This article is devoted to sequences ${(u_n)}_n$ of subharmonic functions in ${ℝ}^N$, with finite order, whose means $J_{u_n}(r)$ (over spheres centered at the origin, with radius r) satisfy such a condition as: $\forall r>0$, $\exists A_r>0$ such that $J_{u_n}(r)\le A_r$, $\forall n\in 𝐍$. The paper investigates under which conditions one may extract a pointwise or uniformly convergent subsequence.

Let ${𝕊}^1$ and $𝔻$ be the unit circle and the unit disc in the plane and let us denote by $𝒜({𝕊}^1)$ the algebra of the complex-valued continuous functions on ${𝕊}^1$ which are traces of functions in the Sobolev class $W^{1,2}(𝔻)$. On $𝒜({𝕊}^1)$ we define the following norm $$\parallel u\parallel ={\parallel u\parallel }_{L^{\mathrm{\infty }}({𝕊}^1)}+{\left({\iint }_{𝔻}{|\nabla \tilde{u}|}^2\right)}^{1/2}$$ where is the harmonic extension of $u$ to $𝔻$. We prove that every isomorphism of the functional algebra $𝒜({𝕊}^1)$ is a quasitsymmetric change of variables on ${𝕊}^1$.

Let $D$ be an integral domain with quotient field $D$. Recall that $D$ is Schreier if $D$ is integrally closed and for all $x,y,z\in D\setminus \{0\}$, $x|yz$ implies that $x=r\cdot s$ where $r|y$ e $s|z$. A GCD domain is Schreier. We show that an integral domain $D$ is a GCD domain if and only if (i) for each pair $a,b\in D\setminus \{0\}$, there is a finitely generated ideal $B$ such that $aD\bigcap bD=B_v$ and (ii) every quadratic in $D[X]$ that is a product of two linear polynomials in $K[X]$ is a product of two linear polynomials in $D[X]$. We also show that $D$ is Schreier if and only if every...

Si considera l’equazione astratta $B{A}_{1}u+A_{0}u=h$, dove $A_i$ $(i=0,1)$ e $B$ sono convenienti operatori lineari chiusi fra spazi di Banach, $A_i$ non è necessariamente invertibile, e $A_0$, $A_1$ non commutano con $B$. Si studiano esistenza ed unicità delle soluzioni. Si indicano alcune applicazioni a certe equazioni differenziali degeneri o singolari.

The following results are proved: (i) if $\alpha$, $\beta \in {ℝ}^{+}$ and $A$ is a sectorial operator, then the set $\{t^{\alpha }{A}^{\beta }(t+A);t>0\}$ is bounded; (ii) the same set of operators is R-bounded if $A$ is R-sectorial.

Sia $K\subset \subset {ℝ}^n$ un compatto, $f\ge 0$ una funzione analitica all'intorno di $K$, ed $m$ la massima molteplicità in $K$ degli zeri di $f$; si prova che la potenza $f^{\lambda }$ ($\lambda \in ℂ$, $Re\lambda >\mathrm{—}\frac{1}{m}$) è integrabile in $K$. L'estensione meromorfa dell'applicazione $\lambda \to f^{\lambda }$ da $Re\lambda >0$ a tutto $ℂ$ (con valori in ${𝒟}^{\prime }(K)$ anziché in $L^1(K)$) era già stata provata in [1] e [2].

In questa Nota, dati uno spazio metrico perfetto $X$ ed un suo sottoinsieme $K$ chiuso e raro, si dimostra l'esistenza di una funzione continua $f:X\to [0,1]$ tale che $int(f^{-1}(t))=\mathrm{\varnothing }$ per ogni $t\in [0,1]$, $f(x)=0$ per ogni $x\in K$ e $f(y)=1$ per qualche $y\in X\setminus K$. In particolare, ciò permette di dare risposta simultaneamente a due questioni poste in [2]. Si mettono in evidenza, poi, ulteriori conseguenze di tale risultato.

Dato un cono $V$ aperto non vuoto, convesso, regolare e affinemente omogeneo in uno spazio vettoriale reale $W$ di dimensione finita si prova che per ogni $v$ appartenente a $V$ esiste un diffeomorfismo $E_v:W\to V$ che soddisfa le condizioni seguenti E1) $E_v(0)=v$; E2) $det(d{E}_v(y))={\mathrm{\Phi }}_V{(E_v(y))}^{-1}$ per ogni $y$ appartenente a $W$ ove ${\mathrm{\Phi }}_V:V\to {𝐑}^{+}$ è la funzione caratteristica di $V$.

Siano $d$, $g$, $t$ interi con $0\le t\le g$; se esiste in ${\mathbb{P}}_3(ℂ)$ una curva connessa, non singolare di grado $d$ e genere $g$, allora esiste in ${\mathbb{P}}_3(ℂ)$ una curva irriducibile di grado $d$, genere aritmetico $g$ e $t$ nodi.

Ogni logica $L$ genera canonicamente la $L$-equivalenza ${\equiv }_L$e la $L$-immersione $\underset{𝐿}{\to }$ proprio come la logica del primo ordine genera l’equivalenza elementare $\equiv$ e l’immersione elementare $\lesssim$. Astraendo da $L$, è interessante studiare in sè relazioni d’equivalenza e di immersione generali tra strutture. Mostriamo che esiste una corrispondenza biunivoca tra relazioni d’equivalenza con la proprietà di Robinson e relazioni d’immersione con la proprietà di Amalgamazione Forte ($A{P}^{+}$). Caratterizziamo algebricamente...

We consider the Grassmannian $𝔾r(k,n)$ of $(k+1)$-dimensional linear subspaces of $V_n=H^0({\mathbb{P}}^1,{𝒪}_{{\mathbb{P}}_1}(n))$. We define ${𝔛}_{k,r,d}$ as the classifying space of the $k$-dimensional linear systems of degree $n$ on ${\mathbb{P}}^1$, whose bases realize a fixed number $r$ of polynomial relations of fixed degree $d$, say $r$ syzygies of degree $d$. Firstly, we compute the dimension of ${𝔛}_{k,r,d}$. In the second part we make a link between ${𝔛}_{k,r,d}$ and the Poncelet varieties. In particular, we prove that the existence of linear syzygies implies the existence of singularities on the...

Inspired by a result from Leibov, we find that the supremum defining the $BLO$ norm in $[0,1]$ is actually attained by a specific sub-interval of $[0,1]$ for $f\in VLO([0,1])$

Si considerano due spazi $S_{\mu }$ e $S_{\nu }^{\ast }$, Riemanniani e a metrica eventualmente indefinita, riferiti a sistemi di co-ordinate $\mathrm{\varnothing }$ e ${\mathrm{\varnothing }}_{\nu }^{\ast }$; e inoltre un doppio tensore $T_{\mathrm{\cdots }}^{\mathrm{\cdots }}$ associato ai punti ${\mathrm{\varnothing }}^{-1}(x)\in S_{\mu }$ e ${\mathrm{\varnothing }}^{\ast -1}(y)\in S^{\ast }$. Si pensa $T_{\mathrm{\cdots }}^{\mathrm{\cdots }}$ dato da una funzione ${\tilde{T}}_{\mathrm{\cdots }}^{\mathrm{\cdots }}$ di $m$ altri tali doppi tensori e di variabili puntuali $x(\in {\mathrm{\Re }}^{\mu })$, $t\in \mathrm{\Re }$ e $y(\in {\mathrm{\Re }}^{\nu })$; poi si considera la funzione composta $${\widehat{T}}_{\mathrm{\cdots }}^{\mathrm{\cdots }}(x,t,y)={\tilde{T}}_{\mathrm{\cdots }}^{\mathrm{\cdots }}[\underset{1,\mathrm{\dots },m}{\underbrace{{\breve{H}}_{\mathrm{\cdots }}^{\mathrm{\cdots }}(x,t,y),\mathrm{\dots },{\breve{H}}_{\mathrm{\cdots }}^{\mathrm{\cdots }}(x,t,y)}},x,t,y].$$ Nella Parte I si scrivono due regole per eseguire la derivazione totale di questa, connessa con una mappa $\widehat{ℰ}$ $(={\widehat{ℰ}}_t)$ fra $S_{\nu }^{\ast }$ e $S_{\mu }$; una è a termini generalmente non covarianti...

Let $F\subset C^{\ast }(X)$ be a vector sublattice over $ℝ$ which separates points from closed sets of $X$. The compactification $e_{F}X$ obtained by embedding $X$ in a real cube via the diagonal map, is different, in general, from the Wallman compactification $\omega (Z(F))$. In this paper, it is shown that there exists a lattice $F_z$ containing $F$ such that $\omega (Z(F))=\omega (Z(F_z))=e_{F}X$. In particular this implies that $\omega (Z(F))\ge e_{F}X$. Conditions in order to be $\omega (Z(F))=e_{F}X$ are given. Finally we prove that, if $\alpha X$ is a compactification of $X$ such that $C{l}_{\alpha X}(\alpha X\setminus X)$ is $0$-dimensional, then there is an...

The object of this note is to study certain 2-dimensional $\lambda$-adic representations of $Gal(\overline{Q}/Q)$; fixed $p_1,\mathrm{\dots },p_n$ distinct primes, we will consider representations $\rho :G\to G{L}_2(A)$, given by the matrix which are unramified outside $p_1,\mathrm{\dots },p_n,\mathrm{\infty }$ and the residue characteristic of $\lambda$, which are a product of $m$ representations over finite extensions of the ring of Witt vectors of the residue field and which are reducible modulo $\lambda$. In analogy with the theory of the modular representations, we will introduce the analogue of Mazur's Hecke...

Si supponga che l'anello $𝐑$ ammetta una decomposizione come prodotto subdiretto $𝐑=\underset{\widetilde{\alpha \in 𝚲}}{\times }{𝐑}_{\alpha }$ di anelli ${𝐑}_{\alpha }\ne \mathrm{}$, tali che per ${𝐒}_{\alpha }=𝐑\cap {𝐑}_{\alpha }$ si abbia ${Ann}_{{𝐑}_{\alpha }}{𝐒}_{\alpha }=\mathrm{}$ ($\forall \alpha \in A$), e sia $𝐒={\oplus }_{\alpha \in 𝐀}{𝐒}_{\alpha }$. Si scelga un $𝐑$-modulo (destro) $𝐌$ che sia libero da torsione rispetto ad $𝐒$, cioè ${Ann}_{𝐌}𝐒=\mathrm{}$; allora $𝐌$ può essere rappresentato come prodotto subdiretto irridondante $𝐌\cong \underset{\widetilde{\alpha \in 𝚲}}{\times }{𝐌}_{\alpha }$ degli ${𝐑}_{\alpha }$-moduli ${𝐌}_{\alpha }$ liberi da torsione rispetto ad ${𝐒}_{\alpha }$. Si fa uno studio di un subprodotto generale di una classe $𝐂$ di $𝐑$-moduli ${𝐌}^{(𝐢)}$ $(i\in I)$, dove $𝐂$ è determinato per mezzo di epimorfismi e relazioni. ...

Se $X$ ed $Y$ sono spazi topologici, un'applicazione $f:X\to Y$ dicesi semicontinua [3] se, per ogni insieme aperto $V$ di $Y$, $f^{-1}(V)$ risulta un insieme semiaperto di $X$. Qui si studiano tali applicazioni e si dànno tre loro caratterizzazioni.

Si costruiscono curve di genere $g=4n\mathrm{—}3$, $n\ge 3$ che hanno $2^{n-3}(2^{n-2}-1)$ fasci semicanonici $L$ tali che $h^0(L)=4$. Per $n+3$ si dimostra che gli $L$ sono molto ampi.

The range of the bigenus $P_2$ is one of the unsolved problems concerning smooth complex projective regular threefolds of general type with $p_g=0$: The examples in the literature have $P_2\le 5$. In the present paper we present a non-singular threefold with $p_g=q_1=q_2=0$; $P_2=6$; the bicanonical map is stably birational.

In this paper we determine all squarefree completely lexsegment ideals which have a linear resolution. Let $M_d$ denote the set of all squarefree monomials of degree $d$ in a polynomial ring $k[x_1,\mathrm{\dots },x_n]$ in $n$ variables over a field $k$. We order the monomials lexicographically such that $x_1>x_2>\mathrm{\dots }>x_n$, thus a lexsegment (of degree $d$) is a subset of $M_d$ of the form $L(u,v)=\{w\in M_d:u\ge w\ge v\}$ for some $u,v\in M_d$ con $u\ge v$. An ideal generated by a lexsegment is called a lexsegment ideal. We describe the procedure to determine when such an ideal has a linear...

Let $G$ be a group and $m$ be an integer greater than or equal to $2$. $G$ is said to be $m$-permutable if every product of $m$ elements can be reordered at least in one way. We prove that, if $G$ has a centre of finite index $z$, then $G$ is $(1+[z/2])$-permutable. More bounds are given on the least $m$ such that $G$ is $m$-permutable.

Se $X$ ed $Y$ sono spazi topologici, una funzione $f:X\to Y$ è detta regolarmente chiusa [5] se essa trasforma ogni insieme regolarmente chiuso di $X$ in un insieme chiuso di $Y$. Si dimostra che una funzione regolarmente chiusa $f:X\to Y$ risulta chiusa se $X$ è normale.

Let $X$, $X^{\prime }$, $Y$ be smooth projective complex curves with $Y$ curve of genus $\ge 1$. Let $d$ be an integer $\ge 3$, let $\underset{¯}{e}=(e_1,\mathrm{\dots },e_r)$ be a partition of $d$ and let $|e|={\sum }_{i=1}^r(e_i-1)$. Let $X\stackrel{𝜋}{\to }{X}^{\prime }\stackrel{𝑓}{\to }Y$ be a sequence of coverings where $\pi$ is a degree 2 branched covering and $f$ is a degree $d$ covering, with monodromy group $S_d$, branched in $n_2+1$ points, one of which is special point $c$ whose local monodromy has cycle type given by $\underset{¯}{e}$. Moreover the branch locus of the covering $\pi$ is contained in $f^{-1}(c)$. In this paper we prove the irreducibility...

Let $\mathrm{\Delta }$ be thick affine building of type ${\tilde{G}}_2$ the Laplace operators of $\mathrm{\Delta }$; associated with a pair $(y_1,y_2)$; is the Poisson transform of a suitable finitely additive measure on the maximal boundary $\mathrm{\Omega }$ of $\mathrm{\Delta }$; by using only the combinatorial structure of $\mathrm{\Delta }$.

Siano: $S$ una superficie algebrica proiettiva complessa non singolare, $K$ un divisore canonico ed $H$ un divisore molto ampio su $S$. Questo lavoro ha per oggetto lo studio dell'indice di autointersezione ${(K+H)}^2$. Si dimostra, innanzitutto, la disuguaglianza $${(K+H)}^2\ge 0$$, nell'ipotesi che la superficie $S^{\prime }$ ottenuta immergendo $S$ mediante il sistema lineare completo $|H|$ non sia uno scroll. Questa disuguaglianza è connessa con alcuni risultati di Sommese e Van de Ven sulla generazione del fascio ${𝒪}_s(K+H)$. La dimostrazione...

We fix $\mathrm{\ell }$ a prime and let $M$ be an integer such that $\mathrm{\ell }|̸M$; let $f\in S_2({\mathrm{\Gamma }}_1(M{\mathrm{\ell }}^2))$ be a newform supercuspidal of fixed type at $\mathrm{\ell }$ and special at a finite set of primes. For an indefinite quaternion algebra over $Q$, of discriminant dividing the level of $f$, there is a local quaternionic Hecke algebra $T$ associated to $f$. The algebra $T$ acts on a module $M_f$ coming from the cohomology of a Shimura curve. Applying the Taylor-Wiles criterion and a recent Savitt's theorem, $T$ is the universal deformation ring of a global...

We study the problem of correctors in the framework of the homogenization of linear parabolic equations posed in a heterogeneous medium $\mathrm{\Omega }$ made of two materials. The first one is located in a set $F_{ϵ}$ of cylindrical parallel fibers periodically distributed with a period of size $ϵ$, and the second one is located in the "matrix" $M_{ϵ}=\mathrm{\Omega }\setminus F_{ϵ}$. The ratio between the conductivity coefficients of the two materials is of order $1/{ϵ}^2$. After writing the homogenized problem, we give a corrector result and prove...

Si dimostra che il funzionale ${\int }_{\mathrm{\Omega }}f(u,Du)𝑑x$ è semicontinuo inferiormente su $W_{loc}^{1,1}(\mathrm{\Omega })$, rispetto alla topologia indotta da $L_{loc}^1(\mathrm{\Omega })$, qualora l’integrando $f(s,p)$ sia una funzione non-negativa, misurabile in $s$, convessa in $p$, limitata nell’intorno dei punti del tipo $(s,0)$, e tale che la funzione $s\mapsto f(s,0)$ sia semicontinua inferiormente su $𝐑$.

The aim of this work is to prove a chain rule and an $L^1$-lower semicontinuity theorems for integral functional defined on $BV(\mathrm{\Omega })$. Moreover we apply this result in order to obtain new relaxation and $\mathrm{\Gamma }$-convergence result without any coerciveness and any continuity assumption of the integrand $f(x,s,p)$ with respect to the variable $s$.

Let $X\subset {\mathbb{P}}^N$ be a smooth irreducible non degenerate surface over the complex numbers, $N\ge 4$. We define the projective genus of $X$, denoted by $PG(X)$, as the geometric genus of the singular curve of the projection of $X$ from a general linear subspace of codimension four. Denote by $g(X)$ the sectional genus of $X$. In this paper we conjecture that the only surfaces for which $PG(X)=g(X)-1$ are the del Pezzo surface in ${\mathbb{P}}^4$, in ${\mathbb{P}}^5$ and a conic bundle of degree 5 in ${\mathbb{P}}^4$. We prove that for $N\ge 5$ if $PG(X)=g(X)-1+\lambda$, $\lambda$ a non negative integer, then $g(X)\le \lambda +1+\alpha$ where...

We study the irreducibility of the Dirichlet polynomial $P_G(s)$ when $G$ is the alternating group on $p+1$ elements with $p$ prime and we prove that $P_G(s)$ is irreducible for infinitely many choiches of $p$.

We consider an SPDE in a Hilbert space $H$ of the form $dX(t)=(AX(t)+b(X(t)))dt+\sigma (X(t))dW(t)$, $X(0)=x\in H$ and the corresponding transition semigroup $P_{t}f(x)=𝔼[f(X(t,x))]$. We define the infinitesimal generator $\overline{L}$ of $P_t$ through the Laplace transform of $P_t$ as in [1]. Then we consider the differential operator $L\phi =\frac{1}{2}\mathrm{Tr}[\sigma (x){\sigma }^{*}(x)D^2\phi ]+⟨b(x),D\phi ⟩$ defined on a suitable set $V$ of regular functions. Our main result is that if $V$ is a core for $\overline{L}$, then there exists a unique solution of the martingale problem defined in terms of $L$. Application to the Ornstein-Uhlenbeck equation and to some regular...

CONTENTSIntroduction.......................................................................................61. Definition of certain special forms...........................................62. Statement of results...................................................................83. Proof of Theorem 2.....................................................................94. Preliminaries for Theorem 1.....................................................105. Further preliminaries for Theorem...

Sia $K\subset \subset {ℝ}^n$ un compatto, $f\ge 0$ una funzione analitica all'intorno di $K$, ed $m$ la massima molteplicità in $K$ degli zeri di $f$; si prova che la potenza $f^{\lambda }$ ($\lambda \in ℂ$, $Re\lambda >\mathrm{—}\frac{1}{m}$) è integrabile in $K$. L'estensione meromorfa dell'applicazione $\lambda \to f^{\lambda }$ da $Re\lambda >0$ a tutto $ℂ$ (con valori in ${𝒟}^{\prime }(K)$ anziché in $L^1(K)$) era già stata provata in [1] e [2].

Siano $X$ una varietà algebrica proiettiva complessa non singolare tridimensionale, $L$ un fibrato lineare ampio su $X$, e $n\ge 2$ un intero. Si prova che, a meno di contrarre un numero finito di $-1$-piani di $X$, il fibrato $K_X\otimes L^n$ è ampio ad eccezione di alcuni casi esplicitamente descritti. Come applicazione si dimostra l'ampiezza del divisore di ramificazione di un qualunque rivestimento di ${𝐏}^3$ o della quadrica liscia di ${𝐏}^4$.

Ogni logica $L$ genera canonicamente la $L$-equivalenza ${\equiv }_L$e la $L$-immersione $\underset{𝐿}{\to }$ proprio come la logica del primo ordine genera l’equivalenza elementare $\equiv$ e l’immersione elementare $\lesssim$. Astraendo da $L$, è interessante studiare in sè relazioni d’equivalenza e di immersione generali tra strutture. Mostriamo che esiste una corrispondenza biunivoca tra relazioni d’equivalenza con la proprietà di Robinson e relazioni d’immersione con la proprietà di Amalgamazione Forte ($A{P}^{+}$). Caratterizziamo algebricamente...

Sfruttando i risultati di [1], si prova che le derivate spaziali $D^{\alpha }u$ di ordine $|\alpha |$ con delle soluzioni in $Q$ di un sistema parabolico quasilineare di ordine $2m$ con andamenti strettamente controllati, sono parzialmente hölderiane in $Q$ con esponente di hölderianità decrescente al crescere di $|\alpha |$.

Let $I=[a,b]\subset ℝ$, let , let $u$ and $v$ be positive functions with $u\in L_{p^{\prime }}(I)$ e $v\in L_q(I)$ and let $T:L_p(I)\to L_q(I)$be the Hardy-type operator given by $$\left(Tf\right)\left(x\right)=v\left(x\right){\int }_a^{x}f\left(t\right)u\left(t\right)dt,x\in I.$$ We show that the asymptotic behavior of the eigenvalues$\lambda$of the non-linear integral system $$g\left(x\right)=\left(TF\right)\left(x\right){\left(f\left(x\right)\right)}_{\left(p\right)}=\lambda \left(T^{*}{g}_{\left(p\right)}\right))\left(x\right)$$ (where, for example,$t_{(p)}={|t|}^{p-1}\mathrm{sgn}(t)$is given by $$\begin{array}{cc}& \underset{n\to \infty }{lim}n{\widehat{\lambda }}_n\left(T\right)=c_{p,q}{\left({\int }_I{\left(uv\right)}^r{)}^{1/r}dt\right)}^{1/r},\text{for}1<p<q<\infty \hfill \\ & \underset{n\to \infty }{lim}n{\check{\lambda }}_n\left(T\right)=c_{p,q}{\left({\int }_I{\left(uv\right)}^{r}dt\right)}^{1/r}\text{for}1<q<p<\infty \hfill \end{array}$$ Here$r=\frac{1}{p^{\prime }}+\frac{1}{p}$, $c_{p,q}$ is an explicit constant depending only on $p$ and $q$, $\widehat{\lambda }(T)=\max (s{p}_n(T,p,q))$, ${\check{\lambda }}_n(T)=\min (s{p}_n(T,p,q))$ where $s{p}_n(T,p,q)$ stands for the set of all eigenvalues $\lambda$ corresponding to eigenfunctions $g$ with $n$ zeros.

In [1] I and II various equivalence theorems are proved; e.g. an ODE $(ℰ)\dot{z}=F(t,z,u,\dot{u})(\in {ℝ}^m)$ with a scalar control $u=u(\cdot )$ is linear w.r.t. $\dot{u}$ iff $(\alpha )$ its solution $z(u,\cdot )$ with given initial conditions (chosen arbitrarily) is continuous w.r.t. $u$ in a certain sense, or iff $(\beta )$ $z(u,\cdot )$ satisfies certain conditions by which $1^{st}$-order discontinuities of $u$ and $\dot{u}$ can be treated satisfactorily. In the case when, for $z=(q,p)$ equation $(ℰ)$ is a semi-Hamiltonian system, equivalent to a system of Lagrangian equations of a general type,...

We prove that the $\mathrm{\Gamma }$-limit in $L_{\mu }^{\mathrm{\infty }}$ of a sequence of supremal functionals of the form $F_k(u)={\mu -\mathrm{ess}\sup }_{\mathrm{\Omega }}{f}_k(x,u)$ is itself a supremal functional. We show by a counterexample that, in general, the function which represents the $\mathrm{\Gamma }$-lim $F(\cdot ,B)$ of a sequence of functionals $F_k(u,B)={\mu -\mathrm{ess}\sup }_B{f}_k(x,u)$ can depend on the set $B$ and wegive a necessary and sufficient condition to represent $F$ in the supremal form$F(u,B)={\mu -\mathrm{ess}\sup }_{B}f(x,u)$. As a corollary, if $f$ represents a supremal functional, then the level convex envelope of $f$ represents its weak* lower semicontinuous envelope. ...

We introduce a new class of spaces, called Hausdorff modulo $ℐ$ or $T_2$ mod $ℐ$ spaces with respect to an ideal $ℐ$ which contains the class of all Hausdorff spaces. Characterizations of these spaces are given and their properties are investigated. The concept of compactness modulo an ideal $ℐ$ was introduced by Newcomb in 1967 and studied by Hamlett and Jankovic in 1990. We study the properties of $ℐ$-compact subsets in Hausdorff modulo $ℐ$ spaces and generalize some results of Hamlett and Jankovic....