We introduce the triangulant of two matrices, and relate it to the existence of orthogonal eigenvectors. We also use it for a new characterization of mutually unbiased bases. Generalizing the notion, we introduce higher order triangulants of two matrices, and relate them to the existence of nontrivially intersecting invariant subspaces of complementary dimensions.

Integral representations of quantum relative entropy, and of the directional second and higher order derivatives of von Neumann entropy, are established, and used to give simple proofs of fundamental, known data processing inequalities: the Holevo bound on the quantity of information transmitted by a quantum communication channel, and, much more generally, the monotonicity of quantum relative entropy under trace-preserving positive linear maps – complete positivity of the map need not be assumed. The latter result was first proved by Müller-Hermes and Reeb, based on work of Beigi. For a simple application of such monotonicities, we consider any `divergence' that is non-increasing under quantum measurements, such as the concavity of von Neumann entropy, or various known quantum divergences. An elegant argument due to Hiai, Ohya, and Tsukada is used to show that the infimum of such a `divergence' on pairs of quantum states with prescribed trace distance is the same as the corresponding infimum on pairs of binary classical states. Applications of the new integral formulae to the general probabilistic model of information theory, and a related integral formula for the classical Rényi divergence, are also discussed.

We investigate whether certain non-classical communication channels can be simulated by a classical channel with a given number of states and a given ‘amount’ of noise. It is proved that any noisy quantum channel can be simulated by a corresponding classical channel with ‘the same amount’ of noise. Classical simulations of general probabilistic channels are also studied.

For a classical channel, neither the Shannon ca-pa-city, nor the sum of conditional probabilities corresponding to the cases of successful transmission can be increased by the use of shared entanglement, or, more generally, a non-signaling resource. Yet, perhaps somewhat counterintuitively, entanglement assistance can help and actually elevate the chances of success even in a one-way communicational task that is to be completed by a single-shot use of a noiseless classical channel.To quantify the help that a non-signaling resource provides to a noiseless classical channel, one might ask how many extra letters should be added to the alphabet of the channel in order to perform equally well without the specified non-signaling resource. As was observed by Cubitt, Leung, Matthews, and Winter, there is no upper bound on the number of extra letters required for substituting the assistance of a general non-signaling resource to a noiseless one-bit classical channel. In contrast, here we prove that if this resource is a bipartite quantum system in a maximally entangled state, then an extra classical bit always suffices as a replacement.

Let [Formula: see text] be a fixed prime, and let [Formula: see text] stand for the exponent of [Formula: see text] in the prime factorization of the integer [Formula: see text]. Let [Formula: see text] and [Formula: see text] be two monic polynomials with integer coefficients and nonzero resultant [Formula: see text]. Write [Formula: see text] for the maximum of [Formula: see text] over all integers [Formula: see text]. It is known that [Formula: see text]. We give various lower and upper bounds for the least possible value of [Formula: see text] provided that a given power [Formula: see text] divides both [Formula: see text] and [Formula: see text] for all [Formula: see text]. In particular, the least possible value is [Formula: see text] for [Formula: see text] and is asymptotically [Formula: see text] for large [Formula: see text].

We propose a notion of graph convergence that interpolates between the Benjamini–Schramm convergence of bounded degree graphs and the dense graph convergence developed by László Lovász and his coauthors. We prove that spectra of graphs, and also some important graph parameters such as numbers of colorings or matchings, behave well in convergent graph sequences. Special attention is given to graph sequences of large essential girth, for which asymptotics of coloring numbers are explicitly calculated. We also treat numbers of matchings in approximately regular graphs. We introduce tentative limit objects that we call graphonings because they are common generalizations of graphons and graphings. Special forms of these, called Hausdorff and Euclidean graphonings, involve geometric measure theory. We construct Euclidean graphonings that provide limits of hypercubes and of finite projective planes, and, more generally, of a wide class of regular sequences of large essential girth. For any convergent sequence of large essential girth, we construct weaker limit objects: an involution invariant probability measure on the sub-Markov space of consistent measure sequences (this is unique), or an acyclic reversible sub-Markov kernel on a probability space (non-unique). We also pose some open problems.

In this short note we observe that recent results of Abért and Hubai and of Csikvári and Frenkel about Benjamini–Schramm continuity of the holomorphic moments of the roots of the chromatic polynomial extend to the theory of dense graph sequences. We offer a number of problems and conjectures motivated by this observation.

We show that if two monic polynomials with integer coefficients have a square-free resultant, then all positive divisors of the resultant arise as the greatest common divisor of the values of the two polynomials at a suitable integer.

We determine minimal Cayley–Hamilton and Capelli identities for matrices over a Grassmann algebra of finite rank. For minimal standard identities, we give lower and upper bounds on the degree. These results improve on upper bounds given by L. Márki, J. Meyer, J. Szigeti and L. van Wyk in a recent paper.

We introduce the matching measure of a finite graph as the uniform distribution on the roots of the matching polynomial of the graph. We analyze the asymptotic behavior of the matching measure for graph sequences with bounded degree. A graph parameter is said to be estimable if it converges along every Benjamini-Schramm convergent sparse graph sequence. We prove that the normalized logarithm of the number of matchings is estimable. We also show that the analogous statement for perfect matchings already fails for d-regular bipartite graphs for any fixed d at least 3. The latter result relies on analyzing the probability that a randomly chosen perfect matching contains a particular edge. However, for any sequence of d-regular bipartite graphs converging to the d-regular tree, we prove that the normalized logarithm of the number of perfect matchings converges. This applies to random d-regular bipartite graphs. We show that the limit equals to the exponent in Schrijver's lower bound on the number of perfect matchings. Our analytic approach also yields a short proof for the Nguyen-Onak (also Elek--Lippner) theorem saying that the matching ratio is estimable. In fact, we prove the slightly stronger result that the independence ratio is estimable for claw-free graphs.

We determine the automorphisms and the continuous endomorphisms of the Einstein gyrogroup in arbitrary dimension. This generalizes a recent result of Lajos Moln\'ar and D\'aniel Virosztek, who have determined the continuous endomorphisms in the three-dimensional case.

Recently, M. Abert and T. Hubai studied the following problem. The chromatic measure of a finite simple graph is defined to be the uniform distribution on its chromatic roots. Abert and Hubai proved that for a Benjamini-Schramm convergent sequence of finite graphs, the chromatic measures converge in holomorphic moments. They also showed that the normalized logarithm of the chromatic polynomial converges to a harmonic real function outside a bounded disc.In this paper we generalize their work to a wide class of graph polynomials, namely, multiplicative graph polynomials of bounded exponential type. A special case of our results is that for any fixed complex number v(0) the measures arising from the Tutte polynomial Z(Gn) (z, v(0)) converge in holomorphic moments if the sequence (G(n)) of finite graphs is Benjamini-Schramm convergent. This answers a question of Abert and Hubai in the affirmative. Even in the original case of the chromatic polynomial, our proof is considerably simpler. (c) 2015 Elsevier Ltd. All rights reserved.

A game is played by a team of two—say Alice and Bob—in which the value of a random variable x is revealed to Alice only, who cannot freely communicate with Bob. Instead, she is given a quantum n -level system, respectively a classical n -state system, which she can put in possession of Bob in any state she wishes. We evaluate how successfully they managed to store and recover the value of x by requiring Bob to specify a value z and giving a reward of value f ( x , z ) to the team. We show that whatever the probability distribution of x and the reward function f are, when using a quantum n -level system, the maximum expected reward obtainable with the best possible team strategy is equal to that obtainable with the use of a classical n -state system. The proof relies on mixed discriminants of positive matrices and—perhaps surprisingly—an application of the Supply–Demand Theorem for bipartite graphs. As a corollary, we get an infinite set of new, dimension dependent inequalities regarding positive operator valued measures and density operators on complex n -space. As a further corollary, we see that the greatest value, with respect to a given distribution of x , of the mutual information I ( x ; z ) that is obtainable using an n -level quantum system equals the analogous maximum for a classical n -state system.

Positive polynomials arising from Muirhead's inequality, from classical power mean and elementary symmetric mean inequalities and from Minkowski's inequality can be rewritten as sums of squares.

Let $v_1$,..., $v_n$ be $n$ vectors in an inner product space. Can we find a natural number $d$ and positive (semidefinite) complex matrices $A_1$,..., $A_n$ of size $d \times d$ such that ${\rm Tr}(A_kA_l)= $ for all $k,l=1,..., n$? For such matrices to exist, one must have $ \geq 0$ for all $k,l=1,..., n$. We prove that if $n<5$ then this trivial necessary condition is also a sufficient one and find an appropriate example showing that from $n=5$ this is not so --- even if we allowed realizations by positive operators in a von Neumann algebra with a faithful normal tracial state. The fact that the first such example occurs at $n=5$ is similar to what one has in the well-investigated problem of positive factorization of positive (semidefinite) matrices. If the matrix $( )$ has a positive factorization, then matrices $A_1$,..., $A_n$ as above exist. However, as we show by a large class of examples constructed with the help of the Clifford algebra, the converse implication is false.

Az absztrakt algebra egyes reszteruletein (a csoportelmeletben, az invariansok elmeleteben, a gyűrűelmeletben, a Lie-algebrak elmeleteben, a hurkok [loop-ok] elmeleteben es az univerzalis algebraban) vegeztunk kutatasokat. Eredmenyeinkről 58 dolgozatban es egy konyvben szamoltunk be. Tobb publikacionk vezető matematikai folyoiratokban (Annals of Mathematics, Journal of the European Mathematical Society, Journal of Algebra, Journal of Group Theory, Journal of Algebraic Combinatorics, Proceedings of the American Mathematical Society stb.) jelent meg. Szamos eredmenyunk kozul itt (1) a Lie-tipusu egyszerű csoportok novekedesi fuggvenyeről szolot (Pyber Laszlo es Szabo Endre), (2) a 3x3-as valos szimmetrikus matrixok diszkriminansanak 5 negyzet osszegekent valo felirasat (Domokos Matyas), es (3) a felső haromszogmatrixok csoportjanak konjugaltosztalyszamara vonatkozo Higman-sejtessel kapcsolatosat (Halasi Zoltan es Palfy Peter Pal) emeljuk ki. Az (1) eredmenyt velunk egyidőben Breuillard, Green es Tao is bebizonyitottak. Ennek jelentős kovetkezmenyei vannak az expander grafok teruleten is. A (2) eredmeny Kummer tobb mint masfel evszazados tetelet erősiti, eddig csak 7 negyzet osszegekent valo feliras volt ismert. A (3) eredmeny a Higman-sejtes egy altalanositasat cafolja, ezaltal ketsegesse teve az eredeti sejtes ervenyet is. A kutatasokba harom tehetseges fiatal kutatot is sikerult bekapcsolnunk az OTKA tamogatasaval. | We have conducted research in various subfields of abstract algebra (in group theory, in the theory of invariants, in ring theory, in the theory of Lie algebras, in the theory of loops, and in universal algebra). Members of the research team published 58 papers and a book. Many of our publications appeared in leading mathematical periodicals (Annals of Mathematics, Journal of the European Mathematical Society, Journal of Algebra, Journal of Group Theory, Journal of Algebraic Combinatorics, Proceedings of the American Mathematical Society, etc.). We just mention here our three most important results: (1) on the growth in simple groups of Lie type (L. Pyber and E. Szabo); (2) decomposing the discriminant of a 3x3 real symmetric matrix into the sum of 5 squares (M. Domokos); (3) a result concerning a generalization of Higman's conjecture on the number of conjugacy classes in the group of upper unitriangular matrices (Z. Halasi and P. P. Palfy). Result (1) has been obtained simultaneously by Breuillard, Green, and Tao; it has important implications for expander graphs. Result (2) improves upon a result of Kummer from the middle of nineteenth century; up till now only a decomposition into 7 squares has been known. Result (3) refutes a generalization of Higman's conjecture, hence making the validity of the original conjecture doubtful. The support of OTKA made it possible to employ three talented young researchers as well.

We recall Vere-Jones's definition of the $\alpha$--permanent and describe the connection between the (1/2)--permanent and the hafnian. We establish expansion formulae for the $\alpha$--permanent in terms of partitions of the index set, and we use these to prove Lieb-type inequalities for the $\pm\alpha$--permanent of a positive semi-definite Hermitian $n\times n$ matrix and the $\alpha/2$--permanent of a positive semi-definite real symmetric $n\times n$ matrix if $\alpha$ is a nonnegative integer or $\alpha\ge n-1$. We are unable to settle Shirai's nonnegativity conjecture for $\alpha$--permanents when $\alpha\ge 1$, but we verify it up to the $5\times 5$ case, in addition to recovering and refining some of Shirai's partial results by purely combinatorial proofs.

Palyazatunk harom, lazan osszefuggő problemakort erint. Tobb kiemelkedő eredmenyt ertunk el a Thom polinomok, es altalanosabban, az ekvivarians obstrukciok elmeleteben. A Thom sorok bevezetese es a Morin szingularitasok Thom polinomjainak kiszamitasa a terulet legfontosabb eredmenyei az utobbi evekben. A geometriai oldalon, komoly előrehaladast ertunk el a hiperkahler modulusterek geometriajanak leirasaban, es sikerult bebizonyitanunk a Batyrev-Materov tukor reziduum sejtest torikus orbifoldokra. Vegezetul, projektunk algebrai eredmenyei kozott megemlitjuk uj 2-karakterisztikai jelensegek felfedezeset az ortogonalis csoport reprezentacioelmeleteben, es a Zamolodcsikov periodicitasi sejtes bizonyitasat Y-rendszerekre. Ezenkivul, uj algebrai egyenlőtlensegeket talaltunk szemidefinit matrixokra, es ezeket felhasznalva megjavitottuk a legjobb ismert also becslest valos linearis funkcionalok szorzatara. | The project deals with three loosely interconnected areas of mathematics. We obtained a number of outstanding results in the theory of Thom polynomials, and more generally, in equivariant obstruction theory. In particular, the introduction of Thom series, and the calculation of the Thom polynomials of Morin singularities are the most important advances in the subject in the last few years. On the more geometric side, we made serious progress in the description of the geometry of hyperkahler moduli spaces, and proved the Batyrev-Materov mirror residue conjecture for toric orbifolds. Finally, the more algebraic results of our project include discovering new characteristic-2 phenomena in the representation theory of the orthogonal group, and proving the Zamolodchikov periodicity conjecture for Y-systems. We also found new algebraic inequalities for semidefinite matrices, and using these, improved the best known lower bound on products of real linear functionals.