Kódelmélet és környéke = Coding theory and its neighbourhood

Kódelméletben hasznos véges projektív terek speciális egyenes-, illetve hipersíkmetszetű ponthalmazainak vizsgálata. Egyes cikkeinknek közvetlen kódelméleti alkalmazása van (itt ezeket soroljuk), mások geometriai ill. algebrai szálakkal kapcsolódnak oda. A polinomos módszer alkalmazásával bebizonyítottuk, hogy PG(2,q) egy olyan ponthalmaza, melyet minden egyenes adott r mod p pontban metsz, legalább (r-1)q+(p-1)r pontú kell legyen, ahol p a karakterisztika, r|q. Következésképp egy 3 dimenziós kód,melynek hossza és súlyai is oszthatók r-rel és minimális távolsága legalább 3, legalább (r-1)q+(p-1)r hosszú kell legyen. Ball, Blokhuis és Mazzocca híres, maximális ívek nemlétezéséről szóló tétele is egyszerűen kijön a tételből. Meghatároztuk két fontos poset, D^{k,n} és B_{m,n} automorfizmus-csoportját.A kérdéskör az insertion-deletion kódokhoz kapcsolódik. A B_{m,n} struktúra automorfizmus-csoportja korábban is ismert volt, de a hosszú bizonyítást 1 oldalasra redukáltuk. Megfogalmaztunk egy sejtést algebrai síkgörbék pontjainak számáról: n-edfokú, lineáris komponens nélküli görbének legfeljebb (n-1)q+1 pontja lehet; (n-1)q+n/2-t sikerült igazolni. Ilyen görbék hatékony kódokat adnak. Bebizonyítottuk, hogy ha egy lineáris [n,k,d]_q kód kiterjeszthető nem feltétlenül lineáris [n+1,k,d+1]_q kóddá, akkor a kiterjesztést lineáris módon is meg lehet csinálni. Eredményünk kiterjesztéséből pedig az következhetne,hogy az MDS sejtés lineáris és tetszőleges kódokra ekvivalens. | In coding theory, it is useful to study point sets of finite projective spaces with special intersection multiplicities with respect to lines and hyperplanes. Some of our papers have immediate application in coding theory (here we list those), the others are linked by its geometrical or algebraical concept. Using polynomial method, we proved that point sets of PG(n,q) intersecting each hyperplane in r mod p points have at least (r-1)q+(p-1)r points, where p is the characteristic and r|q. Hence a linear code whose length and weight are divisible by r and whose dual minimum distance is at least 3, has length at least (r-1)q+(p-1)r. Now the famous Ball-Blokhuis-Mazzocca theorem on the non-existence of maximal arcs becomes a corollary of this result. We determined the automorphism group of two important posets D^{k,n} and B_{m,n}. It was already known for B_{m,n}, but we shortened its long proof to 1 page. This topic is related to insertion-deletion codes. We conjecture that an algebraic plane curve of degree n without linear components can have at most (n-1)q+1 points, we showed that it is at most (n-1)q+n/2. Such curves give efficient codes. We proved that if a linear [n,k,d]_q code can be extended to a not necessarily linear[n+1,k,d+1]_q code then it can be done also in a linear way. From an extension of our results it would follow that the MDS-conjecture is equivalent for linear and arbirtary codes

Quantitative ultrastructural analysis of basket and axo-axonic cell terminals in the mouse hippocampus

Three functionally different populations of perisomatic interneurons establish GABAergic synapses on hippocampal pyramidal cells: parvalbumin (PV)-containing basket cells, type 1 cannabinoid receptor (CB1)-positive basket cells both of which target somata, and PV-positive axo-axonic cells that innervate axon initial segments. Using electron microscopic reconstructions, we estimated that a pyramidal cell body receives synapses from about 60 and 140 synaptic terminals in the CA1 and CA3 area, respectively. About 60 % of these terminals were PV positive, whereas 35-40 % of them were CB1 positive. Only about 1 % (CA1) and 4 % (CA3) of the somatic boutons were negative for both markers. Using fluorescent labeling, we showed that most of the CB1-positive terminals expressed vesicular glutamate transporter 3. Reconstruction of somatic boutons revealed that although their volumes are similar, CB1-positive boutons are more flat and the total volume of their mitochondria was smaller than that of PV-positive boutons. Both types of boutons contain dense-core vesicles and frequently formed multiple release sites on their targets and innervated an additional soma or dendrite as well. PV-positive boutons possessed small, macular synapses; whereas the total synaptic area of CB1-positive boutons was larger and formed multiple irregular-shaped synapses. Axo-axonic boutons were smaller than somatic boutons, had only one synapse and their ultrastructural parameters were closer to those of PV-positive somatic boutons. Our results represent the first quantitative measurement-using a highly reliable method-of the contribution of different cell types to the perisomatic innervation of pyramidal neurons, and may help to explain functional differences in their output properties

Gráfok és algoritmusok = Graphs and algorithms

A kutatás az elvárt eredménnyel zárult: tekintélyes nemzetközi konferenciákon és pubikációkban hoztuk nyilvánosságra az eredményéket, ideértve a STOC, SIAM és IEEE kiadványokat is, valamint egy könyvet is. A publikációk száma a matematikában elég magas (74). Ez nemzetközi összehasonlításban is kiemelkedő mutató a támogatás összegére vetítve. A projektben megmutattuk, hogy a gráfelmelet és a diszkrét matematika eszköztára számos helyen jól alkalmazható, ilyen terület a nagysebességű kommunikációs hálózatok tervezése, ezekben igen gyors routerek létrehozása. Egy másik terület a biológiai nagymolekulákon definiált gráfok és geometriai struktúrák. | The research concluded with the awaited results: in good international conferences and journals we published 74 works, including STOC conference, SIAM conferences and journals and one of the best IEEE journal. This number is high above average in mathematics research. We showed in the project that the tools of graph theory and discrete mathematics can be well applied in the high-speed communication network design, where we proposed fast and secure routing solutions. Additionally we also found applications in biological macromolecules

Cellular architecture and transmitter phenotypes of neurons of the mouse median raphe region

The median raphe region (MRR, which consist of MR and paramedian raphe regions) plays a crucial role in regulating cortical as well as subcortical network activity and behavior, while its malfunctioning may lead to disorders, such as schizophrenia, major depression, or anxiety. Mouse MRR neurons are classically identified on the basis of their serotonin (5-HT), vesicular glutamate transporter type 3 (VGLUT3), and gamma-aminobutyric acid (GABA) contents; however, the exact cellular composition of MRR regarding transmitter phenotypes is still unknown. Using an unbiased stereological method, we found that in the MR, 8.5 % of the neurons were 5-HT, 26 % were VGLUT3, and 12.8 % were 5-HT and VGLUT3 positive; whereas 37.2 % of the neurons were GABAergic, and 14.4 % were triple negative. In the whole MRR, 2.1 % of the neurons were 5-HT, 7 % were VGLUT3, and 3.6 % were 5-HT and VGLUT3 positive; whereas 61 % of the neurons were GABAergic. Surprisingly, 25.4 % of the neurons were triple negative and were only positive for the neuronal marker NeuN. PET-1/ePET-Cre transgenic mouse lines are widely used to specifically manipulate only 5-HT containing neurons. Interestingly, however, using the ePET-Cre transgenic mice, we found that far more VGLUT3 positive cells expressed ePET than 5-HT positive cells, and about 38 % of the ePET cells contained only VGLUT3, while more than 30 % of 5-HT cells were ePET negative. These data should facilitate the reinterpretation of PET-1/ePET related data in the literature and the identification of the functional role of a putatively new type of triple-negative neuron in the MRR

Co-transmission of acetylcholine and GABA regulates hippocampal states

The basal forebrain cholinergic system is widely assumed to control cortical functions via non-synaptic transmission of a single neurotransmitter. Yet, we find that mouse hippocampal cholinergic terminals invariably establish GABAergic synapses, and their cholinergic vesicles dock at those synapses only. We demonstrate that these synapses do not co-release but co-transmit GABA and acetylcholine via different vesicles, whose release is triggered by distinct calcium channels. This co-transmission evokes composite postsynaptic potentials, which are mutually cross-regulated by presynaptic autoreceptors. Although postsynaptic cholinergic receptor distribution cannot be investigated, their response latencies suggest a focal, intra- and/or peri-synaptic localisation, while GABAA receptors are detected intra-synaptically. The GABAergic component alone effectively suppresses hippocampal sharp wave-ripples and epileptiform activity. Therefore, the differentially regulated GABAergic and cholinergic co-transmission suggests a hitherto unrecognised level of control over cortical states. This novel model of hippocampal cholinergic neurotransmission may lead to alternative pharmacotherapies after cholinergic deinnervation seen in neurodegenerative disorders

Véges geometria = Finite geometry

Megmutattuk, hogy négyzet q-ra PG(2,q)-ban 4qlog q és q^(3/2)-q+2q^(1/2) között minden méretű minimális lefogó ponthalmaz létezik, sőt egy kicsit szűkebb intervallum minden értékére q-ban több, mint polinomnyi. Magasabb dimenziós projektív terekben a hipersíkokat r modulo p pontban metsző halmazok méretére bizonyos esetekben éles alsó becslést adtunk, amely a maximális ívek nemlétezésére vonatkozó Ball-Blokhuis-Mazzocca tétel általánosítása. Ez osztható lineáris kódok hosszára az n legalább (r-1)q+(p-1)r alsó becslést adja, ahol r az az érték, amellyel n és minden kódszó súlya is osztható. Megmutattuk, hogy PG(2,q) reguláris szemioválisai csak az oválisok és az unitálok. Segre típusí eredményt sikerült belátni másodrendű kúpok részleges kúpszeletnyalábjaira. Kis minimális lefogó ponthalmazok struktúrájáról azt sikerült megmutatni, hogy ezek minden egyenest 1 modulo p^e pontban metszenek, ahol e osztja h-t, ha q=p^h. Ezen túlmenően, ha a metszet p^e+1 elemű, akkor az GF(p^e) feletti részegyenes. Kis t-szeres lefogó ponthalmazokra az egyenesekkel való metszetekre beláttuk, hogy azok modulo p t-vel kongruensek, ahol t a karakterisztika. Ha q páros, akkor stabilitási eredményt bizonyítottunk PG(2,q) páros halmazaira. Az eredmény négyzet q-ra éles, és B. Segre ívek beágyazásáról szóló híres tételét általánosítja. Megmutattuk, hogy a Q(4,q) általánosított négyszögben nincsenek q^2-1 pontú maximális parciális ovoidok. | It was proven that in PG(2,q), q square, there is a minimal blocking set for any size between 4qlog q and q^(3/2)-q+2q^(1/2), Moreover, for a slightly smaller interval we also proved that the number of nonisomorphic minimal blocking sets of that size is more than polynomial in q. For sets intersecting all hyperplanes in r modulo p points we found a lower bound that is sharp in some cases. The proof generalizes the nonexistence of maximal arcs, due to Ball-Blokhuis-Mazzocca. For divisible linear codes it gives that the length is at least (r-1)q+(p-1)r, where divides the length and the weight of all codewords. We found that in PG(2,q) regular semiovals must be either ovals or unitals. We obtained a Segre type theorem for partial flocks of the quadratic cone. About the structure of small minimal blocking sets we obtained the following: each line intersects the set in 1 modulo p^e points, where e divides h and q=p^h. Furthermore, if the intersection has p^e+1 points, then it is a subline over GF(p^e). We proved that a small minimal t-fold blocking set intersects every line in t modulo p points, where p is the characteristics. For even q-s we proved a stability theorem for sets of even type in PG(2,q). The result is sharp when q is a square, and it generalizes a famous embeddability theorem for arcs, due to B. Segre. We also proved that the GQ Q(4,q) does not have maximal partial ovoids of size q^2-1

Brainstem nucleus incertus controls contextual memory formation

Hippocampal pyramidal cells encode memory engrams, which guide adaptive behavior. Selection of engram-forming cells is regulated by somatostatin-positive dendrite-targeting interneurons, which inhibit pyramidal cells that are not required for memory formation. Here, we found that gamma-aminobutyric acid ( GABA)-releasing neurons of the mouse nucleus incertus (NI) selectively inhibit somatostatin-positive interneurons in the hippocampus, both monosynaptically and indirectly through the inhibition of their subcortical excitatory inputs. We demonstrated that NI GABAergic neurons receive monosynaptic inputs from brain areas processing important environmental information, and their hippocampal projections are strongly activated by salient environmental inputs in vivo. Optogenetic manipulations of NI GABAergic neurons can shift hippocampal network state and bidirectionally modify the strength of contextual fear memory formation. Our results indicate that brainstem NI GABAergic cells are essential for controlling contextual memories