A Spinorial Formulation of the Maximum Clique Problem of a Graph

We present a new formulation of the maximum clique problem of a graph in complex space. We start observing that the adjacency matrix A of a graph can always be written in the form A = B B where B is a complex, symmetric matrix formed by vectors of zero length (null vectors) and the maximum clique problem can be transformed in a geometrical problem for these vectors. This problem, in turn, is translated in spinorial language and we show that each graph uniquely identifies a set of pure spinors, that is vectors of the endomorphism space of Clifford algebras, and the maximum clique problem is formalized in this setting so that, this much studied problem, may take advantage from recent progresses of pure spinor geometry

On Spinors Transformations

We begin showing that for even dimensional vector spaces $V$ all automorphisms of their Clifford algebras are inner. So all orthogonal transformations of $V$ are restrictions to $V$ of inner automorphisms of the algebra. Thus under orthogonal transformations $P$ and $T$ - space and time reversal - all algebra elements, including vectors $v$ and spinors $\varphi$, transform as $v \to x v x^{-1}$ and $\varphi \to x \varphi x^{-1}$ for some algebra element $x$. We show that while under combined $PT$ spinor $\varphi \to x \varphi x^{-1}$ remain in its spinor space, under $P$ or $T$ separately $\varphi$ goes to a 'different' spinor space and may have opposite chirality. We conclude with a preliminary characterization of inner automorphisms with respect to their property to change, or not, spinor spaces.Comment: Minor changes to propositions 1 and

Neural Relax

We present an algorithm for data preprocessing of an associative memory inspired to an electrostatic problem that turns out to have intimate relations with information maximization

k-deformed Poincare algebras and quantum Clifford-Hopf algebras

The Minkowski spacetime quantum Clifford algebra structure associated with the conformal group and the Clifford-Hopf alternative k-deformed quantum Poincare algebra is investigated in the Atiyah-Bott-Shapiro mod 8 theorem context. The resulting algebra is equivalent to the deformed anti-de Sitter algebra U_q(so(3,2)), when the associated Clifford-Hopf algebra is taken into account, together with the associated quantum Clifford algebra and a (not braided) deformation of the periodicity Atiyah-Bott-Shapiro theorem.Comment: 10 pages, RevTeX, one Section and references added, improved content

Optimal growth ofLactobacillus casei in a Cheddar cheese ripening model system requires exogenous fatty acids

Flavor development in ripening Cheddar cheese depends on complex microbial and biochemical processes that are difficult to study in natural cheese. Thus, our group has developed Cheddar cheese extract (CCE) as a model system to study these processes. In previous work, we found that CCE supported growth of Lactobacillus casei, one of the most prominent nonstarter lactic acid bacteria (NSLAB) species found in ripening Cheddar cheese, to a final cell density of 108 cfu/mL at 37°C. However, when similar growth experiments were performed at 8°C in CCE derived from 4-mo-old cheese (4mCCE), the final cell densities obtained were only about 106 cfu/mL, which is at the lower end of the range of the NSLAB population expected in ripening Cheddar cheese. Here, we report that addition of Tween 80 to CCE resulted in a significant increase in the final cell density of L. casei during growth at 8°C and produced concomitant changes in cytoplasmic membrane fatty acid (CMFA) composition. Although the effect was not as dramatic, addition of milk fat or a monoacylglycerol (MAG) mixture based on the MAG profile of milk fat to 4mCCE also led to an increased final cell density of L. casei in CCE at 8°C and changes in CMFA composition. These observations suggest that optimal growth of L. casei in CCE at low temperature requires supplementation with a source of fatty acids (FA). We hypothesize that L. casei incorporates environmental FA into its CMFA, thereby reducing its energy requirement for growth. The exogenous FA may then be modified or supplemented with FA from de novo synthesis to arrive at a CMFA composition that yields the functionality (i.e., viscosity) required for growth in specific conditions. Additional studies utilizing the CCE model to investigate microbial contributions to cheese ripening should be conducted in CCE supplemented with 1% milk fat

Growth of Lactobacillus paracasei ATCC 334 in a cheese model system: a biochemical approach

Growth of Lactobacillus paracasei ATCC 334, in a cheese-ripening model system based upon a medium prepared from ripening Cheddar cheese extract (CCE) was evaluated. Lactobacillus paracasei ATCC 334 grows in CCE made from cheese ripened for 2 (2mCCE), 6 (6mCCE), and 8 (8mCCE) mo, to final cell densities of 5.9 × 108, 1.2 × 108, and 2.1 × 107 cfu/mL, respectively. Biochemical analysis and mass balance equations were used to determine substrate consumption patterns and products formed in 2mCCE. The products formed included formate, acetate, and d-lactate. These data allowed us to identify the pathways likely used and to initiate metabolic flux analysis. The production of volatiles during growth of Lb. paracasei ATCC 334 in 8mCCE was monitored to evaluate the metabolic pathways utilized by Lb. paracasei during the later stages of ripening Cheddar cheese. The 2 volatiles detected at high levels were ethanol and acetate. The remaining detected volatiles are present in significantly lower amounts and likely result from amino acid, pyruvate, and acetyl-coenzyme A metabolism. Carbon balance of galactose, lactose, citrate, and phosphoserine/phosphoserine-containing peptides in terms of d-lactate, acetate, and formate are in agreement with the amounts of substrates observed in 2mCCE; however, this was not the case for 6mCCE and 8mCCE, suggesting that additional energy sources are utilized during growth of Lb. paracasei ATCC 334 in these CCE. This study provides valuable information on the biochemistry and physiology of Lb. paracasei ATCC 334 in ripening cheese

Cartoon Computation: Quantum-like computing without quantum mechanics

We present a computational framework based on geometric structures. No quantum mechanics is involved, and yet the algorithms perform tasks analogous to quantum computation. Tensor products and entangled states are not needed -- they are replaced by sets of basic shapes. To test the formalism we solve in geometric terms the Deutsch-Jozsa problem, historically the first example that demonstrated the potential power of quantum computation. Each step of the algorithm has a clear geometric interpetation and allows for a cartoon representation.Comment: version accepted in J. Phys.A (Letter to the Editor

The general classical solution of the superparticle

The theory of vectors and spinors in 9+1 dimensional spacetime is introduced in a completely octonionic formalism based on an octonionic representation of the Clifford algebra \Cl(9,1). The general solution of the classical equations of motion of the CBS superparticle is given to all orders of the Grassmann hierarchy. A spinor and a vector are combined into a $3 \times 3$ Grassmann, octonionic, Jordan matrix in order to construct a superspace variable to describe the superparticle. The combined Lorentz and supersymmetry transformations of the fermionic and bosonic variables are expressed in terms of Jordan products.Comment: 11 pages, REVTe

Octonionic representations of Clifford algebras and triality

The theory of representations of Clifford algebras is extended to employ the division algebra of the octonions or Cayley numbers. In particular, questions that arise from the non-associativity and non-commutativity of this division algebra are answered. Octonionic representations for Clifford algebras lead to a notion of octonionic spinors and are used to give octonionic representations of the respective orthogonal groups. Finally, the triality automorphisms are shown to exhibit a manifest \perm_3 \times SO(8) structure in this framework.Comment: 33 page

Covariant Quantization of Superstrings Without Pure Spinor Constraints

We construct a covariant quantum superstring, extending Berkovits' approach by introducing new ghosts to relax the pure spinor constraints. The central charge of the underlying Kac-Moody algebra, which would lead to an anomaly in the BRST charge, is treated as a new generator with a new b-c system. We construct a nilpotent BRST current, an anomalous ghost current and an anomaly-free energy-momentum tensor. For open superstrings, we find the correct massless spectrum. In addition, we construct a Lorentz invariant B-field to be used for the computation of the integrated vertex operators and amplitudes.Comment: 30 page