arXiv subjects
Publications and source records attributed to Leonid Gurvits.
We give new lower and upper bounds on the permanent of a doubly stochastic matrix. Combined with previous work, this improves on the deterministic approximation factor for the permanent. We also give a combinatorial application of the lower bound, proving S. Friedland's "Asymptotic Lower Matching Conjecture" for the monomer-dimer problem.
The X-ray transient MAXI J1836-194 is a newly-identified Galactic black hole binary candidate. As most X-ray transients, it was discovered at the beginning of an X-ray outburst. After the initial canonical X-ray hard state, the outburst evolved into a hard intermediate state and then went back to the hard state. The existing RATAN-600 radio monitoring observations revealed that it was variable on a timescale of days and had a flat or inverted spectrum, consistent with optically thick synchrotron emission, possibly from a self-absorbed jet in the vicinity of the central compact object. We observed the transient in the hard state near the end of the X-ray outburst with the European VLBI Network (EVN) at 5 GHz and the Chinese VLBI Network (CVN) at 2.3 and 8.3 GHz. The 8.3 GHz observations were carried out at a recording rate of 2048 Mbps using the newly-developed Chinese VLBI data acquisition system (CDAS), twice higher than the recording rate used in the other observations. We successfully detected the low-declination source with a high confidence level in both observations. The source was unresolved (<=0.5 mas), which is in agreement with an AU-scale compact jet.
We examine a two-person game we call Will-Testing in which the strategy space for both players is a real number. It has no equilibrium. When an infinitely large set of players plays this in all possible pairings, there is an equilibrium for the distribution of strategies which requires all players to use different strategies. We conjecture this solution could underlie some phenomena (like pecking orders) observed in animals.
Let $A \in Ω_n$ be doubly-stochastic $n \times n$ matrix. Alexander Schrijver proved in 1998 the following remarkable inequality per(\widetilde{A}) \geq \prod_{1 \leq i,j \leq n} (1- A(i,j)); \widetilde{A}(i,j) =: A(i,j)(1-A(i,j)), 1 \leq i,j \leq n. We use the above Shrijver's inequality to prove the following lower bound: \frac{per(A)}{F(A)} \geq 1; F(A) =: \prod_{1 \leq i,j \leq n} (1- A(i,j))^{1- A(i,j)}. We use this new lower bound to prove S.Friedland's Asymptotic Lower Matching Conjecture(LAMC) on monomer-dimer problem. We use some ideas of our proof of (LAMC) to disprove [Lu,Mohr,Szekely] positive correlation conjecture. We present explicit doubly-stochastic $n \times n$ matrices $A$ with the ratio $\frac{per(A)}{F(A)} = \sqrt{2}^{n}$; conjecture that \max_{A \in Ω_n}\frac{per(A)}{F(A)} \approx (\sqrt{2})^{n} and give some examples supporting the conjecture. If true, the conjecture (and other ones stated in the paper) would imply a deterministic poly-time algorithm to approximate the permanent of $n \times n$ nonnegative matrices within the relative factor $(\sqrt{2})^{n}$. The best current such factor is $e^n$.
This paper has been withdrawn by the author as one of the coauthors needs institutional permission.
An upper bound on the ergodic capacity of {\bf MIMO} channels was introduced recently in arXiv:0903.1952. This upper bound amounts to the maximization on the simplex of some multilinear polynomial $p(λ_1,...,λ_n)$ with non-negative coefficients. Interestingly, the coefficients are subpermanents of some non-negative matrix. In general, such maximizations problems are {\bf NP-HARD}. But if say, the functional $\log(p)$ is concave on the simplex and can be efficiently evaluated, then the maximization can also be done efficiently. Such log-concavity was conjectured in arXiv:0903.1952. We give in this paper self-contained proof of the conjecture, based on the theory of {\bf H-Stable} polynomials.
We study multivariate entire functions and polynomials with non-negative coefficients. A class of {\bf Strongly Log-Concave} entire functions, generalizing {\it Minkowski} volume polynomials, is introduced: an entire function $f$ in $m$ variables is called {\bf Strongly Log-Concave} if the function $(\partial x_1)^{c_1}...(\partial x_m)^{c_m} f$ is either zero or $\log((\partial x_1)^{c_1}...(\partial x_m)^{c_m} f)$ is concave on $R_{+}^{m}$. We start with yet another point of view (via {\it propagation}) on the standard univarite (or homogeneous bivariate) {\bf Newton Inequlities}. We prove analogues of (univariate) {\bf Newton Inequlities} in the (multivariate) {\bf Strongly Log-Concave} case. One of the corollaries of our new Newton(like) inequalities is the fact that the support $supp(f)$ of a {\bf Strongly Log-Concave} entire function $f$ is discretely convex ($D$-convex in our notation). The proofs are based on a natural convex relaxation of the derivatives $Der_{f}(r_1,...,r_m)$ of $f$ at zero and on the lower bounds on $Der_{f}(r_1,...,r_m)$, which generalize the {\bf Van Der Waerden-Falikman-Egorychev} inequality for the permanent of doubly-stochastic matrices. A few open questions are posed in the final section.
Let ${\bf K} = (K_1, ..., K_n)$ be an $n$-tuple of convex compact subsets in the Euclidean space $\R^n$, and let $V(\cdot)$ be the Euclidean volume in $\R^n$. The Minkowski polynomial $V_{\bf K}$ is defined as $V_{\bf K}(λ_1, ... ,λ_n) = V(λ_1 K_1 +, ..., + λ_n K_n)$ and the mixed volume $V(K_1, ..., K_n)$ as $$ V(K_1, ..., K_n) = \frac{\partial^n}{\partial λ_1...\partial λ_n} V_{\bf K}(λ_1 K_1 +, ..., + λ_n K_n). $$ Our main result is a poly-time algorithm which approximates $V(K_1, ..., K_n)$ with multiplicative error $e^n$ and with better rates if the affine dimensions of most of the sets $K_i$ are small. Our approach is based on a particular approximation of $\log(V(K_1, ..., K_n))$ by a solution of some convex minimization problem. We prove the mixed volume analogues of the Van der Waerden and Schrijver-Valiant conjectures on the permanent. These results, interesting on their own, allow us to justify the abovementioned approximation by a convex minimization, which is solved using the ellipsoid method and a randomized poly-time time algorithm for the approximation of the volume of a convex set.
Let $p$ be a homogeneous polynomial of degree $n$ in $n$ variables, $p(z_1,...,z_n) = p(Z)$, $Z \in C^{n}$. We call such a polynomial $p$ {\bf H-Stable} if $p(z_1,...,z_n) \neq 0$ provided the real parts $Re(z_i) > 0, 1 \leq i \leq n$. This notion from {\it Control Theory} is closely related to the notion of {\it Hyperbolicity} used intensively in the {\it PDE} theory. The main theorem in this paper states that if $p(x_1,...,x_n)$ is a homogeneous {\bf H-Stable} polynomial of degree $n$ with nonnegative coefficients; $deg_{p}(i)$ is the maximum degree of the variable $x_i$, $C_i = \min(deg_{p}(i),i)$ and $$ Cap(p) = \inf_{x_i > 0, 1 \leq i \leq n} \frac{p(x_1,...,x_n)}{x_1 ... x_n} $$ then the following inequality holds $$ \frac{\partial^n}{\partial x_1... \partial x_n} p(0,...,0) \geq Cap(p) \prod_{2 \leq i \leq n} (\frac{C_i -1}{C_i})^{C_{i}-1}. $$ This inequality is a vast (and unifying) generalization of the Van der Waerden conjecture on the permanents of doubly stochastic matrices as well as the Schrijver-Valiant conjecture on the number of perfect matchings in $k$-regular bipartite graphs. These two famous results correspond to the {\bf H-Stable} polynomials which are products of linear forms. Our proof is relatively simple and ``noncomputational''; it uses just very basic properties of complex numbers and the AM/GM inequality.
R. Pemantle conjectured, and T.M. Liggett proved in 1997, that the convolution of two ultra-logconcave is ultra-logconcave. Liggett's proof is elementary but long. We present here a short proof, based on the mixed volume of convex sets.
Based on the VLBI Space Observatory Programme (VSOP) observations at 1.6 and 5 GHz, we find that the luminous high-redshift (z=3.215) quasar PKS 1402+044 (J1405+0415) has a pronounced 'core--jet' structure. The jet shows a steeper spectral index and lower brightness temperature with the increase of the distance from the core. The variation of brightness temperature is basically consistent with the shock-in-jet model. Assuming that the jet is collimated by the ambient magnetic field, we estimate the mass of the central object as ~10^9 M_sun. The upper limit of the jet proper motion of PKS 1402+044 is 0.03 mas/yr (~3c) in the east-west direction.
We derive here the Friedland-Tverberg inequality for positive hyperbolic polynomials. This inequality is applied to give lower bounds for the number of matchings in $r$-regular bipartite graphs. It is shown that some of these bounds are asymptotically sharp. We improve the known lower bound for the three dimensional monomer-dimer entropy. We present Ryser-like formulas for computations of matchings in bipartite and general graphs. Additional algorithmic applications are given.
Let $p(x_1,...,x_n) = p(X), X \in R^{n}$ be a homogeneous polynomial of degree $n$ in $n$ real variables, $e = (1,1,..,1) \in R^n$ be a vector of all ones . Such polynomial $p$ is called $e$-hyperbolic if for all real vectors $X \in R^{n}$ the univariate polynomial equation $P(te - X) = 0$ has all real roots $λ_{1}(X) \geq ... \geq λ_{n}(X)$ . The number of nonzero roots $|\{i :λ_{i}(X) \neq 0 \}|$ is called $Rank_{p}(X)$ . A $e$-hyperbolic polynomial $p$ is called $POS$-hyperbolic if roots of vectors $X \in R^{n}_{+}$ with nonnegative coordinates are also nonnegative (the orthant $R^{n}_{+}$ belongs to the hyperbolic cone) and $p(e) > 0$ . Below $\{e_1,...,e_n\}$ stands for the canonical orthogonal basis in $R^{n}$. The main results states that if $p(x_1,x_2,...,x_n)$ is a $POS$-hyperbolic (homogeneous) polynomial of degree $n$, $Rank_{p} (e_{i}) = R_i$ and $ p(x_1,x_2,...,x_n) \geq \prod_{1 \leq i \leq n} x_i ; x_i > 0, 1 \leq i \leq n , $ then the following inequality holds $$ \frac{\partial^n}{\partial x_1...\partial x_n} p(0,...,0) \geq \prod_{1 \leq i \leq n} (\frac{G_{i} -1}{G_{i}})^{G_{i} -1} (G_i = \min(R_{i}, n+1-i)) . $$ This theorem is a vast (and unifying) generalization of as the van der Waerden conjecture on the permanents of doubly stochastic matrices as well Schrijver-Valiant conjecture on the number of perfect matchings in $k$-regular bipartite graphs . The paper is (almost) self-contained, most of the proofs can be found in the {\bf Appendices}.
Consider a homogeneous polynomial $p(z_1,...,z_n)$ of degree $n$ in $n$ complex variables . Assume that this polynomial satisfies the property : \\ $|p(z_1,...,z_n)| \geq \prod_{1 \leq i \leq n} Re(z_i)$ on the domain $\{(z_1,...,z_n) : Re(z_i) \geq 0, 1 \leq i \leq n \}$ . \\ We prove that $|\frac{\partial^n}{\partial z_1...\partial z_n} p | \geq \frac{n!}{n^n}$ . Our proof is relatively short and self-contained (i.e. we only use basic properties of hyperbolic polynomials). As the van der Waerden conjecture for permanents, proved by D.I. Falikman and G.P. Egorychev, as well Bapat's conjecture for mixed discriminants, proved by the author, are particular cases of this result. We also prove so called "small rank" lower bound (in the permanents context it corresponds to sparse doubly-stochastic matrices, i.e. with small number of non-zero entries in each column). The later lower bound generalizes (with simpler proofs) recent lower bounds by A.Schrijver for the number of perfect matchings of $k$-regular bipartite graphs. We present some important algorithmic applications of the result, including a polynomial time deterministic algorithm approximating the permanent of $n \times n$ nonnegative entry-wise matrices within multiplicative factor $\frac{e^n}{n^m}$ for any fixed positive $m$ .
We show that for an m-qubit quantum system, there is a ball of radius asymptotically approaching kappa 2^{-gamma m} in Frobenius norm, centered at the identity matrix, of separable (unentangled) positive semidefinite matrices, for an exponent gamma = (1/2)((ln 3/ln 2) - 1), roughly .29248125. This is much smaller in magnitude than the best previously known exponent, from our earlier work, of 1/2. For normalized m-qubit states, we get a separable ball of radius sqrt(3^(m+1)/(3^m+3)) * 2^{-(1 + γ)m}, i.e. sqrt{3^{m+1}/(3^m+3)}\times 6^{-m/2} (note that κ= \sqrt{3}), compared to the previous 2 * 2^{-3m/2}. This implies that with parameters realistic for current experiments, NMR with standard pseudopure-state preparation techniques can access only unentangled states if 36 qubits or fewer are used (compared to 23 qubits via our earlier results). We also obtain an improved exponent for m-partite systems of fixed local dimension d_0, although approaching our earlier exponent as d_0 approaches infinity.
Let $p(x_1,...,x_n) =\sum_{(r_1,...,r_n) \in I_{n,n}} a_{(r_1,...,r_n)} \prod_{1 \leq i \leq n} x_{i}^{r_{i}}$ be homogeneous polynomial of degree $n$ in $n$ real variables with integer nonnegative coefficients. The support of such polynomial $p(x_1,...,x_n)$ is defined as $supp(p) = \{(r_1,...,r_n) \in I_{n,n} : a_{(r_1,...,r_n)} \neq 0 \}$ . The convex hull $CO(supp(p))$ of $supp(p)$ is called the Newton polytope of $p$ . We study the following decision problems, which are far-reaching generalizations of the classical perfect matching problem : {itemize} {\bf Problem 1 .} Consider a homogeneous polynomial $p(x_1,...,x_n)$ of degree $n$ in $n$ real variables with nonnegative integer coefficients given as a black box (oracle) . {\it Is it true that $(1,1,..,1) \in supp(p)$ ?} {\bf Problem 2 .} Consider a homogeneous polynomial $p(x_1,...,x_n)$ of degree $n$ in $n$ real variables with nonnegative integer coefficients given as a black box (oracle) . {\it Is it true that $(1,1,..,1) \in CO(supp(p))$ ?} {itemize} We prove that for hyperbolic polynomials these two problems are equivalent and can be solved by deterministic polynomial-time oracle algorithms . This result is based on a "hyperbolic" generalization of Rado theorem .
It is proved that the roots of combinations of matrix polynomials with real roots can be recast as eigenvalues of combinations of real symmetric matrices, under certain hypotheses. The proof is based on recent solution of the Lax conjecture. Several applications and corollaries, in particular concerning hyperbolic matrix polynomials, are presented.
We prove that the mixed discriminant of doubly stochastic $n$-tuples of semidefinite hermitian $n \times n$ matrices is bounded below by $\frac{n!}{n^{n}}$ and that this bound is uniquely attained at the $n$-tuple $(\frac{1}{n} I,...,\frac{1}{n} I)$. This result settles a conjecture posed by R. Bapat in 1989. We consider various generalizations and applications of this result.