On a question of D. Serre

2020 ◽  
Vol 26 ◽  
pp. 97
Author(s):  
Luigi De Rosa ◽  
Riccardo Tione
Keyword(s):  
Convex Function ◽  
Negative Answer ◽  
Henri Poincaré ◽  
The Matrix ◽  
Divergence Free ◽  
Matrix Fields ◽  
Open Question

In this paper we give a negative answer to the question posed in D. Serre (Ann. Inst. Henri Poincaré C Anal. Non linéaire 35 (2018) 1209–1234, Open Question 2.1) about possible gains of integrability of determinants of divergence-free, non-negative definite matrix-fields. We also analyze the case in which the matrix-field is given by the Hessian of a convex function.

scholarly journals IDEAL PROJECTIONS AND FORCING PROJECTIONS

2014 ◽  
Vol 79 (4) ◽  
pp. 1247-1285 ◽  
Author(s):  
SEAN COX ◽  
MARTIN ZEMAN
Keyword(s):  
Set Theory ◽  
Negative Answer ◽  
Large Cardinals ◽  
Normal Ideal ◽  
List Type ◽  
Type Number ◽  
Inner Model ◽  
Image Position ◽  
Open Question ◽  
Woodin Cardinal

AbstractIt is well known that saturation of ideals is closely related to the “antichain-catching” phenomenon from Foreman–Magidor–Shelah [10]. We consider several antichain-catching properties that are weaker than saturation, and prove:(1)If${\cal I}$is a normal ideal on$\omega _2 $which satisfiesstationary antichain catching, then there is an inner model with a Woodin cardinal;(2)For any$n \in \omega $, it is consistent relative to large cardinals that there is a normal ideal${\cal I}$on$\omega _n $which satisfiesprojective antichain catching, yet${\cal I}$is not saturated (or even strong). This provides a negative answer to Open Question number 13 from Foreman’s chapter in the Handbook of Set Theory ([7]).

Hierarchical Divergence-Free Bases and Their Application to Particulate Flows

2003 ◽  
Vol 70 (1) ◽  
pp. 44-49 ◽  
Author(s):  
V. Sarin ◽  
A. H. Sameh
Keyword(s):  
Stokes Equations ◽  
Equations Of Motion ◽  
Navier Stokes ◽  
Particulate Flow ◽  
Particulate Flows ◽  
Navier Stokes Equations ◽  
Flow Problems ◽  
Rigid Particles ◽  
The Matrix ◽  
Divergence Free

The paper presents an algebraic scheme to construct hierarchical divergence-free basis for velocity in incompressible fluids. A reduced system of equations is solved in the corresponding subspace by an appropriate iterative method. The basis is constructed from the matrix representing the incompressibility constraints by computing algebraic decompositions of local constraint matrices. A recursive strategy leads to a hierarchical basis with desirable properties such as fast matrix-vector products, a well-conditioned reduced system, and efficient parallelization of the computation. The scheme has been extended to particulate flow problems in which the Navier-Stokes equations for fluid are coupled with equations of motion for rigid particles suspended in the fluid. Experimental results of particulate flow simulations have been reported for the SGI Origin 2000.

Analyticity and metric transitivity on the torus

1998 ◽  
Vol 18 (3) ◽  
pp. 717-723
Author(s):  
SOL SCHWARTZMAN
Keyword(s):  
Vector Field ◽  
Periodic Orbit ◽  
Differential Equations ◽  
Entire Plane ◽  
Transitive Flow ◽  
Divergence Free Vector ◽  
Divergence Free ◽  
Open Question ◽  
Set Of Points

Suppose we are given an analytic divergence free vector field $(X,Y)$ on the standard torus. We can find constants $a$ and $b$ and a function $F(x,y)$ of period one in both $x$ and $y$ such that $(X,Y)=(a-F_y,b+F_x)$. For a given $F$, let $P$ be the map sending $(x,y)$ into $(F_y(x,y),-F_x(x,y))$. Let $A$ be the image of the torus under this map and let $B$ be the image under this map of the set of points $(x,y)$ at which $F_{xx}F_{yy}-(F_{xy})^2$ vanishes. For any point $(a,b)$ in the complement of the interior of $A$, the flow on the torus arising from the differential equations $dx/dt=a-F_y(x,y)$, $dy/dt=b+F_x(x,y)$ is metrically transitive if and only if $a/b$ is irrational. For any point in $A$ but not in $B$ the flow is not metrically transitive. Moreover, if $a/b$ is irrational but the flow on the torus is not metrically transitive and we use our differential equations to define a flow in the entire plane (rather than on the torus), this flow has a nonstationary periodic orbit. It is an open question whether a point $(a,b)$ in the interior of $A$ can give rise to a metrically transitive flow.

scholarly journals Subcellular localization and function of melanogenic enzymes in the ink gland of Sepia officinalis

1997 ◽  
Vol 323 (3) ◽  
pp. 749-756 ◽  
Author(s):  
Anna PALUMBO ◽  
Anna DI COSMO ◽  
Ida GESUALDO ◽  
Vincent J. HEARING
Keyword(s):  
Endoplasmic Reticulum ◽  
Subcellular Localization ◽  
Rough Endoplasmic Reticulum ◽  
Polyclonal Antibodies ◽  
Sepia Officinalis ◽  
Complex Organization ◽  
The Matrix ◽  
And Function ◽  
Golgi Network ◽  
Open Question

The ink gland of the cuttlefish Sepia officinalis has traditionally been regarded as a convenient model system for investigating melanogenesis. This gland has been shown to contain a variety of melanogenic enzymes including tyrosinase, a dopachrome-rearranging enzyme and peroxidase. However, whether and to what extent these enzymes co-localize in the melanogenic compartments and interact is an open question. Using polyclonal antibodies that recognize the corresponding Sepia proteins, we have been able to demonstrate that peroxidase has a different subcellular localization pattern from tyrosinase and dopachrome-rearranging enzyme. Whereas peroxidase is located in the rough endoplasmic reticulum and in the matrix of premelanosomes and melanosomes, tyrosinase and dopachrome-rearranging enzyme are present in the rough endoplasmic reticulum–Golgi transport system, at the level of trans-Golgi cisternae, trans-Golgi network and coated vesicles, and in melanosomes on pigmented granules. These results fill a longstanding gap in our knowledge of the melanin-producing system in Sepia and provide the necessary background for dissection at the molecular level of the complex interaction between melanogenic enzymes. Moreover, the peculiar and complex organization of melanin in an invertebrate such as Sepia officinalis is surprising and could provide the basis for understanding the process in more evolved systems such as that of mammals.

scholarly journals A linear complementarity problem involving a subgradient

1988 ◽  
Vol 37 (3) ◽  
pp. 345-351 ◽  
Author(s):  
J. Parida ◽  
A. Sen ◽  
A. Kumar
Keyword(s):  
Convex Function ◽  
Linear Complementarity Problem ◽  
Complementarity Problem ◽  
Nonlinear Complementarity Problem ◽  
Linear Complementarity ◽  
Existence Of A Solution ◽  
Nonlinear Complementarity ◽  
The Matrix ◽  
Square Matrix

A linear complementarity problem, involving a given square matrix and vector, is generalised by including an element of the subdifferential of a convex function. The existence of a solution to this nonlinear complementarity problem is shown, under various conditions on the matrix. An application to convex nonlinear nondifferentiable programs is presented.

scholarly journals The last problem of Harald Bohr

1989 ◽  
Vol 47 (1) ◽  
pp. 133-152 ◽  
Author(s):  
Jean-Pierre Kahane
Keyword(s):  
Dirichlet Series ◽  
Negative Answer ◽  
Lindelöf Hypothesis ◽  
Open Question

AbstractThe first and last papers of Harald Bohr deal with ordinary Dirichlet series and their order (or Lindelöf) function μ(σ) (= inf{α;f(σ + it) + 0(|t|α)}). The Lindelöf hypothesis is μ(σ) = inf(0, ½ − t) when an = (−1)n. Are there ordinary Dirichlet series with −l < μ′ (σ) < 0 for some σ? A negative answer would imply Lindelöf's hypothesis. This is the last problem of Harald Bohr. This paper gives (1) a review on Bohr's results on ordinary Dinchlet series; (2) a review on results of the author and of Queffelec on “almost sure” and “quasi sure” properties of series with the solution of a previous problem of Bohr; (3) the following answer to the last problem: μ′(σ) can approach − ½, and necessarily μ(σ + μ(σ) + ½) = 0. The characterization of the order functions of ordinary Dirichlet series remains an open question.

scholarly journals RANK LOGIC IS DEAD, LONG LIVE RANK LOGIC!

2019 ◽  
Vol 84 (1) ◽  
pp. 54-87
Author(s):  
ERICH GRÄDEL ◽  
WIED PAKUSA
Keyword(s):  
Finite Fields ◽  
Polynomial Time ◽  
Expressive Power ◽  
Matrix Rank ◽  
Original Definition ◽  
The Matrix ◽  
Prime Fields ◽  
Solvability Problem ◽  
Image Position ◽  
Open Question

AbstractMotivated by the search for a logic for polynomial time, we study rank logic (FPR) which extends fixed-point logic with counting (FPC) by operators that determine the rank of matrices over finite fields. WhileFPRcan express most of the known queries that separateFPCfromPtime, almost nothing was known about the limitations of its expressive power.In our first main result we show that the extensions ofFPCby rank operators over different prime fields are incomparable. This solves an open question posed by Dawar and Holm and also implies that rank logic, in its original definition with a distinct rank operator for every field, fails to capture polynomial time. In particular we show that the variant of rank logic${\text{FPR}}^{\text{*}}$with an operator that uniformly expresses the matrix rank over finite fields is more expressive thanFPR.One important step in our proof is to consider solvability logicFPSwhich is the analogous extension ofFPCby quantifiers which express the solvability problem for linear equation systems over finite fields. Solvability logic can easily be embedded into rank logic, but it is open whether it is a strict fragment. In our second main result we give a partial answer to this question: in the absence of counting, rank operators are strictly more expressive than solvability quantifiers.

scholarly journals Klein's Trace Inequality and Superquadratic Trace Functions

2020 ◽  
Author(s):  
Mohsen Kian ◽  
Mohammad W. Alomari
Keyword(s):  
Convex Function ◽  
Matrix Algebra ◽  
Trace Inequality ◽  
Superquadratic Function ◽  
The Matrix ◽  
Superquadratic Functions ◽  
Positive Matrices

We show that if $f$ is a non-negative superquadratic function, then $A\mapsto\mathrm{Tr}f(A)$ is a superquadratic function on the matrix algebra. In particular, \begin{align*} \tr f\left( {\frac{{A + B}}{2}} \right) +\tr f\left(\left| {\frac{{A - B}}{2}}\right|\right) \leq \frac{{\tr {f\left( A \right)} + \tr {f\left( B \right)} }}{2} \end{align*} holds for all positive matrices $A,B$. In addition, we present a Klein's inequality for superquadratic functions as $$ \mathrm{Tr}[f(A)-f(B)-(A-B)f'(B)]\geq \mathrm{Tr}[f(|A-B|)] $$ for all positive matrices $A,B$. It gives in particular improvement of Klein's inequality for non-negative convex function. As a consequence, some variants of the Jensen trace inequality for superquadratic functions have been presented.

scholarly journals Multivariate semi-markov matrices

1972 ◽  
Vol 13 (1) ◽  
pp. 107-113 ◽  
Author(s):  
Marcel F. Neuts ◽  
Peter Purdue
Keyword(s):  
Probability Theory ◽  
Probability Distribution ◽  
Convex Function ◽  
Maximal Eigenvalue ◽  
Partial Derivatives ◽  
The Matrix ◽  
Stieltjes Transforms ◽  
Quantities Of Interest

Finite matrices with entries pij Fij (x1,…, xk), where {pij} is stochastic and Fij(.) is a k-variate probability distribution are discussed. It is shown that the matrix of k-variate Laplace-Stieltjes transforms of the Pij Fij(x1, …, xk) has a Perron-Frobenius eigenvalue which is a convex function in k variables in a suitably defined region. The values of the partial derivatives near the origin of this maximal eigenvalue are exhibited. They are quantities of interest in a variety of applications in Probability theory.

scholarly journals Positive Solutions of Transport Equations and Classical Nonuniqueness of Characteristic curves

2021 ◽  
Vol 240 (2) ◽  
pp. 1055-1090 ◽  
Author(s):  
Elia Brué ◽  
Maria Colombo ◽  
Camillo De Lellis
Keyword(s):  
Positive Solutions ◽  
Continuity Equation ◽  
Vector Fields ◽  
Superposition Principle ◽  
Negative Answer ◽  
New Class ◽  
Open Question ◽  
Invent Math ◽  
Lagrangian Flow ◽  
Selection Of

AbstractThe seminal work of DiPerna and Lions (Invent Math 98(3):511–547, 1989) guarantees the existence and uniqueness of regular Lagrangian flows for Sobolev vector fields. The latter is a suitable selection of trajectories of the related ODE satisfying additional compressibility/semigroup properties. A long-standing open question is whether the uniqueness of the regular Lagrangian flow is a corollary of the uniqueness of the trajectory of the ODE for a.e. initial datum. Using Ambrosio’s superposition principle, we relate the latter to the uniqueness of positive solutions of the continuity equation and we then provide a negative answer using tools introduced by Modena and Székelyhidi in the recent groundbreaking work (Modena and Székelyhidi in Ann PDE 4(2):38, 2018). On the opposite side, we introduce a new class of asymmetric Lusin–Lipschitz inequalities and use them to prove the uniqueness of positive solutions of the continuity equation in an integrability range which goes beyond the DiPerna–Lions theory.

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