Máté Matolcsi

Cell-specific STORM super-resolution imaging reveals nanoscale organization of cannabinoid signaling

A major challenge in neuroscience is to determine the nanoscale position and quantity of signaling molecules in a cell type– and subcellular compartment–specific manner. We developed a new approach to this problem by combining cell-specific physiological and anatomical characterization with super-resolution imaging and studied the molecular and structural parameters shaping the physiological properties of synaptic endocannabinoid signaling in the mouse hippocampus. We found that axon terminals of perisomatically projecting GABAergic interneurons possessed increased CB1 receptor number, active-zone complexity and receptor/effector ratio compared with dendritically projecting interneurons, consistent with higher efficiency of cannabinoid signaling at somatic versus dendritic synapses. Furthermore, chronic Δ9-tetrahydrocannabinol administration, which reduces cannabinoid efficacy on GABA release, evoked marked CB1 downregulation in a dose-dependent manner. Full receptor recovery required several weeks after the cessation of Δ9-tetrahydrocannabinol treatment. These findings indicate that cell type–specific nanoscale analysis of endogenous protein distribution is possible in brain circuits and identify previously unknown molecular properties controlling endocannabinoid signaling and cannabis-induced cognitive dysfunction.

Tiles with no spectra

Abstract We exhibit a subset of a finite Abelian group, which tiles the group by translation, and such that its tiling complements do not have a common spectrum (orthogonal basis for their L 2 space consisting of group characters). This disproves the Universal Spectrum Conjecture of Lagarias and Wang [Lagarias J. C. and Wang Y.: Spectral sets and factorizations of finite Abelian groups.J. Func. Anal. 145 (1997), 73–98]. Further, we construct a set in some finite Abelian group, which tiles the group but has no spectrum. We extend this last example to the groups ℤ d and ℝ d (for d ≥5 ) thus disproving one direction of the Spectral Set Conjecture of Fuglede [Fuglede B.: Commuting self-adjoint partial differential operators and a group theoretic problem. J. Funct. Anal. 16 (1974), 101–121]. The other direction was recently disproved by Tao [Tao T.: Fugledes conjecture is false in 5 and higher dimensions. Math. Res. Letters 11 (2004), 251–258].

Fuglede’s conjecture fails in dimension 4

In this note we modify a recent example of Tao and give an example of a set Omega subset of R-4 such that L-2(Omega) admits an orthonormal basis of exponentials {1/vertical bar Omega vertical bar 1/2 e(2 pi i )}(xi epsilon Lambda) for some set Lambda subset of R-4, but which does not tile R-4 by translations. This shows that one direction of Fugledes conjecture fails already in dimension 4. Some common properties of translational tiles and spectral sets are also proved.

A generalized Pauli problem and an infinite family of MUB-triplets in dimension 6

We exhibit an infinite family of triplets of mutually unbiased bases (MUBs) in dimension 6. These triplets involve the Fourier family of Hadamard matrices, F(a, b). However, in the main result of this paper we also prove that for any values of the parameters (a, b), the standard basis and F(a, b) cannot be extended to a MUB-quartet. The main novelty lies in the method of proof which may successfully be applied in the future to prove that the maximal number of MUBs in dimension 6 is three.

Minimal positive realizations of transfer functions with nonnegative multiple poles

This note concerns a particular case of the minimality problem in positive system theory. A standard result in linear system theory states that any nth-order rational transfer function of a discrete time-invariant linear single-input-single-output (SISO) system admits a realization of order n. In some applications, however, one is restricted to realizations with nonnegative entries (i.e., a positive system), and it is known that this restriction may force the order N of realizations to be strictly larger than n. A general solution to the minimality problem (i.e., determining the smallest possible value of N) is not known. In this note, we consider the case of transfer functions with nonnegative multiple poles, and give sufficient conditions for the existence of positive realizations of order N=n. With the help of our results we also give an improvement of an existing result in positive system theory.

Constructions of Complex Hadamard Matrices via Tiling Abelian Groups

Applications in quantum information theory and quantum tomography have raised current interest in complex Hadamard matrices. In this note we investigate the connection between tiling of Abelian groups and constructions of complex Hadamard matrices. First, we recover a recent, very general construction of complex Hadamard matrices due to Dita [2] via a natural tiling construction. Then we find some necessary conditions for any given complex Hadamard matrix to be equivalent to a Dita-type matrix. Finally, using another tiling construction, due to Szabó [8], we arrive at new parametric families of complex Hadamard matrices of order 8, 12 and 16, and we use our necessary conditions to prove that these families do not arise with Dita’s construction. These new families complement the recent catalogue [10] of complex Hadamard matrices of small order.

A superadditivity and submultiplicativity property for cardinalities of sumsets

For finite sets of integers A1,…,An we study the cardinality of the n-fold sumset A1+…+ An compared to those of (n−1)-fold sumsets A1+…+Ai−1+Ai+1+…+An. We prove a superadditivity and a submultiplicativity property for these quantities. We also examine the case when the addition of elements is restricted to an addition graph between the sets.

Towards a classification of 6 × 6 complex hadamard matrices

Complex Hadamard matrices have received considerable attention in the past few years due to their application in quantum information theory. While a complete characterization currently available [5] is only up to order 5, several new constructions of higher order matrices have appeared recently [4, 12, 2, 7, 11]. In particular, the classification of self-adjoint complex Hadamard matrices of order 6 was completed by Beuachamp and Nicoara in [2], providing a previously unknown non-affine one-parameter orbit. In this paper we classify all dephased, symmetric complex Hadamard matrices with real diagonal of order 6. Furthermore, relaxing the condition on the diagonal entries we obtain a new non-affine one-parameter orbit connecting the Fourier matrix F6 and Diţăs matrix D6. This answers a recent question of Bengtsson et al. [3].

A lowerbound on the dimension of positive realizations

A basic phenomenon in positive system theory is that the dimension N of an arbitrary positive realization of a given transfer function H(z) may be strictly larger than the dimension n of its minimal realizations. The aim of this brief is to provide a nontrivial lowerbound on the value of N under the assumption that there exists a time instant k/sub 0/ at which the (always nonnegative) impulse response of H(z) is 0 but the impulse response becomes strictly positive for all k>k/sub 0/. Transfer functions with this property may be regarded as extremal cases in positive system theory.

Order Bound for the Realization of a Combination of Positive Filters

In a problem on the realization of digital filters, initiated by Gersho and Gopinath, we extend and complete a remarkable result of Benvenuti, Farina and Anderson on decomposing the transfer function t(z) of an arbitrary linear, asymptotically stable, discrete, time-invariant single-input-single-output system as a difference t(z)=t1(z)-t2(z) of two positive, asymptotically stable linear systems. We give an easy-to-compute algorithm to handle the general problem, in particular, also the case of transfer functions t(z) with multiple poles, which was left open in a previous paper. One of the appearing positive, asymptotically stable systems is always one-dimensional, while the other has dimension depending on the order and, in the case of nonreal poles, also on the location of the poles of t(z). The appearing dimension is seen to be minimal in some cases and it can always be calculated before carrying out the realization