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This paper investigates the existence of $m$-stiff configurations in the unit sphere $S^{d-1}$, which are spherical $(2m-1)$-designs that lie on $m$ parallel hyperplanes. We establish two non-existence results: (1) for each fixed integer $m > 5$, there exists no $m$-stiff configuration in $S^{d-1}$ for sufficiently large $d$; (2) for each fixed integer $d > 10$, there exists no $m$-stiff configuration in $S^{d-1}$ for sufficiently large $m$. Furthermore, we provide a complete classification of the dimensions where $m$-stiff configurations exist for $m=2,3,4,5$. We also determine the non-existence (and the existence) of $m$-stiff configurations in $S^{d-1}$ for small $d$ ($3 \leq d \leq 120$) with arbitrary $m$, and also for small $m$ ($6 \leq m \leq 10$) with arbitrary $d$. Finally, we conjecture that there is no $m$-stiff configuration in $S^{d-1}$ for $(d,m)$ with $d\geq 3$ and $m\geq 6$.

Maximin share (MMS) allocations are a popular relaxation of envy-free allocations that have received wide attention in the fair division of indivisible items. Although MMS allocations of goods can fail to exist, previous work has found conditions under which they exist. Specifically, MMS allocations of goods exist whenever $m \leq n+5$, and this bound is tight in the sense that they can fail to exist when $m = n+6$. The techniques used to obtain these results do not apply to the mixed manna setting, leaving the question of whether similar results hold for the general setting. This paper addresses this by introducing new techniques to handle these settings. In particular, we are able to answer this question completely for the chores setting, and partially for the mixed manna setting. An agent $i$ is a {\em chores agent} if it considers every item to be a chore and a {\em non-negative agent} if its MMS guarantee is non-negative. In this paper, we prove that an MMS allocation exists as long as $m \leq n+5$ and either (i) every agent is a chores agent, or (ii) there exists a non-negative agent. In addition, for $n \leq 3$, we also prove that an MMS allocation exists as long as $m \leq

In this paper we study the existence and non-existence of minimizers for a type of (critical) Poincaré-Sobolev inequalities. We show that minimizers do exist for smooth domains in $\mathbb{R}^d$, an also for some polyhedral domains. On the other hand, we prove the non-existence of minimizers in the rectangular isosceles triangle in $\mathbb{R}^2$.

This paper considers the foundational question of the existence of a fundamental (resp. essential) matrix given $m$ point correspondences in two views. We present a complete answer for the existence of fundamental matrices for any value of $m$. Using examples we disprove the widely held beliefs that fundamental matrices always exist whenever $m \leq 7$. At the same time, we prove that they exist unconditionally when $m \leq 5$. Under a mild genericity condition, we show that an essential matrix always exists when $m \leq 4$. We also characterize the six and seven point configurations in two views for which all matrices satisfying the epipolar constraint have rank at most one.

We consider the problem of existence of constrained minimizers for the focusing mass-subcritical Half-Wave equation with a defocusing mass-subcritical perturbation. We show the existence of a critical mass such that minimizers do exist for any mass larger than or equal to the critical one, and do not exist below it. At the dynamical level, in the one dimensional case, we show that the ground states are orbitally stable.

A mathematical model for collision-induced breakage is considered. Existence of weak solutions to the continuous nonlinear collision-induced breakage equation is shown for a large class of unbounded collision kernels and daughter distribution functions, assuming the collision kernel $K$ to be given by $K(x,y)= x^α y^β + x^β y^α$ with $α\le β\le 1$. When $α+ β\in [1,2]$, it is shown that there exists at least one weak mass-conserving solution for all times. In contrast, when $α+ β\in [0,1)$ and $α\ge 0$, global mass-conserving weak solutions do not exist, though such solutions are constructed on a finite time interval depending on the initial condition. The question of uniqueness is also considered. Finally, for $α<0$ and a specific daughter distribution function, the non-existence of mass-conserving solutions is also established.

Let $(X,\mathcal{B},μ)$ be a standard probability space. We give new fundamental results determining solutions to the coboundary equation: \begin{eqnarray*} f = g - g \circ T \end{eqnarray*} where $f \in L^p$ and $T$ is ergodic invertible measure preserving on $(X, \mathcal{B}, μ)$. We extend previous results by showing for any measurable $f$ that is non-zero on a set of positive measure, the class of measure preserving $T$ with a measurable solution $g$ is meager (including the case where $\int_X f dμ= 0$). From this fact, a natural question arises: given $f$, does there always exist a solution pair $T$ and $g$? In regards to this question, our main results are: (i) Given measurable $f$, there exists an ergodic invertible measure preserving transformation $T$ and measurable function $g$ such that $f(x) = g(x) - g(Tx)$ for a.e. $x\in X$, if and only if $\int_{f > 0} f dμ= - \int_{f < 0} f dμ$ (whether finite or $\infty$). (ii) Given mean-zero $f \in L^p$ for $p \geq 1$, there exists an ergodic invertible measure preserving $T$ and $g \in L^{p-1}$ such that $f(x) = g(x) - g( Tx )$ for a.e. $x \in X$. (iii) In some sense, the previous existence result is the best possible. For

We use semi-analytic techniques to evaluate the burst sensitivity of designs for the EXIST hard X-ray survey mission. Applying these techniques to the mission design proposed for the Beyond Einstein program, we find that with its very large field-of-view and faint gamma-ray burst detection threshold, EXIST will detect and localize approximately two bursts per day, a large fraction of which may be at high redshift. We estimate that EXIST's maximum sensitivity will be ~4 times greater than that of Swift's Burst Alert Telescope. Bursts will be localized to better than 40 arcsec at threshold, with a burst position as good as a few arcsec for strong bursts. EXIST's combination of three different detector systems will provide spectra from 3 keV to more than 10 MeV. Thus, EXIST will enable a major leap in the understanding of bursts, their evolution, environment, and utility as cosmological probes.

The synergy of GLAST and the proposed EXIST mission as the Black Hole Finder Probe in the Beyond Einstein Program is remarkable. With its full-sky per orbit hard X-ray imaging (3-600 keV) and "nuFnu" sensitivity comparable to GLAST, EXIST could measure variability and spectra of Blazars in the hard X-ray synchrotron component simultaneous with GLAST (~10-100GeV) measures of the inverse Compton component, thereby uniquely constraining intrinsic source spectra and allowing measured high energy spectral breaks to measure the cosmic diffuse extra-galactic background light (EBL) by determining the intervening diffuse IR photon field required to yield the observed break from photon-photon absorption. Such studies also constrain the physics of jets (and parameters and indeed the validity of SSC models) and the origin of the >100 MeV gamma-ray diffuse background likely arising from Blazars and jet-dominated sources. An overview of the EXIST mission, which could fly in the GLAST era, is given together with a synopsis of other key synergies of GLAST-EXIST science.

This paper asks, "Do classics exist in megaproject management?" We identify three types of classic texts: conventional, Kuhnian, and citation classics. We find that the answer to our question depends on the definition of "classic" employed. First, "citation classics" do exist in megaproject management, and they perform remarkably well when compared to the rest of the management literature. A preliminary Top Ten of citation classics is presented. Second, there is no indication that "conventional classics" exist in megaproject management, i.e., texts recognized as definitive by a majority of experts. Third, there is also no consensus as to whether "Kuhnian classics" exist, i.e., texts with paradigmatic clout. The importance of classics seems to be accepted, however, just as work to develop, discuss, and consolidate classics is seen as essential by megaproject scholars. A set of guidelines is presented for developing classics in megaproject management research.

The problem considered in this paper is to find when the non-central Wishart distribution, defined on the cone $\bar{\mathcal{P}_d}$ of semi positive definite matrices of order $d$ and with a real valued shape parameter, exists. We reduce this problem to the problem of existence of the measures $m(n,k,d)$ defined on $\bar{\mathcal{P}_d}$ and with Laplace transform $(\det s)^{-n/2}\exp \tr(s^{-1}w)$ where $n$ is an integer and where $w=\mathrm{diag}(0,...,0,1,...,1)$ has order $d$ and rank $k.$ We compute $m(d-1,d,d)$ and we show that neither $m(d-2,d,d)$ nor $m(d-2,d-1,d)$ exist. This proves a conjecture of E. Mayerhofer.

In this paper we study existence and uniqueness of rational normal curves in $\PP^n$ passing through $p$ points and intersecting $l$ codimension two linear spaces in $n-1$ points each. If $p+l=n+3$ and the points and the linear spaces are generic, one expects the curve to exist, but this is not always the case. Our main result precisely describes in which cases the curve exists and in which it does not exist.

Based on holographic arguments Tanaka and Emparan et al have claimed that large localized static black holes do not exist in the one-brane Randall-Sundrum model. If such black holes are time-dependent as they propose, there are potentially significant phenomenological and theoretical consequences. We revisit the issue, arguing that their reasoning does not take into account the strongly coupled nature of the holographic theory. We claim that static black holes with smooth metrics should indeed exist in these theories, and give a simple example. However, although the existence of such solutions is relevant to exact and numerical solution searches, such static solutions might be dynamically unstable, again leading to time dependence with phenomenological consequences. We explore a plausible instability, suggested by Tanaka, analogous to that of Gregory and Laflamme, but argue that there is no reliable reason at this point to assume it must exist.

We show that the quantum superluminal communication based on the quantum nonlocal influence must exist if a basic principle of science is still valid, which states that if we have demonstrated the existence of something real, we can confirm its existence.

Let $G$ be a complex affine algebraic group and $H, F \subset G$ be closed subgroups. The homogeneous space $G / H$ can be equipped with structure of a smooth quasiprojective variety. The situation is different for double coset varieties $\dcosets{F}{G}{H}$. In this paper we give examples showing that the variety $\dcosets{F}{G}{H}$ does not necessarily exist. We also address the question of existence of $\dcosets{F}{G}{H}$ in the category of constructible spaces and show that under sufficiently general assumptions $\dcosets{F}{G}{H}$ does exist as a constructible space.

Consider a given space, e.g., the Euclidean plane, and its decomposition into Voronoi regions induced by given sites. It seems intuitively clear that each point in the space belongs to at least one of the regions, i.e., no neutral region can exist. As simple counterexamples show this is not true in general, but we present a simple necessary and sufficient condition ensuring the non-existence of a neutral region. We discuss a similar phenomenon concerning recent variations of Voronoi diagrams called zone diagrams, double zone diagrams, and (double) territory diagrams. These objects are defined in a somewhat implicit way and they also induce a decomposition of the space into regions. In several works it was claimed without providing a proof that some of these objects induce a decomposition in which a neutral region must exist. We show that this assertion is true in a wide class of cases but not in general. We also discuss other properties related to the neutral region, among them a one related to the concentration of measure phenomenon.

Since the pioneering work of Ghys, Langevin and Walczak among others, it has been known that several methods of dynamical systems theory can be adopted to study of foliations. Our aim in this paper is to investigate complexity of foliations, by generalising existence problem of time averages in dynamical systems theory to foliations: It has recently been realised that a positive Lebesgue measure set of points without time averages only appears for complicated dynamical systems, such as dynamical systems with heteroclinic connections or homoclinic tangencies. In this paper, we introduce the concept of length averages to singular foliations, and attempt to collect interesting examples with/without length averages. In particular, we demonstrate that length averages exist everywhere for any codimension one $\mathscr{C}^1 $ orientable singular foliation without degenerate singularities on a compact surface under a mild condition on quasi-minimal sets of the foliation, which is in strong contrast to time averages of surface flows.

The recent hard X-ray surveys performed by INTEGRAL and Swift have started to reveal the demographics of compact sources including Super-Massive Black Holes hosted in AGNs and have proven invaluable in tracking explosive events as the death of massive stars revealed by Gamma-Ray Bursts up to cosmological distances. Whereas the observations have contributed significantly to our understanding of the sources populations in the Local Universe, it has also become evident that revealing the processes that drive the birth and evolution of the first massive stars and galaxies would have required a further big step in both sensitivity and capability to study transient phenomena since their very beginning and covering different wavebands simultaneously. Therefore, after its decennial history as a proposed hard X-ray survey mission, EXIST has now turned into a new, more advanced concept with three instruments on board covering the IR/optical and X-ray/soft gamma-ray bands. The EXIST new design (Grindlay 2009a) is therefore much improved in its capability for prompt study of GRBs (with autonomous determination of the redshift for many of them) and broadband spectral studies of SMBHs and transi

The Energetic X-ray Imaging Survey Telescope (EXIST) mission concept is optimized for study of high-z GRBs as probes of the early Universe. With a High Energy Telescope (HET) incorporating a 4.5m^2 5-600keV (CZT; 0.6mm pixels) detector plane for coded aperture imaging a 90deg x 70deg (>10% coding fraction) field of view with 2' resolution and <20" (90% conf.) positions for >5 sigma sources, EXIST will perform rapid (<200sec) slews onto GRBs. Prompt images and spectra are obtained with a co-aligned soft X-ray telescope (SXI; 0.1 - 10keV) and with a 1.1m optical-IR telescope (IRT) simultaneously in 4 bands (0.3 - 0.52micron, 0.52 - 0.9micron, 0.9 - 1.38micron, and 1.38 - 2.3micron). An initial image (100s) will yield prompt identification within the HET error circle from a <2" prompt SXI position; or from VIS vs. IR dropouts or variability. An autonomous spacecraft re-point (<30") will then put the GRB on a 0.3" x 4" slit for either R = 3000 (for AB <21) or R =30 (for AB ~21-25) prompt spectra over the 0.3 - 0.9 micron and 0.9 - 2.3 micron bands. This will provide onboard redshifts within ~500-2000sec for most GRBs, reaching z ~20 (for Lyman-alpha breaks) if such

The Energetic X-ray Imaging Survey Telescope (EXIST) is a proposed next generation multi-wavelength survey mission. The primary instrument is a High Energy telescope (HET) that conducts the deepest survey for Gamma-ray Bursts (GRBs), obscured-accreting and dormant Supermassive Black Holes and Transients of all varieties for immediate followup studies by the two secondary instruments: a Soft X-ray Imager (SXI) and an Optical/Infrared Telescope (IRT). EXIST will explore the early Universe using high redshift GRBs as cosmic probes and survey black holes on all scales. The HET is a coded aperture telescope employing a large array of imaging CZT detectors (4.5 m^2, 0.6 mm pixel) and a hybrid Tungsten mask. We review the current HET concept which follows an intensive design revision by the HET imaging working group and the recent engineering studies in the Instrument and Mission Design Lab at the Goddard Space Flight Center. The HET will locate GRBs and transients quickly (<10-30 sec) and accurately (< 20") for rapid (< 1-3 min) onboard followup soft X-ray and optical/IR (0.3-2.2 micron) imaging and spectroscopy. The broad energy band (5-600 keV) and the wide field of view (~90