Generalized shuffles related to Nijenhuis and TD-algebras

Shuffle and quasi-shuffle products are well-known in the mathematics literature. They are intimately related to Loday's dendriform algebras, and were extensively used to give explicit constructions of free commutative Rota-Baxter algebras. In the literature there exist at least two other Rota-Baxter type algebras, namely, the Nijenhuis algebra and the so-called TD-algebra. The explicit construction of the free unital commutative Nijenhuis algebra uses a modified quasi-shuffle product, called the right-shift shuffle. We show that another modification of the quasi-shuffle product, the so-called left-shift shuffle, can be used to give an explicit construction of the free unital commutative TD-algebra. We explore some basic properties of TD-operators and show that the free unital commutative Nijenhuis algebra is a TD-algebra. We relate our construction to Loday's unital commutative dendriform trialgebras, including the involutive case. The concept of Rota-Baxter, Nijenhuis and TD-bialgebras is introduced at the end and we show that any commutative bialgebra provides such objects.Comment: 20 pages, typos corrected, accepted for publication in Communications in Algebr

Изучение терапевтических эффектов излучения с длиной волны 670 нм при различных типах диабетического макулярного отека

The purpose of this study was to investigate the therapeutic effects of 670 nm irradiation in patients with diabetic macular edema. In several studies, positive effects of red/near-infrared irradiation showed in a range of ocular diseases such as macular degeneration, macular edema, and retinitis pigmentosa. This study was conducted on forty five eyes of 26 diabetic patients with macular edema between the ages of 51 and 80.Measurement of visual acuity and slit lamp examination, funduscopy, and optical coherence tomography were performed in all subjects. None of the patients had proliferative retinopathy. We used a portable LED device (Warp 10, Quantum Devices) for treatment. Patients held this device at a distance of 3 cm from their eyes for 240 seconds for three months. Full ophthalmic examinations were repeated 1, 2, and 3 months after treatment.After 3 months, the mean visual acuity improved from 0.44 ± 0.38 log MAR to 0.27 ± 0.24 log MAR and vision increased by 1.52 ± 1.16 lines post treatment (р<0.001). The mean central macula thickness decreased from 381.49 ± 144.40 μm to 359.72 ± 128.84 μm (р=0.050). In patients with mild and moderate nonproliferative diabetic retinopathy, the mean central retinal thickness decreased 52.06 ± 67.78 μm and 39.27 ± 44.69 μm, respectively, but patients with severe type showed an increase of 34.93 ± 65.65 μm in the mean central retinal thickness (р<0.001). Also, the severity of macular edema had no effect on final outcomes (р>0.05). Photobiomodulation can positively affect diabetic macular edema, especially in patients with mild to moderate diabetic retinopathy.Целью данной исследовательской работы являлось изучение терапевтических эффектов излучения с длиной волны 670 нм у пациентов с диабетическим макулярным отеком. Ряд предыдущих исследований свидетельствует о положительном эффекте красного инфракрасного излучения при некоторых заболеваниях глаз, таких как макулодистрофия (дегенерация желтого пятна), макулярный отек и пигментный ретинит. Наше исследование было проведено на 45 глазах у 26 больных сахарным диабетом в возрасте от 51 до 80 лет с макулярным отеком. Всем пациентам были проведены определение остроты зрения, осмотр глаз щелевой лампой, фундоскопия и оптическая когерентная томография. Ни у одного из пациентов не было пролиферативной ретинопатии. Для лечения нами был применен портативный светодиодный прибор (Warp 10, Quantum devices). Пациенты держали светодиод на расстоянии 3 см от глаза в течение 240 сек в течение 3 мес. Все офтальмологические исследования были повторены через 1, 2 и 3 мес после проведения лечебной процедуры. Через 3 мес средняя острота зрения улучшилась с показателем логарифма минимального угла разрешения 0,44 ± 0,38 до 0,27 ± 0,24, что показало увеличение показателя остроты зрения на 1,52 ± 1,16 после лечения (р<0,001). Средняя центральная толщина сетчатки в области макулы уменьшилась с 381,49 ± 144,40 мкм до 359,72 ± 128,84 мкм (р=0,050). У пациентов с легкой и умеренной непролиферативной диабетической ретинопатией средняя толщина сетчатки уменьшилась до 52,06 ± 67,78 и 52,06 ± 67,78 мкм, соответственно, а у пациентов с тяжелой ретинопатией наблюдалось увеличение на 34,93 ± 65,65 мкм (р<0,001). Помимо того, степень макулярного отека не повлияла на окончательный результат лечения (р>0,05). Фотобиомодуляция была эффективной при диабетическом макулярном отеке, в частности, у пациентов с легкой и умеренной диабетической ретинопатией

Rota-Baxter algebras and new combinatorial identities

The word problem for an arbitrary associative Rota-Baxter algebra is solved. This leads to a noncommutative generalization of the classical Spitzer identities. Links to other combinatorial aspects, particularly of interest in physics, are indicated.Comment: 8 pages, improved versio

Amino Acid Features of P1B-ATPase Heavy Metal Transporters Enabling Small Numbers of Organisms to Cope with Heavy Metal Pollution

Phytoremediation refers to the use of plants for extraction and detoxification of pollutants, providing a new and powerful weapon against a polluted environment. In some plants, such as Thlaspi spp, heavy metal ATPases are involved in overall metal ion homeostasis and hyperaccumulation. P1B-ATPases pump a wide range of cations, especially heavy metals, across membranes against their electrochemical gradients. Determination of the protein characteristics of P1B-ATPases in hyperaccumulator plants provides a new opportuntity for engineering of phytoremediating plants. In this study, using diverse weighting and modeling approaches, 2644 protein characteristics of primary, secondary, and tertiary structures of P1B-ATPases in hyperaccumulator and nonhyperaccumulator plants were extracted and compared to identify differences between proteins in hyperaccumulator and nonhyperaccumulator pumps. Although the protein characteristics were variable in their weighting, tree and rule induction models; glycine count, frequency of glutamine-valine, and valine-phenylalanine count were the most important attributes highlighted by 10, five, and four models, respectively. In addition, a precise model was built to discriminate P1B-ATPases in different organisms based on their structural protein features. Moreover, reliable models for prediction of the hyperaccumulating activity of unknown P1B-ATPase pumps were developed. Uncovering important structural features of hyperaccumulator pumps in this study has provided the knowledge required for future modification and engineering of these pumps by techniques such as site-directed mutagenesis

Video Data Compression by Progressive Iterative Approximation

In the present paper, the B-spline curve is used for reducing the entropy of video data. We consider the color or luminance variations of a spatial position in a series of frames as input data points in Euclidean space R or R3. The progressive and iterative approximation (PIA) method is a direct and intuitive way of generating curve series of high and higher fitting accuracy. The video data points are approximated using progressive and iterative approximation for least square (LSPIA) fitting. The Lossless video data compression is done through storing the B-spline curve control points (CPs) and the difference between fitted and original video data. The proposed method is applied to two classes of synthetically produced and naturally recorded video sequences and makes a reduction in the entropy of both. However, this reduction is higher for syntactically created than those naturally produced. The comparative analysis of experiments on a variety of video sequences suggests that the entropy of output video data is much less than that of input video data

Renormalization: a quasi-shuffle approach

In recent years, the usual BPHZ algorithm for renormalization in perturbative quantum field theory has been interpreted, after dimensional regularization, as a Birkhoff decomposition of characters on the Hopf algebra of Feynman graphs, with values in a Rota-Baxter algebra of amplitudes. We associate in this paper to any such algebra a universal semi-group (different in nature from the Connes-Marcolli "cosmical Galois group"). Its action on the physical amplitudes associated to Feynman graphs produces the expected operations: Bogoliubov's preparation map, extraction of divergences, renormalization. In this process a key role is played by commutative and noncommutative quasi-shuffle bialgebras whose universal properties are instrumental in encoding the renormalization process

Exponential renormalization

Moving beyond the classical additive and multiplicative approaches, we present an "exponential" method for perturbative renormalization. Using Dyson's identity for Green's functions as well as the link between the Faa di Bruno Hopf algebra and the Hopf algebras of Feynman graphs, its relation to the composition of formal power series is analyzed. Eventually, we argue that the new method has several attractive features and encompasses the BPHZ method. The latter can be seen as a special case of the new procedure for renormalization scheme maps with the Rota-Baxter property. To our best knowledge, although very natural from group-theoretical and physical points of view, several ideas introduced in the present paper seem to be new (besides the exponential method, let us mention the notions of counterfactors and of order n bare coupling constants).Comment: revised version; accepted for publication in Annales Henri Poincar