0806.1372v2 [cs.IT] UPDATED) - This paper considers a spectrum sharing based cognitive radio (CR)
communication system, which consists of a secondary user (SU) having multiple
transmit antennas and a single receive antenna and a primary user (PU) having a
single receive antenna. The channel state information (CSI) on the link of the
SU is assumed to be perfectly known at the SU transmitter (SU-Tx). However, due
to loose cooperation between the SU and the PU, only partial CSI of the link
between the SU-Tx and the PU is available at the SU-Tx. With the partial CSI
and a prescribed transmit power constraint, our design objective is to
determine the transmit signal covariance matrix that maximizes the rate of the
SU while keeping the interference power to the PU below a threshold for all the
possible channel realization within an uncertainty set. This problem, termed
the robust cognitive beamforming problem, can be naturally formulated as a
semi-infinite programming (SIP) problem with infinitely many constraints. Thi...
Feed Source: arxiv.org
3D binary anti-commutative operadic Lax representations for harmonic oscillator. (arXiv:0806.1349v2 [math-ph] UPDATED) - It is explained how the time evolution of the operadic variables may be
introduced by using the operadic Lax equation. The operadic Lax representations
for the harmonic oscillator are constructed in 3-dimensional binary
anti-commutative algebras. As an example, an operadic Lax representation for
the harmonic oscillator in the Lie algebra sl(2) is constructed.
...
Feed Source: arxiv.org
Positivity for toric vector bundles. (arXiv:0805.4035v2 [math.AG] UPDATED) - We show that an equivariant vector bundle on a complete toric variety is nef
or ample if and only if its restriction to every invariant curve is nef or
ample, respectively. Furthermore, we show that nef toric vector bundles have a
nonvanishing global section at every point, and deduce that the underlying
vector bundle is trivial if and only if its restriction to every invariant
curve is trivial. We apply our methods and results to study, in particular, the
vector bundles M_L that arise as the kernel of the evaluation map on sections
of L, when L is an ample line bundle. We give examples of twists of such
bundles that are ample but not globally generated.
...
Feed Source: arxiv.org
Perturbations of Matter Fields in the Second-order Gauge-invariant Cosmological Perturbation Theory. (arXiv:0804.3840v4 [gr-qc] UPDATED) - Some formulae for the perturbations of the matter fields are summarized
within the framework of the second-order gauge-invariant cosmological
perturbation theory in a four dimensional homogeneous isotropic universe, which
is developed in the papers [K.Nakamura, Prog.Theor.Phys., 117 (2007), 17.]. We
derive the formulae for the perturbations of the energy momentum tensors and
equations of motion for a perfect fluid, an imperfect fluid, and a signle
scalar field, and show that all equations are derived in terms of
gauge-invariant variables without any gauge fixing.
...
Feed Source: arxiv.org
The second largest component in the supercritical 2D Hamming graph. (arXiv:0801.1608v3 [math.PR] UPDATED) - The 2-dimensional Hamming graph H(2,n) consists of the $n^2$ vertices
$(i,j)$, $1\leq i,j\leq n$, two vertices being adjacent when they share a
common coordinate. We examine random subgraphs of H(2,n) in percolation with
edge probability $p$, so that the average degree $2(n-1)p=1+\epsilon$. Previous
work by van der Hofstad and Luczak had shown that in the barely supercritical
region $n^{-2/3}\ln^{1/3}n\ll \epsilon \ll 1$ the largest component has size
$\sim 2\epsilon n$. Here we show that the second largest component has size
close to $\epsilon^{-2}$, so that the dominant component has emerged. This
result also suggests that a {\it discrete duality principle} might hold,
whereby, after removing the largest connected component in the supercritical
regime, the remaining random subgraphs behave as in the subcritical regime.
...
Feed Source: arxiv.org
Curves without automorphisms and integral invariants of Calabi-Yau three-folds. (arXiv:0807.0492v2 [math.SG] UPDATED) - This paper has been withdrawn by the author, due a crucial mistake in proof
of lemma 4.2.
...
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Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras. (arXiv:0808.2032v3 [math.RT] UPDATED) - We construct an explicit isomorphism between blocks of cyclotomic Hecke
algebras and (sign-modified) Khovanov-Lauda algebras in type A. These
isomorphisms connect the categorification conjecture of Khovanov and Lauda to
Ariki's categorification theorem. The Khovanov-Lauda algebras are naturally
graded, which allows us to exhibit a non-trivial Z-grading on blocks of
cyclotomic Hecke algebras, including symmetric groups in positive
characteristic.
...
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On boundedness, existence and uniqueness of strong solutions of the Navier-Stokes Equations in 3 dimensions. (arXiv:0810.0318v2 [math.GM] UPDATED) - In this paper we consider the Navier-Stokes Equations in 3 dimensions in the
vorticity formulation in the absence of the external forces. We derive upper
bounds on L_{infinity} norm of omega and use them together with the Local
Existence and Uniqueness results to show Global Existence and Uniqueness of the
solution provided that at t=0, L_{infinity} norm of omega is finite, or L_4
norm of omega is finite.
...
Feed Source: arxiv.org
On isometric dilations of product systems of C*-correspondences and applications to families of contractions associated to higher-rank graphs. (arXiv:0809.4348v2 [math.OA] UPDATED) - Let E be a product system of C*-correspondences over N^r. Some sufficient
conditions for the existence of a not necessarily regular isometric dilation of
a completely contractive representation of E are established and difference
between regular and *-regular dilations discussed. It is in particular shown
that a minimal isometric dilation is *-regular if and only if it is doubly
commuting. The case of product systems associated with higher-rank graphs is
analysed in detail.
...
Feed Source: arxiv.org
How to Fully Exploit the Degrees of Freedom in the Downlink of MISO Systems With Opportunistic Beamforming. (arXiv:0809.0536v2 [cs.IT] UPDATED) - The opportunistic beamforming in the downlink of multiple-input single-output
(MISO) systems forms $N$ transmit beams, usually, no more than the number of
transmit antennas $N_t$. However, the degrees of freedom in this downlink is as
large as $N_t^2$. That is, at most $N_t^2$ rather than only $N_t$ users can be
simultaneously transmitted and thus the scheduling latency can be significantly
reduced. In this paper, we focus on the opportunistic beamforming schemes with
$N_t<N\le N_t^2$ transmit beams in the downlink of MISO systems over Rayleigh
fading channels. We first show how to design the beamforming matrices with
maximum number of transmit beams as well as least correlation between any pair
of them as possible, through Fourier, Grassmannian, and mutually unbiased bases
(MUB) based constructions in practice. Then, we analyze their system throughput
by exploiting the asymptotic theory of extreme order statistics. Finally, our
simulation results show the Grassmannian-based beam...
Feed Source: arxiv.org
Sur l'homologie des groupes orthogonaux et symplectiques \`a coefficients tordus. (arXiv:0808.4035v3 [math.AT] UPDATED) - We compute the stable homology of orthogonal and symplectic groups over a
finite field k with coefficients coming from an usual endofunctor F of k-vector
spaces (exterior, symmetric, divided powers ), that is, for all natural
integer i, we compute the colimits of the vector spaces $H_i(O_{n,n}(k) ;
F(k^{2n}))$ and $H_i(Sp_{2n}(k) ; F(k^{2n}))$. In this situation, the
stabilization is a classical result of Charney. We give a formal framework to
connect stable homology of some families of groups and homology of suitable
small categories thanks to a spectral sequence which collapses in several
cases. By our purely algebraic methods (i.e. without stable K-theory) we obtain
again results of Betley for stable homology of linear groups and symmetric
groups. For orthogonal and symplectic groups over a field we prove a
categorical result for vector spaces equipped with quadratic or alternating
forms and use powerful cancellation results known in homology of functors
(Suslin, Scorichenko, D...
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Algebraic characterization of logically defined tree languages. (arXiv:0709.2962v2 [cs.LO] UPDATED) - We give an algebraic characterization of the tree languages that are defined
by logical formulas using certain Lindstr\"om quantifiers. An important
instance of our result concerns first-order definable tree languages. Our
characterization relies on the usage of preclones, an algebraic structure
introduced by the authors in a previous paper, and of the block product
operation on preclones. Our results generalize analogous results on finite word
languages, but it must be noted that, as they stand, they do not yield an
algorithm to decide whether a given regular tree language is first-order
definable.
...
Feed Source: arxiv.org
Consistency of Equations in the Second-order Gauge-invariant Cosmological Perturbation Theory. (arXiv:0812.4865v2 [gr-qc] UPDATED) - Along the general framework of the gauge-invariant perturbation theory
developed in the papers [K. Nakamura, Prog. Theor. Phys. {\bf 110} (2003), 723;
{\it ibid}, {\bf 113} (2005), 481.], we re-derive the second-order Einstein
equations on four-dimensional homogeneous isotropic background universe in
gauge-invariant manner without ignoring any mode of perturbations. We consider
the perturbations both in the universe dominated by the single perfect fluid
and in that dominated by the single scalar field. We also confirmed the
consistency of all equations of the second-order Einstein equation and the
equations of motion for matter fields which are derived in the paper [K.
Nakamura, arXiv:0804.3840 [gr-qc]]. This confirmation implies that the all
derived equations of the second order are self-consistent and these equations
are correct in this sense.
...
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Homological index formulas for elliptic operators over C*-algebras. (arXiv:math/0603694v2 [math.KT] UPDATED) - We prove index formulas for elliptic operators acting between sections of
C*-vector bundles on a closed manifold. The formulas involve Karoubi's Chern
character from K-theory of a C*-algebra to de Rham homology of smooth
subalgebras. We show how they apply to the higher index theorem for coverings
and to flat foliated bundles, and prove an index theorem for C*-dynamical
systems associated to actions of compact Lie groups. In an Appendix we relate
the pairing of odd K-theory and KK-theory to the noncommutative spectral flow
and prove the regularity of elliptic pseudodifferential operators over
C*-algebras.
...
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An approach to non simply laced cluster algebras. (arXiv:math/0512043v4 [math.RT] UPDATED) - Let $\Delta$ be an oriented valued graph equipped with a group of admissible
automorphisms satisfying a certain stability condition. We prove that the
(coefficient-free) cluster algebra $\mathcal A(\Delta/G)$ associated to the
valued quotient graph $\Delta/G$ is a subalgebra of the quotient $\pi(\mathcal
A(\Delta))$ of the cluster algebra associated to $\Delta$ by the action of $G$.
When $\Delta$ is a Dynkin diagram, we prove that $\mathcal A(\Delta/G)$ and
$\pi(\mathcal A(\Delta))$ coincide. As an example of application, we prove that
affine valued graphs are mutation-finite, giving an alternative proof to a
result of Seven.
...
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Nonlinear Classical Fields. (arXiv:math-ph/0701054v16 UPDATED) - We regard a classical field as a medium. Then additional parameter appear. It
is the local four vector of a field velocity . If the one itself regard as
potential of same field then the self-energies of the fields became finite.
Electromagnetic, mechanical, pionic, and somewhat gluonic fields are regarding
...
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Semi-simple extension of the (super)Poincar\'e algebra. (arXiv:hep-th/0605251v4 UPDATED) - A semi-simple tensor extension of the Poincar\'e algebra is proposed for the
arbitrary dimensions $D$. A supersymmetric also semi-simple generalization of
this extension is constructed in the D=4 dimensions. This paper is dedicated to
the memory of Anna Yakovlevna Gelyukh.
...
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Graded Specht modules. (arXiv:0901.0218v2 [math.RT] UPDATED) - Recently, the first two authors have defined a Z-grading on group algebras of
symmetric groups and more generally on the cyclotomic Hecke algebras of type
G(l,1,d). In this paper we explain how to grade Specht modules over these
algebras.
...
Feed Source: arxiv.org
Dirac operators on noncommutative manifolds with boundary. (arXiv:0901.0123v2 [math.OA] UPDATED) - We study an example of an index problem for a Dirac-like operator subject to
Atiyah-Patodi-Singer boundary conditions on a noncommutative manifold with
boundary, namely the quantum unit disk.
...
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Central Schemes for Porous Media Flows. (arXiv:math/0610454v4 [math.NA] UPDATED) - We are concerned with central differencing schemes for solving scalar
hyperbolic conservation laws arising in the simulation of multiphase flows in
heterogeneous porous media. We compare the Kurganov-Tadmor, 2000 semi-discrete
central scheme with the Nessyahu-Tadmor, 1990 central scheme. The KT scheme
uses more precise information about the local speeds of propagation together
with integration over nonuniform control volumes, which contain the Riemann
fans. These methods can accurately resolve sharp fronts in the fluid
saturations without introducing spurious oscillations or excessive numerical
diffusion. We first discuss the coupling of these methods with velocity fields
approximated by mixed finite elements. Then, numerical simulations are
presented for two-phase, two-dimensional flow problems in multi-scale
heterogeneous petroleum reservoirs. We find the KT scheme to be considerably
less diffusive, particularly in the presence of high permeability flow
channels, which lead to str...
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Bounds of some real (complex) solution of a finite system of polynomial equations with rational coefficients. (arXiv:math/0702558v83 [math.AC] UPDATED) - We discuss two conjectures.
(I) For each x_1, ,x_n \in R (C) there exist y_1, ,y_n \in R (C) such
that
\forall i \in {1, ,n} |y_i| \leq 2^{2^{n-2}}
\forall i \in {1, ,n} (x_i=1 \Rightarrow y_i=1)
\forall i,j,k \in {1, ,n} (x_i+x_j=x_k \Rightarrow y_i+y_j=y_k)
\forall i,j,k \in {1, ,n} (x_i \cdot x_j=x_k \Rightarrow y_i \cdot y_j=y_k)
(II) Let G be an additive subgroup of C. Then for each x_1, ,x_n \in G
there exist y_1, ,y_n \in G \cap Q such that
\forall i \in {1, ,n} |y_i| \leq 2^{n-1}
\forall i \in {1, ,n} (x_i=1 \Rightarrow y_i=1)
\forall i,j,k \in {1, ,n} (x_i+x_j=x_k \Rightarrow y_i+y_j=y_k)
...
Feed Source: arxiv.org
Gorenstein projective dimension with respect to a semidualizing module. (arXiv:math/0611711v2 [math.AC] UPDATED) - We introduce and investigate the notion of $\gc$-projective modules over
(possibly non-noetherian) commutative rings, where $C$ is a semidualizing
module. This extends Holm and J{\o}rgensen's notion of $C$-Gorenstein
projective modules to the non-noetherian setting and generalizes projective and
Gorenstein projective modules within this setting. We then study the resulting
modules of finite $\gc$-projective dimension, showing in particular that they
admit $\gc$-projective approximations, a generalization of the maximal
Cohen-Macaulay approximations of Auslander and Buchweitz. Over a local
(noetherian) ring, we provide necessary and sufficient conditions for a
$G_C$-approximation to be minimal.
...
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On the nonexistence of stationary weak solutions to the compressible fluid equations. (arXiv:0812.4869v3 [math.AP] UPDATED) - In this paper we prove that under some integrability conditions for the
density and the velocity fields the only stationary weak solutions to the
compressible fluid equations on $\Bbb R^N$ correspond to the zero density. In
the case of compressible magnetohydrodynamics equations similar integrability
conditions for density, velocity and the magnetic fields lead to the zero
density and the zero magnetic field.
...
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Operator algebras: an informal overview. (arXiv:0901.0232v1 [math.OA] CROSS LISTED) - In this article we give a short and informal overview of some aspects of the
theory of C*- and von Neumann algebras. We also mention some classical results
and applications of these families of operator algebras.
...
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Numerical Performance of Compact Fourth Order Formulation of the Navier-Stokes Equations. (arXiv:0901.0172v1 [physics.flu-dyn] CROSS LISTED) - In this study the numerical performance of the fourth order compact
formulation of the steady 2-D incompressible Navier-Stokes equations introduced
by Erturk et al. (Int. J. Numer. Methods Fluids, 50, 421-436) will be
presented. The benchmark driven cavity flow problem will be solved using the
introduced compact fourth order formulation of the Navier-Stokes equations with
two different line iterative semi-implicit methods for both second and fourth
order spatial accuracy. The extra CPU work needed for increasing the spatial
accuracy from second order (O(x2)) to fourth order (O(x4)) formulation will be
presented.
...
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Random Current Representation for Transverse Field Ising Models. (arXiv:0812.4834v1 [math-ph] CROSS LISTED) - Recently, a random current representation for transverse field Ising models
has been introduced in \cite{ILN}. This representation is a space-time version
of the classical random current representation exploited by Aizenman et. al.
%It is a space-time version of the classical random current representation
\cite{Ai82, ABF, AF}. In this paper we formulate and prove corresponding
space-time versions of the classical switching lemma and show how they generate
various correlation inequalities. In particular we prove exponential decay of
truncated two-point functions at positive magnetic fields in $\sfz$-direction
and address the issue of the sharpness of phase transition.
...
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Multiple orthogonal polynomials, string equations and the large-n limit. (arXiv:0812.3817v2 [nlin.SI] CROSS LISTED) - The Riemann-Hilbert problems for multiple orthogonal polynomials of types I
and II are used to derive string equations associated to pairs of Lax-Orlov
operators. A method for determining the quasiclassical limit of string
equations in the phase space of the Whitham hierarchy of dispersionless
integrable systems is provided. Applications to the analysis of the large-n
limit of multiple orthogonal polynomials and their associated random matrix
ensembles and models of non-intersecting Brownian motions are given.
...
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The volume of causal diamonds, asymptotically de Sitter space-times and irreversibility. (arXiv:0812.3410v1 [hep-th] CROSS LISTED) - In this note we prove that the volume of a causal diamond associated with an
inertial observer in asymptotically de Sitter 4-dimensional space-time is
monotonically increasing function of cosmological time. The asymptotic value of
the volume is that of in maximally symmetric de Sitter space-time. The
monotonic property of the volume is checked in two cases: in vacuum and in the
presence of a massless scalar field. In vacuum, the volume flow (with respect
to cosmological time) asymptotically vanishes if and only if future space-like
infinity is 3-manifold of constant curvature. The volume flow thus represents
irreversibility of asymptotic evolution in spacetimes with positive
cosmological constant.
...
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Saint George International Learning Center - Language is the proof of human?s intellectual. Learning languages at Saint George International will improve your ability of foreign languages, such as English, Italian, Spanish, French, and German. That website focuses at business language education. London English courses are the most popular program for people who want to learn business English in London. The [ ]...
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Math is no longer scary and boring - Traditional learning is often boring for children. But learning is no longer boring at Score Educational Centre. It provides fun learning study for the children. Moreover, Score Educational Centre helps children age 4-14 to gain goals and reach academic potential in school subjects, especially math.Math is often scary and boring to children. However, the children [ ]...
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Destum Partners : Life Sciences consulting and advisory services - Destum Partners is a consulting and advisory company for the Life Science Industry. It provides recent modern technology to make an analytic service and advisory service. Their services include primary research, market analytics, strategic partnering, divestitures, and finally acquisition.
Destum Partners stated that they will begin every project by conducting a thorough market and competitive [ ]...
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Pi : a dessert that half of its height is 1.57 - Quite an old joke I heard from one TV serial long time ago (I forgot what’s the title though) :
Q : What is a dessert that half of its height is 1.57
A : A piece of pi(e).
lol… quite funny isn’t.
Well, we all know, perhaps since we’re at elementary school, that Pi or ? is [ ]...
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Find the most suitable loans for you - Purchasing a property or a car requires a lot of money and sometimes we do not have it. Our attention is turned to borrowing loans of money, but sometimes it turns out that the lenders charged a very high interest and some of them are untrustworthy. However, now you can compare personal loans offers so [ ]...
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Disabled students get bullied most in school - Bullying has always become a problem that happens among schoolmates. Although we have tried to prevent it, but it always occur no matter whether it is just in the form of jokes or serious allusions that despise a group of people. A brawl can break out if the bullied students fight back for equal treatments [ ]...
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Dyslexia students still need help - Hundreds of people are suffering from dyslexia, but they do not get any help to recover from this problem. That is what happens to students who study in Texas. Dyslexia is a learning disability that has the symptoms of difficulty in reading, pronouncing words and therefore they cannot absorb the lessons taught by the teachers. [ ]...
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Discrimination in US education system - Education system in United States is believed by many US citizens to have prejudiced attitudes towards the white, male students, especially those in public schools. Although the government has constantly reminded the citizens that they will have ?zero tolerance? for misbehavior in public schools, US citizens view that this rules apply more harshly [ ]...
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Can statistics predict the future? - Competition for resources and power between countries can result in war, which will adversely affect people in those countries. However, we sometimes see that the outcome of war is unpredictable; the more powerful countries lose to the supposedly weaker countries. A scientist in the University of Georgia then comes up with a theory that [ ]...
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Save the environment using mathematics models - Weather extremes have become a universal problem that everyone has to take part in order to prevent it from occurring. In further research to this serious issue, researchers at the Department of Energy’s Oak Ridge National Laboratory are trying to develop climate models to identify these weather extremes and their relationships with other climate extremes. [ ]...
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