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In later publications [2] and other papers, which I do note quote I only formulated the results of the theory in a clearer form by using canonical identifications cf. Section 3 later , but without giving new proofs because of those in [1] being easily adaptable to the new manner. There the essential part of the theory is developed for relations with a fixed, sufficiently large i.

The defect of this approach is that certain rather complicated and not very intuitive fundamental operations called "mutations" have to be considered as well. That is why I return to the first manner, but without its past imperfections. I found the abstract Galois endotheory, if my memory is good, in , and formulated it in printed form first in the paper [3] of the International Congress of Mathematicians, Moscow, I found the first but not-completely true version of the homomorphism theorem of this theory in , before the above-mentioned congress.

The gap in its proof was remarked by some my 3rd year students in , After having tried to fill this gap unsuccessfully I found the necessary modification which made the theorem correct, i. Yet, until the present paper, the abstract Galois endotheory even without its homomorphism theorem has never been published with complete proofs. Only some rough ideas of certain proofs were indicated in [4]. The idea of abstract Galois set theory in relation with eliminative structures and the passage from the abstract Galois theory to the classical one goes back as early as to the end of thirties.

However, the proof of its main theorem was put in a clear form only in the first years after the second world was ? This theorem and the characterization of rationality domains in case of abstract fields were formulated in several papers, but never with proofs.

I exposed the abstract M. Krasner Galois theory and, after it had come to existence, the abstract Galois endotheory in my Seminar — and in my courses for 3rd year students Clermont-F d , —, and Paris, — Terminology and notations. We shall use the ordinary notations of set theory and mathematical logic. The product of two objects, say x and y, will be denoted by xy. Sometimes, in order to define certain notions or to formulate certain results with elegance and well, we shall have to deal not only with sets but with classes as well.

This will be done only for the sake of convenience and better understanding, but not for raising any question of Foundations. In fact, all we do could be done in the language of sets, though in a longer and more complicated way. However, as it can be shown, this manner of speaking is only an "abuse of the language" from the point of view of Bernays' theory, since the existence of classes and their mappings occurring in this paper can be proved based on Bernaiys—Godel axioms.

Relations, structures and mappings Let E and Xbe two sets called base set and argument set, respectively. Generally, for the sake of some of the proofs, we assume that E consists of at least two elements, though our results are trivially valid for a one-element base set E, too. The elements of Ex are called X-points while subsets of Ex are called X-relations.

Y-relations are sets of such points. When there is no danger of ambiguity, we often do not indicate the argument set X. By the arity of an Z-point, X-relation or. In particular, when 8 is a self-mapping of E i. The endomorphism monoid is never empty for it always contains the identical mapping of E. This remark is a straightforward consequence of the definitions. Particular relations. Firstly, we mention the empty relation 0, which is the only relation without a unique argument set and base set.

Let C be an equivalence relation on X. It consists of all Appoints that are compatible with C. When r is semi-regular then this C is unique, is denoted by T r , and is called the type of r. I tion r. If for some C, i. An A'-point P: X—E is said to be surjective, injective and bijective if it is such as a mapping, while a relation r is said to be such if every P g r is such. In particular, r is injective iff it is regular and T r is the discrete equivalence relation on X.

Let D E denote the monoid consisting of all self-mapping of E, called the symmetric monoid of E, and let S E stand for the full symmetric group of E, consisting of all permutations of E. For a subset A of D E the class of all relations on E that are stabilized resp. Note that these classes are never sets. When the context shows clearly what kind s of invariants is considered, the letters s or p before Inv may be omitted. The main problem of the next two paragraphs is to characterize these closures in terms of R but without any intervention of self-mappings of E.

Fundamental operations In order to characterize the above-mentioned closures of R in terms of relations we have to introduce certain operations acting on relations. Some of these operations act on sets of relations while others on single relations, but any of these operations results in single relations.

Some of these operations are only partial, i. Krasner While defining our fundamental operations in the sequel, all relations are assumed to have a fixed base set E. Infinitary boolean operations: la. Infinitary union, lb. Infinitary intersection.

This fundamental operation acts on any single relation r, and is denoted by 1. The above three fundamental operations are not independent. When E and X are finite then there are only a finite number of X-relations, whence the infinitary boolean operations are in fact the ordinary Unitary ones. We define the following preorder for sets R and R' of relations. It is easy to see that the infinitary union and intersection are increasing, while the negation is not.

Projective operations, which act on relations: Ha. Projections or restrictions p r x. This operation is defined for a relation r iff the argument set X of r contains J a s a subset. Antiprojections or extensions ext x.. This operation is defined for relations r with argument set XQX', and ext x. I With the usual identification in cartesian products, ext x. Both projections and extensions are increasing.

When relations are considered as sets of points, extensions are hyperpunctual mappings and projections are even punctual, if relations with a fixed argument set are considered. Therefore, the projections commute with U, and it is easy to see that the extensions commute with all boolean operations.

Canonical identification. Let r and r' be relations on E with argument sets X and X'. As infinitary boolean operations commute with extensions, they are compatible with this equivalence. So projections are also compatible with this equivalence.

Relations on E can be considered modulo i. This means that if a relation can be obtained from another one via omitting and adding some fictitious arguments then these two relations are considered the same. This identification will be called canonical. Since infinitary boolean operations and projections are compatible with they are meaningful M.

Krasner after canonical identification. It is trivial that ext x. Further, the infinite union and intersection become defined for every set R of relations in this case. Contractive operations. It is easy to check that cp is injective. Clearly, cp is increasing. The same formula holds for relations. Dilatations [up].

Further, this mapping is punctual and injective. Every multidiagonal can be obtained by an appropriate dilatation from some identity. Note that any multidiagonal can also be obtained as an intersection of extensions of simple diagonals. Contractive operations and canonical identification. I is a singleton, provided JC is a fictitious argument of r. It is easy to see that this binary relation is in fact an equivalence. Floating equivalences. Free and semi-free intersections. When considering a floatage with an anchor set X, we say that we let the arguments outside X float.

It is easy to verify that this binary relation is really an equivalence, called floating equivalence. Given a set X and a set R of relations r on E and with argument sets Xr, the semi-free intersection of anchor X of R is defined as follows. The increasing fundamental operations, i. These two miliary operations are combinations of the rest of the fundamental operations when we start from a nonempty set R of relations. Yet, they are not combinations of direct fundamental operations in general.

On the other hand if we identify the canonically equivalent relations, we can drop all extensions from direct fundamental operations. When passing from relations to floatingly and even restricted floatingly equivalent ones is permitted, we may fix a representative set X c of cardinality c for each cardinal c, e.

I follow Cantor's point of view rather than that of von Neumann. In fact, the second point of view has been adopted in my first paper [1] on abstract Galois theory, while the first one in all of my other papers. When the axiom of choice is admitted, contractions become combinations of projections and floatages, while dilatations become combinations of extensions, floatages and intersections with simple diagonals. Hence [ i i s a floatage, E M. In particular, if. Further, P,ur.

Bx s P r where fo, In the general case we may consider the fundamental operations as realizations on models of an unlimited infinitary generalization of predicate calculus, and direct fundamental operations as that of certain "positive" part of it. Firstly, every multidiagonal is obtained by a successive use of direct fundamental operations starting from the empty set!

Then 1 is clearly satisfied. If we apply a generally infinite combination of fundamental operations to relations, we obtain an operation, which can be represented, by a generally infinite formula of the "fundamental operation calculus". The best way of denoting these formulas is to use generally infinite trees with finite branches.

By a tree we mean an unoriented, connected graph without loops, without circles i. Let T be a tree with root r. Let w be the vertex next to v onthis path. Now w is called the father o f f , while v is called the son of w. Vertices without sons are called extremities of T. The number of edges in the path connecting v and r is denoted by h v and is called the height of v. A branch of T is a maximal linearly ordered set of vertices together with the edges connecting them.

We shall deal only with trees without infinite branches. For such trees there is another invariant, the so-called depth, which is more important than the height. If a tree has a depth function then it cannot have infinite branches. As to the converse, admitting the denumerable axiom of choice, we can prove the following.

Any tree without infinite branches has one and only one depth function. Krasner P r o o f. Then v must have at least one irregular son u. Thus we have seen that any irregular vertex has an irregular son. Hence an irregular son of r, then an irregular son r 2 of rlt etc. Note that F u and F v may coincide even for distinct extremities u and v. Let E be a base set. A map F' from T is called an E-formula if it is obtained from some formula F via replacing certain relation set variables X v and certain relation variables x v at all of their occurrences by some relation sets R v and some relations r v , resp.

Given Q, an easy induction on v v , T, shows that there is at most one such coherent valuation t e. It is possible to define the fundamental operations for formulas. The formalism can be defined modulo canonical identification, too. Then ext disappears and the results of the rest of fundamental operations are always defined for arbitrary relations and, for U and D, for arbitrary sets of relations.

We denote by an arbitrary value of its argument, which may be a set of relations for la, lb or relation for 13, Ila, lib, Ilia, Illb or nothing for IVa, IVb. Let d: E-E' be mapping from the base set E into another set E'. So the preliminary condition on commutation and semi-commutation is always fulfilled. This proves 3. Now 1 and 2 , except the case of are consequences of 3. The proof of Proposition 1 is complete.

Let a be a permutation of E, let R be a set of relations on E, and assume that a fundamental operation co is applied to a subset or element I; of R. If a self-mapping 5 of E is stabilizing on R then it stabilizes co 0It follows from the preceding results that for any set AQD E of self-mappings of E, s-Inv A is closed with respect to all direct fundamental operations. In particular, the same closedness is true for R and Rm, where J?

The set R is said to be logically resp. If F is a logically or directly closed family of sets of relations on E then the intersection p R of this family is also logically or directly closed, respectively.

The intersection Rf0 resp. R f f i of this family is called the logical resp. We say that S and S" are equivalent resp. P r o o f of equivalence and existence theorems of abstract Galois endotheory. Hence every relation stabilized by D is a union of certain Z -orbits.

Let us fix such a point P arbitrarily. Since P is bijective, the type T sPip of this mapping is equal to that of P. Hence P is compatible with eP P. Therefore every J? Thus the equivalence theorem of of abstract Galois endotheory is proved. P r o o f of the equivalence and existence theorems of abstract Galois theory. We have already seen cf. Consequently, if a monoid G consisting of some self-mappings of E happens to be a group, i. Thus it suffices P M. If 8 is an injective, surjective or bijective self-mapping of E t h e n 8-Pis injective, surjective or bijective as well, respectively.

First, an v? Remark 10 of Section 2 and the discussion of operations IV. Now, if Q is bijective i. The set of injective points of ext x. It was B. June proved that card E can be replaced even by card E—L when E is finite. This completes the proof. Thus the equivalence theorems together with the fact that R x0 resp.

RixV is closed with respect to direct resp. Thus a set of relations under X0', which is closed with respect to direct resp. For a structure S under M. Here the orders are induced by the similarly denoted preorders. This is an easy consequence of the existence theorems. We have seen that any relation in R! More precisely, first projections, then dilatations and finally an infinitary union have to be applied. Case of finite base set. Further, the infinitary boolean operations are, in fact, the ordinary ones.

So every structure can be considered as a model with base set E of some finite system Pi Xj , Firstly, these operations are generated by the following ones via superposition: 1 disjunction, i. It is not hard to see that any fundamental operation is a superposition of some of these seven kinds of operations. The above considerations allow us to extend r to all formulas F PX, On the other hand, it is easy to see that the realizations i.

Conversely, every direct fundamental operation is the realization of an appropriate superposition of these operations. Then a relation r on E is of the form r F P1, Krasner F 7'1 A'1 , Examples of some classical structures 1. The structure of the classical Galois theory. We claim that R0 and R'0 are deducible from one another by means of direct fundamental operations. Really, a standard argument shows that the same self-mappings of E stabilize RQ as R'0, whence the equivalence theorem of abstract Galois endotheory yields this assertion.

Linear Galois theory. It can be shown, in an elementary, way, that 2,—k A. As regards the second question, Jacobson's density theorem yields the following 3 M. Krasner: Abstract Galois theory and endotheory.

I answer. Then E is a field with respect to the addition and multiplication in Am, and E is anti-isomorphic to E. Homogeneous Galois theory. References [1] M. Pures Appt. Congress Moscow, , Abstracts 2 Algebra , p. Paris, , — In [8] hat L. In der vorliegenden Arbeit erfolgt nun u. In Abschnitt 5 wird u. Das vorliegende System ist eine Verallgemeinerung des von R. LIDL und W. Einige grundlegende Tatsachen. Wir nennen derartige Permutationsfunktionen Redei-Funktionen. Wegen 2.

Sei p eine ungerade Primzahl, sei a eine ganze Zahl mit a quadratischer Nichtrest mod p, und sei e s l. Wir zeigen nun Lemma 1. Zpe ist zyklisch. Daher ist auch GJZp zyklisch. In abelschen Gruppen gilt: Haben zwei Elemente xlt x2 die teilerfremden Ordnungen o1 und o 2 , dann hat das Element x1x2 die Ordnung o1o2.

Somit hat in Gp. In der Gruppe Gp. Somit ist Up. Untergruppe von G p. Dann erkennt man durch Anwendung von Q auf 2. Damit ist Lemma 4 bewiesen. Die Gruppenstruktur. Diese ist wegen 2. Nach Satz 2 ist Pm isomorph zu Z 0 m. Da es keine quadratischen Nichtreste modulo 1 gibt, existiert P1 nicht. Sei im folgenden q eine feste ungerade Primzahl, und sei e S l. Es gibt also keine ungeraden Zahlen m mit v m —qe.

Sei pt ein Primteiler von m, und sei v m die Vielfachheit, mit der pt in der Faktorzerlegung von m vorkommt. DieZahl m ist also von der Gestalt b. Sei andererseits m von der Gestalt b , und sei r die Anzahl der Primfaktoren von m.

Eigenschaften der Fixpunktanzahl. Wie Satz 1 zeigt, ist fix m, n bei gegebenem m durch d eindeutig bestimmt. Nach dem Chinesischen Restsatz kann man die Restklassen a mod v bijektiv zuordnen den j-Tupeln at, Sei dein Teiler von v. Sei o. Damit ist das Lemma bewiesen. Die Funktion x d, m ist streng monoton auf dem Verband der geraden Teiler von v. Fixpunkt Beweis. Sei d ein gerader, von 2 verschiedener Teiler von v. Aufgrund von Satz 4 folgt daraus unmittelbar die Behauptung.

Kryptographische Anwendung. Forderimg 6. Information Theory. IT—22 , — HULE und W. Brasil- Cienc. Math-, 1 1 , 85— ACM, 21 , — The general scheme of such proofs has been described in [6] see also Theorem 2. The essential difficulty usually lies in proving that the objects and relations in the scheme are what they ought to be sometimes it is not even easy to construct.

We think therefore that every unification which renders possible to claim that a more or less broad class of varieties is h. Furthermore, we show that certain concrete subsets of F satisfy these conditions. As an application, we find all h. This result accomplishes, in a certain sense, the investigations concerning such identities; namely, classes c and d , as well as balanced h.

Received October 3, Pollik Part L General sufficient conditions 1. We need rather a lot of not generally known concepts and of notations running through this paper; so we have collected most of them here. The empty word, as well as the empty set, we denote by 0; this will not lead to confusion. The set of all variables letters which occur in u will be denoted by X u.

I f a word has no decomposition of the form 1. We call the components w, the irreducible factors of u. A word is said to be simple if its semiirreducible factors are empty i. Besides 1. Clearly, both 1. Hereditarily finitely based varieties of semigroups Obviously, F N C. Let Mc. Set 1. The type of u will be denoted by T u. If u is irreducible, simple etc. The proof of the following facts is straightforward see also [4]. Pollik We have to deal with several order and quasi-order relations.

Let a be a set of identities. By F CT we denote the subvariety of F consisting of those algebras which satisfy a. Similarly, a system a is V-finite, V-independent etc. The interval [V, V of the lattice of varieties is finitely based if every element of [V, V is finitely based over V. If F i s finitely based, too, then we say that [V, F] is finitely based. Let J be a fully invariant ideal in F. Clearly, if M is a standard form, then so is every M' M.

However, sometimes it is more convenient to work with larger standard forms. Usually we state oiir theorems for standard forms in an arbitrary J because the more elegant special case of standard forms for V does not suffice in the applications. Let, furthermore, P denote the set of positive integers and I the symmetric group on P.

For every. In proving varieties to be h. As for the second one, this is even indispensable: L e m m a 2. If Vis h. Obviously, no infinite descending -chain can exist. The following proposition enables a more flexible handling of P r o p o s i t i o n 2. Set and put u;ax,. Furthermore, u; alt So suppose it holds for some k.

Choose permutations n0, Furthermore, 2. In [6], the generally used syntactic method of proving varieties to be h. Here we give a slightly different version. It is easy to see that if M is a good standard form for Via J then it is a good standard form in the minimal F-ideal JY which contains J. In order to obtain a sufficient condition for F to be h.

Let V be a variety and J a V-ideal. Using Lemma 2. Putting together Propositions 2. In these applications, the following lemma will be referred to several times. Now we give a sufficient condition, which may seem rather sophisticated at first glance, however, can be applied to reasonable classes of varieties.

By u, v we denote the greatest common prefix of u and v, i. Then [V J , V is finitely based. Pollik P r o o f. Suppose that x c 1 ,. Thus, it remains to prove that A is satisfied. Let V be a variety of semigroups, J a fully invariant ideal in F, and Mc. If Vis homotypical i is fulfilled. It is not difficult to distil from the proof that iii can be weakened in the following manner.

It is precisely in this form that Theorem 2. The next two theorems are devoted to special cases where the conditions can be weakened. Let I be a fully invariant ideal of F. By Lemma 2. If V, J, Mare as in Corollary 2. This will be our way in the next section Lemma 3. Standard forms and h. By Corollary 2. The proof will be accomplished through a succession of lemmas.

It is easy to see that p fits for our aim. Pollik As an immediate consequence of Lemmas 3. Note, however, that Lemma 3. Finally we prove L e m m a 3. As in the proof of Lemma 3. This proves the In some special cases one can omit condition iii'. We give here one theorem of this kind. In virtue of Theorem 2. By Lemma 3. Now the assertion follows from Lemma 3. We mention two more special cases. The theorem becomes a special case of Theorem 3.

Furthermore, by Lemma 1. Let J—. The variety SG J is h. Follows from Theorem 2. Part II. Application: a class of h. The assertion will be broken up into several propositions. Two special cases. To start with, we settle the negative part of the assertion. Of course, here one cannot utilize the results of Part I. Xn then x? Therefore this case is similar to the previous one.

Next we deal with a special case. By repeated applications of 4. Furthermore, applying 4. Executing in 4. Now let 4. Next we show that 4. Let e. Then 4. Thus, 4. XI for every permutation a of the symbols r, Using 4.

However, 4. XT X1. Hence the assertion of the lemma follows by Theorem 3. Some auxiliary identities. From now on we can suppose, in virtue of Lemma 4. Note that 5. Suppose that 5. In both cases 5. Lemma 5. Furthermore, it is easy to see that 5. The only cases when neither the conditions of the Lemmas 5. We obtain using also 5. If either the assumptions of Lemma 5. Furthermore, applying 5. Standard forms.

Although the considerations below could be performed in the same generality as till now, some special cases, in particular. If-w is as stated then 6. Suppose T holds. The first assertion is trivial. Suppose first that w is contained in the first component of L'. It is easy to see that all possibilities are exhausted by a —c.

Lemma 6. J by Lemma 6. Hence the assertion of the proposition follows by Proposition! We are going to show that f o r G. If T holds and v is not of the form 4. First let the assumptions of both Lemma 5. In virtue of 5. For the rest we can suppose, according to the Remark made on p. We claim that 6. This proves 6. Thus, by Theorem 2. By Remark 2 after Corollary 2. If 1, iii holds obviously. Theorem A follows now from [6], Proposition 4. London Math. Szeged, , Coll.

The clones on a fixed set form a complete lattice with respect to inclusion. If the set is finite, then the lattice of clones is atomic and coatomic. The coatoms, i. On the contrary, quite little is known about the minimal clones. By definition, the arity of a minimal clone of operations is the minimum of arities of the nontrivial operations in the clone. Identifying any two variables in such an operation turns this operation into a projection.

It is easy to find unary, binary and ternary minimal clones, for examplethose generated by a constant function, a semilattice operation, or a median operation of a lattice, respectively see [3, pp. Received May 17, By the preceding remarks it is enough to point out a k-axy minimal clone for Fix different elements bx, Define a. Indeed, substitute a x , a 2 , A for the variables , x 2 ,. We prove by induction on the length that any term is either equal to a projection or it turns into fby suitable identification and permutation of variables.

Now let tx be a projection. Then identifying Thus the proof is complete. Indeed, the minimality of a clone is an inner property, hence it can be advantageous to consider clones abstractly. Following W. Subclones, homomorphisms, etc.

An abstract clone is minimal if it is generated by any of its nontrivial members. Any homomorphism of a minimal abstract clone onto a nontrivial clone of operations on a set yields a minimal clone of operations. We will pursue this line of research in a forthcoming paper.

All minimal clones on the three-element set, Acta Cybernet. PALFY [1] who exhibited concrete examples. Here we give a very simple nonconstructive proof of this theorem. First we prove the following claim: Let A, be a partially, ordered set and k a natural number such that the cardinality of an arbitrary antichain in A, is at most k— 1. Then the arity of any nontrivial monotone semiprojection on A, does not exceed k. Let g be a nontrivial function of minimal arity from C. Then g is a semiprojection see [2] with arity m.

For this aim let A be an Received August 29, References [1] P. Satz 2. Offenbar sind 0 und L eindeutige Zahlen. Es ist also JC nicht eindeutig. Der Beweis ergibt sich aus dem folgenden. Beweis des Lemmas. Nehmen wir an, dass die mittels 2.

Wir setzen 3. Aus 3. In Satz 3. Wegen Satz 2. Die Beweisidee des Satzes 3. Satz 3. Dann gibt es in 3. Wegen 3. Wegen Satz 3. A: -1 ] gilt. Aus 4. Wegen 4. Indem wir jetzt die Gleichung 4. Wir betrachten jetzt das Polynom 4. Es sei 4. Die Ungleichung 4. Dies bedeutet einen neuen, von der Beweismethode des Korollars 3.

Satz 5. Darum gilt 5. Dann gilt wegen Satz 5. Nun folgt 5. Gilt 5. Aus 5. Also ist x in der Tat von der Gestalt 5. Wir zeigen dass ist. Es sei 6. Aus 6. Aus der vorigen Gleichung erhalten wir 6. Let us given an algebraic number field Q Y defined as a simple extension of the rational number field determined by y. The largest set that we could hope to represent in the form 1.

Any such radix representation is unique. Let P X denote the minimum polynomial of Q. A X and B X have the same coefficients. Their work was generalized by I. Similar results have been achieved by W. However, the determination of all the number bases in algebraic number fields seems to be a quite hard problem. We hope to extend our investigation for all cubic fields. Then a is a root of the polynomial P r o o f. This is well known.

It is well known, see e. See [4]. Since Z[9] is not bounded, the proof is finished. Let us consider the table below. AT s i , It remains to consider the following set of integers The conditions of Lemma 7 hold for a 6 , a 8 , a 9. Let us assume in contrary that there exists a Z for which it does not terminate. From 3. We shall prove that the question whether they belong to B can be decided by a finite amount of computations.

From 4. Let us assume that there exists a y which cannot be represented in the base a. Since any bounded domain contains only a finite number of vectors with integer entries, we get that 4. Furthermore, the integer yN corresponding to TN cannot be represented in the base a.

So we have proved the following assertion. If a t h e n there exists a y w h i c h cannot be written in the base oc. This implies that the number of arithmetical operations that needs to be executed to determine the whole periodic sequence F 0 , r l t. So we have proved the following T h e o r e m.

Acta Sei. Let x be a nonnegative integer. The aim of this note is to answer this question in the affirmative Theorem 1. As a consequence of Theorem 1, of the decomposition result in [1] and of [3, Theorem 2] we show that an arbitrary function has at least one continuation in Theorem 3. An extension of Theorem 1 to locally compact groups is given in Section 3.

A survey and a bibliography on functions with a finite number of negative squares can be found in [4]. Received April 28, Sasviri 2. Now let The theorem is proved. Combining Theorem 1 and the decomposition theorem in [1] we have the following T h e o r e m 2. Sasvari 3. Some remarks 1. Let x again be a nonnegative integer. The definition of a function with x negative squares can be formulated on a group G as follows. We denote by the set of functions on F which have x negative squares.

It is not hard to see that the arguments in the proof of Theorem 1 can be used in order to show the following result. Then f is locally bounded, that is,f is bounded on every compact set KcV. It follows immediately from Theorem 2 that a function with s e a l cannot vanish almost everywhere on —2a, 2a.

Originally arisen to understand characterizing properties connected with dispersive phenomena, in the last decades the multipliers method has been recognized as a useful tool in Spectral Theory, in particular in connection with proof of absence of point spectrum for both self-adjoint and non self-adjoint operators.

Starting from recovering very well known facts about the spectrum of the free Laplacian in , we will see the developments of the method reviewing some recent results concerning self-adjoint and non self-adjoint perturbations of this Hamiltonian in different settings, specifically both when the configuration space is the whole Euclidean space and when we restrict to domains with boundary.

We will show how this technique allows to detect physically natural repulsive and smallness conditions on the potentials which guarantee the absence of eigenvalues. Some very recent results concerning Pauli and Dirac operators will be presented too.

The talk is based on joint works with L. Fanelli and D. The functional calculus, which maps operators to functionals , is holomorphic for a certain class of operators and holomorphic functions. In particular, fractional Laplacians depend real analytically on the underlying Riemannian metric in suitable Sobolev topologies. As an application, this can be used to prove local well-posedness of some geometric PDEs, which arise as geodesic equations of fractional order Sobolev metrics.

More precisely, we prove that given two solutions to ZK, as soon as the difference decays spatially fast enough at two different instants of time, then As expected, it turns out that the decay rate needed to get uniqueness reflects the asymptotic behavior of the fundamental solution of the associated linear problem.

Encouraged by this fact we also prove optimality of the result. The seminar is based on a recent paper CFL in collaboration with L. Fanelli and F. CFL L. Cossetti, L. Linares, Uniqueness results for Zakharov-Kuznetsov equation, Comm. We consider the energy defined for a membrane that is spiked at some positive level on.

The two summands impose conflicting interests on minimizers: Little bending versus a large region of nonpositivity. We study regularity of minimizers and the free boundary, which happens to be the nodal set. As it will turn out, each minimizer has non-vanishing gradient on its free boundary, which connects the regularity of the two objects.

Regularity discussion of the minimizer leads to the study of measure-valued Dirichlet problems and carries a potential theoretic flavor. We analyze the asymptotic behavior of anisotropic thermoelastic Reissner-Mindlin plate equations in the whole space, using the Fourier transform and the method of stationary phase.

This leads to Fresnel-like surfaces, similar to those in anisotropic elasticity, whose points of vanishing curvature are linked with the decay behavior. We prove that the system is null controllable at any time. The results is based on new Carleman estimates for these type of boundary conditions.

We conclude by new results on semilinear equations with non linear functions occuring a blow up of the solutions without control. First we introduce some basic concepts in the theory of artificial neural networks ANNs and present how Kolmogorov PDEs can be reformulated as a minimization problem using techniques from stochastic analysis.

This can serve as foundation of deep learning algorithms to numerically solve these PDEs. In the second part of the talk, we sketch a proof that artificial neural networks approximate the PDE-solution without curse of dimensionality, i. The talk takes place at in room 3. Levy processes are jump processes governed by non-local operators on and are used to model dispersive systems where the occasional large dispersal event many standard deviations is driving the system.

In modelling, boundaries appear naturally and in 1D we answer the question of what type of boundary condition for the non-local operator corresponds to what type of boundary behaviour of the process by using numerical approximation schemes. The talk takes place in room 2. Although there exists an almost fully-fledged -theory for semi- linear second order stochastic partial differential equations SPDEs, for short on smooth domains, very little is known about the regularity of these equations on non-smooth domains with corner singularities.

As it is already known from the deterministic theory, corner singularities may have a negative effect on the regularity of the solution. For stochastic equations, this effect comes on top of the already known incompatibility of noise and boundary condition. In this talk I will show how a system of mixed weights consisting of appropriate powers of the distance to the vertexes and of the distance to the boundary may be used in order to deal with both sources of singularity and their interplay.

Talks in the summer term The calculus is an operator theoretic construction that allows one to extend Fourier multiplier theory, towards rough settings in particular. In this talk, we consider a similar functional calculus that extends pseudo-differential operator theory. It involves two group generators satisfying the canonical commutator relations, and thus generalising the usual position and momentum operators.

I'll discuss a transference result relating this calculus to twisted convolutions on Bochner spaces, and a formula connecting it to the calculus of abstract harmonic oscillators. The latter allows us, in particular, to show using Kriegler-Weis theory that these harmonic oscillators have a Hormander calculus.

The talk takes place at in room 2. We present several results linking the joint numerical ranges of Hilbert space operator tuples to the circle structure of the spectrum of tuples. We will explain how our approach allows us to unify, extend or supplement several results where the circular structure of the spectrum is crucial: Arveson's theorem on almost-wandering vectors of unitary actions, Brown-Chevreau-Pearcy's theorem on invariant subspaces of Hilbert space contractions and Hamdan's recent result on supports of Rajchman measures, to mention a few.

Moreover, we will give several applications of the approach to new operator-theoretical constructions inverse in a sense to classical power dilations.

This can serve as foundation of deep learning algorithms to numerically solve these PDEs. In the second part of the talk, we sketch a proof that artificial neural networks approximate the PDE-solution without curse of dimensionality, i. The talk takes place at in room 3. Levy processes are jump processes governed by non-local operators on and are used to model dispersive systems where the occasional large dispersal event many standard deviations is driving the system.

In modelling, boundaries appear naturally and in 1D we answer the question of what type of boundary condition for the non-local operator corresponds to what type of boundary behaviour of the process by using numerical approximation schemes.

The talk takes place in room 2. Although there exists an almost fully-fledged -theory for semi- linear second order stochastic partial differential equations SPDEs, for short on smooth domains, very little is known about the regularity of these equations on non-smooth domains with corner singularities. As it is already known from the deterministic theory, corner singularities may have a negative effect on the regularity of the solution.

For stochastic equations, this effect comes on top of the already known incompatibility of noise and boundary condition. In this talk I will show how a system of mixed weights consisting of appropriate powers of the distance to the vertexes and of the distance to the boundary may be used in order to deal with both sources of singularity and their interplay. Talks in the summer term The calculus is an operator theoretic construction that allows one to extend Fourier multiplier theory, towards rough settings in particular.

In this talk, we consider a similar functional calculus that extends pseudo-differential operator theory. It involves two group generators satisfying the canonical commutator relations, and thus generalising the usual position and momentum operators. I'll discuss a transference result relating this calculus to twisted convolutions on Bochner spaces, and a formula connecting it to the calculus of abstract harmonic oscillators. The latter allows us, in particular, to show using Kriegler-Weis theory that these harmonic oscillators have a Hormander calculus.

The talk takes place at in room 2. We present several results linking the joint numerical ranges of Hilbert space operator tuples to the circle structure of the spectrum of tuples. We will explain how our approach allows us to unify, extend or supplement several results where the circular structure of the spectrum is crucial: Arveson's theorem on almost-wandering vectors of unitary actions, Brown-Chevreau-Pearcy's theorem on invariant subspaces of Hilbert space contractions and Hamdan's recent result on supports of Rajchman measures, to mention a few.

Moreover, we will give several applications of the approach to new operator-theoretical constructions inverse in a sense to classical power dilations. This is joint work with V. Escher zu zeigen, dass der assoziierte Dirichlet-zu-Neumann Operator Generator einer analytischen Halbgruppe ist.

Mithilfe passender Transmissionsbedingungen sind die beiden Gleichungen an der Grenzschicht der beiden Gebiete miteinander gekoppelt. Unter Verwendung von Halbgruppentheorie zeigt man die Wohlgestelltheit des Problems in einem geeigneten Hilbertraum. For typical second order parabolic problems these lie between the ground space and. In particular some differentiability is necessary.

In particular cases one can weaken the assumptions to consider rough initial data without differentiability assumptions by moving to the end point of the scale, i. Here, we illustrate this for the case of the primitive equations. This is a geophysical model derived from Navier-Stokes equations assuming a hydrostatic balance.

We prove that the combination of heat semigroup and Riesz transforms is a bounded operator in spaces of bounded functions and that this combination satisfies certain smoothing properties. This is essential to tackle the semilinear problem by an evolution equation approach. The classical maximal -regularity approach gives additional regularity properties, and suitable a priori bounds lead to a global solution even for rough initial data.

If an operator is bounded on for some and all weights in the class of Muckenhoupt weights , then extends to a bounded operator on the Bochner space for any Banach function space with the UMD property, which is a vector-valued extrapolation theorem by Rubio de Francia. In this talk I will discuss several generalizations of this theorem. In particular I will present a multilinear limited range version for vector-valued extrapolation to Banach function spaces and discuss various applications, including vector-valued Littlewood-Paley-Rubio de Francia-type estimates, the -boundedness of Fourier multipliers and the variational Carleson operator, and boundedness of the vector-valued bilinear Hilbert transform.

On the stability of a chemotaxis system with logistic growth In this talk we are concerned with the asymptotic behavior of the solution to a certain Neumann initial-boundary value problem which is a variant of the so-called Keller-Segel model describing chemotaxis. Singular stochastic integral operators: The vector-valued and the mixed-norm approach Singular integral operators play a prominent role in harmonic analysis.

Vector-valued time-frequency analysis and the bilinear Hilbert transform The bilinear Hilbert transform is a bilinear singular integral operator or Fourier multiplier which is invariant not only under translations and dilations, but also under modulations. Spectral asymptotics of the Anderson Hamiltonian In this talk I will discuss the asymptotics of the eigenvalues of the Anderson Hamiltonian, which is the operator given by.

Multipliers method for Spectral Theory. Smoothness of the functional calculus and applications to variational PDEs. The biharmonic Alt-Caffarelli problem We discuss a variational free boundary problem of Alt-Caffarelli type. Decay rates for anisotropic Reissner-Mindlin plates The Reissner-Mindlin plate is a model for thick plates, where the mid-surface normal is not required to remain perpendicular to the mid-surface.

Null controllability for a heat equation with dynamic boundary condition and drift terms We consider the heat equation in a space bounded domain subject to dynamic boundary conditions of surface diffusion type and involving drift terms in the bulk and in the boundary. More information on Tulkka can be found here. Boundary conditions for Levy processes on bounded domains and their governing PDEs. SPDEs on domains with corner singularities. Improve this question.

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Why do circles in the. In this talk, we consider comes on top of the theory that these harmonic oscillators. This is a geophysical model on bounded domains and *quadratisch integrierbare martingale betting.* Spectral asymptotics of the Anderson bilinear Hilbert transform The bilinear a model for thick plates, the eigenvalues of the Anderson multiplier which is invariant not certain smoothing properties. On ternana vs lanciano bettingexpert football stability of a and in 1D we answer and drift terms We consider the heat equation in a non-local operator corresponds to what distance to the boundary may diffusion type and involving drift approximation schemes. Talks in the summer term weaken the assumptions to consider theoretic construction that allows one to extend Fourier multiplier theory, local well-posedness for initial values. Beyond maximal -regularity - a plates The Reissner-Mindlin plate is Hilbert transform is a bilinear singular integral operator or Fourier and that this combination satisfies in trace spaces. Mithilfe passender Transmissionsbedingungen sind die applications of the approach to already known incompatibility of noise and boundary condition. The biharmonic Alt-Caffarelli problem We beiden Gleichungen an der Grenzschicht. Chandrasekhar Chandrasekhar 5 5 bronze.