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We improve the best known upper bound on the density of a planar measurable set A containing no two points at unit distance to 0.25442. We use a combination of Fourier analytic and linear programming methods to obtain the result. The estimate is achieved by means of obtaining new linear constraints on the autocorrelation function of A utilizing triple-order correlations in A, a concept that has not been previously studied. [ABSTRACT FROM AUTHOR]

For a set A of points in the plane, not all collinear, we denote by tr (A) the number of triangles in a triangulation of A, that is, tr (A) = 2 i + b - 2 , where b and i are the numbers of boundary and interior points of the convex hull [A] of A respectively. We conjecture the following discrete analog of the Brunn–Minkowski inequality: for any two finite point sets A , B ⊂ R 2 one has tr (A + B) ≥ tr (A) 1 / 2 + tr (B) 1 / 2. We prove this conjecture in the cases where [ A ] = [ B ] , B = A ∪ { b } , | B | = 3 and if A and B have no interior points. A generalization to larger dimensions is also discussed. [ABSTRACT FROM AUTHOR]

By improving upon previous estimates on a problem posed by L. Moser, we prove a conjecture of Erdős that the density of any measurable planar set avoiding unit distances is less than 1/4. Our argument implies the upper bound of 0.2470. [ABSTRACT FROM AUTHOR]

Abstract: We use the connection between positive definite functions and the character table of the symmetric group S6 to give a short new proof of the nonexistence of a finite projective plane of order 6. For higher orders, like 10 and 12, the method seems to be inconclusive as of now, but could be a basis of further research. [ABSTRACT FROM AUTHOR]

We present a new approach to the problem of mutually unbiased bases (MUBs), based on positive definite functions on the unitary group. The method provides a new proof of the fact that there are at most d + 1 MUBs in Cd, and it may also lead to a proof of non-existence of complete systems of MUBs in dimension 6 via a conjectured algebraic identity. [ABSTRACT FROM AUTHOR]

A 1-avoiding set is a subset of $$\mathbb {R}^n$$ that does not contain pairs of points at distance 1. Let $$m_1(\mathbb {R}^n)$$ denote the maximum fraction of $$\mathbb {R}^n$$ that can be covered by a measurable 1-avoiding set. We prove two results. First, we show that any 1-avoiding set in $$\mathbb {R}^n (n\ge 2)$$ that displays block structure (i.e., is made up of blocks such that the distance between any two points from the same block is less than 1 and points from distinct blocks lie farther than 1 unit of distance apart from each other) has density strictly less than $$1/2^n$$ . For the special case of sets with block structure this proves a conjecture of Erdős asserting that $$m_1(\mathbb {R}^2) < 1/4$$ . Second, we use linear programming and harmonic analysis to show that $$m_1(\mathbb {R}^2) \le 0.258795$$ . [ABSTRACT FROM AUTHOR]

Abstract: This short note gives an upper bound on the measure of sets such that has no solutions in . [Copyright &y& Elsevier]

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 L2 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.: Fuglede's conjecture is false in 5 and higher dimensions. Math. Res. Letters 11 (2004), 251–258]. [ABSTRACT FROM AUTHOR]

Abstract: The open problem of determining the exact value of the n-th linear polarization constant c n of has received considerable attention over the past few years. This paper makes a contribution to the subject by providing a new lower bound on the value of sup∥y∥=1∣〈x 1, y〉⋯〈x n , y〉∣, where x 1,…, x n are unit vectors in . The new estimate is given in terms of the eigenvalues of the Gram matrix [〈x i , x j 〉] and improves upon earlier estimates of this kind. However, the intriguing conjecture c n = n n/2 remains open. [Copyright &y& Elsevier]

He present work deals with estimations of the n-th linear polarization constant c(H)n of an n-dimensional real Hilbert space H. We provide some new lower bounds on the value of sup║y║=1 │〈x1,y〉...〈xn,y〉, where x1, ... , xn are unit vectors in H. In particular, the results improve an earlier estimate of Marcus. However, the intriguing conjecture c(H) n= nn/2 remains open. [ABSTRACT FROM AUTHOR]

Exact and simple calculation of Thomas rotation and Thomas precessions along a circular world line is presented in an absolute (coordinate-free) formulation of special relativity. A straightforward derivation of the Fermi–Walker equation is also given. Besides the simplicity of calculations the absolute treatment of spacetime allows us to make a clear conceptual distinction between the phenomena of Thomas rotation and Thomas precession. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give a characterization in Hilbert spaces of the generators of C0-semigroups associated with closed, sectorial forms in terms of the convergence of a generalized Trotter''s product formula. In the course of the proof of the main result we also present a similarity result which can be of independent interest: for any unbounded generator A of a C0-semigroup etA it is possible to introduce an equivalent scalar product on the space, such that etA becomes non-quasi-contractive with respect to the new scalar product. [Copyright &y& Elsevier]

Presents a finite-step algorithm for the positive realization of a rational transfer function. Purpose of the positive realization problem of a given rational transfer function of a discrete time-invariant linear system; Proposal of a method for reducing the pole order of the dominant pole for primitive transfer functions; Description of the structure of the algorithm.

Provides a nontrivial lowerbound on the value of the dimension N under the assumption that there exists a time instant at which the impulse response is zero. Transfer functions with the property which may be regarded as external cases in positive system theory; Basic phenomenon in positive system theory that the dimension N of an arbitrary positive realization of a given transfer function may be strictly larger than the dimension of its minimal realization.

Abstract: We give an improvement of the best known lower bound for the supremum of autoconvolutions of nonnegative functions supported in a compact interval. Also, by means of explicit examples we disprove a long standing natural conjecture of Schinzel and Schmidt concerning the extremal function for such autoconvolutions. [ABSTRACT FROM AUTHOR]

Abstract: We observe that successive applications of known results from the theory of positive systems lead to an efficient general algorithm for positive realizations of transfer functions. We give two examples to illustrate the algorithm, one of which complements an earlier result of [L. Benvenuti, L. Farina, An example of how positivity may force realizations of ‘large’ dimensions, Systems Control Lett. 36 (1999) 261–266]. Finally, we improve a lower-bound of [B. Nagy, M. Matolcsi, A lower-bound on the dimension of positive realizations, IEEE Trans. Circuits Syst. I 50 (2003) 782–784] to indicate that the algorithm is indeed efficient in general. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]